Preface to the first edition
the two most fundamental concepts in our world because all else is unimaginable without assuming that space (or time) exists. It is therefore not surprising that the sophisticated Euclidean
Space and
time
model of space was
a common
are
already existed more than 2000 years. For centuries it belief by scientists and philosophers alike that the Eu-
clidean structure of space was one of the very few eternal truths. It was only at the beginning of the 20th century that this belief was shat-
special and general relativity. Today, Einstein's theory of general relativity is completely established, and there are many textbooks which explain it at all levels of mathematical sophistication. What is missing, however, is a modern textbook on general relativity for mathematicians and mathematical physicists with emphasis on the physical justification of the
tered with the introduction of Albert Einstein's theories of
mathematical framework. This book aims to fill this gap. Knowledge of physics is not assumed. While physical and heuristic
proofs. The book is also suitable as an introduction to pseudo-Riemannian geometry with emphasis on the intuition for geometrical concepts. arguments
The
are
given, they
physical
are
not used
as
substitutes for any
theme of the book
general relativity typically start with a more or less pseudo-Riemannian geometry. In such textbooks of some knowledge special relativity is usually assumed, and the reader is expected to accept the geometrical framework presented on trust. This approach is very economical but obscures the extent to which classical general relativity succeeds in describing our universe, and also where it those attemptmay fail. This is a point that is of particular relevance to of view it is Rom a to important to physical point quantise gravity. ing realise which parts of the theory reflect genuine physical insights, and which are dispensible. One way this can be achieved is through a criti-
Modern textbooks
on
formal introduction to
cal introduction that stresses foundational matters. There
textbooks
taking
this
approach, and
I
hope
are no
modern
to fill this gap with my -book.
V111
Preface to the first edition
One of the most
exciting aspects of general relativity is the predicBig Bang. Such predictions gained weight through the singularity theorems pioneered by Penrose. In various textbooks on general relativity singularity theorems are presented and then tion of black holes and the
used to argue that black holes exist and that the universe started with a big bang. To date what has been lacking is a critical analysis of what
really predict-' We give a proof of a typical singularity theorem and use this theorem to illustrate problems arising through the possibilities of "causality violations" and very weak "shell crossing singularities". These problems add weight to the point of view that the singularity theorems alone are not sufficient to predict the existence of physical singularities. these theorems
The mathematical theme of the book In order to gain both a solid understanding of and good intuition for any mathematical theory, one,should try to realise it as a model of a familiar non-mathematical concept. Physical theories have had an especially important impact on the development of mathematics, and conversely various modern physical theories require rather sophisticated mathematics for their formulation. Today, both physics and mathematics are so complex that it is often very difficult to master the theories in both subjects. However, in the case of pseudo-Riemannian differential geometry or general relativity the relationship between physics and mathematics is especially close, and it is therefore possible to profit from an interdisciplinary approach. Euclidean geometry had its origins as the description of shapes in physical space. It is generally considered a mathematical discipline rather than a physical theory, because it is possible to derive it from a small set of physical postulates, which can alternatively be viewed as mathematical axioms. Since the concept of space is basic to our everyday experience, Euclidean geometry combines mathematical rigor with intuitiveness a combination which has proved to be extremely fruitful for both mathematics and physics. Riemannian geometry is abstracted from the study -
of surfaces in Euclidean space and inherits much of the intuitiveness of Euclidean geometry. Hence Riemannian geometry is very well developed,
and even
growing number of geometers have branched out to Lorentzian or pseudo-Riemannian geometry. In my experience, these fields (and
a
Since I had written this passage a review article (Senovilla 1998) which a very similar theme has been pointed out to me. This article pro-
has
vides many very illuminating examples of spacetimes as well as discussions which reinforce our sceptical approach towards the physical interpretation of singularity theorems.
Preface to the first edition
even
Riemannian
geometry)
appear
quite abstract
to the
Ix
majority of
students. A careful mental
analysis
of space,
time, and free fall
(classical) physical concepts
-
the most funda-
-
leads almost
automatically
to
Lorentzian geometry. With respect to Lorentzian geometry, we are therefore in a similar situation as ancient geometers were with respect to
Euclidean geometry. What's more, virtually no physical background is required for this approach. Since Riemannian geometry comes to play in the
study of submanifolds representing an instant in time, it is completely straightforward to extrapolate pseudo-Riemannian geometry from the special and physically motivated cases of Lorentzian and Riemannian
geometry. While
some
modern textbooks present
pseudo-Riemannian geometry
(and general relativity) to mathematicians (an example of this is that by O'Neill (1983)), they have not motiv,ated the geometry from basic properties of
space and time. Instead
stract mathematical
theory.
To
they have developed
it
as
an
ab-
that the mathematical
description physical concepts, all definitions have a justification in this book. This approach also leads to a careful treatment of the structural aspects of the mathematics. ensure
mirrors the
How to read this book This book is not
designed
at page 1 and then to read
who take this
approach
so on
that it is necessary for the reader to start until she or he arrives at page 424. People
will very
likely give
14! The material is ordered in such used the
as a
reference
theory
that
a
shortcut.
ning of
a
a
urged As
It is
way
are
page
be
reader who is interested in space and time, and so to follow the guides in the margins, which provide
example, the text in the belonging to the shortcut:
margin denotes the
indicated
by --->2, where
corresponding footnote. The larly. Having understood the
begin-
denotes the page p. 222] the page
p. 111
number where the last shortcut passage ended and number where the present passage will end. Additional the footnotes
they reach
to allow the text to
unfortunate fact that many parts of to the preliminaries are not of imme-
an
passage
up before as
an
logically belong
diate interest to the reader is
source.
a
explanations
in
2 refers to the number of the
end of shortcut passages is marked simimaterial leading to Einstein's equation it
is then not difficult to return to the parts that have been skipped on an earlier reading. In addition, hints are -given at the beginning of most
sections
'
as
to what is
important and should be read
Explanations referring
to the
guide
in the
margin.
.
P. 111 -2
[I
p.
1
222]
X
Preface to the first edition
This
and
book,
with its 424 pages is meant to cover both general relativity differential geometry. It is therefore clear that
pseudo-Riemannian
important topics had to be omitted. mathematicians, the most important omissions are certainly some topics peculiar to Riemannian geometry, such as the Hopf-Rinow theorem (O'Neill 1983, Theorem 5.21) and the Myers theorem (O'Neill 1983, Theorem 10.24). Because these results are contrary to intuition one should obtain for Lorentzian (or general pseudo-Riemannian) geometry and since they are not needed for the description of space and time, they have been omitted from this book. Physicists may find that the presentation of this book is only loosely linked to other physical theories. This loose linkage is possible since the theory of space and time is fundamental to any other physical theory. some
For
The book is therefore accessible to mathematicians and
physicists alike. applications to astrophysics may wish to consult the book by Weinberg (1972). Weinberg's approach is opposite to the one used in this book, and personally I believe that it should ideally be read after the reader has a solid knowledge of the conceptional aspects of relativity as presented in this book. Most other books on general relativity also present the "Kerr solution", which is supposed to model the exterior of a rotating black hole. It has been omitted since it is not essential to unders,tanding general relativity. Moreover, it is well described in other books. People interested in this solution should probably first read Chap. 12 of the book by Wald (1984). The purely mathematical aspects of this solutions are clearly presented in O'Neill's Physicists
book
who
are
interested in
(1995).
Acknowledgements The reader will
undoubtedly
notice that this book
owes
much to excellent
text books and survey articles. For the
I wish to mention
philosophical aspects of this book especially the classic book by Weyl (1923) and the
by Ehlers (1973). freely used material which appears elsewhere (O'Neill 1983; Wald 1984; Beem and Ehrlich 1981; Hawking and Ellis 1973; Sachs and Wu 1979; Karcher 1994; De Felice and Clarke 1990; Abraham and Marsden 1978; Garabedian 1986) without always acknowledging this
survey article
1 have also
fact. I
warmly thank
mended the book
Bernd
project
Wegner, who encouraged me and recomSpringer-Verlag. I wish especially to thank introducing me to relativity but also for read-
to
only for ing through the whole manuscript and for his
Volker Perlick not
ments.
many
important improve-
Preface to the first edition
xi
This book is dedicated to two Australian way to
relativity students who on their gaining their doctorates courageously stood up against the immoral behaviour of their supervisor and the highhandedness of their university.
G6ttingen, 19th July
1999
M. Kriele
Table of Contents
1.
Local 1.1
theory Space 1.1.1
1.1.2
of space and time
.......................
I
..............................................
1
Affine space The fundamental theorem in affine geometry and doubly ruled surfaces
.................................
.....................
3
...........................
14
1.2
Euclidean geometry Absolute space and absolute time
1.3
Non-relativistic particles Galilei's theory of relativity
1.4
Einstein's
1.1.3
.....................
17
......................
20
1.2.1
..........................
special theory of relativity Causality in special relativity Length contraction and time dilatation Relativistic particles and photons
..................
1.4.1 1.4.2 1.4.3 2.
Analysis 2.1
2.1.1 2.1.2 2.2
on
39
..........
40
...............
43
................................
47
.........................................
48
Construction of manifolds Partition of
unity
.....................
54
.............................
57
Vector bundles and the tangent bundle 2.2.1 Construction of the tangent bundle
................
61
.............
63
The derivative of maps between manifolds Tensors and tensor fields 2.2.2
2.3
2.4 2.5
2.3.1
Algebraic preliminaries:
2.3.2
Tensor fields
Vector fields and
2.7
tensors
................
69 84
The lemma of Poincar6
........................
2.5.2
The theorem of Frobenius
2.5.3.
Orientable real manifolds on
87 94 102
.....................
106
......................
109
real manifolds
..................
Connections and projective structures Examples of connections
112
.................
121
.............................
132
2.7.1
The Levi-Civita connection
2.7.2
The
Weyl
........
...................................
Integration
67 68
..................................
ordinary differential equations
Differential forms
2.5.4 2.6
.......
............................
2.5.1
22 27
..................
manifolds
Manifolds
2
connection
....................
132
.........................
135
XIV
Table of Contents
2.8
Curvature
2.9
Applications to Weyl Variation of geodesics
.........................................
2.8.1
3.
4.
structures
................
...............................
143
Space and time from a global point 3.1 Light rays: the conformal structure
of view
...........
151
...................
151
3.2
Inertial observers: the
3.3
Compatibility: Weyl
3.4
Reduction to the Lorentzian structure
projective
structure
Pse udo-Riemannian manifolds
structure
.............
........................
........................
171
4.2
The volume form and the
.....
175
.........
176
............
184
Hodge operator Curvature of pseudo-Riemannian manifolds 4.3.1 2-dimensional pseudo-Riemannian manifolds Submanifolds
star
.....
......................................
4.4.1
Hyperquadrics 4.4.2 Umbilic and totally geodesic submanifolds 4.4.3 Warped products Isometries and Killing vector fields Length and energy functionals 4.6.1 Variation of length and energy 4.6.2 Conjugate and focal points 4.6.3 Existence of focal points
...............................
.......
.............................
4.5 4.6
5.
General
Matter
5.2
Some
5.3
5.4
6.
6.2
.......................
.................
216
....................
227
241
.....................................
255
............................................
255
specific
matter models
perfect fluid
.........................
264
.............................
264
The collisionless gas The electromagnetic field
..........................
Einstein's
equation
The
The Einstein
267
.................................
268
equation
as a
cosmology
Homogeneity and isotropy The initial value problem
equation 271
.......................
279
.........................
287
...........................
287
infinitesimally isotropic spacetimes
................
294
..............................
297
6.3
Geodesics and redshift
6.4
The age of the universe and the A simple model for the universe
6.5
of Einstein's
system
partial differential equations
for
265
......................
Lagrangian formulation
Robertson-Walker 6.1
205 209
5.2.2
of
204
213
The
5.3.1
202
....................
5.2.1
5.2.3
191
193
.......................
relativity
5.1
160 166
Existence of Lorentzian and Riemannian manifolds
4.4
158
.................
4.1
4.3
137 142
big bang we
live in
..............
.............
300 304
Table of Contents
7.
Spherical symmetry
XV
..................................
7.1
Pseudo-Riemannian manifolds with
7.2
The Schwarzschild solution
spherical symmetry
308
..........................
315
Experimental tests for the Schwarzschild solution. Quasi-linear hyperbolic systems of equations in two independent variables The initial value problem for spherically symmetric perfect fluid spacetimes with non-interacting electromagnetic fields Static perfect fluid stars 7.2.1
7.3
7.4
7.5 8.
9.
.........................
328
.............
337 348
.............................................
357
................................
358
8.2
Cluster and limit
8.3
Achronal submanifolds and
curves
.............................
Cauchy developments
Singularity theorems 9.1 Energy conditions Closed trapped surfaces 9.2 The singularity theorem of Hawking and Penrose 9.3 9.3.1 Applications of the singularity theorem 9.3.2 General problems with Theorem 9.3.1 9.4 Singularities and causality violations
.......
365
374
..................................
383
..................................
384
.............................
389
.......
390
..........
395
...........
397
..................
397
9.4.1
The G6del solution
...........................
9.4.2
Newman's
...........................
405
example Strength of singularities and cosmic censorship A simple, 3-dimensional example 9.5.1
397
.........
409
...............
412
................................................
425
.....................................................
429
References Index
322
.............................
Causality 8.1 Causality conditions
9.5
307
.
List of
Figures
1.1.3
Additivity of f Additivity of k Multiplicativity
1.1.4
A ruled surface
1.1.5
Proof of Theorem 1.1.2
1.1.6
Proof of Theorem 1.1.2
1.2.1
A
1.2.2 1.3.1
Absolute space, absolute time Parallaxis effect
1.3.2
Tower
1.3.3
1.4.1
Relative space, absolute time A flash of light at times to 0, tI
1.4.2
A
1.1.1
1.1.2
1.4.3 1.4.4
curve
in
8
........
.....................................
of k
10
.....................................
11
-
-
first
case
second
spacetime diagram
a
example
....................
13
.......................
18
........................
19
.....................................
23
......................................
23
t2
=
..................
consisting of linked oscillations Superposition of waves Future light cone and Galileian relativity. The observer moving with spatial velocity U measures a different centre OV of the flash of light and therefore different radii dj, d2 front
25 27
................
28
...............................
29
.....................................
29
wave
wave
12
.................
case
.........................
for its
9
.................................
30
1.4.7
Michelson-Morley experiment, at rest relative to the ether Michelson-Morley experiment, moving relative to the ether Length contraction ..................................
41
1.4.8
Time dilatation
.....................................
41
1.4.9
Twin
1.4.10
The
2.0.1
The torus T2
2.1.1
A
1.4.5 1.4.6
paradox twin paradox
......................................
in
43
.......................................
48
universe
locally homeomorphic
to R
.............................
R 2 which is not
submanifold of R 2
2.1.2
A manifold M
2.1.3
The construction of
2.1.4
The
2.8.1
The immersed surface in Theorem 2.8.1
3.4.1
C
a
and
finally
a
M6bius band
of Lemma 2.1.7
The world lines from
tially
42
.............
cylindrical
a
topological space which is but fails to be Hausdorff
proof
30
x
...
52
....................
53
............................
................
to y of two atoms which
at rest with
51
are
respect to each other
59 138
ini-
......
167
XVIII
4.6.1
List of
Figures
A broken curve
4.6.2
A
of
curve
lightlike geodesic can be smoothed out by a arbitrarily small length minimising the distance between two spacelike
submanifolds Z1 and Z2 4.6.3
Conjugate points
the
on
5.1.1
A localised congruence
5.1.2
Transformation of the
7.2.1
.......................
216
.............................
217
sphere
.......................
232
...............................
258
density
mass
in
special relativity
.
.
Schwarzschild spacetime in Schwarzschild coordinates Schwarzschild spacetime. Radial null geodesics are the
....
7.2.2
const and Y const. The region straight lines X Schwarzschild coordinates is shaded by =
=
............
The size of
central star in Schwarzschild spacetime
a
....
8.1.1
A strip of two-dimensional Minkowski space where future and past boundaries are identified
8.1.2
Misner's spacetime (S' x R, 2dtdW + td W2) A gedanken experiment to disprove causality violation
.
...................
.............
8.1.3 8.1.4
8.1.5
small
8.2.2 8.2.3
319 324
359 360
361 A spacetime which is causal but fails to be strongly causal 364 A spacetime which is strongly causal. An infinitesimally
with 8.2.1
317
cov-
ered 7.2.3
263
The
perturbation of the chronology violation
proof of Lemma
A limit
curve
-y of
a
8.2.1
metric results in
...
spacetime
a
............................
365
............................
366
sequence of
curves
7
..............
367
An example where a limit curve from x to y is not a cluster curve for a sequence of points from x to y 367 A spacetime which is causal but fails to be strongly causal 368 .......
8.2.4 8.2.5
Assume that
b1c
is rational and
a/b
is irrational. Then
the
projection of the line with slope c/a from torus depicted in the figure is a dense curve 8.3.1
is
cluster
of -y The definition of null boundary and achronal every
curve
a
8.3.2
The
Cauchy
9.5.1
The
singularity
curve
horizon for
a
R 2 to the -y. Hence
.....................
boundary
structure of
at
A the
given by Vo + tq
direction and that at B
375
spacetime. The case kI, k2 < suppressed. The sin-
(Wo/i9y)-l (o9q/Oy)
is
singularity B the singularity
368
set which fails to be achronal.. 381
0. The y-component of spacetime is gularity A is given by I + t
the
..
=
=
0 and
0. Observe that
light cone degenerates in the ydegenerates in the x-direction.
Hence there exist future directed timelike
ing from the singularity and
cosmic
curves
censorship
emanat-
is violated 416
List of
postulates
1.3.1
Galileian
1.4.1
Invariance of the future
3.1.1
Existence of
3.1.2
Light
relativity
........................................
light
cones
conformal structure
........................
25 32
.......................
152
...............................................
153
a
3.2.1
rays Existence of inertial observers
3.2.2
Law of inertia
3.3.1
Compatibility
3.4.1
No second clock effect
5.1.1
5.1.2
Tensorial. character of energy momentum Infinitesimal conservation law
5.3.1
Gravitation is determined
...........................
158
...........................................
158
with the causal structure
..................
...................................
................
............................
by
a
2nd-order
pde
............
160 168 259 264 269
1. Local
theory
of space and time
This book is not meant to be read in the order the
presented. Please follow the guide in the margins or skip material as proposed in the italic text at the beginning of most sections. material is
P*
chapter we will develop those aspects of space and time which can be locally observed, say in a laboratory. We will start with Euclid's description of space and then incorporate time in to the picture. The path we take is rather historical. It starts with intuitive but surprisingly complicated concepts (Newton's theory of absolute space and absolute time) and ends with the not so intuitive but mathematically simpler theory of special relativity. The guiding principle of this book will not be mathematical simplification, but the solution of problems occurring In this
in earlier theories.
description in this chapter seems to be global and extrapolations which are not validated by any experiments and which are not generally true. In the following chapters we will take up this point again, and show that the description given in this chapter should be considered infinitesimally rather than globally. This is the theme of The mathematical
leads to
the book.
1. 1
Space
In this section
we
consider space and introduce Euclidean geometry. This
material is assumed to be familiar to the reader and is therefore in
a
presented
rather concise way.
Readers who wish to learn the essentials of the
quickly and do not mind skipping guide in the margins.
M. Kriele: LNPm 59, pp. 1 - 1, 1999 © Springer-Verlag Berlin Heidelberg 1999
some
theory
mathematical
of space and time can use the
proofs
1
U
p.
3]
Local
1.
theory
of space and time
1.1.1 Affine space In this section
we
introduce
scription of space. Affine of 0 E R7 are ignored.2
affine
space
as
our
most
elementary de-
just R' where the special properties
is
space
basic experience that we can uniquely describe any point in space three real numbers. This seems to be the idea of Descartes (1637)
It is
by
a
who
developed analytic geometry as an example of his Discours de la plausible to identify R 3 with (physical) R3 contains a distinguished point 0 whereas space apparently does space, 3 Hence not. by using R as a description of space we introduce a mathematical structure which has no physical counterpart. This would lead M6thode. While it is therefore
to constructions which cannot be realised in space. For
instance, there is 3 negative of a vector v e R but there is no way to assign the negative to a point in space. As another example, addition of vectors has no direct interpretation in terms of points in space. If we want to have a reliable description of space with the property that all phenomena exhibited in this description are mirrored by physically verifiable phenomena,
the unique
we
have to abstract from these additional structures.
We will
meaning
isolate those structures of R 3 which have
now
in terms of space. Given two
points
we can
x, y
an
intuitive
construct
an
which points from x to y. This arrow induces a map from space to space. We just move the arrow (without rotating) such that its untipped end coincides with a given point z. The point z is then mapped to the arrow v
It appears that as long definition is independent of the path which
tip of the z.
In
z
-4
R3, z
arrow.
this
+ y
parallel transport Observe that y
x.
-
-
as we
we use
just given by the
is -
x
don't rotate to
move v
map
stands here for the
v
Ry-x: R3 arrow v
this
-
from
x
--->
to
R3,
which is
not a point. Thus the geometric interpretation is different from a simple addition of vectors. In order to separate the concepts involved we define
the concept of real affine space. We do not yet know what "rotating" should terms
-
following of
our
so
far
we
have
definition
(for
simply
mean
physical picture
a
the n-dimensional
case)
reflects
naive notion of
additional
in mathematical
in mind.
However, the some properties
parallel transport and does not introduce The set of points is denoted by A'. properties.
any
Definition 1.1.1. An n-dimensional real affine space is a set A' and v E Rnj of bijective maps such that the a collection fRv: A' --> A' :
following
holds.
N R,+,, 2
=
Readers who
R,
R,,
Vv,
already have
and connections considered
o
as a
(cf.
Sect.
manifold
w
some
2.6)
G
Rn,
knowledge
together
of manifold
theory (cf. Chap. 2)
take affine space to be the usual R7 with the flat connection.
can
1.1
(ii) for all Rvx,, (x)
R,(x) by
We denote
allow for
see
A"
there exists
+
or v
x
v
+
and v.,,Y
x
below that this definition is
more
unique v,,,y
a
R' such that
c
y.
=
We will
E
x, y
Space
affine spaces than there
by
or
y-x
x--y'.
restrictive that it does not
so
vector spaces.
are
Remark 1. 1. 1. Of course, space is 3-dimensional. We work with general n for several reasons. Firstly, most of the theory we will be
dimension
with the exception of developing does not depend on the dimension a key-result whose proof, however, is too complicated to be presented in this book (cf. Theorem 5.3. 1). Secondly, it is often convenient to compare the theory with lower dimensional analogues which are easier to visualise. -
It is therefore advisable to formulate it in
analogues
a
way which encompasses these
and also shows the restriction of the
exist modifications of Einstein's
dimensions
general theory
analogy. Thirdly, there of relativity to higher
("Kaluza-Klein theories").
Another way to generalise the theory would be to allow for complex vector spaces as well. We refrain from doing so since the main result of Section 1.1.2 Note that for
only x
E
holds for affine spaces
An,
u,
x
R' the associative law
v c
(U
+
R.
over
+
V)
(X
=
U)
+
+
V
holds. It is easy to see that all n-dimensional real affine spaces are isomorphic, and can be realised by R' in the following way. Choose E
A' and define
any
o
tify
An with
Rv(x)
=
0,, (A'
An
0,,:
R',
--
and define for
oo(Rv(oo- (x))). Clearly, x,,y
these definitions
are
Oo(x) OO(A'), v
x
E
x
F-+
Observe that
=
recover
the structure of R'
1.1.2 The fundamental theorem in affine
doubly
o.
Con-
by
iden-
geometry
ruled surfaces
In this section
we
be needed in the
results
of affine geometry which will 1-4.1. This section is very technical first reading.
present
proof of
and should be omitted
some
Theorem
on
Let o, xj,..., xk E An and a',..., the barycentre with masses a 1 3
vo,x. Now idenRn the bijection
independent of the arbitrarily chosen point
versely, choosing an o E An we can tifying o with the zero vector 0.
and
=
c
ak
E
R such that
Ek
j=1
a'
=
1. Then
k a
Section 1.1.2 is needed for the proof of Theorem 1.4.1 which is central to our interpretation of the Michelson-Morley experiment. However, the reader is strongly advised against reading this part now.
[
[P_
7 311]
4
1. Local
theory
of space and time
k
al X, +a 2X2+
-
-
-
+a kXk
:
0
`:::
+
Ea
(xi
-
0),
i=1
is
independent
of
defined via the
and therefore
o
an
hand side and
right
affine invariant. The
binations" where the real factors add to 1. is
set of
a
X11
...
is k
7
xk
symbol + is applied to "linear comAn affine subspace Bof A'
only
can
be
1
2 k a Xl+a X2+ points f x +01 Xk : Eki=1 a i 1, where are pairwise different, fixed points. The affine dimension of B
1. It follows that
an affine subspace is an affine space. An affine subspace of dimension 1 is called an affine line. We call points lying on a single line collinear. Observe that lines are the smallest sets which are invariant under parallel transport. -
Lemma 1. 1. 1. Let x, y, z E An. Then x, y, z lie on an only if there exists a A E R such that x y + A(z-y).
affine line if
and
==
Proof
x
R with
lies
x
=
the line
on
generated by
Oy+(l -O)z
=
z
y,
if and
only
if there exists
y+O(y-y) + (1 -0)(z-y)
=
then
f
is called
A collineation is collinear points into Consider
given by
a
an
biJ, ection
n
f (x) If A
=
is
A' which maps any three
--
line 1 and three points X 1) X2 i X3 A(X2 -X1) is denoted by
X3 -X1
G
affine transformation.
f: A collinear points. a
0
y+ (I -0)(z-y). I
Definition 1.1.2. An affine map is a map f: A' -4 An, A(x-o) + b, where A is a linear map, o E An, and b c Rn.
bijective
an
on
1. Then the number A
=
X3-XI
X2-XI
The the
following lemma is the classical theorem of Thales. It will be used in proof of the fundamental theorem in affine geometry (Theorem 1.1.1
below). Lemma 1.1.2. Let a
H1, H2, H3
line which intersects these
C Rn
be
parallel hypersurfaces and 1 hypersurfaces. Let xi(l) Hi n 1. Then
be
=
X3(l)-XI(l) X2(l)-X1(l)' does not
Proof.
depend
Denote
on
1.
by R
the
subspace
of Rn which is the associated vector
space to the affine space
H1 (and since H1, H2, H3
H2, H3).
quotient
We consider the
space A
nIft
are
defined
parallel also by
to
1.1
This space has Rn / fJ given by to the
a ir
5
y-xEfl.
ifandonlyif
x-y
Space
natural affine structure with associated vector space Jf (x z) where 7r, f denote the proj ections 7r (z)
(x)
=
-
-
equivalence classes. We have
7r(X3(l))-7r(X1(l)) ::_-';f(X3(l)-X1(l)) Jf
X3(l)-Xl(l) (X2 (1) (X2(l)-XI(l)
-XI
(1))
X3(l)-XI(l) f (X2 X1 X2(l)-Xl(l) X3(l)-Xl(l) (7 (X2 (1)) 7F (X 1 (1))) X2 (1) -X1 -
-
which
implies
that
(1) -XI X2(l)-XI(l)
(X3 (1)) -7r (XI 7r (X2 (1)) IT (X 1 (1)) Ir
X3
only depends
ir(xi(l))
that
Hi
x, y E
==> 7r
-
the projected values. Now it is sufficient to observe independent of I since all points in Hi are equivalent:
on
is
(x)
=
7r
(y).
I
It is easy to see that all bijective, affine maps are collineations. Conversely, the fundamental theorem in affine geometry asserts that any
collineation must be affine: Theorem 1.1.1. Let A' be 0
E
A. Let
points
f:
An
--+
into collinear
invertible linear map
An be
an a
affine
over
space
R with
n
> 2
and
fix
bi ection which takes each three collinear
points. Then there f such that f (x)
=
exists
a
f(x-o)
point b E An and for all x E An.
an
+ b
The
proof is elementary but lengthy and requires some preparatory lemWe will follow (Berger 1987, p. 52-55) where one can also find a version of this theorem which holds in the complex case. Observe that the following proof makes heavy use of the assumption n > 2. The themas.
orem
does not hold for
points
into collinear
n
=
I since in this
any map maps collinear
case
points.
Lemma 1. 1.3. Let o, x,.... iXk E R, and
An, f
be
a
collineation,
k
x==o+EA'(xi-o)EA
n
i=1
Then there exist
fil.... Ak I
E R
such that k
fW
=
f(o)
+
E Ai ff (xi) i=1
-
f (0))
-
A
Ak E
6
1. Local
Proof.
For k
theory
of space and time
1 the claim is clear
=
by the
Assume now, the assertion is true for all
definition of
1,
m
M+1 X
Then
we
.
.
k
,
collineation.
a
-
11.
For
M
Ai (Xi-0)
+
0
=
.
X,
let
=
0
Ai (Xi-0).
+
have X
=
X/ + Am+, (XM+1 -0)
(1.1.1)
p"
by induction hypothesis there are real numbers y' f (x') f (o) Eim. 1 M" (f (xi) f (o)). We define also and
=
-
with
-
Y
=
z
=
A'+'(xm+,-o),
+
0
2
Y+
(1.1.2)
X
2
The triples I z, x', y 1, 1 y, o, xm+ 1 1, and f z, o, x I consist each of collinear points. This is clear for the first triple and follows from Lemma 1.1.1 for the second triple. To see this for the third triple observe that y-o 1 x1. z x + 12 x' is the centre of the parallelogram defined by o, y, x, x' 2y and therefore the intersection of the line connecting y with x' and the line connecting o with x. Since each of these three triples consists of collinear points there exist a, 0, -/ such that =
-
This
=
f (Z)
=
af (X')+ (I
fW
=
Of (o)+ (I
f (Y)
=
f (o)+-Y(f (x.+i)-f (0)).
-
-
a) f (y),
O)f (z),
implies
fW
=
Of (o)+(I
=
'3(f (0) (I
(I
-
-
-
-
O)f (Z)
f (0))
+
(I
-
'3) ((af W) + (I )3) (c, (f (x')
-
0) (f (Z)
-
-
00 f (y))
f (o))
+
(1
-
f (0))
-
f (0))
+
(I
-
/i
A
(f (Xi)
a)-Y(f (X-+1)
-
-
f (0))
f (0)))
M+1 =
E 'U
(f (Xi)
-
f (0))
+
f (0)
-
f (0)
+
a) (f (y)))
M
0) (01
+
+
f (0)
f (0) +
f (o)
Space
1.1
Lemma 1.1.4. Let o, xl,. Xn E An such that f X1 -0i iXn-01 is a basis of R'. If f is a collineation then f f (xi) f (o), if (Xn)-f(O)j is ...
-
...
also
Proof
ol
basis
a
is
of R'.
Let -3- E An be any point and let x f basis of Rn there exist ' E R such that
Since
=
a
Lemma 1.1.3
that there exist
implies
pl,...'
n
x-o
E
=
fxl-o,...
7
Xn-
Enj I V(xi-o).
R such that
k
:
-
f (0)
=
fW
-
f (0)
=
EA
(f (Xi)-f (0))
-
j=1
Since 1--
was
arbitrary the
assertion follows.
Lemma 1.1.5. A
bijection f affine lines.
is
a
collineation
if
and
only if
it 'Maps
affine
lines onto
Proof.
Let x, y E An and denote by 1 the line spanned by these points. a point on the line spanned by f (x), f (y). We have to show
Let i be that
z
=
f-'( )
G
1. If this
was
not true than the vectors z-x,y-x
would be
linearly independent. But then Lemma 1.1.4 would imply that f (z) f (x), f (y) f (x) where linearly independent as well. Contradiction to the construction of I f (z) -
-
=
Lemma 1.1.6. Let
parallel
f
be
collineation. Then f maps parallel lines into
a
lines.
otherwise 1, 1 be two parallel lines (which do not coincide nothing to prove). Since they are parallel they span a plane P rather than a 3-dimensional subspace of An. This plane is mapped into a plane P'. In order to see this consider a line i such that the lines 1, i intersect and span P. It is clear that any line which intersects both 1 and i is contained in P. Moreover, any point y E P lies on a line I which intersects both I and 1. Let P' be the plane generated by the (intersecting) lines f (1) and f (1). f (y) lies on the line f (1) which intersects f (1) and f (i). Hence f (T) (and therefore f (y)) lies
Proof.
Let
-
there would be
in P'.
Having established that f (P) 0. If there f (1) n f (1)
show that
=
would lie in both 1 and V which is
is
subset of
a plane we only have to point z c f (1) n f (1) then f -'(z) impossible since both lines are parallel.
a
was a
I
Lemma 1. 1. 7. Let k: R
and k (a +,3)
=
--->
k (a) + k (,3)
R
for
an
k (a) k ( 3) automorphism, i. e., k (ao) id oz,,8. If k 7 0 then k
all real numbers
=
=
theory of
1. Local
space and time
-
k (0 + 0) k (0) + k (0) implies k (0) 0. Assume, there is an Proof. k (0) 0 0. Then with a 4 k(a) k(o) k(oz)k(ola) 0 for all 0 and k must vanish. Hence k(a) =A 0 Va 34 0. k(l) k(l 1) k(l)k(l) implies 1. By induction we obtain k(n) n for all natural numbers. k(l) k(-n) k(O) k(n) -k(n). Similarly, we have k(l/n) k(O n) =
=
=
=
=
=
=
-
=
-
=
1/n. For n, m E Z we have now k(n/m) for all rational numbers. a < 0 implies
1/k(n) is
proved any positive number
2 we
7
number. Then there exists
k(-y 2)
have
=
n/m k(a) :5 k(,3)
since for
Let
-y be any
and the lemma
=
k(-y)k(-y)
> 0.
now
monotonically increasing sequence ai 'Y a monotonically decreasing sequence of Hence ai Oi which k(ai) < k(-y) < k(0j) a
-->
of rational numbers and likewise
rational numbers
implies k(-y)
Oi
--->
-y.
=
=
I
-y.
=
Observe that this lemma would be false if z
would be
i-->.
above)
a
counter
This is
example.
is not true for affine spaces
f:
had
we
why
replaced
R
theorem 1.1.1
by
C
as
stated
(as
the field C.
over
R', v -4 f(v) f (o + v) f (o). an automorphism k: R --> R such that proof k (A) f (v) + k (y) f (w) holds for all A, M G R and v, w c R'. f (Av + I-tw) We will use constructions based on parallel lines in order to represent vectors such as v + w, (A + A)v, Attv. Since f maps parallel lines into parallel lines (Lemma 1.1.6) these constructions will be preserved by f and can therefore be used in order to prove linearity and multiplicativity Proof of
Theorem 1. 1. 1.
The idea of
Let
R'
--->
=
-
is to construct
f, k.
of
We will first show that
fis
additive.
f
(o+
(o o
+
V
+
o
o,
of
1.1.1.
+
W)
(0)----Ie- f(lo + V)
Additivity of f
1w that contains
parallel
lines
o
+
=
v
and of
1, that
mapped analogously
f(o+V+W)-f(o)
=
contains
o
+
w
(cf. Figure
1. 1. 1).
f (o + v), f (o + w). Hence f(o+V+W)-f(0+V)+f(o+V)-f(o) into
are
is constructed
w) f(V+W) +
V
Let v, w E R' and consider the lines 1,, 1w spanned by o, o + v and + w. The point o + v + w is the intersection of the parallel translation
Since v
W) (o +
f
Fig.
+
W
lines
parallel f (o), f (o
from
we
know that
+
=
1. 1
Space
+ v) f(w) + f(v). Here we have used the f (o) connecting f (o) with f (o + w) and f (o + v) with f (o + v + w) are identical since they correspond to opposite sides of a parallelogram in a plane. Now we show that there is a well defined automorphism k: R --> R
f (o
+
w)
f (o)
-
f (o
+
=
-
fact that the vectors
such that
f(Av)
k (A)
=
f(v)
for all
v
E
Rn and A E R. We first fix
a
and consider the line 1
vector
R through o spanned by v. Denote by gi: 1 the map o + AV --> A and by gf the i--> Since + ttf(v) map f (o) f /-t. (1) maps the line through o which is spanned by v into the line through f (o) which is spanned by f (o + v) f (o) the map k: R R is well defined the From through relationship f (Av) k(A)f(v). v
-->
-
=
f (o) we see
f(v)
+ k (A)
that k is
f (o)
given by k (A)
+
=
f(Av) o
gf (1)
f (o f
+ 1
o
gi
Av)
f (gi
'(A))
(A).
f
/,00-
f (0
+
W)
W O+W
AV AV 0 0
Fig.
0
+ AV + [tv
+ ttv /,V
X" , AV +
+ AV +
/-tV)
of k
Additivity
1.1.2.
(*0
f (0) f (0 + AV) f (0 + /,tv) f (0
In order to prove additivity of k we use the fact that (A+M)v Av+,uv be constructed using parallel lines (cf. Figure 1.1.2) Let w E R' be =
can
linearly independent from points
o,
that the
the lines
o
+ w,
o
v
and consider the
+ Av. This
triangle defined by the parallely translated so (We simply parallely translate
triangle
o is mapped into generated by its sides
point
o
can
+ /-tv.
be
indicated in the figure). Since this translation preserves the vectors defined by the sides of the triangle we have obtained a geometric construction of the point o + Av + Aw. Since this construction only employs intersection points and lines it is as
p rallel
preserved by the
k(A)f(v)
+
k(A
map
k(p)f(v) +
p)
=
=
=
f
.
Hence
we
obtain
f ((A + /-t)v)
=
f (Av)
+
f (ttv)
and therefore
gf (1)
o
f
1 o
gi
(A
+
/-t)
==
gf (1)
o
f (o
f((A + tt)v)) gf (1) (f (o) + k (A) fi(v) + k (M) f(v)) gf (1) (f (o) +
+
(A
+
tz)v)
10
Local
1.
theory
of space and time
gf (1) (f (o) +
(k (A)
+ k (p))
f(v))
=
k (A) + k (y).
f
f (0 o
f (0) (0
A> 0
f 1(o
+
AW)
+ AW
W w 0+W
0 + 01
W)
+
0 + o4
A[IV f (o + V) f (o (0, + /\V) AV) ff (0
'+U Vlf
()o + pv)
'v 0 + /_tV 0+
\v + AV
f f(o+ (0 + A/,tV)
V
-
Fig.
1.1.3.
Multiplicativity of k
proof of multiplicativity is similar and employs a slightly difgeometrical construction (cf. Figure 1.1.3) which is justified by Lemma 1.1.2. The configuration in the first part of Figure 1.1.3 lies in a plane whence hypersurfaces are simply lines. Denote by H2 the line which connects o + v with o + w, by I-I, its parallel translation through o, and by H3 its parallel translation through o + Av. Further denote the line through o and o + v by 1 and the line which connects o with o + w by 1'. Using the notation of Lemma 1.1.2 we have The
ferent
(0 + AV)-0 (0 + V), 0
A
X3(1)-X1(0 X1 (1) X2 (1) -
implies that the intersection of H3 and 1' is really depicted in the figure. We apply this lemma a second time where the three parallel hypersurfaces H2, H1, H3' are now given by the line connecting o + /-tv with o + w, its parallel translation through o, and its parallel translation through o + Aw. It follows that the intersection of H3' with 1 is o + y(Av) o + A/-tv. Since this construction only em1 loys intersections and parallel lines it is preserved by f and we obtain f (Apv) k(A)k(p)f (v). This implies Hence Lemma 1.1.2 o
+ Aw
as
=
k(Att)
=
=
Hence k is
metrically o.
gf (1)
o
+
Aix)
=
gf (1) (f (o) +
gf (1) (f (o) + k (A) k (y) f(v))
really
our case
=
&yv))
k (A) k (y).
automorphism of the real line. One can geoautomorphism neither depends on v nor on this automorphism is trivially well defined since
an
show that this
However in
f (o
1. 1
Space
11
already know that the only non-zero automorphism of R is the identity. This also implies f (,\v) \f (v) for all X E R, v E Rn Hence the theorem is proved. I we
=
-
We will
now
turn
attention to
our
special subsets of affine
space which
become important in the proof of Theorem 1.4.1. Definition 1.1.3. Let U C R 2 be
an open set and x: U --, An be a C' map such that at each point (s, t) E U the differential Dx(s, t) is injective. Then x is called an immersed surface. If x is also injective
then it is
simply called
surface.
a
A surface should be envisaged by its image, a two-dimensional, smooth subset. An immersed surface may have self-intersections. Since lines have such a fundamental meaning in affine geometry, surfaces which are generated by lines are of special interest.. Definition 1.1.4. A ruled surface is zed
by
sation
a surface which can function of the form x(s, t) c(s) + tw(s). Such is called a ruling of the surface. a
=
Fig.
1.1.1. The
Example different
(i) (ii)
be parametria
parameteri-
1.1.4. A ruled surface
simplest ruled surfaces
those which admit two
are
rulings.
A trivial
example would be any plane. slightly more sophisticated example is given by the rotational hyperboloid. Let c(s) (COS(S), Sin(S), O)T be the unit circle in A
==
R' and consider is
a
Xhyp(S,t)
ruled surface and
t sin (s), sin (s) + t 2
(X hy P)2
-
(X3hy P)2
COS =
surface is described
C(S)
=
(8), t) T.
a
different ruling.
t(6(8)
+
(0, 0, I)T). Clearly, (COS(S) =
Since it satisfies the equation
I it must be
by -Zhyp (8 t) 7
is
+
explicitly given by Xhyp(S,t) a
rotational
=
C(S)
(xlhy P)2 +
hyperboloid. The
+ t (_ 6(8) +
x -
(1, 0, 0) T)
same
which
Local
1.
12
(iii) A c(s)
theory
third
=
of space and time
is
example
(s, 0, 0)
'
and
w
given by the hyperbolic paraboloid. Let 1, ks)T. Then Xpar(S7 t) 2 (0, 1_k1-2
(s)
=
v
+1
8-
X 3ar and Xpar parameterises a p I hyperbolic paraboloid. We can interchange x par and Apar to obT tain a different ruling of the same surface, ;r-par (8) t) (0, S, 0) +
c(s)
+
satisfies
tw(s)
kxlarx2ar p P
=
=
I
(1,0,kS)T. a surface which admits two different M is a subset of affine transformation rotational hyperboloid, or a hyperbolic paraboloid.
Theorem 1.1.2. Let M C A' be
rulings.
Then
either
plane,
a
-
a
up to
an
-
Proof. It is easy to see that any surfaces M C A' with two rulings can locally be embedded into A'. One just has to consider -a line 11 of the first ruling which intersects a line l'2 of the second ruling. Choose another line l'3 of the second ruling which also intersects 11. Then all three lines further line of span a 3-dimensional affine subspace. At least locally, any the first ruling must intersect both l'2 and l'3 whence it is contained in the same affine subspace. Since M is generated by the lines of the first ruling we have proved the assertion.
1.1.5. Proof of Theo-
Fig. rem
If there
are
any two
1.1.2
-
first
generators of the first ruling which lie
then the ruled surface must be this
Hence
plane. linearly independent. There
case
in
we can assume
a
plane
that any
possibilities. parallel to a single plane or any three generators are linearly independent. In the first case let 111 12,13 be different generators of the first ruling which are all parallel to a single plane. We can now find linear coordinates I X ,X2,X31 such that the x1-Axis coincides with 11 and the x 2-axis is 2 parallel to 12. By choosing the origin appropriately, 11 is given by x 3 3 a2, X1 +a 3X2 0, a', and 13 is given by x X 0, x 0, 12 by x1 3 contains which be P Let 11. Then there R. a where a 1,a2,a E two-plane 2 SX3 0. P is that x + such R Any generator V exists an s E given by both 12 and intersect must P in contained is which second of the ruling intersection the for obtain We points: 13-
two
generators
are
Either there exist three generators which
are
are now
two
all
=
=
=
=
=
=
=
1. 1
1.1.6. Proof of Theo-
Fig.
1.1.2
rem
jPn12j:
X
jPn13j:
X
=
0,
=
a
1
It follows that 1' has the
2
1
X
a
X
8,
Since the ruled surface is a
eters s, t
2
X
s,
=
a
a2a3s
0
tained
case
,
3
a
second
parameter form
-als al
1/
a
==
2
2
3
2
X
s,
-
1.
3
a
13
Space
+ t
W
-S
a2
al)
_
:
t ER
al
_
1', we have obEliminating the param-
such these lines
generated by
x(s, t) parameterisation (s, t) 3 -a3a 2 2, whence the surface we obtain xlx of it.
-->
=
must be
a
hyperbolic paraboloid. For the second
case we
f x1, x2, X31
choose linear coordinates
thatthe X3 -Axis coincides with 13 , the X2-axis is
such
parallel to 12, and the x1
-
11. We can chose the origin 0 of the coordinate system parallel 0. Then such that it lies in 13 and such that 12 lies in the plane X3 X2 there exist numbers al, a2, a3 C R such that 13 01, JX : X1 a3 P be X2 Let x3 a and X1 : a2,X3 11 : a', 1. fx 12 01, fX X2 exists Then there 0. to not and is contains which 13 parallel plane 2 0. Any line 1' of the second sx an s c R such that P is given by x1 axis is
to
=
=
=
=
=
=
=
=
=
=
-
family which
11 and 12- We calculate
lies in P must intersect
JP
n
111: x1
fP
n
121:
2 =
sa
3
2
2 ,
X
a
,
3
X
a
1
This
1
I X
=
a
a
2 ,
X
3
=
,
X
=
0.
gives the line a'
a'
alls 0
+ t
2 -
alls
sa
2 -
-a
3
a
tGR
14
1. Local
It follows that
(s, t)
theory
(s, t)
-a
a
that this
1, (X 3)2 ( 2)2+ (X3)2 =
X
-
1, the
(X 2)2 that
cone
_
our
we
eliminate
3X1X2
2
+
a
xIx3
1x2x3
a
_
+
I a
a3x 2
=
0.
quadric. We could use the Gram-Schmidt-procedure to show quadric is affinely equivalent to a hyperboloid. But since any
in R 3 is
quadric
parameters of the ruled surface. If
are
obtain the equation
we
This is
of space and time
X3
==
affinely equivalent either
the two-dimensional
1, the rotational
=
(X 1) 0,
2
+
or a
(X2)
2 _
X3
to the
pseudo-hyperbolic
hyperboloid,
=
-
(x')
(X 1)2 +
space,
2
+
(X2)
plane, we can infer without affinely equivalent to a
(X2)2 + (x 1)2 +
-
2
0, the hyperbolic paraboloid
surface must be
1.1.3 Euclidean
sphere
2
+ -
(X3) (XI)
2
+
any further calculation rotational hyperboloid.
geometry
Euclidean geometry gives the local model of space. In the following sections we will obtain models of space & time which incorporate Euclidean geometry as description of space. Unless otherwise stated, here and in the following space has dimension n 1. We assume that Eu-
clidean geometry is known to the reader and a
therefore only
summarise
few facts.
In affine space, we have no definition for "length" or "angle". Since these are fundamental concepts for our perception of space, we must endow affine space with an additional structure. The first scientific and ex-
perimentally well tested description ing
and axiomatisation of space involv(ca. 340b.C.-
these notions culminated in the "Elements of Euclid"
In modern
270b.C.).
terminology, Euclid's theory
of space
tified with Euclidean geometry. The central object of Euclidean geometry is the scalar Definition 1.1.5. A scalar
product V
such that
for
any u, v,
w
E
V
--+
(u,v)
F-4
V, A,
A u, V) (i) (U Av + AW) V, U), (ii) (u, V) (iii) (u, U) > 0, OV) u, U) 0 == > U 0 =
,
=
=
hold.
=
x
+
on a
U
product. a
map
(1.1.4)
(U,V)
(u, W),
be iden-
real vector space V is
R
It c R the
can
properties
Space
1.1
We a
define
can now
scalar
Euclidean space
an
as an
affine space
15
equipped with
product.
Definition 1.1.6. An Euclidean space is
(i) A "
is the
(n
-
pair (A",
a
real
1) -dimensional,
where
space with associated
affine
vector space R n- 1,
(") (*C)R1-1 0: An'
A map
is
scalar
a
product
An-1 is
-->
an
R".
on
isometry if and only if
O(Yl)-0(X1)i 0(Y2)-0(X2))R71-1 for
An-1
all Y1) X1 7 Y2 7 X2 c:
The
notions
physical
initions x, y E
It is
wish to capture with
we
"angle". only depend on
A` should
plausible
mathematical def-
our
The distance between two points
"distance" and
are
YI-Xli Y2-X2)Rn-1
:--::
the connecting vector u y-x. x and x + \u is X times =
to demand that the distance of
the distance between
definition dist (x ,
x
y)
and X
x
+
Hence, given
u.
Y Rn-1
-
scalar
a
V(X_y1X_y)R11-1
:=
product, the to be
seems
a
reasonable choice. It is vectors
in order to
clear, however, that needs
one
map Rn
a
x
Rn
measure
of the
real number. The
directions u,
by
Z (u,
v)
=
(
arccos
between two
angle
symmetric in both vector is multiplied by a
entries and remains
defined
unchanged if one angle between two
the
R which is
-->
v
may therefore be
(U,V)R"-l IIUIIR"-l 11V11R11-1
priori clear that a scalar product is indeed the appropriate additional structure for defining lengths and angles. See (Weyl 1923, 19) It is not
for
a
a
theoretical
1.1.1. A map
Proposition ture
(A n-1, (7 *)RII-1)
A: R n-1 and
(Au, Au)R11-1
0: An-1
invariant
Rn- ' and
-4
of the usage of scalar
justification
points
if
only if there
I such that o, b E An-
U) U)Rn-1 for
:=
An-1 leaves the Euclidean
__
and
all
products.
u E
exist
V)(x)
struc-
linear map
a
A(x-o)
=
+b [p.
R".
I
Proof. Observe first that all
u
E R
for all v,
O(Yl)
n-1
w
-
c
application of Au, Au)R11-1 u v + w implies (Av, Aw) R11
an
to the vector
R n-1
.
:--
=
Hence the map
0(X1)i0(Y2)
-
In the
proof
of
O(x)
=
A(x-o)
-
I
(U7 U)R11-1 for (V W) R11 =
I
+ b satisfies
0(X2))R11-1
(A(yi-o) =
(A(y,
=
(Y1
-
x t-->
-
-
X1, Y2
Proposition
A(x1-o), A(Y2-0)
x1), A(Y2
1.1.1
-
-
A(X2_o))R11-1
X2))R11-1 X261-1 -
*
we
appeal
14
1]
-4
to Theorem 1.1.1.
-
p. 16
16
1.
Local
theory of
space and time
Conversely, any map V) which preserves the Euclidean structure preserves particular the affine structure. Hence Theorem 1. 1. 1 implies that there
in
linear map A and a point b such that O(x) A(x-o) + b for all Since 0(y) O(x) A(y x) it is clear that A must satisfy
is
a
x
E A.
=
=
-
Au, Au)R,,-'
(ul U)R,,,-'
=
-
for all
u c
R".
P.
Remark 1.1.2. At first
sight our definition of a Euclidean space may general. The reader may feel that in space there is a subset of physically distinguished scalar products: Let e be a vector which we use as measuring stick defining unit length and E a plane which contains el. Using a pair of compasses we can construct a line 1, C E which is orthogonal to el and therefore also seem
to be too
vector e2 of the
length as el but perpendicular to el. We may plane E,, by rotating,e-2 around el and a third plane E,, by rotating el around e2. The intersection E,1 n Ee2 is a line orthogonal to el and e2. Using again or pair of compasses we can construct a third vector e3 which is of unit length and orthogonal to el and e2- Our distinguished scalar product is now given by (ei, ej)R3 6ii' It follows from the Theorem of Pythagoras that the length of a vector u is given by JUIIR3- We can use a pair of compasses to approximately (but arbitrarily well) divide the circle into a fixed number of arcs thereby introducing an approximate measure of angle. From the definition of the cosine it is clear that (up to a constant factor depending on the number of arcs) the size of an angle is given by the definition above. However, this introduction of the standard scalar product is based on procedures which are intuitive but which cannot be defined in mathematical terms without having a scalar product in the first place. In fact, if we had started with any given scalar product (., -) and had defined a
now
construct
a
same
second
=
(i)
a
rotation
as a
linear map which leaves the scalar
product
invari-
ant and
(ii)
a
pair of
containing same
then using
a
length our
compasses
given as
construction
1.1.2. Let
Euclidean spaces. which satisfies
(O(Yl) for
-
a
e
device which for each given plane E produces all vectors e' c E with the
e,
lowing proposition gives Proposition
as
vector
a
we
would just have recovered
(An-1, (*1')R"1-1)
Then there exists
V)(X1)i'0(Y2)
all X1i Yli X21 Y2 E
The fol-
mathematical explanation of this fact.
An-1.
-
an
V)(X2))R11-1
and
affine
=
Yl
R11-1) map
-
0: A'-'
X1) Y2
-
-,
X2 R-l
be two
An-1
1.2 Absolute space and absolute time
Proof. Choose tively
points o,b
any
be
Corollary is
a
basis
(-)R11-1)
ei. Then 1) We define the linear map A by A i b desired is the isomorphism. A(x-o) +
(respectively
-
O(x)
=
is often referred to
0
A'-' and let lei,..., e,,_11 (respec-
E
orthonormal basis with respect to =
the affine map The map
an
17
as an
Euclidean
transformation.
*) *61-, be a scalar product of Rn-1. Then there 6ij, where en-11 of Rn-1 such that ei, ej )RI-1
1.1.1. Let
lei,
=
6ij is the Kronecker
I
for
i
0
otherwise
=
j,
symbol.
Today, Euclidean geometry is-often taught as a prime example for a closed and consistent mathematical theory. This obscures the fact that angles and distances are physically measurable and that therefore Euclidean geometry can be falsified as a physical theory. (For instance, one of the most influential philosophers since the time of enlightenment, Kant (1781), wrongly considered space as given "a priori"). In modern times, Carl Friedrich GauJ3 (1777-1855) seems to have been the first to realise the possibility that Euclidean geometry may not be the correct description of our world though the legend that he tried to verify Euclidean geometry by measuring the angles between three mountain summits is not true (Osserman 1995, page 66). He has developed a non-Euclidean geometry in which the parallel axiom does not hold but did not publish it. This geometry was also independently discovered by the Hungarian mathematician Jdnos Bolyai (1802-1860). Later in this book we will conclude that space should be described by geometries which are far more general than those considered by GauB and Bolyai. -
1.2 Absolute space and absolute time In this section
we
present the "naive" model of
space and time.
We
complicated it really is. We will also give a short account of Newton's theory of particles which is the main physical justification of this spacetime concept. will take
Time
seems
care
to show how
to have
striking similarities with
space but nevertheless to be
something which is very different. Like space time is a continuum. However, space is a 3-dimensional continuum while time is 1-dimensional. Moreover,
we
can
freely
move
in space but
merely
drift in time. It is
18
1.
Local
theory
of space and time
--------------
-----------_-------
movement
inovement
in tirrie
in
space-
tP1
time
t--------
Fig. 1.2.1. A curve in spacetime diagram
often
practical to treat space and time in spacetime diagrams are used to describe Hermann Minkowski
nobody
has
ever
unified
manner.
movements
(1864-1909) (Minkowski 1909)
experienced
space without time
This observation is borne out
following
a
For
instance,
(cf. Fig. 1.2.1).
has
pointed
out that
time without space. space & time in the
or
by characterising
way.
Definition 1.2.1. A primitive spacetime is set. The points time are called events.
of
a
space-
Of course, this definition does not tell anything about the relation of or even allows to distinguish between these concepts.
space and time
In order to do
so we must supplement the primitive spacetime with a geometrical structure. In the preceding section we have recalled that space can well be described by (n I)-dimensional Euclidean space. The fact that time is I-dimensional indicates that spacetime can be considered as an ndimensional affine space which is foliated by (n I)-dimensional subspaces each of them carrying a Euclidean structure. Any foliation with affine hyperspaces corresponds to a linear map T: R' -4 R, where x, y E A' are in the same hyperspace if and only T(x-y) 0. Denote by Ex fy E An : 7-(y-x) 01 the affine hyperplane through x and let o'o/ E An. Then the vector spaces associated with all these affine hyperplanes E., (x E An) are identical. In fact, they are given by -F-1(0) 0 1. Hence we only have to specify one single Euclidean f v E Rn : -r (v) scalar product .),-, (0) on the vector space -r- 1 (0) in order to get a -
-
=
=
=:
=
foliation of
with one
our
(n
-
l)-dimensional
Euclidean spaces. This is in accordance of space does not change from
experience that the geometry
instant of time to another.
The map -r can be interpreted between to events x and z is just
as a
T(z
world clock: The time difference -
x).
Observe that
defined up to a factor. This factor corresponds to the which time is measured. We still need to link events in different
spond
to the
same
point
in space. The
is
uniquely physical unit in T
hypersurfaces which simplest way to do so is to
corre-
intro-
1.2 Absolute space and-absolute time
duce
line
as a
x
by -r(t)
t
second structure
+ Rt
as
the
a
vector t and
interpret all points lying
in space at different times. If
point
same
I then the time difference between
=-
Definition 1.2.2. A Newton
spacetime
is
and y
x
=
we
x
where t E Rn is =
the
+ tt is
just
t.
quadruple
a
(1.2.6)
distinguished vector, -r: R' ---),_ (O) is a scalar product on
R
a
1, and (.,
on
normalise
(An, t',r'
,r(t)
19
linear map such that
a
the vector space
This definition is just the content of Isaac Newton's of absolute time and absolute space.5
7-'(0)..
(1642-1727) theory
1.2.2. Absolute space, abso-
Fig.
lute time
We
see
that
is fibred
spacetime
twice, By lines parallel
hyperspaces
of the form
may appear
quite cumbersome but
in
a
geometrical
One
as
defining
notion of rest, and think of
of v
o
it
is
some
to t and
by
fixed event. This structure
captures
naive
our
point of view
way.
think of t
can
where
E,+tt
-r
as
pair (-r, t) induces A +V and -r(,U) 0.
time. The
a
a
time axis and therefore
an
absolute
defining an absolute notion of instant projection -:: R' -->, R', v 1--* 6, where
=
=
A map which leaves the structure of a Newton spacetime invariant a spatial Euclidean transformation as given in Proposition
consists of 1.1.2 and
a
spacetime translation. 1.2.1. A map
Proposition
(An, t,,T, (.' 1 -r- (0)
A:
-4
-F-
1
(0)
and points o, b
(i) O(x) A(x-o) (ii) (Au, Au),-, (0)
+
=
5
V): A' --+ An leaves the Newton spacetime if and only if there exist a linear map
invariant
In the next section
=
we
E
An such that
T(x-o)t + b and u, u),-, (0) for all
will discuss
an
u
E T_
1
(0)
-
improved spacetime model
named after Galileo Galilei who lived before Newton. The is that Calilei
reason
which is for this
emphasised different points than Newton, points which are more important to us nowadays. However, what will be referred to as a Galilei spacetime also incorporates ideas due to Newton.
20
1.
Local
theory
of space and thne
It is easy to check that maps of this form are isomorphisms of NewConversely, observe that any affine map '0 which maps any affine hyperplane E,, into some affine hyperplaneE,,, is necessarily of the form
Proof
ton structures.
O(x) where A is
a
linear map of
That A satisfies
follows from
Proposition
(0)
into
-F(x-o)v itself,
=
+ b E
v
R'*
(u,u),-,(O)
=
1.1.1 and the fact that
0
and o, b c A'.
for all
-r-'(0)
E
u
restricted to Ex is
invariant of the Newton
an
implies 0 (o + t) 0 (o) this equation is equivalent to v
o+
t. But
1
T
+
an
spaces.
Since the vector t is t- o
-
(Au,Au),-,(O)
isometry of Euclidean tion
A(x-o)
=
t
Observe that the choice of
is
o
spacetime the
t and therefore
-
(-r (t) v
+
equa-
b)
-
b
t.
irrelevant,
it
always
can
be absorbed
by
b.
We call the set of all
isomorphisms 0
Newton group JV. Given a Newton spacetime respect to which T (1, 0,
we can
find
of the Newton
a
basis
spacetime the
lei,. e,,J of R' with 0) T, and (u, v) (0) ..,
t
=
.
=
(1, 0,
I
Eni,j=i 6ijuiVi.
1.2.1 Non-relativistic
particles
Here
we very briefly indicate elementary aspects of Newton's theory of particle mechanics. We will only touch on those features which are necessary for later sections. This section is included for the benefit of
mathematicians.
A particle is
thought
to be
a
exterior structure. This is of
small material
course a
object without
interior
gross idealisation of many
or
macro-
scopic objects, but for
some purposes surprisingly good. Billiard balls typical examples. On the other hand, one cannot neglect the internal structure of a football. It will be noticeably deformed when hit. This contributes to its springiness and at the same time shows that the particle model is not adequate. An American football has a shape which are
contributes to its movement when it rolls
particle description would
be
on
a
flat surface.
bad approximation. Newton observed that even if all its structure
material ment in
object does
carry
a
Again,
a
neglected,
a
a
can
be
parameter which characterises its
spacetime. This parameter is its
mass.
move-
1.2 Absolute space and absolute time
Definition 1.2-3. A non-relativistic particle with mass (m,,y) where m E R+ and -y: t --> - (t) G An satisfies T( -y is called its world line in
curve
21
is
m
(t))
=
a
pair
1.
The
'
spacetime.
It has been first
expressed by Galilei (cf. Sect. 1.3) that under ideal particle which is not subjected to any external force moves straight line. 7
conditions
along
a
a
Definition 1. 2.4. A non-relativistic inertial
particle (m, 7) which satisfies It is clear that x
t(t
+
7('Y)
+
Y)
is
a non-
relativistic
some
0.
=
Of
special
interest to
derstand under confined to in the
particle
0.
non-relativistic particle is inertial if and only if -Y(t) point x E A' and some constant vector U with
a
for
=
compact subset of spacetime. It is best to think of collisions
a
sense
collisions of inertial particles. We unparticles any interaction of them which is
us are
collision of
a
of
colliding
billiard balls. But
we
explicitly
allow that par-
ticles break up or stick together. Since the collision is confined in space and time it is possible to speak in connection with a collision of incoming
and outgoing inertial particles. Let (Tni, ^ i)i=1,_k denote the incoming the outgoing inertial particles. Then particles and (m 3, 7
inertial the
following
laws
experimentally well justified.'
are
I:k
(i)
Conservation
of
(ii) (iii)
Conversation
of spatial of kinetic
Conservation
mass.
Eli momentum. I:k i= 1
Mi
=
j,
i=1
M/.,Yf 3
Mi7i
energy.
k
1
1
E2
M,
Mi
1
oli"07_1(0)
=
i=1
1: 2
i( j, /
M
j=1
It is easy to see that these laws are invariant with respect to isomorin the Newton group A(. It is clear that most particles do not move along straight lines. In
phisms this
case an
to take
a
external
force
must act
on
the
particle
in order to force it
different path.
Definition 1.2.5. A
(time dependent)
force field
_P
is
a
map F: An
7-1(0). We
only need -r( (t))
> 0 in order to
its future. The normalisation
guarantee that the particle moves into I synchronises each particle with
-r( (t))
=
the world clock t. 7
8
It is not
absolutely clear whether Galilei really meant straight lines or more complicated curves which take into account the shape of the earth. These laws are intimately linked to the homogeneity of space and time. This is the content of the Noether Theorem. For further details cf. any textbook on
(theoretical)
mechanics,
1. Local
22
of space and time
theory
given force field f equation
In
a
particle
a
-y
according
moves
to the differential
my= F.
(1.2.7)
In
particular, vanishing force implies that -y is an inertial particle. According to the physical interactions under consideration a particle may also carry a variety of other parameters besides m. As an example consider an electrical field f : A' ---+ -F 1 (0). Every particle (m, -y) carries -
another parameter q which determines the force with which the electrical on the particle, f qf.
field acts
=
1.3 Galilei's
theory of relativity
In this section
drop
of the structure of Newton spacetime in theory of relativity. We also argue that his revolutionary given the paradigms of the time. we
some
order to arrive at Galilei's
theory Galilei's
was
of
theory
has been motivated
relativity
not feel that the earth
moves
into any
by cosmology.
We do
preferred direction. It is therefore
plausible to believe that the earth is at rest and that all objects at the sky are moving around it: The sun rises in the East and during the course of a day moves to the West, and there are analogous descriptions of the movements of the moon and the stars. It was already well known that planets are not moving along strictly circular orbits. In the traditional cosmology of the Greek astronomer Claudius Ptolemeaus (ca. 100-160) this was accounted for by an elaborate construction using epicycles. It was a revolutionary act of Nicolaus Copernicus (1473-1543) to assert that the
around the
did
so
sun
sun
is the centre of the universe and that the earth is
just like
any other
in order to arrive at
a
planet
or
star
moving
(Copernicus 1543).
He
model in which movements would be theo-
retically more uniform and which would therefore be in better accordance with the teaching of the ancient Greek philosophers Pythagoras (ca. 570 b.C.-500
b.C.)
and Platon
1984)). However, (using more epicycles
his model
of serious
(i)
If
than
(ca.
428 b.C.-347
was
not
b.C.) (cf. (Kanitscheider
only technically
Ptolemeaus)
more
complicated
but also encountered
a
number
problems.
Copernicus
was right one should be sphere of fixed stars. If
able to discover
a parallaxis sphere and the earth rotate both around the sun with different velocities, then one should observe different angles a, 0 between two neighbouring stars according to the time of the year. (Cf. Fig. 1.3.1). (ii) Some passages in the bible seem to contradict the theory of Copernicus. In particular, it states that Joshua stopped the sun for a few
effect at the
the fix star
hours. This statement would not make have moved before.
sense
if the
sun
would not
1.3 Galilei's
(iii)
The model of
Copernicus
sun
the
other exception
was
around the
in the
sky
seems
to be
moves more
theory
of
relativity
23
inhomogeneous. While all planets circle definitely moves around the earth. No known. According to Ptolemeaus, every object is
moon
around the earth. Hence the traditional system homogeneous on a large scale and therefore to be
advantageous.
(iv)
The laws of mechanics
Imagine
(being
a
stone
seem
falling
fixed to the
to contradict
from the top of
ground)
would
move
a
Copernicus' hypothesis. tower. Since the tower
together
with the
earth,
would not expect the freely falling stone to hit the ground at the foot of the tower. However, exactly this is everyday experience. one
(Cf. Fig. 1.3.2).
Fig.
1.3.1. Parallaxis effect
Fig.
1.3.2. Tower
example
Problem
(i) has been addressed by Copernicus himself. He just assumed sphere of fixed stars is so large that the parallaxis effect cannot be measured. Ironically, the true radius is orders of magnitudes larger than the radius he proposed. (He was just concerned with making the effect unobservable). The other three objections have been answered by that the
Galileo Galilei Calilei
was
(1564-1642)
church authorities effect that
some
60 years later.
least successful with Problem
(for instance,
(ii).
While he could quote
Aurelius
Augustinus (354-430)) to the bible literally when it comes to
one should not interpret the questions of physics, the establishment remained unconvinced. One of the reasons has been the fear to set a precedence. If people started to doubt any part of the writing they could as well start to be sceptical about other parts which are closer to the main doctrine. Hence there was a major threat to the whole building of Christian belief. The theory of Copernicus was put on the index and Galilei after having written -
1. Local
24
a
theory
of space and time
brilliant but rather defiant
matter
mild
-
semi-popular
book
(Galilei 1632)9
sentenced to house arrest. He obtained this
was
punishment
after
a
public but
on
the
comparatively
insincere abdication of his scientific
assertions.
Galilei solved Problem
(iii) by careful observation (Galilei 1610).
telescope had just been invented and Galilei was as a scientific tool. He observed that the planet
one
of the first to
The
use
it
Jupiter also has moons show that the to and used this observation cosmological system of Ptolemeaus of the universe was not more homogeneous than the system of Copernicus. On the other hand, since it was believed that beyond the moon the world was filled with a medium very different from air, many philosophers doubted the accuracy of the telescope. They claimed therefore that it was doubtful that the telescope which was acknowledged to work well on earth could be trusted when applied to the position of planets. Galilei argued that the telescope was accurate with respect to all the known phenomena in the sky and that it was therefore justified to
use
it
as a
scientific tool.
of inertia which asserts that a constant movement had no influence on physical processes. Galilei supplemented this law with the important physical assertion that complicated velocities can be decomposed into simpler ones. According to this law the stone would keep its initial tangential velocity while falling down and therefore come to rest at the foot of the tower regardless of the velocity of the earth. It can be argued that this solution of the problem was the most revolutionary act in natural sciences and started physics as a scientific discipline in the modern sense. Recall that everyday experience seems to point against Galilei's law of inertia: If we set a wagon into motion it will certainly come to a stop after some while. Moreover, there was a generally accepted physical theory by Aristotle (384 b.C.-322 b.C.) which explained this experimental fact. (The wagon has an initial impetus which is responsible for the movement and which is used up during the motion.) Galilei gave many examples to make his law of inertia plausible and to show that it is a law for a limiting case without friction. For instance, he claimed that a stone falling from the mast top of a smoothly sailing ship would also reach the ground at the foot of the mast-10 Galilei solved Problem
(iv) by asserting
a
law
-
9
10
This book is
a
literary and physical
master
piece.
Even
today
it is well worth
reading! As compelling this example may appear to us, at the time there were some good reasons to doubt it. Since the velocities involved are rather small it would be difficult to verify Galilei's claim experimentally. Also, while the wind is blowing into the same direction. it is conceivable that the just blown to the right position. (To value the merit of such counter arguments one has to be aware that at this time, good, quantitative physics has not yet been available). Some of these arguments have already been answered by Galilei, who, for instance, circumvented the wind argument by
ship
moves
stone is
1.3 Galilei's
theory
Galilei realised that his law of inertia is not notion of absolute rest. Instead he
postulated
a
of
relativity
25
compatible with the principle of
fundamental
relativity. Postulate 1.3.1 move
the
are
(Galileian relativity).
relative to each other with constant
For any two observers which all physical processes"
velocity
same.
It follows that the vector t in the definition of Newton
defines absolute rest does not have of the a
originality
of Galilei that
spacetime which another sign
physical meaning. (It is Newton thought he had to a
re-introduce
concept which had already been shown to be superfluous). While we have lost the notion of absolute space we can still retain
absolute time. obtain the
Spacetime is then fibred by hyperplanes following simpler structure of sPacetime.
Definition 1.3.1. A Galilei spacetime is
(An, where
Rn
7-:
product
on
R is
--
a
T,
non-zero
the vector space
(., .)
_
,
a
(0)
const and
t
we
triple
)
(1.3.8)
,
linear map and
is
a
scalar
T-1(0).
.40.00.."M
Fig.
1.3.3. Relative space, abso-
lute time
The linear map
-r
defines
a
world clock
time difference between to events
model
by defining 7-(y
-
x)
to be the
and y exactly as in the Newtonian above. In contrast to Newton's spacetime we do not have
presented
x
-
the vector field t at
our disposal and therefore there is no absolute rest replaced "absolute space" by a distinguished family of systems" or "inertial observers". The notion of "Rest" can only
space. We have
"inertial
be defined relative to
an
Definition 1.3.2. Let
(i) 'T
"inertial observer":
(An,,F, (., .),-,(0))
A non-relativistic observer is
( M)
=
I
be
a curve
a
Galilei spacetime.
7: t
->
-y(t)
E An
such that
-
claiming that the physics in a cabin of a smoothly sailing ship would exactly the same as on earth. Strictly speakingl'he only considered mechanical processes.
be
26
1.
Local
(ii) A -/(t) (iii) A
theory
of space and time
non-relativistic inertial observer -y is
==
+
x
tt, where
R', -r(t)
t E
non- relativistic
a
curve
of the form
1.
=
observer y is at rest with respect to
relativistic inertial observer
-y(t)
=
x
if A(t)
+ tt
a non-
t.
Hence
given a non-relativistic inertial observer 7(t) x + tt we obtain splitting of spacetime into space and time relative to -Y. Physically, this amounts to regarding the observer -y as being at rest. We can also a
interpret
t
as a
observer -y
relative time axis. Relative to the
non-
have thus recovered the structure of
we
Notice, however, that this one non- relativistic
is
a
relativistic inertial
Newton spacetime.
only possible by arbitrarily distinguishing following defi-
inertial observer. This motivates the
nition.
Definition 1.3.3. Let t is called
R' be
E
a
vector with
T(t)
1. The
=
For any via the
given reference frame (t, -F) we obtain a map Y + A where -F(V) unique decomposition v =
1.3.1. A
Proposition time
A:
(An',r'
T-1(0)
pair (t, -F)
non-relativistic reference frame.
a
(0),
T
>
---
a
(i) O(x) A(x-o) 1, and (ii) T(v) (iii) (Au, Au),-,(O) =
An
0:
map
--+
invariant
if
vector
Rn,
+
v
R'
=
G
-r(x-o)v
+
and
A'
leaves the there
only if
and points o, b
are
cz
F-+
Rn- I
0.
Galilei spacea linear map
An such that
b,
=
(u, u),-,(O) for
=
all
u
Cz
T-1(0).
Proof. It is straightforward to check that maps of this form are isomorphisms of Galilei spacetimes. Conversely, observe that any affine map 0 which maps each affine hyperplane E,, into some other affine hyperplane E,,, is necessarily of the form
O(x) where A is Since
V)
A'. Hence
,T
a
=
linear map of
preserves we
(A (X
which in turn
r we
A(x-o) -
1
T
(0)
have
+
into
T(x-o)v itself,
v
-r(O(x)-O(o))
+ b
and o, b c An. T(x-o) for all x,
E R' =
o
E
obtain
-
0)
+
T(X-O)V
implies -F(v)
=
+
b-b)
=
-r(x-o)-F(v)
=
7-(x-o)
1.
The third property follows since A must preserve the Euclidean scalar I product The Galilei group 9 is the group of maps which leaves the Galilei spacetime invariant.
1.4 Einstein's
special theory
It should be noted that the Galilei
Newton's
of
theory
spacetime
of
is
relativity
compatible
27
with
described at the end of Sect. 1.2. The
particles well accepted as the correct model of space and time for more than 200 years. However, in the 19th century a theory of electro-magnetism emerged which, together with this spacetime model, was incompatible with Postulate 1. 3. 1. Still, scientists continued to think that the postulate would hold for mechanical processes. Galilei spacetime
1.4 Einstein's
as
was
special theory
of
relativity
We start with a discussion of the fundamental Michelson-Morley Experiment which indicates that the velocity of light has an absolute value c. These findings indicate that the set of all possible light rays form a further invariant of nature. We will see that this leads to the structure of a Minkowski spacetime (Theorem 1.4. 1), or, equivalently, to Einstein's special theory of relativity. We use the results from the two preceding sections to show that there is no need for additional structures in spacetime. In Sect. 1.4.2 we give a short discussion of some consequences of special relativity such as the "Twin paradox" (which, of course, is not paradoxical at all). The proof of the fundamental Theorem 1.4. 1 requires section 1. 1. 2
to
ti
t2
Fig.
1. 4. 1. A flash of
light
at times to
=
0,
t17 t2
In the 17th century two
In 1676
Olaf
important properties of light emerged.
R&mer discovered that the
by noticing that the there of the moons of Jupiter. did this
was a
velocity of light is finite. He yearly oscillation in the periods
physicist Christian Huygens (1629-1695) developed a theory of light (Huygens 1690). In a very superficial way, we may light as an analogon to water waves.12
The Dutch wave
view
paragraphs should not be taken too seriously by the overly simplified version of the wave theory of light just enough in order to understand the Michelson-Morley experiment presented below. Moreover, today the theory of quantum electro dynamics provides a much deeper understanding.
The
following
two
reader. We give -
an
28
1.
Local
In water cle
theory
waves
of space and time
each
thereby inducing
a
"drop
of water"
individually
similar movement
(with
a
moves
in
small time
a
cir-
delay)
neighbouring drops. All these moving drops together form a wave (cf. Fig. 1.4.2) Since each drop is influenced by the neighbouring drops these time delays accumulate and the whole wave seems to move. If two different waves meet then (in a very rough approximation) they simply linearly superpose each other.13 This will result in a characteristic (and often complicated) pattern, the "interference pattern". In particular, this superposition will result in a much larger wave if both waves are synchronised and in the other extreme they may cancel, Diffraction experiments indicate that this crude picture qualitatively also applies to light for which, however, matters are mathematically simpler. Again using a very rough model, one may think of the electrical field E at each point as oscillating up and down with respect of a fixed direction. The influence of neighbouring points gives rise of to a wave as described above. The wave length \ is the distance between two consecutive maxima and very small. It specifies the colour of light. If two waves are superimposed then the result may be brighter if they are synchronised. In the other extreme, the waves may even cancel altogether if the setup is arranged such that maxima and minima (of the same size) are superimposed. In this case the result is darkness. (cf. Fig. 1.4.3). In order to explain the wave nature of light one used to believe that space is filled with a substance called "ether" which plays the same r6le as the water for the water waves. An important problem would then be of
to determine the movement of the earth with
Fig.
1.4.2. A
respect
to the ether.
consisting of linked
os-
directionally dependent,
in
wave
cillations
Since the
velocity
of
light should
not be
the non-relativistic reference frame connected with the ether
a
flash of
light should propagate in concentric spheres (cf. Fig. 1.4.1). The corresponding picture in spacetime would be a cone. To be more precise, consider the non-relativistic reference frame of the ether, given by the pair (t, -r). Let o be the event at which the flash of light is emitted and For water waves, this linear superposition is in fact mation.
a
rather bad approxi-
1.4 Einstein's
1.4.3.
Fig.
Superposition of
special theory of relativity
29
waves
Future
light
Galileian
rela-
Fig.
1.4.4.
cone
and
tivity. The observer moving with spatial velocity V measures a
different centre
OV of the flash of light and therefore different radii di, d2 for its wave front
E,,
Ix
=
have
a
can
v
v
An
:
be
-r(x-o) 01 be the instant of time defined by o. We of A with spatial hyperspaces. Each vector f Eo+ttjtGR =
n
uniquely decomposed
into
0. A
light
-r(v)t+V where
-r(,U)
spatial velocity
6 describes the
==
the
G
foliation
respect to the
CO+
=
to
a
C-
spatial and temporal components, ray which is sent out at
curve o,
+
reference frame fixed to the ether
future light
y
=
An
:
R(t + cl a
in A n
x .
flash of light
C
EO with
Hence with
corresponds
cone
IICI12_1(0)(,T(Y_O))2
=
Y_01 ))7-1(0) Y-0
I
7-
(Y X) -
-> 0 -
I
C + X-0 of fuspacetime. The fact that the field x --> C: with Galilei to transformations invariant not are cones respect light the would enable reference frame of the to one measure (cf. Fig. 1.4.4) earth with the of the the ether. This was movement to ether, Le, respect in
=
ture
the aim of Albert Abraham Michelson
(1852-1931) and Edward Williams Morley (1838-1923) (Michelson 1881), (Michelson and Morley 1887) in their famous interference experiment (cf. Figs. 1.4.5, 1.4.6). A light ray is
partially reflected
at
a
half silvered mirror H. The part of the
light
ray which is not reflected at H is reflected at a mirror M and then partially reflected at H before reaching the observer 0. The part of the ray
which is
immediately reflected at H is reflected by a mirror M' and then passes through H to arrive at the observer 0. The distance
(partially)
between M and H is 1 whereas the distance between H and M' is 1'. Both
light rays have the same intensity when they arrive at 0. Here they produce an interference pattern which allows to measure the difference
30
Local
1.
theory
of space and time
0
H
lamp
1
H,," H 11 H H11
lamp
M
H MMU
0
M/
M, MIMI
Fig. 1.4.5. Michelson-Morley experiment, at rest relative to the
Fig. Michelson-Morley experiment, moving relative to the
ether
ether
1.4.6.
of the distances which each of the fact that the
wave
light ray has travelled (Here one makes use lengths of visible light are extremely small
and that the superposition effect allows to measure the distance which a light ray has travelled at an accuracy of half a wave length). Since the
laboratory is relativity the
at rest with
HM and the
velocity of the earth. Let c (E R+ be the modulus of the light in the ether, Y be the velocity of the earth relative to
velocity
of the
the ether and
the
ct,
1 +
jjVjj,-1(0)tj.
the distance Ct2
:--=
1
-
7--
1
The other part of the same
Y
(cf. Fig. 1.4.6).
in time tj and
_
1/2
2
21/c 11,U112,r_,(O)IC2
+
cover
(1.4.9)
'
to M' and in
Jjyjj2_j(0)tj2.
gives
21'1c
2t'
(1.4.10) C2
(0)
We
The first part of the distance
cover
light ray travels in time t' from H H, thereby covering the distance
time back to
2ct' This
11
M,
earth, according to Galileian depend on the angle between
If it travels from M to H in time t2, it will jj'Ujj-r-1(O)t2- These equations imply
tl + t2
the
to the
first that HM
assume
ray will travel from H to
light ==
respect
interference pattern should
are
interested in the time difference zAt
Since the number
I I 61 I
=
2t-
(tl + t2)
for both
paths.
is very small we only need to calculate the , (0) Ic time difference to second order in I JUj Jr 1 (0) 1c. -
1
-
2
At
=
tl + t2
-
2t'
---
I
-
11
1
-
JjUjj2_1(0)1C2 Ir
-
C
1
11,6112
Ir
_I
(0) /C2
1.4 Einstein's
.j(jj,yjj2_j(0)/C2)j/ I 11,611 2(0) /C2
21+
2
C
of
special theory
relativity
31
j/
_
7'
-
-r
2
(_1
jj
j(jjyjj2_j(0)/C2 + jjjyjj2_j(0)/C2j Jjqj2_j(0)/C2
+
2
C
2
7'
2
1
1
(jj (_1 (jjUjj2_j(0)1C2) +
C
2
+
jjUjj2_j(0)1C2j
T
1
11,U112 -1(0)
2JI
C
C2
C
1 + J1. This
I
11,UI12_1(0)
2
C2
+
gives
a
displacement
C, At
zAZI1
2
(I jjUjj2_j(0)/C2)
X
where 1'
+
2JI
(0)
(
::Z
=
A
A
C
2
per
C
wave
length
JJV112 I + 2
21(o)
C
A of
)
-
depends crucially on the length difference accurately enough. In order to overcome this difficulty Michelson and Morley turned the whole setup by 7r/2 and It follows that the interference
51 which cannot be measured
measured the interference difference. For the rotated setup we must set 2t' ,At tj t2 and interchange 1, 1'. An analogous (but in the details =
-
-
calculation
slightly different)
C
zAZ_L
The relative
1
jjqJ2_1 (0)
2(-61)
A
C2
C
At -Z
gives 7'
+
-
,
A
displacement depends
on
61
21
ljqJ2
A
C
C2
only
up to second order and is
given by AZ
If
one
assumes
spectral
;:
-zz
-
AZ_L
sun
5.461.
1(0)
1
-
2
rests relative to the
10-'Om and has I
experiments had
61) 1
ether,
a negative possible explanation of this negative outcome
uses
21m then
0.4 which is well in the range which
all such
However,
AZI,
that the
line with A
obtain AZ
A
=
can
the
one
Hg
would
be observed.
outcome.
is that the earth rests
with respect to the ether. But the earth circles around the sun which itself rotates in our galaxy. Since in the course of the year the earth changes its
velocity
direction relative to these other
that all year round the
neglected. light
it is inconceivable
can be velocity explanation would be that light moves like particles Galileian relativity would apply to light as well. Since
Another
and that therefore the
velocities,
of the earth relative to the ether
was
from earth bound
sources
and the observations have been
32
Local
1.
theory
of space and time
made
on earth, this assumption would explain the negative outcome of experiment. The Michelson-Morley experiment has therefore been
the
repeated using
light
again with negative results. Now one could starlight is reflected at mirrors fixed to the earth, the light reflected light should be viewed as being produced on the earth. This would explain even negative results for star light in the framework of Calileian relativity (Hasse 1995). While this explanation is conceivable, it would demand a new theory of reflection. It is much simpler to assume that the velocity of light is independent of the movement of its source. This is the traditional interpretation which we will adopt in this book. It has been given further support by many consequences of the resulting theory (for instance, the possibility of obtaining huge argue that
star
as soon
as
-
the
amounts of energy from nuclear fission and nuclear
Since
fusion).
interpretation of the experiment of Michelson and Morley is in contradiction to Galilei's theory of relativity we have to reconsider the foundations of spacetime. In order to do so we start with our new
insight
our
about the nature of
future
light cones is an following we will chose c
=
2.99792458
light propagation,
invariant structure of units such that
c
i.e. that the set of
possible spacetime. Here and in the
(In
1.
=
the
SI-system,
one
has
108M/S.)
-
Postulate 1.4.1
(Invariance of the future light cones). Spacetime can be identified with An together with an field of future light cones C C + x-o, X E An.
invariant
=
We start the
investigation of this postulate, by first determining all maps light cone structure invariant. To simplify the dischoose again a non-relativistic reference frame (-r, t) and de-
which leave this future cussion note
we
by (-) the induced projection
of Rn to
-r-
1
(0). Defining
the bilinear
form 770: Rn
we can
C.,
=
x
Rn
R,
(u,v) i-4,q(u,v)
--F(u)-r(v)
=
ly
E
An
:
77(y-x, y-x)
light
cone
and
=
0 and
C1
Cx-
the past
Cx
y E
X
light
cone
transformation which leaves the field of the future
must also leave the field of
form qo is
a
light
Minkowski metric
as
cones x
-*
This
means
that there is
an
Cx
at
:
-r(y-x)
01.
It is clear that
x.
light
<
cones
invariant
invariant. The bilinear
defined below.
Definition 1.4.1. A Minkowski metric 77 is Rn with signature (-, +, ..., +).14 14
+
write
We call Cx the a
-4
a
constant bilinear
"orthonormal basis"
as
defined
form
directly
on
below.
1.4 Einstein's
of
special theory
33
relativity
Jeo,..., en-11 of R' is called an orthonormal. basis with re-60,, and i7(ej, ej) 6ij for all i, i E tl,.. n if 77(eo, e,) n 11. f 0,.
A basis
spect
to q
11,
E
a
=
=
..,
A Lorentz transformation is
,q(u, v)
-
-,
-
=
q (Lu, Lv) for all
u,
v E
a
linear map L: Rn -- R' such that of all Lorentz transforma-
Rn. The set
tion is the the Lorentz group and denoted now easy to find invariant. Let a E R
It is
by O(x)
a
class of maps which leave the light b G Rn, 0 E An, and 0: Rn
+ b. It is immediate that
isomorphisms
are
of this
structure
Rn be
O(Cy)
n and that therefore the transformation 0 satisfies y E R requirement. The following theorem due to Alexandrov
that all
cone
\ 101,
aL(x-o)
=
by O(n, 1).
CO(y) our
given
for all
invariance
(1950) implies
form.
special
be a bijective map such --+ Rn lightlike vectors, 77 (x y, x y) 0 and analogously for 0'. Then there 0 = > ?7(0(x)-O(y), O(x)-O(y)) exist an L E O(n, 1), an a E R \ 101, an 0 E A', and a b E Rn such that TP--TT]1 5 aL(x-o) + b for all x E An. O(x)
Theorem 1.4.1. Let that
0 and 0
map
n
> 3
lightlike
and
0:
Rn
vectors into
-
-
=
=
=
-
1
There
are
several
proofs of
this result. For
instance, Benz (1992)
that the theorem follows from the fundamental theorem of
His
proof rests
on
Laguerre
ge-
who gave a particularly elegant affine results in geometry which have been given
ometry. We will follow Alexandrov
proof.
shows
(1975)
in Sect. 1.1.1.
0 and by Proof of Theoreml-4-1. Let y E Q,. Then 77(y-x,y-x) 0. The last equality implies assumption q(O(y)-O(x),O(y)-O(x)) 0(y) G Ck(x) and therefore that 0 maps generators of the light cone Cx into the light cone C,6(,,). Now assume that P is a two-plane which intersects C., in two genera1., tors lx lX. We will show that O(P) is also a two-plane. Since for any y 1,, and Py which is parallel to the cone C. intersects P in generators 1. 1',, (and similarly for y' E 1.,), P is ruled by two different families of parallel generators. Since 0 maps generators into generators O(P) must also be a surface with two different rulings and, by Theorem 1.1.2, be affinely equivalent to either a plane, a rotational hyperboloid, or a hyperbolic paraboloid. Each generator in P of one family intersects all generators of the other family. Since in a rotational hyperboloid the generators at opposite points of the circle c(s) lie in parallel planes, O(P) cannot be affinely equivalent to a rotational hyperboloid. To see that O(P) cannot be affinely equivalent to a hyperbolic paraboloid note that in a hyperbolic paraboloid all generators of a given family are parallel to a single 2-plane and that this property is affinely invariant. Consider 0(y) c 10(x) =
=
=
15
The
proof of this theorem requires the material presented proof is essential for the following.
The theorem but not its
in Sect. 1.1.2.
P. 34
34
1
Local
-
and fix any C,6(y) which
along lo(,,)
theory
of space and time
2-plane Q. Then there are exactly two generators 11, 12 Of are parallel to Q. If we now (parallely) translate the cone
we see
that the generators of the translated
cone
which
are
parallel to Q must also be parallel to 11 and 12. This implies that O(P) is generated by a family of parallel lines. Hence the hyperbolic paraboloid degenerates to a plane. We can now show that 0 is an affine map. Let 1 be any line and x E I and consider two different planes P1, P2 which contain 1 and intersect Q, \ fxJ. The intersections of these planes with Cx consist of two generators each. Then O(Pi) and O(P2) are also planes and their intersection 0(l) a line. It follows from Theorem 1.1.1 that 0 is affine. Since the property
?7(y-x,y-x)
0 is translation invariant is linear. Let t be
vector with
a
77(e,e) Then 0
0
=
be
0
77(t, t)
=:>,q(O(y)-O(x),O(y)-O(x)) =
generality
-1 and let
n(e, t)
1,
=
=,q(ct, et) implies 0(e)) -77(0(t), 0(t))
=
without loss of
we can
c
be
that
vector with
0.
=
(1.4.11) andtherefore
=
=:
assume
a
a,
n(o(e), 0(t))
=
Any vector v can Equations (1.4.11)
0.
decomposed into v v,c + vtt, where e satisfies and v, vt E R. We obtain q(o(v), 0(v)) aTI(v, v) which implies that 1 0 leaves the quadratic form associated with q invariant. But then it must also leave Tj invariant. I =
=
[I
p.
401 Definition 1.4.2. Let 77 be called Minkowski spacetime.
Minkowski metric.
a
The pair
(An, n)
is
Using orthonormal bases it is easy to see that all Minkowski spacetimes are isomorphic, i.e., we can speak of "the" Minkowski spacetime. In a Minkowski spacetime there is no designation of future and past.
(Observe that we needed the 1-form -F in order to define the future direction.) Observe that the set C,\ I ol consists of two connected components, say C0+ \ f ol and Co- \ f ol. We may now choose C (respectively, CO-) as the set of events in Q, to the future (respectively, past) of o (including o). Hence C0+ is the set of all events which can be reached by a light ray with
in
source
o.
By continuity, this
at any other event
x
E
An where
definition coincides with v
E Rn
(i) (ii)
a
w)
q (v,
> 0
or
we
vector
v
w)
can
E
< 0
w
alternatively Rn with
v)
for all
for all
==
(x-o)
=
w
G
E
C
< 0.
C
Then
we
and q (v,
and 77 (v,
w)
C0+-
+
previous definition
our
vector with q (v,
either 71 (v,
Hence a
be
also determines the future direction
C.+
Observe that this
in the
w) < 0
> 0
for all
for all
define the future direction
q(v, v)
<
0. Since it is
Let
case
have
more
with vectors than with connected components of chooses this alternative definition.
light
CoCo-.
w G
w c
by singling
practical
out
to work
cones one
usually
1.4 Einstein's
ts an
all v, w E Given o
equivalence
Let
a
35
class
[v],
time orientation
setCo-
)5
relativity
[v].
E A is the set
is the
of
(A',,q) a Minkowski spacetime. A time orienta[v] of vectors in Rn such that 77(v, w) < 0 for
Definition 1.4.3. Let tion
special theory
==
we
say that the
C f o + w C,, q (w, v) :5 01 fo+w EQ, : 77(w,v) >01. E
=
:
be the invariance group of the
light
future light cone at light cone
and the past
structure and P be
cone
the group of Poincar6 transformations, P fx --> L(x-o) + b : b c orientation Given 0 L an E [v], we call P+ P+ n P (m, 1) 1. Rn, 0 E An, =
=
the group of time orientation preserving Poincare transformations. The discussion above suggests to reduce the group )5 to a subgroup 73+ by asserting that the elements of 13+ map future light cones into future light cones.
(An, 77) JbE
Lemma 1.4.1. Let
J5+
entation. Then
Minkowski
a
b(C,+,)
=
=
spacetime and [v] a time CO+v)(0) I is a subgroup of
ori-
Proof. Clear.
The
Lemma 1.4.2.
there is
only if there
is
R
transformation 0 is an element of P if and only if and a 0 E P such that 0 ao; 0 E )5+ if and E R+ \ 101 and a 0 E P+ such that 0 ao.
\ 101
=
=
an a
This follows
Proof. A
G
an a
directly
form Theorem 1.4.1.
it is conceivable that there exist other fundamental invariants
priori
of space and time which would restrict the group even further. We will now show that because of the validity of Euclidean geometry and the
principle
of Galileian
relativity
1.4.1. Fix
Proposition
1.3.1 this is not the
a non-
relativistic, inertial observer
and consider its associated Newton spacetime each
x
E A let
E
=
yE
An
:
case.
F(y-x)
=
(An, t,,r,
t
o
+ tt
(0)).
For
-->
01.
metricq which generates the light cones -* o + tt. as Further, this Minkowski by metric is unique up to a multiple. (ii) Let x cz An The map 0 E )5 restricted to the Euclidean space E.,, (., .),-,(0) is an isometry if and only if 0 E P.
(i)
There exists
a
Minkowski
the observer t
measured
.
(iii) Let P' be a subgroup of the Poincar6 group P such that E., there exists a 0 (a) for each Euclidean isometry 0: E with
(b) for
01
All
0
E
a
0
(E
P'
V),
each non-relativistic observer t
there is
(c)
=
E
P' with
O(o
+
R+t)
=
o' + tt' with
o' +
P' preserve the time orientation.
R+V,
77(t', V)
< 0
36
1. Local
Then P'
theory of
space and time
'P+.
=
Proof. (i):
The adapted Minkowski metric q is just given by q (u, v) Observe that T(w) -7-(u)-r(v) + -77(,T, w) for all W E Rn. (0) (ii): This follows directly form Lemma 1.4.2. (iii): We will first show that for any two different vectors eo, eo with -1 and 77(eo, e0 ) < 0 there is a Lorentz transfor71(eo, eo) q(eO, eO) mation L which leaves spanf eo, e'01 and its q-orthogonal complement spanf eo, e'01 Jf w E A : q(w, eo) q(w, eo) 01 invariant. There is a vector v c R' with 0. Since q (eo, v) we have eo I I eo + v and q (eo, v) 77(v, v) > 0 and the vector el vlV'rq(v, v) is well defined. By definition it satisfies 71(eo, el) 0 andq(el, el) 1. We complete the set of linearly vectors to an 77-orthonormal basis f eo, independent f eo, el I el, e,,,
=
=
-
=:
=
=
=
=
=
=
=
.
of R'. Let L be the Lorentz transformation defined
Leo
(eo
+
Lei
(el
+
Lei=ei
v)1 1 JJvJJR_ JJVJJR`-180)1V1
.
.
7
by
2
-
eo)
IIVIIRI-
2 I
,and
ViEf2,...,nJ.
It maps eo into eO and leaves the subspaces spanf t, 61 1, spanf eil (i E 2, ., n 11) invariant. This transformation is called a Lorentz boost. -
..
We will
now show that the group G generated by all Newton transformations with respect to the inertial frame (eo,,q(eo, .)) and all Poincar6 transformations 0 of the form O(x) o + L(x-o) where L is =
Lorentz boost coincides with the group P+ of time orientation preserving Poincar6 maps. It is clear that G is a subgroup of P+. To show the converse Let o, o' G An and f eo, two oren 1 1, f eo.... e'_lJ n thonormal bases with respect to q. We have to show that the Poincare a
.
transformation V) which maps i is
a
actly
into o' and
(respectively, e ) with 77 (6k 6k) -1) and 77(ek,t) -- 0 (respectively, 77(el,t)
vector ek
=
-
i
renumber the basis vectors such that ek let 01 be the map x -4 x + (o'-o). If eo tor
this
v
E
+ ei into o' + ei for all
o
of maps in JV and G. In each basis there is
composition
one
o
-
Rn with
let
77(v, eo) be
=:
0, 0
<
71(v, v)
<
ex-
(respectively, 0). We can
eo If eO Co then there is a veceo 1, and eO 11 eo + v. In
eO, el
<
1
=
=
-
element of the Newton group with respect to non-relativistic inertial frame (eo, 77(eo, .)) which maps o to o' and el to case
v/JJvJJ,-i(o).
01
an
Consider the Poincar6 transformation
02(X)
==
o
+
L(x-o)
where L is Lorentzian boost which maps eo into eO and leaves the plane spanjeo,e'01 invariant. Observe that and
f (02) -1 (eo1)1 (02)
-
I
(ell
(02)
-1
(enl- 1)
are
both orthonormal
with respect to q and that
spanfol (el),
.
.
.
,
bl (e, 1) 1
=
spanf (02)
1
(02) _l(en'_l)J n
1.4 Einstein's
=
Hence there is element non- relativistic
Ix
:
special theory of relativity
71(x
o,
-
eo)
=
37
01.
of the Newton group with respect the to (e0,,q(eo, .)) which leaves o' invariant and
03
inertial frame
1 n 03 0 01 (ei) (02) _1(6s (i 1) This implies 0 02003001Finally we can show that P+ is the only subgroup of P which satisfies
satisfies
=
=
-
(a), (b), (c)
.
-E0, be unique transformation Op
Let
-
0: E0
Euclidean isometry. Then
an
P which leaves t invariant,
E
0 extends to a 0 (x) V) (X 0) + =
Hence the Newton group Ar is a sub-group of P'. Let o' E A' and let V be a vector with Since -F(t') I and
-
-F(x-o)t.
=
By assumption there is a 0 E P which satisfies 0(o + R+t) In particular, the associated Lorentz transformation maps We have
seen
above that there
(t, -r)
such that
are
Newton transformations
77(t', V) =
< 0.
o' + R+t'.
t into
R+V.
01, 03 with
V) 02 0 03 0 01 where a 02 is the Poincar6 corresponds to a Lorentz boost which maps t into R+ V and leaves the spaces spanf t, t'j, and its 77-orthogonal complement ' spanf t, V1 invariant. Hence 02 is the composition of maps provided by assumption (a) and (b). Since V was arbitrary we obtain that P' contains respect to
=
,
transformation that
all Lorentzian boosts and all elements of the Newton group associated I (t, -r). Consequently, P+ P.
with Let
=
(t, -r)
be
a non- relativistic
reference frame
and E, be
a
hyperspace
in
spacetime which represents an instant of time. If we assume the axioms of Euclidean geometry for E,, then, to be consistent, we have to assume that this structure is invariant with respect to any transformation which is an isomorphism of our physical structure. Hence (ii) of Proposition 1.4.1
implies that
we
must restrict to the Poincar6 group P. Preservation of
time orientation reduces this group to
P+. By Proposition
1.4.1
(iii),
the axioms of Euclidean geometry, and Postulate 1.3.1, the Poincar6 group cannot be further reduced. Hence we conclude that space and time
are
well described
by
a
Minkowski spacetime
together
with
a
time
orientation.
The
(arbitrary) non-relativistic inertial frame (t, -r) we have started -1. Recall that t was the velocity vector of the 77(t, t)
with satisfies
inertial observer t
=
+ tt who
was supposed to be at rest. Since the isomorphisms of Minkowski spacetime, any other inertial observer t 1-4 o'+ tt' who can be supposed to be at rest must be linked to t --> o + tt by a time orientation preserving Poincar6 transformation 0. If we denote the associated linear transformation by Lo then V E R+Lo(t) holds." In particular, all admissible observers V must satisfy 77(t, V) < 0. -->
o
Poincar6 transformations
16
Observe that t' not
are
the
0 Lp(t) in general since -r which preserved by Lorentz transformations.
is used to normalise t, is
1. Local
38
theory of
Definition 1.4.4. Let
(A',,q)
be
a
Minkowski 8pacetime with time ori-
[v].
entation
(i)
space and time
A
(special-relativistic) infinitesimal observer is a vector t E Rn -1, 77(v,t) < 0. 77(t,t) A (special-relativistic) observer is a curve -y: t (ii) -1(t) such that the velocity vector (t) is a special- relativistic infinitesimal observer with
=
-
for all
(iii)
A
t.
(special-relativistic)
tt E An where
x
inertial observer is
E A' and t is
a
a
curve
special- relativistic
-/: t
-4
x
+
infinitesimal
observer. For any special-relativistic infinitesimal observer t" E Rn and any any otl E An there is a time orientation preserving Poincare transformation
b which
maps t
1-4
o
+ tt to o" + tt" for all t.
The rest space with respect to (t, -r) at the event o E An coincides with Eo f x E An : 77(x-o, t) 01. Let 0 E P+ such that its associated =
=
Lorentz transformation maps t to V. Since 0 maps E into the set E' 01 this set must be interpreted as the rest space f x1 E An : q(x'-o', V) =
=
with respect to the special-relativistic inertial observer t --> o' + tt'. In general, this space does not coincide with the non-relativistic rest space
E,,,. Hence
we
arrive at the
Definition 1.4.5. Let t be
following a
definition.
special- relativistic infinitesimal
(i) The infinitesimal rest space with respect to 1W G Rn : 'q(t', W) 01 C Rn. (ii) The affine rest space containing o' E An given by o' + t" C An.
observer.
t is the set V
I :=
=
The affine rest space inherits the Euclidean scalar 71 (u,
to
V) (v,
w
E
W) ')
with respect to t is
product (u, v)
-
Similarly, the time difference Att, (x, y) of two events x, y with respect the special-relativistic infinitesimal observer V is given by t where
(unique)
t E R is the
number such that
x
+ tt' lies in the affine rest space
of V whichcontains y. The original non-relativistic observer t
-->
o
+ tt is also
a
relativistic observer and for this observer the non-relativistic and
specialspecial-
relativistic definitions for rest space with associated Euclidean scalar product and time difference coincide. The relativity Postulate 1.3.1 im-
plies that for every other special-relativistic inertial observer o' + tt' and time differences lengths and angle should be measured by by Att, (-, -) In conclusion, a Minkowski spacetime together with a time orientation contain all the geometric information of spacetime. This geometric structure is mathematically simpler and more elegant than the structure -
of
a
Galilei
spacetime.
1.4 Einstein's
Remark
1.4-1. The geometric
Minkowski
1905)
(1909).
special theory of relativity
structure of
spacetime
But before him Albert Einstein
39
discovered
was
by
(1879-1955) (Einstein
had realised that absolute time does not exist and
came
to
an
equivalent but less elegant description of spacetime. His work contains the main physical discovery which justifies to speak of Einstein's special theory of relativity. An important precursor of Einstein was Hendrik Antoon Lorentz (1853-1928) whose explanation of the Michelson-Morley experiment anticipated the length contraction17. 1.4.1
Causality
We start with
in
special relativity
terminology
some
Definition 1.4.6. Let
(A',,q)
which will be
justified below.
be the Minkowski spacetime.
A vector w is called spacelike, if j7 (w, w) > 0, timelike if 77 (w, w) < 0. A vector w is called causal 0, and lightlike (or null) if 77(w, w) if it is timelike or lightlike.
(i)
==
(ii)
Let
[t]
directed
A
(iii)
be
a
time orientation. A causal vector
(past directed) if q (u, t)
<
0
u
is called future
-
-y is called
spacelike (resp., timelike, lightlike (or null), causal, future directed, past directed) if all its velocity vectors are spacelike (resp., timelike, lightlike (or null), causal, future directed, past
Let
w
be
there is
curve
directed). a
an
vector and x, y E A' with y
that the events
respect
x
and y
=
x
+
t, in particular, these
to
==
x
+
infinitesimal observer t such that w
lie in the
events
are
w.
If
q(w, t)
same
w
is
=
spacelike then implies
0. This
affine rest space with at the same time
taking place
with t. Hence there cannot be any causal process which links vice
x
to y
or
versa.
On the other either
definiteness
if
hand,
wlV-- (w, w) i--+
x
w
y-x is timelike and future directed then
=
-wIVI-77(w, w)
that
assume
inertial observer t
or
+t
wl _-,q(w, w W
-) 11 Vf___(W
is
an
infinitesimal observer. For
is in the time orientation. The
connects
x
with y. Hence the event
x
I
must take
definitely place before the event y. This motivates definitions above (see also Postulate 8.0.1). This discussion also as a
causal
Corollary
(i)
implies
that the field of light
(A', n)
be
cones x
our
1-4
causality Cx
serves
boundary. 1.4.1. Let
a
Minkowski spacetime.
of all events y E x which can be causally influenced by taking place at x are given by J+(x) := fy E An : y can processes be reached from x by a causal, future directed curvel.
17
His
The set
interpretation
was
different from
Einstein's, however.
40
Local
1.
theory of
0i) aj+(X) Above
have
we
space and time
C:.
=
that the most important
seen
ingredient
of
is the fact that the
of
our
discussion
the elements of
spacetime isomorphisms just preserving Poincar6 group P+. The are
the time orientation
main step to P+ is contained in Theorem 1.4.1 in connection with Postu-
arrive at
late 1.4.1. One may object that the constancy of the velocity of light is a rather awkward postulate. However, it is closely linked to the funda-
our
causality.
Theorem 1.4.2. Let
An
Let
0: causality:
An be
_-,
and consider the Minkowski space (A',,q). bijective map such that 0 and 0' both respect
J+ (x)
* an
y E
> 3
n
a
Then there exist b E R' such that I
allows to
Alexandrov
obtained in
The
following theorem which has also been replace the light cone structure by (1975) fundamental postulate by the assumption of causality.
mental notion of
O(x)
0(y)
J+ (O(x)) for all
E
0(n, 1), aL(x-o) +
L E
=
an a
b
R
E
for all
x, y E An.
\ 01,
an o
An, and
E
a
An.
E
x
p. 40
Proof. This theorem is a corollary to Theorem 1.4.1. We only have to lightlike vectors are mapped into lightlike vectors by 0 and
show that
We will first prove that for y E J+ (x), x 0 is equivalent to the assertion ,q(y-x,y-x) Z1 i Z2 E
If
J+ (x)
q(y-x, y-x)
Clearly,
ray.
n J-
(y) =
we
any two
,q(x-y,x-y)
< 0
have either zi E J+ (Z2)
0 then
then
J+ (x)
points
on
J+(x)
It follows that U must contain
spacelike Now
vector we
holds. Since
w.
n
a
n J-
light
J-(y)
points
It is clear that z1
(y)
is
ray
[I
p.
Z2 E
part of an
Z1 i Z2 which
J+ (Z2) and
"for all
J+ (Zl) a
" -
single light
causally related.
are
contains
must also hold. But this is
,q(0(x)-0(y), OW-0(y)) Analogously
or
condition
:4=>
If
open set U C An.
are
Z2
connected via
J+ (Zl)
a
-
will prove the theorem. If x-y is lightlike then A (x, y) A(x, y) is formulated entirely in terms of causal relation-
ships, A(O(x), 0(y))
p. 40
a
y the
4
A(x,y)
=
equivalent =
to
0.
for
1
451
philosophically more appealing to demand causality than inlight cones but it should be noted that our original version to the actual experiments motivating special relativity.
It may be
variance of the is closer
1.4.2 Length contraction and time dilatation Since there
are no
that
absolute time
a
or
absolute space it should not come are observer-
in space and time- differences
lengths surprise dependent. To simplify notation at
let
JJwJJ
==
-\/ (_ww)
for any
spacelike
1.4 Einstein's
special theory
of
relativity
41
Fix an event x E A' and an infinitesimal observer t and conrod which rests in the affine rest space x + t 1 of t. If with respect to the observer its endpoints are given by x and x + f at a time to, then it will sweep out the subset vector
sider
w.
a
S
C:
z
A'
z
X
=
+ M +
/-tt, A
E
[0, 1],
E
RI
Lt
t'
Lt
K
O
Lei
Fig.
in
1.4.7.
Length
contraction
Fig.
1.4.8. Time dilatation
spacetime (cf. Fig. 1.4.7). We complete It, ei 111f Ii to an orthonorf t, ei, e,,- I I of A'. Now consider a second infinitesimal ob=
mal basis
.
t' who
.
.
,
(relative to t) with the velocity v II UII -ei. Let L be a Lorentz transformation which maps t into t' and leaves spanf t, el I invariant. We have V Lt _JIVI, Lei (ei + I I VII t) / _IVI (t + v) / I 12, server
moves
=
=
and
VIT
=
V-1
=
-
-
Vi E f 2,.. n Lei ei 11. The rest spaces relative to t' are all parallel to Ltj-. In order to determine the length of the rod with respect =
-
.,
to t'
we
must calculate the
S n
(x + Lt-L )
length
of
11 yllt..
ei + =
X
+
01
F_
a .
,
G
[01 IV III
V
It follows that
observer t' is
one
11 'Il
=
of the
reasons
V/1
-
11,61
-
Iltll
<
11 11.
Hence the infinitesimal
shorter
length than t. This Lorentz contraction why initially many physicists found special general
measures a
hard to understand. It should be remarked that an investigation of 3-dimensional objects shows that the Lorentz contraction is more
relativity
like
a
rotation. In
particular,
a
moving sphere looks like
a
sphere
at rest.
42
1.
Local
There is
spacetime,
a
a
theory of space and
time
similar effect with respect to time, the time dilatation. In
time interval At with T
(cf. Fig. 1.4.8).
=:
x + tt +
respect t G
a :
[0,./At], a
In order to calculate ,At' t +
Tn(x+RLt)=
to t is
we
set
t-L I
must consider the subset
t
V
x+t
==
,
V-1-11 m -
E
given by the
E
V-1--11,Y112
[0, /-At]
spacetime. It follows that At' J_-JJVTJAt < At. This is the for the twin "paradox": Consider two twins, one of them staying at home, the other one travelling with high velocity away, and then, after some years coming home. Afterwards the twin who had travelled will be younger. Let t be the infinitesimal observer associated with the first twin, t, be the infinitesimal observer of the second twin at her outward journey
of
=
reason
and t, be the infinitesimal observer of the second twin
journey. As
a
consequence of time
outset and return with
the time
respect
lapse with respect
to
dilatation, the travelling
during her return lapse between
the time
twin will be shorter then
to the twin at rest. Hence after her return
the twin who has travelled will be younger than the other one. If the time interval and the involved velocities are large enough the effect can
be
spectacular.
This effect is
purely geometrical and
ation. The fact that the situation is not
has nothing to do with accelercompletely symmetric, i.e. that
1A Einstein's
the twin who had travelled inertial observer is
changes
special theory of relativity
43
her direction and therefore is not
an
needed to get her back but has nothing to do with the effect. To make this point clearer imagine that spacetime would
only
cylindrical. To be concrete, let y be an spacelike and consider the hyperspaces Fx be
and
Fy
=
Iz
EE
A
77(z-y, v)
:
=
01.
event such that =
f
E
z
These two
A
:
is
shown that
just the
locally
set M where
it is
(A',,q). (We will not chapter). If the twins
impossible
Fx and Fy
hyperspaces
she would
(by
y-x is =
01
enclose
assume
identified. It
to differentiate between
a
that
can
(M,,q)
be
and
prove this fact. It will become clear in the next
lived in
(M, 77)
instead of
travels would not need to turn back. As
Fx)
are
=
77(z-x, v)
M which is bounded in the direction of v. Now
region spacetime
v
identification)
(A',,q),
soon as
the sister who
she arrives at
Fy (or
be at the other
hyperspace and therefore on the other side of her sister. Just travelling on she would 2 then M is just an ordinary eventually meet her sister again. If n and visualise the whole it is to cylinder setup (cf. Fig. 1.4.10). easy our
=
t
Lt 0
Fig. 1.4.10. The twin paradox in a cylindrical universe
-1.4.3 Relativistic We give,
particles
and
photons
brief outline of the elementary concepts of photons, paranalogues have been introduced in the,section on the non-relativistic theory of particles (cf. p 20). The content of this section will be used to motivate definitions in Chap. 5. Part of the physical discussion in Chap. 6 also requires this section. a
ticles and their collisions. Their non-relativistic
The
following particle.
is the direct
analogue of Definition
1.2.3 for
a
non-relativistic
1.
44
theory of
Local
space and time
special-relativistic particle with mass m is a pair -1. The a curve satisfying (rn,,y) curve -y is called its world line in spacetime. A special-relativistic inertial particle is a particle -y which satisfies
Definition 1.4.7. A
where
=
m
c
R+ and -y is
0.
Unlike in the non-relativistic case, we have now only one law which covdenote the incoming and (m'., 7j')j=l... .11 ers collisions. Let 3 Then conservation of momentum is excollision. of a particles outgoing
pressed by
the
single equation k
Y mi" Yi
M/j'Y
_
(1.4.12)
i
j=1
i=1
Choose any infinitesimal observer t and denote the projection to the orthogonal complement of t by.: v -* Y. Then the momentum m splits into
spatial
and
temporal parts
follows:
as
M
(t
M,Y
Conservation of the
momentum takes the form
spatial
Ek
1
Tni
Ej=l -1--71i 1 _F M
v
"i 112
page 21.
a
-
3
IN
which is
+
(1.4.13)
ly
modified form of the non-relativistic conservation law
(ii)
on
conservation takes the form
Energy k
1
-
Mi
-7
T,
M
i
VFIF___1 1 jll
(1.4.14)
j=1
Since 1
M
M
+ 2
M11,6112
+
0(11,U112)
approximation the non-relativistic law of mass Eli m'i holds exactly we also equality I:ki= 1 mi
we recover as an
vation. If the an
=
approximation
conserrecover
of the conservation law for kinetic energy. The relaonly more transparent and
tivistic version of the conservation laws is not
economical,
it also leads to
Example 1.4.1. tains helium
If
one
important applications:
bombards lithium 7Li with
2He according
to the nuclear reaction
111 +7 Li
--
22He
+ energy.
hydrogen 'H
one
ob-
1.4 Eiristein's
of
special theory
45
relativity
The weights of 1 mol (i.e, 6.02213 1023) atoms of hydrogen (respectively, lithium, helium) are 1, 00783g (respectively, 7.01601g, 4,00260g). It follows that the final product is lighter by about 0, 01864g/mol hydrogen atoms. According to the energy conservation theorem above, this mass difference corresponds to the energy E Pz 1, 864 10-5kg C2 ';Z: 1.67525 1012j. This energy is huge in 1, 864 (2.9979) 10" m kg/s comparison to the amount of material involved. The energy amount of an adult is about 8000 U per day or 2920000 U per year. It follows that 1.67525 .1012j would last a human being for more than 500 years. -
-
-
-
-
-
Observe that the amount of energy which can be obtained from nuclear huge because the velocity of light c is extremely large. In this
fusion is
book
we
chose natural units where
These units
are
appropriate
a
length
unit is defined
to discuss relativistic effects
level but obscure the fact that velocities in with respect to
as
c-time unit.
on a
everyday live
theoretical
are
negligible
c.
photon is (classically) characterised by its velocity and its frequency. hil elementary quantum mechanics one has the relationship E 6.62608 .10-34j S is the Planck constant and v the frequency where h A
From
=
=
photon. These quantities uniquely determine the momentum p of hv be the energy photon. Let t be an infinitesimal observer and E of the photon as measured by t. If 6 is the spatial velocity of the photon relative to t then its momentum is completely determined and given by E(t +,6111cll). Any other infinitesimal observer -C measures the light p -,q(t',p)1h. The frequency of visible light light ranges frequency v' of the
the
=
==
=
from about 4
-
1 014 oscillations
oscillations per second
(ultra
per second
violet
light).
(infra
red
light)
to 8
-
1014M
[p. I
40
11
p. 47
Analysis
2.
on
manifolds
p. 45 p -1
only be expected to be a good description locally. We will assume that special relativity is an exact "infinitesimal" description, i.e. that it holds as a first order approximation near any given point. A rigorous formulation of this idea requires manifolds and tensor fields which will be introduced in this chapter. This chapter contains much more material than is necessary for the understanding of the following Chap. 3. While in an ideal world, all this material would be standard knowledge of mathematicians and theoretical physicists, we give a self-contained treatment for those readers who still have to learn about analysis on manifolds. We will be a little more general than necessary. Instead of using the field R we will develop the theory for both fields R and C and write K if a statement is valid in both cases. This generality is not needed for the main purpose of the book, i.e., for presenting the theory of spacetime. However, both physicists who go on to study gauge theory and mathematicians who are interested in differential geometry as a mathematical discipline will benefit from this generality. It is also instructive to see which concepts depend on the real structure. Writing K instead of R if possible will not introduce any additional difficulty. While everything presented here will be used somewhere in the book, readers primarily interested in space and time may want to skip material where possible and come back later to it when needed. The
Special relativity
can
minimal amount the reader should know in order to pass
on
[I
to the
chapter is 1. The definition of manifolds: Sect. 2.1 up to Sect. 2.1.1; 2. The tangent bundle: Sect. 2.2; 3. Tensors and tensor fields: Sect. 2.3 up to including Definition 2.3.7, Sect. 2.3.2; 4. Connections: Sect. 2.6 up to including Definition 2.6.2; 5. Examples of connections: Sect. 2.7.
next
There is
a
conceptional problem with
the
theory developed
so
far. From
local experiments we implicitly extrapolated a structure of spacetime which has only been tested in a small part of the small spacetime region our
which is inhabitated
only
by human beings. The Michelson Morley experiment spacetime is nowadays a good description
indicates that Minkowski
of the
spacetime
structure of
oratory. A weak form of the guide in the margins analysis on manifolds.
The
a
(comparatively small)
philosophical
assumes
M. Kriele: LNPm 59, pp. 47 - 149, 1999 © Springer-Verlag Berlin Heidelberg 1999
earth-bound lab-
Copernican principle
that the reader has
no
states
knowledge
of
P. p.
1
J4]
1554]
48
2.
that
our
Analysis
nianifolds
on
position in spacetime is in no way special. In particular, at any spacetime one would observe the same physical laws. Nat-
other event in
ural sciences which go impossible without
be
beyond the mere cataloging of phenomena would adopting this principle. Hence we feel justified to
extrapolate that every event in the universe has
time structure is well described
We will
see
that this
a
by
neighbourhood
whose space-
Minkowski spacetime.
extrapolation is very different from the naive posglobally the structure of a Minkowski space-
tulate that the universe has time.
As
a
first step
we
will have to find out how to connect
our
local
Minkowski spacetimes. To give a simple example which exhibits part of the problem consider the torus T' which can be obtained from A' via the identification is
a
+ aei for all
c- Z, x E A', where feiliE 1_-,nj equip Tn in a natural way with a induced by An. While locally there is no possibility to
x
-
x
fixed standard basis of R1. We
Minkowski metric
differentiate between different
(Tn' 77) (cf. Fig. 2.0.1).
and
a
can
(A', 77),
both spaces
Fig.
We will
are
globally
2.0.1. The torus T
very
2
develop the mathematical techniques needed to globalise given by our collection of Minkowski spacetimes. First note that we cannot even expect that locally Minkowski spacetime is an exact description. But it is reasonable to expect that Minkowski spacetime is the better an approximation the smaller the subset of events we are considering.is. This means that we will have to formulate the theory infinitesimally. In Sect. 2.1 we will generalise (part of) the structure of A' to manifolds which may be thought of as a collection of local A's. We will now
the structure
then construct tangent spaces to approximations of it (Sect. 2.2).
a
manifold which
are
the infinitesimal
2.1 Manifolds In this section
we
will localise part
foundation for calculus.
of the
structure
of A
and
lay
the
2.1 Manifolds
One of the most important structures of R' is given
by
49
the collection
of all open subsets, because this collection is needed when one defines limits, the most basic notion of analysis. Since A' can be thought of as
special properties of the vector 0 been forgotten about there straightforward way to define open sets in A'. The subset U C A' is C R' such open if and only if there is a point o E A and an open set R' with the is
a
thatU=fo+v: vE0J. We will
localise the
now
topological
the structure of A' which tells
collection
of
r
The collection T
7-
A', i.e., the part are
of
open.
subsets
(i) 0 E r, M E 7-; (ii) U, V c- r =* U n V E -r; (iii) if A is an index set and Ua U E
structure of
which subsets of An
topological space (M, -F) is a set M together with of M which satisfies the following properties.
Definition 2. 1. 1. A a
us
is called the
and is closed
apointxEM if
if
M
\U
E
-F
for each a
topology of G -F.
E
A then
UaEA Ua
G 7--
M. A set U C M is open
A set V C M is
a
if neighbourhood of
there is anUE-r withxEUCV.
It is clear that the collection of open sets of Kn
(and therefore
also of An)
justifies the definition of a topological space. On the other hand, a general topological space may have properties which are quite pathological. This can be seen from the following two extremal examples. Let M be any set and define -rfine to be the set of all subsets of M and 'Tcoarse f 0, MI. Then A Tfine) and (MiTcoarse) are topological spaces. Definitions which can be stated purely in terms of open sets carry over to topological spaces. satisfies properties
(i)-(iii)
Definition 2.1.2. Let
of Definition 2. 1. 1. This
(M,,T) and(M, fl
be
topological
spaces. 1
f : M --+ 1 1 is called continuous if f (1 ) E 7 for all bijective, continuous map whose inverse is also continuous is called a homeomorphism. (ii) A subset U C M is compact if for every collection of sets WaIaEA with Ua E T and U C UaEA Ua there are finitely many U,,(1), Ua(k)
(i2
A map
-
U E T. A
.
withUc
.
uki=AW-
is connected if U, V E -r with U n V 0 or necessarily of the form U M, V V 0. For the topological spaces we are interested in M, U (cf. Definition 2.1.4 below) this is equivalent to the requirement that M. any two points can be connected by a continuous curve [0, 1] A collection subsets basis the is a of open of topology WaJaEA (iv) V* -r if for every open set V there is a subset B C A with UbEB Ub
(iii)
A
.
topological
0 and U
U
V
space
M
are
(M, -r)
=
=
=
--->
=
50
2.
Analysis
on
manifolds
A subset U C V is called dense
where t! is the closure if V of U, i.e., the smallest closed set containing U. (vi) A collection of open subsets fUa'jae_-A is a sub-basis if all finite intersections of sets U,,, form a basis of the topology -r.
(p)
Lemma 2.1.1. Let M be
a set and jUaja(EA be any collection of subM. Then there is uniquely defined of M which satisfies UaEA Ua topology -r of M such that jUafaEA together with the empty set 0 are a sub-basis of T.
sets
=
I
Proof.
This follows
from the definition of
immediately
a
I
topology.
We
can now describe those topological spaces which cannot be locally distinguished from A'. Let (M, T) be a topological space which is Hausdorff, i.e has the property that for any two different points x, y E M there are neighbourhoods U of x, V of y which satisfy U n V 0. The topological space (M,,r) is locally indistinguishable from A' (considered as a topological space) if each x E M has an open neighbourhood U such that there is a homeomorphism W: U -* V c R'. 2 The pair (U, o) is called a topological chart of M. Since for each open subset 1 C U the restriction of o to 1 is also a homeomorphism onto its image, we have =
the
following compatibility property. Let
(U, W)
and
0. Then the
(1 , 3)
map
be
topological charts
o
p-:
p(U)
n
( (a)
of
(M, -r) with UnO p(U) n ( (a) is
--+
a
homeomorphism. We would like to have not a
structure which allows
there is not
an
is
only
a
topological
structure
on
M but also
the tools of
analysis. Unfortunately, independent definition of a "differentiable space" which the definition of a topological space. To get an idea how us
to
use
analogous to this difficulty can be overcome we can view the charts (U, W) as a way to induce the local topological structure of R' on M. To be more precise, let M be any set, WiliEl be a set of subsets of M with Uj, Ui c-I M, and pj: Uj > Vi bijective maps onto open subsets of R'. We can now attempt to define a topology on M by using the sets f oj-'(Wj) : Wj C Uj is openj as a sub-basis for the topology of M. In order to get a topology consistent with a local description we have now also to demand the compatibility property above. Still, the resulting topological space could fail to be Hausdorff (cf. Fig. 2. 1. 1). Since this is a local property, we will demand it in addition. We have now defined a topological structure on M which is locally indistinguishable from the topological structure of R1. This definition
2
can
be carried
Recall from the definition of the x --+ x
-
o
is
a
homeomorphism.
over
to the differentiable structure.
topology
of A' that the map
V),:
X,
--+
R,
Manifolds
2.1
Fig.
identify
51
A topological space locally homeomorphic to
2.1-1.
which is
R but fails to be Hausdorff
Definition 2.1.3. Let I be
(i
c
index set and
an
A Ck-atlas of M is
which is
a Hausdorff. 1) of local homeornorphisms such that
(M,,r)
be
collection
(i) each Uj is open and connected,' (ii) each x E M is is contained in some Uj, (iii) for each i, j with Uj n Uj V - 0 the map oj (Uj n Uj) is a Ck -diffeomorphism.
oj
o
a
topological Uj
oj:
M
oj (Uj n
:
oj
C
space
-4
Kn
Uj)
pairs (Uj, pj) are called charts of M. A chart (Uj, oj) is centered at 0. Two charts A Oa) (a if x E Uj and Wi (x) 1, 2) are called compatible if they satisfy the compatibility condition (iii). Two atlasses are compatible if each pair of charts in their union is compatible. A Ck_ atlas A is called maximal if any Ck -atlas containing A coincides with The
x
E M
=
=
7
A. Remark 2. 1. 1. In the
erty that Wi
(Wj)-1
case
K
=
Ck -atlas (k
C every
>
1)
has the prop-
analytic, i.e., is locally given by its Taylor series. This follows immediately from the fact that C-differentiable functions are analytic. o
For technical
reasons
ogy of M has
a
(cf.
Sect.
2.1.2)
we
countable basis. This
will also demand that the means
that there
are
topolcountably
such that any open set VV is the union of sets Vi.
fVjjjeN
many sets
is
(M, -r) be a connected topological space which is Hausdorff and which has a countable basis. (M, -r) together with a maximal Ck -atlas is called a Ck -manifold. A C'-manifold is also called We will often refer to smooth manifolds simply as a smooth manifold.
Definition 2.1.4. Let
manifolds. A subset N C M is x
E
N there is
W(U)
n
ly
K'
E
submanifold
a
is
Observe that
a
manifold of M
chart :
an
m-dimensional submanifold
(U, W) of M
y'+'
often called
=
a
centered at
x
y' 01. An (n hypersurface. ...
subset N C M
=
can
=
be
a
of M if for
such that -
each
W(N n U) l)-dimensional
manifold without
=
being
a
sub-
(cf. Fig. 2.1.2).
The
following lemma guarantees that a manifold is determined by any (not necessarily maximal) Ck -atlas (k > 1) which is compatible with the given maximal Ck -atlas. In practice, it is therefore sufficient to work with any
given atlas. It
can
be shown
(Hirsch 1976)
that any maximal
2.
52
Analysis
Ck -atlas
contains
manifolds
on
Fig.
2.1.2. A manifold M c R 2which is
not
submanifold of R2
a
subatlas which is C'. Hence for most questions it
a
loss of
generality to consider only smooth (i.e., C"O) manifolds.' On the other hand, it should be noted that Co-manifolds are really more general. We will not consider such manifolds in this book. is
no
Lemma 2.1.2. Let A be
Ck -atlas. Then there
a
is
a
unique, maximal
Ck -atlas containing A.
Proof. Let
B be the set of
are
Clearly, compatible. To see
with set
x
E
01 (U
Ck -charts which
A C B.
chart in A.
Any
this let
U. Then the maps o o n V, n V2). It follows
Ck. That 01
0
-1
(02)
is
Ck
are
compatible which each
two charts
(Vi, 01) and (V2, b2) in B x G V, n V2 and (U, o) be a chart in A (0j)-1 and 02 (p-1 are Ck in the open by composition that 02 (01) -1 is also o
0
be shown in the
can
to prove that B is maximal and
same
way. It remains
unique. The first assertion follows from
the definition of B. Assume that B' is another atlas every chart of B' is
B
by the definition
as
well.
compatible with each
chart of
containing A. Since A, it must belong to
of B. Hence B' C B and the second assertion follows
I
Example 2.1.1. Consider the cylinder which can be obtained by identifying opposite sides of the rectangle [a,, bi] x (a2, b2)
f (a,, y) As charts
we can
can
a
:
y c
x
-
-
(b2
(a2, b2)
good
al x
2
-
a2), Y)
-4R
[a2, b211
for
x
for
x
<
(a2, b2) bj bj
--
R
2
2al 2al
2
(X,Y).
in fact be shown that there is
it is not
-
(x, Y)
(al, bl) (x1y)
It
I (bi, y)
(a2, b2))
(x
3
-
bi x
(XI Y 02:
[a2, b211
take the maps
([a,, bij
oj:
y (E
:
idea to restrict to
important technical tools do
always an analytic subatlas. However, analytic atlasses because then some
not work
(cf.
Sect.
2.1.2.)
2.1 Manifolds
Consider
edges
(long) rectangular strip
a
of paper. The two shorter in two ways. Either one obtains
be
53
opposite
glued together a cylinder (cf. Example 2. 1. 1) or a figure which looks like a cylinder with a twist. This Wbius band can mathematically be constructed as follows (cf. Fig. 2.1.3). can
identify Fig.
2.1.3. The construction of
Example
(-2a, 0)
with
a
flip
M6bius band
(M6bius band). Let V, (0, 2a) x (-b, b) C R 2 and V2 (-b, b) C R 2. We define on VI U V2 an equivalence relation
2.1.2 x
=
=
-
by I
(X ,x2)_(Xl
(X1, X2)_(Xl
-
-
3a, X2)
if
_X2)
if
a,
and the manifold M
jection V, Wi
=
U
V2
-*
(ir_1)IUi (i
E
on
C
(a, 2 a)
E
(0, a)
x
x
(- b, b),
(- b, b)
by M (Vi U V2)/-. Denote the canonical proI M, (x ,x2) -, [(XI, X2 )] by,7r and let Ui ir-'(Vi), =
=
11, 21).
Since manifolds have tiable maps
(XI, X2) (XI, X2)
a
Then
J(Ul, 01), (42, W2)J
differentiable structure
we
is
an
can
atlas of M.
define differen-
manifolds.
M, N be C1 -manifolds and k < 1. A map f : M Ck -differentiable if for every two charts A, Oa), (Vb, Ob) of M, N the composed map Ob 0 f 0 (W)- I is a Ck -map. A C' -differentiable map is called smooth. The set of all Ck -differentiable maps from M to N is denoted by Ck (M, N).
Definition 2.1.5. Let N is
for Oa
The maps g: M every chart Pa, o
g: Km
--+
maps from M Ck (K-, M).
Kn
-+
Oa)
the
composed
--->
M
maps 9
0
are
Ck -differentiable if
(Wa)- I:
K'
--
K' and
Ck -differentiable. The set of all Ck -differentiable and from K' to M are denoted by Ck(M, KI) and
are
to K'
K' and h: K'
54
Analysis
2.
on
manifolds
It is easy to
see
to any
atlas. For N
given
that this definition is satisfied if =
K',
M
=
f
Ck with respect
is
K', the definition coincides with
the usual definition of
differentiability in elementary analysis. Recall from analysis that a continuously differentiable map F: K' Km has rank r at x E K' if the subspace DF(x)K' C K' has the dimension Let
r.
Nbea Ck -differentiable map and x G M. If (Ul ol), x and (Viol), (V2) 02) charts centered at
f:M--
(421 W2) f(x), then the rankof the mapsolof oWl-1 at Vl(x) andV)20fo 02_ I at (P2(X) coincide. We can therefore speak of the rank of f at x and the charts centered at
are
following definition
is
independent
Definition 2.1.6. Let M. Let
be
(U, W)
(any)
of N centered
chart
(i) (ii)
has rank is
W(x),
(iii)
is
is
an
local
if 0
f: a
Proof
Since
if D(o
map and
(V, 0)
be
x
E
(any)
f
f o W-' has rank r if D(o o f o W-1) is x
and
x
o
f
o
p(x),
at an
o-')
injective linear
is
a
map
surjective linear
local
at
x
if D(o
o
f
o
o-')
is
a
bijective
-+
r
at y.
immersion (respectively, a submersion, x) then it is also an immersion (respectively, diffeomorphism) at any y E W.
diffeomorphism a
o
x
Ck -differentiable
N be a Ck -differentiable map of rank r at neighbourhood W of x such that for each y E W
M
the map f has rank ry > In particular, if f is
submersion,
The map
diffeomorphism
M. Then there is
local
of
a
M centered at
o(x).
Lemma 2.1.3. Let E
x
N be
W(x),
linear map at
x
f (x).
submersion at
an
map at at
(iv)
at
---*
chart
immersion at
an
at
r
at
M
f:
of the chosen charts.
an
at
D(oof oW-1)
is continous the existence of r
a a
linearly indepen-
dent vectors in D (o o f o o- 1) (,p(x))Kn implies that for y close enough to x there are also r linearly independent vectors in (DO o f o r 1) (,P(y))Kn. Hence the rank of f cannot fall in
a sufficiently small neighbourhood of given point. For the second statement observe that immersions, suband local diffeomorphisms all have maximal rank. p. 47 11 _ZT-11 mersions, F a
I
p. 61
2.1.1 Construction of manifolds
plays a fundamental r6le from infinitesimal assumptions. In this section we show that the inverse function theorem and also similar theorems carry over to manifolds. A special case (Proposition 2.1.1) allows a construction of submanifolds without specifying
In
analysis,
the inverse
function
theorem
because it allows to draw local conclusions
an
Atlas.
2.1
Manifolds
55
We will
who is not
skip this The
occasionally use the results of this section. But the reader primarily interested in analysis on manifolds is advised to
section and to return to it when needed.
following lemma
is
a
consequence of the inverse function theorem.
Lemma 2.1.4. Let U C a
Kn, V C K' be oPen, x E U, and f : continuous, differentiable map with constant rank r in U.
U
--
V be
Then there exist
(i) an open neigbourhood 0 C U of x, (ii) a homeomorphism 0: 0 f y E Kn : I y I < 11, (iii) an open neighbourhood V D f (U) of f (x), (iv) and a homeomorphism 0: Iz c KI : IzI < 11 -->
such that
f
=
Oop, oo where
MY',-
ly n)
-4
f)
(yi.... Y" 0'...' 0).
=
I
Proof. Let E be the (n r)-dimensional vector space E Iv Cz Kn and F Kn be an r-dimensional vector space with Kn C Df (x)v 01 =
-
=
f e,+,,. enj be a basis of E and f el,. ..' e,j be a basis of F. 4 For each y C Kn let A'(y) be the ith component of y with respect to the e basis f el, The vectors r 1) are a basis fi Df (x) ej (i E f 1, , en 1. E(DF. Let
=
.
.
.
of the r-dimensional vector space Df (x)Kn C K'. We choose linearly is a basis of K". independent vectors f,+l.... f, such that f fl, , f,,, I .
For
z
E K'
let
y'f (z)
.
.
be the ith component of
z
with respect to this
basis. We define
A'f (Z)
0
1-tf (Z)
0
A (Y)
+I(Y)
E
Kn,
,(Z)
f0
=
An (Y)
E
Kn.
0
e
The map g: U
-4
IK',
continuously differentiable
is
by
y
and
-4
[t
o
Dg(x):
the inverse function theorem there is
f (y) Kn a
+ A (y) -4
Kn is invertible. Hence
neighbourhood 0
such that g is a diffeomorphism from 1 onto the open set is an c > 0 such that B,(x) := Iz E Kn : Iz g(x)l < Ej C define g -'(B, (x)) and the bijective map
c U
g(Z ).
-
U
---
BI (0)
C
Kn'
Y
(g (Y)
This numeration of basis vectors of E 8 F may more practical later on in the proof.
prove to be
-
of
x
There
g(U).
We
g (x)).
seem
slightly
odd but will
Analysis
2.
56
The rank of for all y
for all
are
both
implies that dim(Df (y)Kn) bijective for all y E U and Dg(y)(v) the maps Df (y): F -+ Df (y)F and --4
K' (D
f01
C
v
o
=
=
Df (y)F be
show that the map h
y'+',..., yn. We write K' Kn. Since f (y) hQ it f (y)
E
K' E) K`
101
r
f
o
0-1
Kn
the inverse to the
does not
Kr E) K` and
variables any
=
is
Let vy: Kr E)
bijective. now
F
E
v
U which
on
Dg(y)
Df (y)F
/-t:
latter map. We will
manifolds
is constant
f
U. Since
E
I-t(Df (y)v)
on
ED
-!A(y)
-
Iu
o
depend
v
=
f (x)
the
on
V1 (D V2 for
!A(x))
(D
we
have
eDf(y)v
Djhj-!,Aof(Y)ED-,!A(Y)--! PO f (x) E) + D2h, ittof (y)(3)-i cx(y) -1
vy
o
p
o
Df (y)
into this
D2h,.I,,Of (Y)E)i,\(Y)- i./,Of(x)E)iL,\(x) where ay is the linear map
ay: Kr ED
Df (y)v
Av.
o
equation implies Av
=
ay
o
/-t
o
Df (y)v,
evy
-
Djhj-,!,UOf(Y)(D-,1A(Y)--! tto f (x) E)
is invertible and A maps K' onto =
0 in order to prove
D2h
=
-1 A (x) ,
f 0 j Kn-r
0. For
E
v
F
0. The map 0 and and therefore ay o ft o Df (y)v coincided on F with Dg(y) and is therefore bijective. In particular,
A(v)
have
pol)f /-t o Df F
==
Since can
--+
y
to show that ay
only
o
given by
,,pof (y)E)!A(y)
Since y have
we we
101 -4K',
o
P
-i.\(x)
c
Inserting Df (y)
0
A (x)
write
We
=
=
101
Kr
we
which is the domain of 7y. Hence ay vanishes. h(y) does not depend on yr+1.... yn
have proven that
h(yi)
identify
instead of
h(yi
(D
Y2)-
K' with Kr ED K'
r
and write
w
W1 E) W3.
=
be the canonical basis of K` and let
jbj,...,b,-rj spanjfr+j,...,f,,j We define the map
V): B, (0) This
C
be the linear map which is defined
-r:
K`
by -r(bi)
=
0 by
K'
--4
K'
ZI ED Z3
-'
h(ZI
+ A (f (X))) +
'T
(Z3)
-
implies 0
Pr
0
O(Y)
-=
0
0
=
V)
0
=
-0 E
Pr( g(Y)
g(x))
Pr(1P
-IP
0
Pr(P
we
0
f (Y)
0
f (Y)
-
A
0
0
fW
f (X))
+
1A (Y) -1AW)
Let --,
fi.
=
-
h(/,t
o
f (y)
-
p
o
f (x)
+
A(f (x)))
57
Manifolds
2.1
=
-
h(I-c
o
f (y)).
f
6
-
projection of tt to the span of the first r canonical basis 0 h o tt. Equation D2h by A. Then we have h o 0 0 0 (Y) h in which turn f IP (Y) gives Pr ( 1 A (f (y))) implies f (y) We still have to show that V) is a homeomorphism. But this follows I since both h and T are homeomorphisms.
Denote the
vectors of K'
=
=
Corollary 2.1.1. Let Mbe an n-dimensional, N be an m-dimensional N be a Ck -differentiable map which has conCk -manifold, and f : M stant rank in a neighbourhood of x E M. Then there exist charts (U, v), (V, 0) centered at x, f (x) such that W(x) 0, O(f (x)) 0, and --
=
=
0
0
f
0-
-
1 :
Kn
(xi
Km'
_
I ....
Xn)
F
---
(XII
...
I
xr, 01
...
10).
Proposition 2.1.1. Let Mbe an n-dimensional, N bean m-dimensional N be a Ck -differentiable map which has conCk -manifold, and f : M stant rank in a neighbourhood of x E M. Let y E f (M). Then the set f -'(y) is a closed (n r) -dimensional submanifold of M. -4
-
Proof.
Let
E
x
f
(y)
and take charts
as
given by Corollary
2. 1. 1. Then
have
we
p(U
n
f -1(y))
=
=
The assertion follows
p(U) o(U)
directly
n n
(0 fz
o
f
o
c- K
V-')-'(O)
n
:
ZI
=
...
=
Z'
from the Definition 2.1.4 of
=
a
01.
submanifold. I
Proposition
2.1.1 is
a
powerful
tool which is used to construct manifolds.
R, x i--> IX12. Example 2.1.3. Consider the submersion f: R' \ 101 M which O,in radius of > the that sphere p Proposition 2.1.1 implies -I of submanifold is coincides with the set f (p2) an (n I)-dimensional of atlas down an write himself herto for reader or R'. The directly try -+
-
the
sphere
of radius p. It is much
2.1.2 Partition of
more
work.
unity
we provide a method of localisation using functions practical if one wants to define a global object using charts. The prime example is the definition of integration in Sect. 2.5.4. This R. method works only for real manifolds: K This subsection is somewhat technical. The reader may therefore want to skim (or skip) this section on first reading and to return to its proofs when the results of this section are needed.
In this section
which is
=
58
Analysis
2.
manifolels,
on
The aim of this section is to construct
an
atlas and for each chart
a
function with support in this chart such that
(i) (ii)
point there
at any
the
sum
only finitely
are
of all functions
We start with two
)lVi
sequence
)1Vj+j
C
Proof. all
a
and
of
a
manifold. If
f)/Vilic-N
open sets
UiEN)/Vi
jUjjjErj
Let
by (i)) equals
I.
topological lemmas.
Lemma 2.1.5. Let M be exists
(which
many charts which intersect
is well defined
be
a
"::
M is not compact then there
with compact closure such that
M'
countable basis of the
topology of
M such that
Rj- are
compact. Let W1 := U1. Since the closure of this set is compact there is a k, E N with Wj- C Uk, Uj =:: VV2. We proceed now by ini= 1 duction.
Assume,
we
Ui. Then there
have is
already constructed )/Vj, kp+l > kp such that WP C-
a
.
_
Lemma 2.1.6. Let M be cover
all
of M.
.
a
manifold. And lualaEA
Then there exists
a
.
,
)IV,,
Uk,,+l j=1
where
Ui
)/Vp )/VP
be open sets which
countable collection
jVjjjEN of open
sets such that
(i) each Vj lies in some Ua and has compact closure, (ii) each Vj intersects only finitely many Vi. ("') M Uj E N Vj' ::-::
Proof. We will first show that countable. Let 0
=
I Oi IiErq
be
a
we
atlas of M. For each Vc there
an
that
Vc
U Zj Oi,,.
=
Let Oi be
index
c(j)
can
restrict to the
countable basis and let are
countably
Since 0 is countable
so
many sets
is the set
that A is
case
f (V,, Oc) I cc- c Oi,,
E
be
0 such
j0j,cjjEN,cEc
C 0.
of these sets and choose for each j C N an E C with Oi c V,(j). Then the collection j(0j, wc(j))IjEN is a a
re-numbering
countable atlas of M. Since Oi is
homeomorphic
to Kn there is for each
dense sequence fXi,j IiEN of points in 0j. For each (i, j) C N x N let a(i, j (E A) be an index with xi,j E Uqjj). Then the countable subset fUa(ij)jijEN Of fualaEA covers all of M. It follows that we can assume
j
a
without loss of Let Wo
=
generality
in Lemma 2.1.5.
by finitely
Ual, (k) N
x
n
many
O/ViliEri be the sequence of sets constructed The set )lVk+l \ Wk is compact and can be covered Ual(k), Ua.,,,(,,) (k) for every k c N. We set Vk'j =
...
(Wk+2 \ Wk-1)
N is countable.
property
(iii)
which defines
Property (i)
a
countable
family
of sets since
is clear from the definition of
follows from the fact that the sets
Finally, property (ii) by finitely many Vk,jM.
that A is countable.
0 and let
follows because each set
)/Vk+l \ Wk Wk is only
Vk,j
cover
and
all of
intersected
I
Manifolds
2.1
Lemma 2.1.7. Let M be
Then there is
(i) h(x) (ii) h(x)
r,
Fig.
real
a
manifold
C' -function h: M
a
1 0
--+
and U, V open sets with with
59
R
C
V
-
[0, 1]
for all x E a, for all x E M \ V.
r2
rl
2.1.4. The
proof of Lemma 2.1.7
Proof. We consider first the special
r2
case
but different radii: Let 0 < r, < r2 and Ix (E Rn: jxj < r2j. The map Br,(0)
of two balls with
B,,(0)
=
Ix
(E
origin
Rn
jxj
:
0 E Rn
rij,
<
=
gri,r2: R+
is C' and has the
((t-ri)(t-r2)
O
exp
R+,
--
t
-->
1
< t < r2
otherwise
Irl, r2l
support
for
-
It follows that t
', 9r,,12 (s) ds fr,12 gri,r2 (s)ds
jri,r2 (t) is also C'. The
r
properties
j,r1,r2 (t)
j,,,,2 (t)
cz
(0, 1)
jr1,r2 (t) imply
12:
[0, 1]
is well defined and satisfies E
:::::
G
for all t G
0 for all t E
[0, rij, (ri, r2),
[r2) 00)
that the C'-function
h,,
x
for All t
---+
R+,
X
SUPP(h,i,r2)
1-`
=
gri, r2
(11 X I I )
Br2(0), hr,,r2 (X)
=
1 for all
B, (0). Consider
now
open sets
U,
V with
R
C
V and let
fVjjjEN
collection of open sets as provided by Lemma 2.1.6. For each now construct a smooth function hj which satisfies
j
be
we
a
will
60
2.
Analysis
manifolds
on
(i) hj (x) E [0, 1] for all x E M, I for all x E U n Vj, (ii) hj (x) (iii) supp (hj) c V n Vj. =
Let
U n Vi and
E
x
and Wx c V n
Br2(X)(0)
(IlVx, (,ox )
There
Vj. (PX(WX)'
C
h3.: IlVx
--+
X
are
be
chart centered at
a
R+,
hr, (x),r2 (X)
--*
y
xi,
...
I
0
=
PX (Y)
0
for y E
XK
Wx
otherwise
is well defined and smooth. Since U n
points
with Wx (X)
The map
0
many
x
positive numbers ri (x) < r2 (X) such that
is compact there exist
Vj
finitely
such that the open sets -
1
(Br, (x,) (0))jk=1,...'K K
Hence the map hj (Y) :=: 1 f1k=1(1 hiX (y)) is also well defined and smooth. Clearly, we have hj(y) c [0, 11 for all y c- M. Since U n
cover
Vj.
-
0 for all y hx, (y) \ V n Vj If y E U 1. implies hj(y)
E M
=
M
-
n
Vj
\
Vn
Vj
we
-
have also
then there exists
an
hj (y)
=
.,
xi with h3X
0 for all y G I which (y) =
=
The function
h(x)
Ej' , hj(x)
is well defined since for each
x
all
finitely many hj(x) vanish. h satisfies supp(h) c V, J'(x) > 0 for all x E M, and h(x) > 1 for all x E a. Hence in order to prove the lemma we x c M : h(x) < 1/21. only need to normalise h appropriately. Let U This set is open and its closure is contained in M U. Hence by the same construction there exists another smooth function h with h(x) > 0 for but
all
X
M,
E
and
h
h(x)
> I for
all
x
E
1,
and
do not vanish both at any
h(x)
then x
E
h(x)
M
\
> I
V then
h(x)
=
by
each
h(x) +
h(x)
=
h(x) e
[0, 1]
for all
0 which in turn
0 and therefore
h(x)
=
E M has a neighbourhood of finitely many fa, I for all x. fa(X)
point
x
Observe that
~
x
E
M. If
implies h(x)
x =
E
U
1. If
1
0.
Definition 2.1.7. A smooth partition of unity is f fa: M - [0, 11 IaC-A such that
(i)
\ a.
=
Further, h(x)
and therefore
C M
given point. Hence
h(x) is well defined and smooth.
supp(h)
a
set
which is
of functions
only
intersected
the support
(") EaEA
=
A partition of unity is subordinate to an open covering every a (E A there is a b (E B with SUPP(fa) C Ub-
fUbjbEB if for
2.2 Vector bundles and the
tangent bundle
61
(Existence of a partition of unity). If jUbjbEB is covering of a real manifold M then there exists a countable partition of unity ffjjjEN which is subordinate to jUbjbEBTheorem 2.1.1 an
open
For each
Proof.
neighbourhood exists
a
of
sequence
M let
E
x
be
b(x)
with
Ub(x) 1Xj1jEN and x
an
index with
x
G
and
Ub(x)
Lemma 2.1.6
ab(x)
be
a
implies that there
C
Ub(x).
a
countable collection of open sets
1Vjj
such that
(i) each Vj lies in some Ub(x.,) and therefore Vj (ii) each Vj intersects only finitely many Vj; ("') M UjEN Vj*
C
Ub(xj);
=
We
apply
can now
to find
a
the
same
argument
to the collection of open sets
countable collection of open sets
(i) each Wk lies in some Vj(k); (ii) each )/Vk intersects only finitely ("') M UkEN )/Vk
many
tWkIkEN
Vj
such that
Wj;
=
*
[0, 1] with By Lemma 2.1.7 there is for each k a function hk: M Since each Vj M for all 0 x E 1 for all x G Wk, hk (X) \ Vj(k) hk (X) 0 for have M each for we x chk (X) intersects at most finitely many Vi well is that the This k. x sum hk (X) all but finitely many implies have in each lies we Since smooth. and some x hk (X) defined Wk ---*
=
=
-
=
I for all
x
G
M. Thus
hk (X)
E' 1= 1 hi (x) Property (i) of Definition 2.1.7 is satisfied only finitely many Vi and supp(fk) c Vj(k). directly from the definition of fk. Hence ffkjkEN is
is well defined and smooth.
because each
Vj
intersects
Property (ii) follows a partition of unity. That follows from
supp(fk)
C
it is subordinate to the open
Vj(k)
C
I
Ub(.,.1(,)).
Remark 2.1.2. In Lemma 2.1.7 and Theorem 2.1.1 it restrict to the
eral. This is
so
case
K
=
covering UbjbEB
R. If K
=
C, both results
was
are
necessary to
wrong in gen-
complex- differentiable maps are automatically locally be written as a power series. This would be
because
analytic, i.e., can impossible for the function h
in Lemma 2.1.7.
2.2 Vector bundles and the
tangent bundle
ordinary calculus, the derivative Df of a map f : K' -4 K' is the linear approximation of f, i.e., it is defined by f (x) f (a) + Dfla(x-a) +
In
=
F-p. 5 j p.
62]
62
Analysis
2.
o(Jx
al)
where
manifolds
on
o(Jx
al)/Ix al --> 0 (x -+ a). To study the linear approximation of a map rather than the map itself is certainly one of the most powerful approaches in mathematics and physics. Because of the -
limit
-
-
0 this approach is often referred to as working infinitesa imally. Until the middle of this century people spoke of infinitesimal (or infinitely small) displacements (meaning the vector x-a if it was 'small'). This terminology can lead to misunderstandings but stresses the main idea of analysis. While we will give a modern presentation, it is a good idea to keep the 'infinitesimal way of thinking' in mind. x
--
-
The definition of a linear approximation of a function f rests on the linear structure of Kn. In the general case of a manifold, such a structure is not at hand. But it is possible to define a linear approximation of a map in two to the the
[P.
steps. First, we linearise the manifold itself. This gives rise tangent space TaM at a point a of a manifold M. Then we
linearise the map thereby obtaining between (linear) tangent spaces.
linear map
a
Taf : TaM
--+
Tf (,,)N
We will linearise the manifold M
by attaching an n-dimensional vecpoint x C M. At first one may think that it is sufficient to consider the product M X Kn and to define TxM JxJ x Kn. However, this would introduce a global structure via the (global) product x. In order to keep within the spirit of localisation, we can only demand that such a product exists locally. This motivates the following tor space to M at each
definition. Definition 2.2.1. A k-vector bundle
manifold
M is
a
and submersion
(i)
For each
triple consisting of
7rE: E x
c-
a
(E, 7rE; M) over an n-dimensional (k + n) -dimensional manifold E,
M such that
--
M is
(7rE)_1(x)
a
k-dimensional vector space
over
K,
(ii) for
each
x c
M there is
for each
a
linear
over
-
(7rE)
1
(U)
a
diffeomorphism
7
__4
(7rE) -I(Y)
isomorphism.
M is called the base
the fibre
k
-
and
y E U the restricted map
V)y: Kk is
neighbourhood U
UxK
0: where
a
manifold,
7rE
the
y, E the total space, and
projection, Ey := (7rE)-I(Y) 0 the bundle chart or local
trivialisation. The set of tangent spaces of a Manifold forms again a manifold of a special a vector bundle. While we will construct many special vector bundles
type, and
general
vector bundles
definition is not central to
are
our
of
importance
discussion.
in gauge
theory, their general
2.2 Vector bundles and the tangent bundle
We will often
just speak of the
vector bundle E instead Of
63
(E, 7E, M)
-
Kk. We call (E, 7TE) M) trivial if there exists a local trivialisation of the form 0: M X Kk -- E. In this case, E can be identified with M X Kk. An example of a vector bundle which is not trivial is given by the M6bius band. Notice that this is just
(M&bius band,
2.2.1
Example
band M is also
(Wi)-' (x', 0)
7rm(x)
(XI, X2)
localisation of M
a
a
continued
from
X
o
and i
The
projection
R2 is Wi(x) where pl: R2 E 11, 21 is an index with x E Ui.
o
pi
53)).
page
vector bundle. The bundle
--4
is
M,5bius
given by
the projection It is clear that
this vector bundle is not trivial. Definition 2.2.2. Let E be which
satisfies IrE(U(X)
==
x
a
for
vector bundle. A map
all
x
E
U is called
a
a:
U C M
--->
E
section of the vector
bundle E. A collection
ful,
...
7
9k
I of sections
spanf a, (x) for all
x
E
U is called
,...
following definition
nius
2.5.3).
will
The
play
a
(E, 7TE, M)
be
(F, 7rF, N) of the vector bundle E and submanifold N of M such that
(7E) IF:
IrF
(X) I
role later
dle
(i)
Uk
Ex
=::::
frame of E.
a
Definition 2.2.3. Let
7
such that
F
--4
N
defines
a
on
Theorem of Frobe-
vector bundle. A vector subbun-
consists
a
(cf.
of
a
submanifold
F
of E
vector bundle structure with base
manifold N,
(ii) Fy
is
a
vector
subspace of Ey for
2.2.1 Construction of the
all y
G
N.
tangent bundle
In affine geometry we had distinguished between points x E A' and translations (or vectors) in the associated vector space Kn. A translation a global concept. Notice that a translation is originally moving the point x along the curve yv: [0, 1] -* An, t -+ x + tv to its endpoint. The velocity vector of the curve is v which may be regarded as the infinitesimal (but in this case exact) approximation , An, t --> -y(t), we of -yv. Given an arbitrary smooth curve y: (a, b] take its derviative as its infinitesimal approximation at a given point x - (to). Taking all these velocity vectors at x we obtain a vector space TxAn which is attached to A' at x. This vector space is in a natural way isomorphic to the associated vector space given by the translations. In the following we will transfer these ideas to manifolds. The main
v: x
F-
x
+
thought of
=:
v
as
is
F
p. 62
p.
65]
64
2.
difficulty which
Analysis
we
manifolds
on
have to solve is the lack of take the derivative of
we can
Let M be
associated vector space in
an
a curve.
smooth
manifold, x c M and -y: (-E, E) --+ M be a through x at parameter value 0. Another smooth curve (-e, e) with x (O) is called x-equivalent to -y if d o -y) o This definition is independent of the chosen (W (W Tt 0. dt 10 chart and puts all curves through x with the same velocity into one equivalence class [-y,,] smooth
a
in M which passes
curve
=
=
-
Definition 2.2.4. Let M be
a smooth manifold and x cz M. The space of all equivalence classes [-y,,] is called the tangent space of M at x and denoted by T,,M. Its elements are called tangent vectors.
We must show that TxM has indeed the structure of
Choose any chart
(U, W)
centered at
vector space.
a
This chart induces
x.
a
bijective
map
d (9 x': TM which Let
a
we can use
E
[-y,,]
dt
(W
to induce
[-yx], [px]
K and
K',
---
on T,,M the vector T.,M. Then we define
E
a[-yx]
:=
[AXI
:=
-
-y) 10
space structure of Kn.
(eP)-l (aRP([7x])) x
x
and
[-Yxl
+
(ex
((9x1 OXI)
+
(9x,([-Yxl))
-
This vector space structure is independent of the chosen chart. In let (V, 0) be another chart centered at x and denote (9 ' o ((9P) x
(9'P,w-'.
Then
we
x
have
x
((9x
(a(9xI ([,Y.])) =
('9")
-
1 0
e"
x
=
((9V))-1(9V) x
=
and
-
x
(en
-1
'W-1
o
o
x
x
(90 (I
0
x
(exI,)-,
-1 (aev (eOX) e"([-Yxl))
x
x
x
,
Yxi))
(01(9X, Q -Y-D)
analogously
(ex
(ex10([-Yxl)
+
(ex ) + (9w x
x
1QAxD)
ex (9x
0
0
((9x,P)
-
I
(e'O
0
x
(,g")-l .,e"([-txl) x
x
(e b)-l e b([Mxj) ,
x
x
x
x
x
x
fact, by
tangent bundle
2.2 Vector bundles and the
((gqp,w-I)-I
+
(ex
We call the set TM
vector bundle
TxM for all
on
=
Tfa: (7rTM) The
are
infinitesimal models
[p. I
a
smooth
manifold.
There is
associated with M such that
a
1]
63
p. 65
natural
(7rTM)
(X)
UXEM TxM
-I
and define 7rTM
x.
of M and define the structure of
f ((7rTM) -I (Ua)) Tfa) I
(Ua)
Kn
-4
'/x I
K n'
We choose
an
smooth manifold
a
where Oa (X) 0) 19'p' x
-4
(/ [XD
-
bijections bijections Oa : Ua
are
as
M.
through the atlas
M
+
be understood
can
(TM, 7rTm, M)
x E
I A, (Pa)IaEA
atlas
[_YX D
2.2.1. Let M be
Let TM
Proof.
.
UxcmTxM the tangent bundle of the manifold
:=
M. The vector spaces TxM of M at x.
Proposition
(1YXD) ex0 QYXD)
191,
,
x
(ex
65
X
Kn
-4
(7rTM)
diffeomorphisms linear isomorphisms.
then
-I
(Ua)
such that
(X, V)
i
(Oa.)x:
Kn
-4
-->
(ef"') _1(v)
-
x
TxM,
v
1-4
(ew-)-'(v) x
p. 65
always practical to work with equivalence classes. We will therefore also give an equivalent definition which is better suited for calculations at the cost of being less intuitive. The key observation is that each tangent vector 1-yx] E TxM induces a map
It is not
If
D
E
Coo (U,
K)
:
U is
an
open
neighbourhood
of
x
K,
d
f
-4
at- f
This map has the
0
-Y(O).
following properties.
(i) D[,yl.,,, is K-linear, (ii) for any smooth functions f
D[,yl., (fg)
=
.;
,
g: M
---
K the derivation property
D [,yl. (f) g (x) +
f (x) D [,yl. (g)
holds,
(iii)
for any open neighbourhood U of which coincide in U we have , g
f
This motivates the
following
x
and any two smooth functions
(f
(g).
[I
p.
1
75]
66
2.
Analysis
manifolds
on
Definition 2.2.5. A map
If
v :
C' (U,
c-
K)
U is
:
an
neighbourhood Of xj
open
---->
K
satisfies properties (i)-(iii) above is called a derivation. The vector space of derivations at x with addition and multiplication being defined pointwise, (avx + bw,)(f) a(v.,(f)) + b(wx(f)), is denoted by D ,. which
=
Remark 2.2. 1. The reader may wonder why in our definitions derivations U is an open neighbourhood of xj instead of on If E C' (U, K) :
act
the
simpler
C' (M,
K),
set C'
(M, K)
but in the
For K
-
K
case
R
=:
could indeed have chosen
we
C there exist
=
only
very few
globally
defined differentiable maps in general. In fact, in the extreme case of the complex torus only the constant functions are smooth. However, locally there is
always
Lemma 2.2. 1.
v.,(f)
=
0
for
Let
I +
lvx (1)
a
-
If f :
M
K is constant in
--+
=
a.
=
vx
(f)
2avx (1)
Theorem 2.2.1. Dx is
=
(f /a) 2vx (f ). avx
==
is
a
Let
=
--+
avx
=
(1)
=
(I 1)
av.,
-
=
avx
(1)
-
I
n-dimensional vector space.
an
Proof. We know already that Dx n. only to show that dirn(Dx) and let x': U
neighbourhood of x, then
a
all derivations vx E Dx-
f (x)
Proof.
abundance of smooth functions.
an
vector space and have therefore
(U, W)
be
a
chart centered at
W-'.
K the ith coordinate component of
x
Observe that
K the identity h(y) W(U) h(O) + Enj , hi(y)y', 1 ah(t') dt and y is the ith coordinate where hi is defined by hi(y) := L --5 1 K we in Kn. Applying this identity to the function f W- : W(U) we
have for any h:
--+
=
o
obtain n
f(Y)
=
f (X)
Xi (Y)
+
Hence for any derivation vx E Dx to
(f W-') i W-' (Y)
f
-
-
get
we
n
vX(f)
=
VX
(f (X))
+ X, (X)vX
The first summand vanishes
x'(x)
vx is
uniquely determined by
0. From vx (f)
((f
by
since
=
E (vX (Xi) (f
+
=
-
0-')
-
i
0
0))
W
I
WW
-
i
-
Lemma 2.2.1 and the last term vanishes
Ei'- I (f
the
n
o
p
-'),
o
(x')
W (x) vx
numbers vx (x 1),
.
.
.
,
VX
we see
(Xn).
that I
2.2 Vector bundles and the
Definition 2.2.6. Let M be
be
manifold, (U, p)
a
tangent bundle
a
67
chart, and
vi
p":
Kn
--,
vs.
K, vn
Then x': U
-->
K,
-4
y
p'
is the ith coordinate function with
(p(y)
o
(xl,...,x') a coordinate system. defined by axi(xi) Jj3 is called the
and the collection
(U,W) respect The pointwise basis to
=
(U, W)
CauBian basis associated with GauBian vector fields. We will
Corollary
2.2.1. Let M be
centered at
x
C'(M)
we
write
often simply smooth
a
fields axi are called 0i instead of
and the vector
manifold
and
(U, W)
M. For any derivation v., E D., and any have E
n
V-M
=
n
Evi'9xif
.'9f
=
ST
a
chart E
W-1
axi
i=1
i=1
where D7 is the usual
0
E v%
be
function f
partial
derivative in Kn.
Proposition 2.2.2. The tangent space T ,M is canonically isomorphic to Dx. The isomorphism is given by the map i: TM --+ D,,, iQ-1] ,)(f) A o dtf y(to) and well defined. Proof. Clear by
construction.
dispend with the symbol Dx and always use TxM instead. Our first definition using curves has the advantage to work also in infinite dimensional settings. However, we are only concerned with finite dimensional manifolds and derivations are more practical to work with than equivalence classes of curves. Hence
we can
Definition 2.2.7. Let M be
vx
*
f
:
=
The map
a
manifold,
x
E
M and U be
a
neigh-
For any v., E Tx M and f E C" (U, K) the number df (vx) : vx (f ) is called the derivative of f in direction vx,
bourhood of
x.
=
df :
TM
-->
K,
-->
vx
df (vx)
is the differential
of f
2.2.2 The derivative of maps between manifolds
In the
preceding
section
we
linearise differentiable maps
taining linear
maps
Tx f :
Definition 2.2.8. Let a
differentiable
map.
f: M
Tx M
--->
--+
N between manifolds
Tf (x)
can now
thereby
ob-
N.
manifolds, x E M, and 0: M -4 N be Txo: TxM --> TxN, Txo(v)(f) v(f o V)) is (Or simply the derivative) of V). We will often
M,
Then
called the tangent map denote T,,o by 0.,
have linearised the manifold. We
N be
=
68
2.
Analysis
manifolds
on
Observe that in terms of
given by [-(.,] of the curve 0 o -y ,0
is
equivalence classes of curves, the derivative of which is clearly the linear approximation
[V) -yp(,,)] at O(x).
-4
o
It is instructive to calculate the
.
systems. In these we
tangent
map with
respect
to
a coor-
o) and (a, 3) be charts of M and N respectively, Xn) and (5 1......; k) the associated coordinate charts 0 has the representation T1 o-' and p 7P
dinate system. Let (U, and denote by (x 1, .
.
,
o
==
o
calculate n
,O.V(f)
E vi'9xi- Y
=
-
0)
V'
=
C W
-
V)
0
-
Oxi
i=1
vi'9f
0
(V, aP 0
a
,9T13 =
a.V Hence with respect to
Y
0
V
'9xi
W
a;,7 -if .
coordinate system, the tangent map 0" is just Again we see the tangent map Txo is the
the derivative of the map T1. linearisation of 0 at x. The
following is
an
immediate
corollary of Definition
2.1.6 and Lemma
2.1.3.
Corollary 2.2-2. Let M be a manifold and x E M. A continuously differentiable map 0: M ---+ N is a submersion (respectively, immersion, local diffeomorphism) near x if and only if TxV): T M T b (x) N is surJective (respectively, injective, bijective).
2.3 Tensors and tensor fields Tensor
fields play a central r6le in geometry and physics. Differential forms are not absolutely necessary for the theory of space and time. However, their usage has many advantages and they also provide a very natural way to define integration. In particular, the integral theorems of Stokes and Gaufl have a very simple, common form when stated in terms of differential forms (cf. Theorem 2.5.5). Unfortunately, the introduction of differential forms requires some
technical preparation.
Some readers may
differential forms
on
The tangent bundle of mations
JTMJ.
linearisation
we
therefore wish first reading.
a
to
skip the
sections
dealing
with
manifold is the collection of its linear approxiuse of the simplifications arising from
In order to make
need to express
physical
and geometrical
objects
in terms
of maps which are adapted to the linear structure of T,,M for all x E M. We will see later in the book that the notion of a tensor field provides a
good
section
framework for this we
simply
(here
still rather
introduce tensor fields
as
vague)
idea.'In the present
mathematical concepts.
2.3 Tensors and tensor fields
2.3.1
Algebraic preliminaries:
In linear
algebra,
the concept of
69
tensors
tensor unifies
a
vectors, linear forms,
bilinear forms, linear maps, determinants etc. Let V be an n-dimensional vector space over K. Then its dual space V* is the vector space of all linear maps V ---> K. It is easy to see that V* is isomorphic to V. In fact, if jej,... , enj is a basis of V then the set 10 ...... O'l C V*, defined by
Oi (ej)
=
6 ,,
is
a
basis of V*. It is
defined
uniquely
by f el....
7
e,,, I and
called the dual basis. While the
isomorphism
the choice of basis
of V and V* defined
lei,..., enl,
there is
by
iOi depends on isomorphism tv of
ei
canonical
a
V and V**.
V*,
tv: V
tv(v): f
V
-->
f (V)
E
following we will freely make use of this canonical identification given by tv and write v instead of tv(v). Using this identification we not only can view a vector v as a linear
In the
V**
V
Tap V*
--->
A: V*
V
x
generalised
K but also
and
linear map A: V
a
K, (f, v)
--->
A(f v)
i,
(")-tensor
Definition 2.3.1. An
is
a
V
X
which is linear in each
space
s
VXV*
X
and an
a
linear map V
V is
--4
bilinear map can be
V*
--+
K
copies
or a
0
tensor
is
an
times
r
of order
co-
The
by Tsr(V).
is denoted
s
=
For
x
We say that
its entries.
of
...
r
The most important special cases T,O(V) V*. A bilinear form such
T20(V)
X
times contravariant tensor
(r)-tensors
of all
...
copies
s
variant and
a
as
map
8
0:
V
f (Av). This reasoning following unifying concept. ,
let to the
we are
-->
=
are
Too (V)
as a
scalar
K, To' (V) V, and product is an element of =
element of
an
T11(V).
explanation of the terminology "covariant/contravariant"
see
will need to define the components of a tensor. This in turn requires the introduction of the (natural) "tensor product" & of tensors which generalises the usual product of numbers. Remark 2.3.1 below. First
Definition 2.3.2. Let sor
0
product 0 0
0 (Vil
&
...
)
0
E
LS+p
...
O(Vi
Tsl(V), 0 (V) by
E
,+q
V87 1011 :=
0
we
)
wp)
....
7
W11 Vs,
...
Wi
I
G
I
Tpq(V).
Then
we
define the
Wr, 77,.... 77 q)
...
IWr) O(wj,
wp,
n1,
ten-
70
2.
Analysis
manifolds
on
Observe that the tensor Lemma 2.3.1.
The tensor
Proof. Let 0, 02 03
is not commutative.
product
is associative.
(ri), (r2), (r3)
be tensors of order
7
we
product
81
82
respectively.
83
Then
have
(01
0
02)
(D
Wil (01
03 (V1
...
I
X
and
...
,
.
.
.
1,
....
V"'
I
*
'
'
W1
03(VSI+S2+11
-
V81+821 V81+82+11
rl+'r2+1, Wrl+r2+1,
,
vs, VS1 +1
I
VS1 +32 +S3
I
rl I
...
VS1 +,92
7
7
W
.
.
.
...
I
V81+82+831 ,
W
rl
+r2 +r3
,)rl,
I
)
,
03 (Vs I +S2 +11
01 (vi
+
rl+r2
,,rl+l, X
r, ,
b2) (V1
0
VS1 I VS1 +I I-
I
ri
...
)02 (Vs, +1,
I
VS1 +82 +,93
7
)rl +r2 + 1,
I
-
'
*
ri
I
VS1 +82
,rl+r2+1,.
+r2 +r3
r1+1 1
rl I
...
+r2
I
,rl+r2+r3)
analogously
01(9(V)2
(D
7P3)(V1
Wil (9
...
rl, W +11
I
....
VS1,
I
( )II
?P3)(Vsj+17-
,,ri +
ri ...
I
-
vs 1
I
,rj+1....
.
*
I
,
W
I
1
IVSI+S2+S3)
+r2 +1
rl+r2+r3)
,
rl)
.,
VS1+S2 7 VS1+82+11' rl
I ....
rl+r2
rl
1W
...
+r2'
V)3 (VS1 +S2 +11
Lemma 2.3.2.
.
)VS1+827VS1+S2+11' +r2 +1
ri
i
01 (vi,
X
IVSIIVSI+ll ri
01 (vi
(02
....
+r2 +
1,
.
.
.
,
I
VSI+S2+S3'
Wrl+r2+r3)
U)rl)'02(Vsl+l
....
I
VSI+S2'
) rl
VS 1 +82 +S3
If jej.... enj, 101,
+r2 +1
on I is
I
rl+r2+r3
a
pair of dual bases then
the set
f 0" forms have
a
(9
basis
...
(9 0'. (2) ej, (9
of the
dim(T,,r(V))
=
space
...
(9
T,(V) of
all
(") -tensors. S
in
particular,
Proof. The set of tensors f 0" independent. In fact, let
(9 0i. (9 ej, (9 & ej,, I is be numbers such that ...
n
V)
E V)ij'---j"'0i1
=
j3
.....
we
nrn'.
j'-1
(9
...
(9
0i.
(2) ej, (9
...
(9 ej,
=
0.
linearly
2.3 Tensors and tensor fields
Then 0
:::::::
O(ek,
I
....
e-k,,
011 ,
...
,
1017,)
71
hence the tensors 01'-*1',,, k,
=
...
k
linearly independent. Conversely, we see that for and any v,.... v, E V, 77' E V we have
0
any tensor
are
all
Tr(V)
E
*
,
n
O(Vi
....
Vs, 17,
......r
O(eii
&oil
I
...
I
ei,
0 ej,
oilI... Oil Oil
I
1
ej,,
(VI
....
VS,
nil
7r).
....
T,,(V) is nrn' since there are exactly n' choices of t-tuples (with possible duplication) from a set of n elements. I
The dimension of
ordered
Definition 2.3.3. Let
01,
on
V)
c
Tr (V) and
f ei,
I
e,,
be a. basis
be the associated dual basis. The numbers
0,"
of V and defined by
n
ii
are
the components
In the
physical (and
...
i,' 0',
(9
...
of 0 with respect old
(9 0'. & ej, (9
(9 ej,,
f ei,
to the basis
mathematical)
...
en
literature it is the standard to
use
for contravariant entries upper and for covariant entries lower indices. This provides a checking mechanism for the syntactical correctness of tensor formulas and also
ing
effective notation in the
simplifies
the
(Einstein's
of formulas involv-
interpretation
components. In Remark 2.3.4 below
tensor
summation
will introduce
we
which is
convention)
a
very
prevalent
physics literature and has at its core the difference between Unfortunately, many modern mathematicians indices for all entries on grounds of "aesthetics".
per and lower indices.
lower
Remark 2.3.1. The
terminology "covariant/contravariant"
arose
upuse
in the
19th century and refers to the transformation of tensor components under transformation of a given pair of dual bases f ell Oil Onj. , e,, 1, f .
Let
v
==
T11 (V) fined
I:ni=1 Viei
be
we can
=
write
V,
W
=
I:ni=1 wiO'
E
v
Eni=1 f),Ei, n
W
=
rn _i=l CoiO'. n
W(W)
w
(z-v%)
.
.
V*, and A
invertible linear map. Then f jI, 0' o (A-') are also Aei and
an
by ji
(E
.
.
a
.
.
=
Ei,j Aiei 0 Oj
,
.
For any
w
G
V
.
.
we
n
Cv'A'i-w (ej)
n
Cv'AiWk Ok (e j)
E
n I den 1, f ', pair of dual bases and
ib'w (Aei)
n
.
Cv'A3j wj
,
have
72
Analysis
2.
manifolds
_Tj'=, Aj' wj.
Hence cDi
i.e.,
on
in the
The components of
Similarly,
n
(A)
v
(j
( j 0' A-')
j 0'(A- v)
-
n
j(A-1)jVkoiej k
j(A-)'k Vk.
i,j,k=l
i,j,k==l
I:nk=l(A -1)'vk, k
=
covariantly,
n
v
n
Hence f)'
transform
as
way
n
v
w
the basis vectors ej. for any A C- V* we have
same
antly with respect
the components of
to the transformation
v
transform contravari-
A, i.e., opposite
to the basis
vectors ej.
Another natural operation which is defined for tensors is their "contraction". Definition 2.3.4. Let
of dual bases. The
slot and the 9th covariant
C,rO(vl,
T,' (V) and f el, en; 0 ...... of 0 with respect to the th slot of 0 is defined by
contraction
0' 1 be
a
pair
contravariant
W1....
Vr-1,
dth slot
,Pth slot
n
6-
Vr
-
1, W
....
WS-1)
02,
1
We have to show that this definition is In
fact,
1 1,
if
there exists
and Ok
=
a
.
-
-
I
linear
=
.
-
6k
qro(vi,
-
independent of the choice of basis. pair of dual bases then nj, A: V V --+ with AEj isomorphism ej En 1 Aj j is another
A-'
o
.
.
,
Enj= (A-1) 3 03
=
W1
Vr_ 1,
I
...
.
We calculate
I
n
O(Vl,..., ej,..., Vr-1, W1,
-
-
-,
0i'.
-
-,
W,
-,
n
A1j*(A-')k'O(vj,---, j.... 7Vr-17W17
-
i,j,k=l n
O(Vl)
-
-
-
jji
Lemma 2.3.3. Let V be
T,r (V).
-
-
a
-,Vr-1,W1,
-
-
vector space
-'0 over
K and
G
TqP(V),
Th e n and
(C9
Z (0 Cq+sr
E
2.3 Tensors and tensor fields
This follows
Proof.
immediately
73
I
from the definitions.
Another class of natural operations on tensors are symmetry operations. We introduce below the two most important symmetry operators, symmetrisation sym and anti- symmetrisatio n alt of entries. First some
technical
we
need
a
bijec-
preparation.
Definition 2.3.5. A
permutation of the numbers (1,...,p)
is
tion
f(il'. .,ip) : fil'. .,ipl
up:
=
-
-
11,
1(il'. .,ip)
-
-
-Al
fil'...'W
11'.. 'Al.
=
If orP is a permutation we write a(ij, ip) (i,(j), i,(,)). The set of all permutations of the p integers j1,...,pj is denoted by Sp. A transposition is a permutation which permutes only two consecutive elements and leaves all other elements fixed. =
-
-
Lemma 2.3.4. mutation
The set
group)
and is
of all permutations Sp forms generated by transpositions.
-
.
a
,
group
(the
per-
Proof. That Sp forms a group is clear since the collection of all bijections of a given set forms a group where the composition of maps is the group operation. be any permutation. Starting Let a(ij,...'ip) with the p-tuple (i 1, ip) we can use successive transpositions in order to
move
the index
to the last
i,(p)
position. Assume
now
that
i,,(k)
i
...
I
Since ia(kfio-(k) ,... I ic(p) I it must be , p. 1) k It that 1. follows we can Move i.(k, 1) to
positions k, at one of the positions 1, k 1 by successive transpositions which all leave the last p position k have there finite shown that is induction invariant. we a By positions sequence of transpositions which is equivalent to a.
i,(p)
are
at
.
.
.
-
.
.
.
-
-
Lemma 2.3.5.
There is
a
natural
sign: Sp
of
the
mined
permutation
by sign(-rp)
--+
homomorphism
(1-1, 11, .)
group into the group
=
-1
for
of
two elements which is deter-
all transpositions -rp.
prove this lemma by showing that every permutation a is product of an even number of transpositions (sign(O-) 1) or the product of an odd number of transpositions (sign(a) -1). First we show that the identity permutation id is not the product of an odd number of transpositions. Assume that id is the product of finitely many transpositions and denote by n1k the number of all those
Proof. We
either the
=
=
74
Analysis
2.
manifolds
on
transpositions which interchange the numbers I and k. The number must be
the
since at the end 1 must be
even
and since there
beginning
change
k and 1. If
p
E 1=1 (Epl=k n1k) Let
now a
Since id is the
(-ri
=
=
o
set n1l
o
-ri
...
0
0
...
Irk)
-
of k + 1
product
Hence k and I
we
=
which is the
are
Irk 1
0
0
0
...
of
even
even or
A permutation up acts in
0
tl
=
(Tk)
0
set up 0 (vi,
...
k + 1 must be'even.
tensor
on a
and any 0 E TO(V) p (v,,, (1), V,,, (p)).
vp)
0
...
even.
transpositions. 0 ;F_J (-Fl)-' 0 ;F, 0
are
1
-
Definition 2.3.6. For any we
at
both odd.
natural way
a
n1k
as
numbers and therefore
o, i where Ti,,Tj
...
transpositions the number
both
side of k
other transpositions which inter0 then the number of all transpositions is
-71
11
same
are no
sum
=
the
on
.
.
.
0
TO (V).
c
p
permutation up
E
Sp
,
Lemma 2.3.6. For any permutations -rp, up E T'(V) we have (up o Tp)V) -rp(apV)).
and any tensor
SP
0
=
p
Proof. Let
vi,
vp
V. We calculate
E
-r,) 0 (vi,
(UP
Vp)
(V,P.-,, (1), (V,, (", (1)), 07po (VTT, (1))
V-P .-,, W) V-,, (-,, W) -
(p))
*
Tp (CPO) (V, (1), which
implies
the first
Lemma 2.3.7.
sym:
equality.
The maps 0
To M
T M,
p
1
P!
alt:
E
0
To P, M
Yp M, I
I: "
are
W
up
V)
V)
P!
S7,
1:
sign (up) up 0
ESI,
linear projections.
Proof. We only
prove the lemma for the
operator alt since the proof
for sym is completely analogous. That alt is linear is clear.from the definitions. For given V) E To (V), vectors v1, and any permutation , vp p we have -Fp .
alt,0 (vl,
.
.
.
,
vp)
P!
Y_ E
P!
.
.
sign (up) 0 (v,,,, (1).... Va" (p)) I
Sp
E ,pEST,
sign(up) sign (,Tp) V)(v,,,, -T, (1),
V-P -', W)
2.3 Tensors and tensor fields
=
where
sign(,rp)
alt
have used that R,:
we
-
-
Sp
Sp,
-
,
75
v_gp)),
up
apTp is
1-4
bijection.
a
It
follows that alt
alt
o
0((vj,..., vp)
=
E
-
P!
sign (-rp) alt V)(v -" (1),
-
-
-
1'r', W
,
7-7, E Sp
1 =
P!
o
alt
=
alt
O(vj,..., vp)
-7, E Sp
alt?P(vj,..., vp)
=
and therefore alt
Y
-
alt.
I
Definition 2.3.7. A covariant tensor V) E T ,(V) is called symmetric (respectively, anti-symmetric) if for all s-tuples of vectors (vl,...,v,) and all permutations a,
b (vi,
V')
of (1,
0 (v,' (1),
=
-
-
.
Symmetric analogously.
and
s)
,
the
equality
-V"' (,))
(respectively, 0 (vi, holds.
.
.
.
.
v,)
,
anti-symmetric
=
sign(o-,) 0 (v,,, (1)
7
...
contravariant tensors
7
va" (S)))
defined
are
p. 65
I
The set of all
r
times contravariant and
TXM, where
E
M, form
x
a
s
times covariant tensors
vector bundle which
generalises
on
the tangent
bundle. 2.3.1. Let M be
Proposition set
T ,M
an
U. CMTr(TxM) of
=
n-dimensional, smooth manifold. The
all
(r) -tensors
carries
natural vector
a
.
bundle structure.
Proof.
Let
associate
0,:
a
(U,, o,,,)
be
an
atlas of M. Then with each chart U,,
we can
map
U
T,,r(TxM)
-
0.(U)
x
K
n"n'
xEU,,.
W. W,
where
0", (a) ii
are J.
...
GauBian basis lies in at least
the components of the tensor
Oxi ......9xr,,. one
....
TrU,,,
:=
that for each
0,
(TrU,, 0,) forms an atlas pair of indices (a, 0) the map o
V), 1: V)O(TrU,
n
TrUe)
Oy
UxEu,,, Tr(TxM) (a
that the collection
--
....
nj
Ox with respect
It is clear that each tensor
of the sets
isEfl
of TrM
0,(TrU,,
n
we
E c
to the
T,"(TyM) A).
To
see
have to show
TrU,8)
p. 84
76
is
Analysis
2.
We denote the coordinate system associated with y") and let x E U, n UO. For any vector v E Tx M we
diffeomorphism.
a
(UO, oo) by (y', then write
can
manifolds
on
.
.
.
Eni=1 V i a.xi.
v
Eni= , v'0 0.,i
a
(WC, ( 00)-')j
=n
V3.
where
-
v
Oyi In other
words, the column
nents of the vector
DW,,e (va), row
where
=
=
of the components of
the charts must then be related
components
By the
v. I
by w(v)
defined
are
by (w')
with respect to since these
(w") (v,,)
=
w
=
=
-
(wO) (vo) -
argument it follows that the components
same
for all vectors
and
..........
(a) i
I
T,
Ozi ()3) ji ...
related
are
by
,
oki
...
k,
i, oil(0) ilj ...
I
1 jl
...
...
i7.
Hence the components
(U, W,,), (Up, W,3) (y, map 0,,, 0
by
Kn"n" Ox
x
construction of
--,
T,,rM
T,,rM
x
(0.)
we
of
0,, with respect to the charts isomorphism D,,,3. Thus the
linear
is
a
a
is indeed
diffeomor-
a
manifold. That TM is also
M and any chart (U, is a linear isomorphism. c
have TM
Definition 2.3.8. Let M be bundle
a
(WO (y), (D,,3 (W,3)))
bundle follows since for any
Tr(T M)
(0,3)
and
related
are
phism. We have shown that TM
By
of the compo-
-
we
(wo), (wO) consisting
vectors
(v,,,), (va) consisting
vectors
with respect to our charts are related by (v,,) have set W,O o, o (WO) 1. Let w be a 1-form. The
v
W,)
a
vector
the map
TOW.
=
and r,
manifold
is called the tensor bundle
s
E
N U
The vector
01.
times contravariant and
of r
s
times covariant tensors.
The tensor bundle
TIOM
is also denoted
by T*M
and called the cotan-
gent bundle of M. The bundle of s-forms is the subbundle
w
Ei
A'M
-#
w(vj, for
-
-
Vi....
-,
of
any set
(i, P
C
f 1,
-
The bundle of s-forms will
-
vj....
vectors
vp)
=
f vi,
-W(V,,...,Vi.... vi7 vp I and any pair I
.
.
.
...
vp)
,
.,Pl. play
a
fundamental r6le in Sect. 2.5
on
dif-
(which may be omitted on first reading). We will give an definition in Lemma 2.5.8. See also Definition 2.3.9 for the
ferential forms
equivalent
I
A'Mof T,M defined by
vector space
analogon As(V).
2.3 Tensors and tensor fields
77
In the rest of this algebraic section we will introduce the space of a special class of tensors which generalises the notion of de-
p4orms,
terminant and
plays
an
important r6le
in the
analysis
manifolds.
on
For instance, it is fundamental to the integration over a manifolds (cf. Sect. 2.5.4 below) and in the theory of differential systems (Bryant,
Chern, Gardner, Goldschmidt, and Griffiths 1991). However, the usefulness of this concept will only become clear in applications and not on this technical algebraic level. The material presented here will not be used before Sect. 2.5. The reader may therefore wish to postpone the study of the rest of this section.
It turns out that the symmetry operator sym is not of particular importance. On the other hand, tensors 0 which satisfy alt oo 0 generalise =
the determinant and therefore, deserve Definition 2.3.9. A form E
To (V) with alt 0 P
AP(V)
=
alt
The tensor
a
special definition.
(of degree p) (or simply p-form) The vector space
of
all
p-forms
is
a
tensor.
is denoted
by
(To(v)). P
product
Definition 2.3.10.
0 induces
product
a
A of forms:
product (or wedge product)
The exterior
A is the
bilinear map A:
AP(V)
x
Aq(V)
--+
(W, 77)
-4
AP+q(V), W
A 77
:=
p!q!
alt(w
0
(p+q)! is not the Remark 2.3.2. The normalisation factor p!ql
TI). only possible
choice and may appear to be rather unnatural. Other factors are also alt(w & 77). We common, in particular the alternative definition w7\,q =
have chosen
our
in formulas to
factor in order to minimise similar combinatorial factors
come
(see
Remark
2.3.3).
Our choice agrees with the choice of Bryant, Chern, Gardner, Goldschmidt, and Griffiths (1991) and (Abraham and Marsden 1978), but is
(Abraham and Marsden 1967) (Kobayashi and Nomizu 1963).
different from the normalisation factor in and the normalisation factor in Lemma 2.3.8.
(i) (u)An) (ii) W A 77
product
is associative and
AP(V),
n E Aq (V), and A E
AA =wA
(77AA)
and
=
hold.
The exterior E
concretely, formulas More
let
w
(_1)pq
77 A
W.
anti-symmetric.
Ar(V).
Then the
78
2.
Analysis
Proof. (i):
manifolds
on
We calculate
(w
7)
A
A A
((p + q) + r)! alt((w A n) & A) (p + q)!r! ((p + q) + r)! (p + # alt ((w & 7) (p + q)!r! p!q! (p + q + r)! alt(w Oq (D A).
=
-
-
0
A)
p!q!r!
Analogously
shows
one
A
w
(n A A)
(p+q+r) I alt (w Oq 0 p!q!r!
_
A)
which proves
associativity.
q)
(ii): (q
=
Let 7-p+q E Sp+q be the permutation given by Tp+q(l + ) P Then we have sign(Tp+q) + p, 1, 1, (_I)pq and , q , q). ...
+
.
.
.
.
Tp+q (W (9
.
n) (V1
=
.
Vp+q)
W
0 77 (Vq+ 17
77 (D
Using these formulas w
A 77
W(V11
...
...
I
Vq)
Vq+pi Vl,.
7
Vp+q)
obtain
we
alt(w Oq)
p!q! I
>"
p!q!
sign (0-,+q) '7p+q(W
(2)
77)
,p+qESI,+q
L
p!q!
sign (Up+ q) sign (Tp+q) '7p+q Tp+q(W
0
77)
,p+qESp+,l 1
(_I)pq
=
p! q!
'PE
sign(Cp+q)9p+q(?7
0
W)
+qESp+q
(_l)pqT,
=
Lemma 2.3.9. Let 1
(i) w A (ii) wl A (iii) wl A
*
*
wl
A wP
...
A wP
...
7
=
A WP
*
A
....
wP E V* and irp E
Sp.
A w", (P) sign(7rp)w"P(1) A F,.,ES,, sign(up)wIz,(1) 0 ...
=
=
W.
0
...
if and only if the 1 -forms
w
0 WIP
1,
-
(P),
wP
are
linearly
dependent.
Proof (i):
Since
to prove this
transpositions generate all permutations, for a transposition
7rP (1 7.... But in this
it is sufficient
equality
case
i'j'...'n)
=
the assertion follows
immediately from Lemma
2.3.8.
2.3 Tensors and tensor fields
The assertion is
(ii):
true for p
clearly
79
2. Assume that it is true
=
for collections of up to q 1-forms wl.... I Wq and denote for any Uq E Sq the permutation (ij, .... iq+1) -4 (Uq(il), '07q(iq) iq+1) by tq+l(gq)Then
we
have
w1A
A
...
w
q
A
I
q!
Wq+1 (V,,.
E
.
.
Vq+l)
,
sign(,Tq+l)(wl
A
A
...
Wq
", +jESq+j W
q+1
(1)
1:
q!
E
sign (-rq+ 1) sign(gq)W
T,I+iESq+l o-qESl
Wq (g
q!
'Y'
Y-
qESq
0
W
q
Wq+1 (Va q-Or,,+1(1)1
1:
W
Sign(tq+l (Uq)
VUqO'rq+1(q),
0
Tq+l)W
1
VT,1+1(q+l))
(9
...
ESq+j
-,,+I
0
I
q+1
(Vlq+
I
(Uq) 07',1+ 1 (1)
Sign(Tq+l)W
1
0
1
'
*
'
7
Vtq+ 1 (Uq) 0r,,+ I (q+ 1) )
...
,T,I+iESq+l (9
(iii): a
If
W
U)q(9
1,
W
q+1
wP
are
linear combinations of
w ).
(q))
(1)
linearly dependent 2
(otherwise we 2,...,p) with wl
wP
w
There exist numbers ai of
we can assume
(i
that
1 w
is
could renumber the i=2
aiw'. Hence
right hand side
the
P
W1
A
...
Aw p
E Wi A U)2 A
...
A wP
i=2
products each of them containing some factor w3 twice. by (i). If the forms w 1, wP are linearly independent then we can complete w' I of V*. Denote the dual basis by f e 1, them to a basis f w 1, 6n I If w' A A wP would vanish then so would wl A A w'. However, we is the
sum
of
Hence all summands vanish
.
,
...
...
have
W1
A
...
A Wn
(el,
e,) all, E
S",
sign(a,,)J1,,,(j) a
a,, E
S"'
jn o,,,,
(n)
.
-
,
-
80
2.
Analysis
manifolds
on
If lei,
Lemma 2.3.10.
en
1, 101'...' Onj
is
a
pair of dual bases then
the set
fO" forms
a
basis
Proof. First
of
we
A
...
A 0"
AP(V)
the space
In
1 I
...
(n). p
particular, dim(AP(V))
show that this set of tensors is
linearly independent.
Let
be numbers such that
77
E
=
ij<
and let k, <
...
p-forms fO" Conversely, A
...
A
O'll ji, <
0'7'(")
il
<
fO"
A
<
i,O"
A
A
0"
0
77(eki
=
all
are
7
...
,
ek,)
77ki
alt(TO(V)) P spans
==
we can
whence the
k,,,
linearly independent.
A
0"'('P)
Sp
E
...
sign(up)O"
A
...
restrict to. those
AP(V) AP(V). Since =
0"'
A
0"
A
...
A
O'P with
ip,
The dimension of are
...
A
A
...
permutations up
...
...
know from Lemma 2'.3.2 and
we
that the set of vectors
for all
Then 0
kp.
<
77j,
...
exactly
(n)
AP(V)
is
(n)
since from
set of
a
n
elements there
p
choices of ordered
t
t-tuples
without
duplication.
I
Definition 2.3.11. The map J: V
x
AP(V) (V, W)
1--> V
is called the interior
Proof. Wp+q-1
=
-i
(wj'...' wp-,)
W:
product. If p
Lemma 2.3.11. Let
satisfies v J (w Aq) all q-forms 7).
AP-'(V)
-->
A
77(Wl
(P
7...
7
we
set
v
J
w
=
0.
v
=
wo. For any vectors wl,...'
sign(Up+q-l)W(WO,W,P+,,-,(l),
ESl,+q-1 X
and
(wo J 71) (Wi,
set
E
1)!q!
7WO'j,+q-1(P-l))
A
we
wp-l)
Wp+q-1)
+q- I
(- 1)Pw
0
W(V' Wi,
The interior product w E AP(V) and 0 E Aq(V). (v -i w) Aq + (- 1)P w A (v Jq) for all p-forms W and
For notational purposes G V we have
(WO J W)
=
-->
-
.
.
i
Wp+q- 1)
77(WO'T,+q-I(P)
W,,,+,,-I(p+q-1))
......
2
"
-11tHisors
11.'.F
I)P 1)!
p!(q X
Sign(Tp+q- 1)LO (W7-T,+,j-j (1)1
....
81
W7-j, +,, -I (P)
-
'rp+q-1ESp+q-1
77(W0IW-rp+q-I(P+1),'-I Ip+q-1(P+'1-1))* W
We consider
all
now
We
Wp+q-l)-
....
E
and tensor fields
of the ordered set
possible permutations divide them into two
can
(wo, wl,
the group SP p+q
subgroups,
where wo is in one of the first p positions and the subgroup Sq where p+q it is in one of the last q positions. Using this notation we can write
wA?7(wo, wl,..., Wp+q-1) 1 =
p!q!
1:
(
sign(6'p+q)U) (W&,,+,, (0)
ID&,+,, (1),
7
-
-
-
,
W&,+,, (p- 1))
&, +, E S,','+ q X
W&p+q(p+q-1))
77(W&I,+q (P)
E
+
sign( p+q)U) (W. ,,+, (0))
10, p+"(P-1))
....
Sq
P+q
X
?7(W- p+q(P)7W' p+q(P+I)I
....
Wr,,+q (p+q-
Consider the first summand. For each the first entry of is
a
by executing unique permutationCp+q-1 w
( p+q
(&p+q)-1(0) E
=
shift wo to
Sp+q-lwith
(- 1) (&p+q)-1(0) W (WO Wlp+ I
X
&p+q
can
IW&p+q(P-1))?7(W&p+q(P)I
W(W&P+q(0)1*
Observe that
Spp+qwe
E
transpositions. Hence there
is the
7(W,,+q (p),
*
*
"
I
WO'p+q (p+q- 1))
Composition Of
WO'Z)+q (P-
7
q
-
1 (&p+q)- (0) transpositions
and
the permutation i
This
each
implies sign(&p+q) of the p possible
one
Up+ q (i)
for i > 0
0
for i
=
0.
(-1)(4+q)-I(0)sign(9p+q)entries of
w we
Since wo
can
be in
obtain
I
E
p!q!
sign(&p+q) W (W&I,+
q
(0) 7 W&I,+,l (1)
1....
W&7,+,,(P-I))
E S"
p+q
X
P -
p!q!
'7(W&p+q(P)l
....
W&p+q(p+q-1))
sign(9p+q-I)W(W0, Wor,,+,I-j (1), X
q(W,P+ll-
I
(P) I
I
-
-
-
WUp+q- I (p+q-
,
W-p+q- I (P- 1))
82
Analysis
2.
manifolds
on
(Wo J w) Aq(wj,...' Wp+q-1).
=
Consider
Sq
p+q
an
now
allows
the second summand. An
us
to
additional factor
transpositions,
a
wo to the first
move
(-l)P
since
E q
E
- p+q
entry of 17. In this case we obtain is the Composition Of (' p+q)_1(0)
'rp+q (i)
for i > 0
0
for i
and the p transpositions which first entry of 17. This and we obtain .1.
for
permutation
i
p!q!
' p+q
analogous argument
wo from
move
implies Sign(' p+q)
sign(' p+q)U)(W_ p+q(0)
7
0,
=
the first entry of -, P+')
(-I)P(-I)(
=
W' p+q(l
I...
1
to the
w
(O)T(07p+q)
M 7'+'J(P_ I))
'q '
Sp+q >(
77
(W' p
(-')P- L
+q
p!q!
W- ,,+,(p+q-1))
(P)
E
Sigll(Tp+q- 1)W (WTI,+q-
7'
1
Wr,,+q- 1 (P))
'rp+q-jESp+q-1 77 (WO,
X
=
W-r,)+q
I
-
(-1)PW
(p + 1) 1...
A
7
W-r,,+,,_j(p+q-1))
(Wo J 77)(Wi,..., Wp+q-1). I
Finally,
we
relate the
theory of p4orms to
the determinant of linear maps. following from linear al-
To motivate the definition below recall the
gebra. Assume that orthonormal basis
have
we
Euclidean scalar
an
f ej,...' e"j.
If
fol'...' onj
product (, -) and
is the dual basis then
an
one
the n-form 01 A A on in order to measure the volume of parallel b, G V one defines the volume of the epipeds. For any vectors bl, A 0 n(bi, parallel epiped spanned by these vectors to be 01 A bn)This number depends on the chosen scalar product but not on the orcan use
...
.
.
.
,
...
.
thonormal basis. The determinant of
defined
as
det(B)
:=
01
This definition of
.
.
,
linear map B: V -* V is often B ej. where bi , b,,) determinant obscures the fact that the deter-
A
a
...
A on
(b 1,
a
=
.
.
.
independent of the choice of scalar product. The following equivalent definition is probably the most natural way to introduce the minant is
concept of
a
determinant.
Definition 2.3.12. Let V be k-dimensional vector space
n-dimensional vector space, W be K, and A: V -4 W a linear map.
an
over
(i)
The pull-back of 0 under A is the defined by A* 0 (vi, (Avi, vp) .
.
.
,
.
.
map A*: .
,
vp).
TO (W) P
--->
a
T' (V) P
2.3 Tensors and tensor fields
(ii)
W. An(V) \ 10} and assume that V det(A) of A is the number defined by A*/-t
Let tz G
minant
83
Then the deter-
=
=
det(A)M.
We have to show that the map det is well defined. First observe that A* maps A,(V) into AP(V) and recall that An is 1-dimensional by Lemma 2.3. 10. Hence A* y must be a multiple of M. If A is any other non-vanishing n-form then there exists a number a 7-1 0 with A ap. Hence we have =
A*A(vl,.
Vn)
=
A*(a/-t)(vl,..., vn)
det(A)M(vj,..., vn)
=a
which
implies that
definition for
our
=
a/-t(Avl,..., Avn) det(A)A(vj,..., vn)
=
det(A)
does not
depend
on
the
chosen n-form. Lemma 2.3.12. Let V be
an
n-dimensional vector space A: V
be
a
linear map and
Let the components
A* (Ol
A
...
A O'P
)
f el....
A3
be
particular
we
n1, fO'7
E ( E
==
V
--
....
defined by Aej
=:
onj be a. pair of dual bases. rnj= 1 A3j ej. Then we have A"'
sign(up)A"0,7, Ul
( P)) Oil A
17P
...
AOiP.
U7,ESP
jl<...<jp
In
e
7
K,
over
have
sign(un)A',, M
det(A)
An
-
a
an
a,,
(n)
E S ,,
Proof. We calculate A*O'l
A
...
A =
O'P (ej,,
.
.
.
ej,)
I
E sign(up)O"(Ae,,,(j,)).....O'P(Ae,p(jp)) orp E Sp
An
sign(o-p)A" 01P 01)
UP
07,
Up E S7,
Remark 2.3.3. We have noted in Remark 2.3.2 above that
torial factors mark
2.3.2)
are
not the
only possible
instead of A,
we
would have had to re-define vj w
in order to preserve Lemma 2.3.11. Another
that the
our
advantage
of
our
simple formula
01
A
...
A
holds for any pair of dual bases
on(e
combina-
choice. If we had chosen A
-
-
-,
en)
=
I
fei1iE{1,...,nj7 joi LE 1,...,nj
-
(cf.
=
p
v
Re_j
w
choice is
Analysis
2.
84
manifolds
on
2.3.2 Tensor fields
Ap.7 5 p. 75
In this section
All
we
and
physical
introduce tensor fields which
geometrical
in terms of tensor fields
U
M
we
are
will encounter
be defined
can
maps of tensor fields.
of
The vector space
Definition 2.3.13.
0:
or
structures
tensor valued maps.
OW
with
T,M
all smooth maps
E
T,(T M)
Vx c M
xEM
by TI(M). These maps are Ox instead of O(x). A (1) -tensor field is a vector field. 0
is denoted
called
(')-tensor
fields.
Often
S
we
will write
(0) -tensor field
A
with
w
P
W(vj'... for
of
any set
vi)
5
...
7
vj....
vp)
=
Ivil
and any pair
fvl,...,vpl
vectors
-w(vi....
...
I
vi'...
(i,j)
VP)
j1,...'pj
C
is
degree p. A differential form of degree p is often simply called a p-form. M be a smooth map. An Let Z be another manifold and f : Z Z is a map 0: Ts'(M) such that O(x) E (')-tensor field along f Z.all x G T" (Tf (x) N) for Vector fields and differential forms along f are defined analogously. M its canonical inclusion If Z is a submanifold of M and f: Z then we also say that 0 is a tensor field along Z. called
a
differential form of
S
--+
(x',
Let
.
.
.
I
be
Xn)
coordinate system and denote
a
by 91,.
..,
an
its
GauBian basis. Its pointwise dual basis is given by the derivatives (dx', dx') (cf. Definition 2.2.7). With respect to these coordinates, any ...
,
(')-tensor
field
S
0
can
be written
Oi31.1
Ow 1
...
1 jj where the
0'.31
...
j.
...
...
...
Z"
j.
as
(X),Oil
(9
...
0
aj,
0 dx-11 (9
...
(D dx3
i,.
j,, n
are
functions.
They
are
the components
(or component
xn). If 0 with respect to the coordinates (x the tensor is a covariant symmetric (0 tensor field we will often drop product sign 0 in order to indicate that the symmetrisation operator sym has been applied,
functions)
of the tensor
,)
sym(O)(x) If N is
a
tensor field
pull-back.
=
1: 1
<
ji
...
second manifold and
T-'(N) P
induces
a
j, :5 nOj,...j, (x) dxjl
f:
M --4N
unique
This tensor field is defined
by
a
...
dxj,
smooth map then each * f 0 E To (M), its
tensor field
P
[ [I
P. 8 p.
2.3 Tensors and tensor fields
f *0., (vl,..., vp)
85
Of ( ,) (Tf (vi),..., Tf (vp)),
:=
Tf : TM --+ TN denotes the derivative of f (Definition 2.2.8). Analogously, each tensor field c TOP (M) induces a unique tensor field f,, c- TOP(N), its push-forward. It is defined by WP) := N is a local of(.,) (wl o Tf,...,wP o Tf). If ?p c- T,(N) and g: M diffeomorphism we define the pull-back of 0 by
where
...,
--
g*,O(vl.... 7vS1W11 :=
0
...
I
Wr)
((g.(V,),
WT (WI)7
-,g-(V '),
(g-,)* pr))
....
This definition agrees with the definition of pull-back above if 0 is a G T,(M) by push forward of a tensor field
covariant tensor field. The a
local
diffeomorphism g is defined by push-forward agree.
g*
(g-')
=
If
s
=
0 then
both definitions of
Lemma 2.3.13. For every local diffeomorphism g: M tensor fields 0, 0 C Tr (M) we have
g*(ao
+
g*(O
00)
=
ag*O
b)
=
g*0 (D g*O,
,O
=
C; g*o
0
g *cg
+
--
N and all
Og*o,
If g fails to be a diffeomorphism, and 0, 0 are covariant vector fields then the first two assertions are still true. Analogous statements also hold for the push-forward. Proof. The lemma follows directly
from the definitions.
Remark 2.3.4. It is often practical to denote a tensor field 0 E T,1' (M) by a, with respect to an unspecified GauBian basis. The components Oal bi
its
...
...
b,.
indices a,.... a, bi b, are "dummy indices" and their only purpose is to give a convenient method for explicitly denoting the entries of the ...
By convention, the tensor is characterised by the core symbol a, cr I .6 However, different general we have Oai Oc, bi b,, di d, 7 Od d, fields of different order can be denoted by the same core symbol
tensor field.
0,
in
so
c
...
tensor
...
since it is clear from the number of
they
must be different
If we
a,,3
...
==
...
E
K, 0,
co-
1
...
and contravariant indices that
objects.
are
two
(') -tensor S
field, and V)
is
an
-tensor field
write
In
some
East
the exact
European countries, some authors use conventions which are C7. opposite of ours, i.e., they have in general 0,1...a, Ocl b, 0 bl d, dl ...
...
dl
...
d,
It
seems
likely that this
variant will fade in the
near
...
future.
86
Analysis
2.
on
(aO
manifolds
,)
+
al
bi
...
...
a,
a0ai bi
b,,
a,+j )ai bi_b,+,
(0 .
jai
...
Cg ,)ai bi
a,.-,
...
...
'Pb1
Oal bi
b_1
...
...
...
...
Since the indices determine the entries
a,V)ai-..ai Oal bi bs bi bg ...
a,
a,.
/a,
b
00al bi
+
b,,
...
aj _i caF%+l
...
ai
bizi
Oai bi
a,
b,
bii
b,;-l cbg+1
...
...
...
ai
...
bi
...
...
explicitly,
Oal bi
...
...
...
...
a_1
b,,-l
we can
write
a,.
...
b,
...
The notation for contraction is often called the Einstein summation If in
vention:
product
a
of tensors
index and somewhere else has to
sum over
We will also a
basis,
for
as a
these indices in the
use
con-
index letter appears as an upper lower index then it is understood that one one
corresponding coordinate expression.
this notation in order to write
tensor with
a
respect
to
instance, n
Aaaa b
0
dXb
=
Aaaa b
0
dXb.
a,b=l
Symmetrisation and anti-symmetrisation
SYM(O)i
.
.....
i"
and
=
denoted
are
alt(O)i
.
.....
i,
=
0[i
by .
.....
i,;]
An
analogous notation is used for contravariant tensors and for the case that symmetrisation or anti-symmetrisation is only applied to a subset of entries. We is not
use
the delimiters
I
to indicate that
a
certain subset of entries
symmetrised (respectively, anti-symmetrised).
For
instance,
Oab[cdJ(ef)gJhJ i(jJkJ1J(mn)Jo) indicates that the entries corresponding to the indices f c, d, hJ are antisymmetrised and that the entries corresponding to the indices f e, f 1, Jj, 1, ol, Im, nj are each symmetrised. This abstract index notation should not be confused with the
ponents of the
tensor field with
mulas appear to be
identical,
to, whereas in the latter
respect
to
a
given
com-
basis. While all for-
in the former case, the tensor is referred
case one
simply has
a
collection of K-valued
functions.
The main
advantages
notation without indices
of the abstract index notation
over
the usual
that tensor operations are easy to remember since tensors look like their components and that even complicated are
contractions and tensor
products can be understood at a glance. disadvantage is that many formulas look unnecessarily clumsy because of the jungle of indices involved. In the physical literature, the abstract index notation is usually preferred and in the mathematical literature the notation without indices is prevalent. In this book, we will multiple
The main
2.4 Vector fields and
both notations.
use
will not
we
reserve
However, unlike
ordinary differential equations
in most of the
physics literature here special ranges. Moreover, we will
certain indices for
sometimes write the summation
2.4 Vector fields and
87
sign
in order to indicate the range.
differential
ordinary
equations
If -y: K --+ M is a curve then for each t E K the vector (t) -= Tt-y(l) is a vector in Ty(t)M. The following question arises naturally: given a vector
field V to
a
M, does there
on
chart, the
answer
is
exist
a
curvey with
given by
(t)
=
V7(t)?
With respect
the fundamental theorem for
ordinary
differential equations which establishes the existence and uniqueness of solutions. Because of its importance we will state it below. However this theorem should
(for mathematicians) 1960, chapter 10) for Theorem 2.4.1
V c
K', j
C
really be treated or a
courses on analysis physicists. See also (Dieudonn6 slightly more general context.
proof
in
a
be C'. For each x
f)
c
j
x
V
E
j
and
Let U c
K',
and
subsets,
(to, xo, yo) of (to, yo)
ODEs.).
theorem for
(Fundamental
K be open
(t,x,y) -->f(t,x,y)
f:j'xUxV-+Kn,
j
in standard
mathematics for
U
x a
x
V there exists
unique
map -y:
(t, y)
neighbourhood
a
F-->
-y(t, x)
E
]Kn
with
d-y (t, y)
f (t, -Y (t, Y), Y)
dt
for all (t, y)
E
j
x
V and
This theorem translates
-y(to, yo)
=
xo.
Further,
straightforwardly
Theorem 2.4.2. Let M be
a
smooth
7 is C'.
to manifolds.
manifold and
V be
a
vector
field
M. For every X E M there exists a subset j C K and a curve x and (t) Vy(t) for all t E Jr. -yx: J --+ M with -yx(O) If VX : 0 there exists a neighbourhood U of jxj x 101 C M x R on
=
=
such that the map F: U -- M, (t, y) 1--4 Ft (y) -yy (t) is well defined. Moreover, the map y -4 Fi(y) is a local diffeomorphism for each i, and =
its inverse is
For t, The
s
curve
given by F -i.
small
enough
-y is called
an
we
have Ft
integral
o
curve
F,
=
Ft+,.
and the map
Ft the flow of V.
Proof Let (U, o) be a chart centered at x. Then the fundamental theorem ordinary differential equations (Theorem 2.4.1) implies that there
for
exists
a
solution
0(t)
of the differential equation
d dt
NO
=
( O-V)'3(t)
88
Analysis
2.
with
13(0)
-/ ' (0)
=
o(x).
=
manifolds
on
Hence 7,,
o-'
=:
o
0
is
integral
an
curve
of V with
X.
That F is smooth and well defined follows from the fact that
integral depend smoothly on parameters (cf. Theorem 2.4.1). The map (Fi)-' is given by (Fi)-'(z) -y_(i) where -y- is the locally unique of -V with curve integral -y-(O) z. Hence Fi is differentiable and has curves
=
=
a
differentiable inverse. The equation
of differential
Ft o F, follows from local uniqueness 'of solutions Ft+, and the fact that both t equations Ft+, (y) and Ft o F, (y) =
-->
are
integral
curves
An
integral
curve
j
-y:
-4
M of
same
a
initial value.
I
vector field V is called maximal if
integral : j --+ M with j C j C K implies J. By the lemma of Zorn and local existence of integral curves
the existence of
j
of V with the
=
an
curve
each
integral curve is contained in some maximal integral curve. The R maximal -integral following Proposition shows that in the case K =
unique.
curves are
K
R then there is
Proposition
2.4.1.
j
unique solution -y
C
K and
a
Let -y,
Proof.
If
=
of - (t)
=
unique maximal subset Vy(t) defined on j. a
be
integral curves of V with -y(O) integral curves coincide on the
=
(O)
=
x.
We must
j of j. In order to do so we will prove that the set IC It E j n j : -y(t) (t)j is both open and closed. Since in the case K R the set jnj is the intersection of two open intervals and therefore show that these
intersection j n
their domains J,
=
=
=
connected, this
set must then coincide with 1C. It is clear that IC is closed.
(U, o)
Let t E K and
be
a
the
chart centered at
=
y(t)
=
(t).
Given local
coordinates, problem finding integral curve solving a system of ordinary differential equations. An application of the fundamentat theorem for ordinary differential equation (Theorem 2.4. 1) proves that there is a unique local integral curve 0 through 0(: ). Hence there is a neighbourhood of t such that -/ on this neighbourhood. I of
of V reduces to
an
=
The
theorem
following
fields
implies that, locally,
Theorem 2.4.3. Let M be vector x
with
field
V(x) 7
such that
(P-
1
(t, X2,
...'
Proof. Let be
There is
a
0.
a
smooth manifold,
Then there is
and the
non-vanishing
vector
integral
N be
an
(n
curves
x
chart
of
V
G
M, and
(U, W) are
V be
a
centered at
given by
t
I)-dimensional submanifold of M through x (i.e., RV(y) (D TyN TyM for all y c N). Let
-
chart of N centered at
an E
a
xn).
which is transverse to V
(V, 0)
all
alike.
are
> 0
such that the map
=
x
and denote the flow of V
by Ft.
2.4 Vector fields and
f (- 6, 0
X
(X 1, X 2'..
0M
M,
n)
FX,
X
.'
is well defined. The differential of
FO
=
ordinary differential equations
df (xl,.
id and
(2.4.1)
(0 -1(X2.... X'))
(2.4.2)
1
at 0 is
f
89
X')
an
)(01
dx
isomorphism
V(X) =A
since
0.
0
a neighbourhood VV of 0 where this map is an diffeomorphism. The pair (U, W) (f (W), (f 1) ju) is therefore a chart centered at x. For any y E U denote by (yo, prN M) E R x N C R x M the unique pair defined by Fy,) (prN M) y. Then we have o o Ft (y) -' 1 o Ft (y) f o Ft+yo (prN M) (t + yo, 0 (prN (y))). Hence the intef gral curves of V are indeed the curves t F--+ o`(tX2'...'Xn). It follows I Ox immediately that o. V
Hence there exists
-
=
=
=
-
=
=
=
Given in
a
vector field
V,
1
-
one can
define the derivative of
a
tensor fiels
direction V.
Definition 2.4.1. Let and
E
x
M, V) be
tensor
a
field,
U be
a
vector
field,
Ft the flow of U. Then
((d) Ft*,O) ,
XU'O(X) is the Lie derivative
(-!!-) dt jt=O
Here sion x
Ft*V)
dt
(x)
It=0
of 0.
is the usual derivative in vector spaces. In
denotes
a
tensor field which is
if t is fixed and small
evaluated at
:=
x.
As
a
enough.
In
defined
particular,
function of t this gives
on a
fact, the
this tensor field
a curve
expres-
neighbourhood can
of
be
in the vector space
T'r (TX M) The Lie derivative
measures
Lemma 2.4.1. Let V be rection V is
a
a
the
vector
derivation, i.e., for
.Cv( O 0 0) -COW + 0)
=
=
change
field.
of
0 along
V.
Then the Lie derivative in di-
any tensor
fields p,,O
the
formulas
CVW 0'0 + W 0 -PVO, -Coo + 40.
hold.
Proof. These formulas follow immediately from the properties of derivatives.
I
90
2.
Analysis
on
manifolds
Theorem 2.4.4. Let
V be vector
U,
fields
and
f
smooth
a
function.
Then
XUf =Uof and .CUV*f =U*Vof -V*Uof. Proof.
The first
equation follows immediately from
d
.Cuf Let
X
d
=
at-It=0
Ft*f (x)
(U, p)
E M and
V C U of
x
and
d
Tt I t=0 f oFt(x)
=
be
a
chart centered at
number
a
of
a
x.
> 0 such that
e
satisfies Ft (y) E U for all for all the
df
==
(t, y)
E
(dt
Ft (x)
-
df (Ux).
It=o
There is
neighbourhood
a
Ft (y) is well defined and
(- 6, E)
x
V. An
application
of
formula to the map t i--+ f o F-t o o-' implies the existence smooth map j: (-,E, c) x o(V) -- R with f o F-t o W- 1 (z) f (z) +
Taylor
=
tj(t, z) for all (t, z) E f (Ft) (y) f (y) + tgt (y) o
=
V. The map gt(y) := j(t, for all y E V and we obtain x
w(y))
satisfies
d
-dt I t=o (Ft*V).,
f
d
d
"
((Ft V)
dt lt=o
f)
==
dt It=o
(((F-t),,V),,
d
d 0
*
dt It=o
e
d dt I t=o 0
V
0
where in the last step
f)X we
(f
*
(V
-
+
0
t9t))
U
gO)X
==
V
t))
0
(dt
0
=
-
dt It=o
(V
*
f)Ft(x)
+
Vx
90
f),
have used
d
0
F-
o
d
(VFt, (x)
-
(V
f)
(((F- t) VFt (x)) f) =dt (VFt (x) (f I t=o
=::
(U
9
f
o
(Ft)
-
1
(-)
It=o
)
=
Vx
9
df
X
(
d
dt lt=o
(Ft)
d =
Vx
9
df
it-lt=o
(F- t) (-)
(V
0
U
0
f),
Theorem 2.4.4 shows that the Lie derivative of V in direction U is the commutator of U and V. This motivates the
Definition 2.4.2. the Lie bracket
if
or
If U,
V
are
their Lie bracket vanishes.
fields
vector
the commutator
of
following
then
U and V.
we
definition.
call
Vector
[U, V] fields
=
CUV
commute
2.4 Vector fields and
Commuting fields
ordinary differential equations
91
are of particular interest since GauBian vector necessarily commuting. The following lemma gives the
vector fields
49,,k, 19.,1
are
to this observation.
converse
(Geometri'c interpretation of the Lie bracket). fields U, V have vanishing Lie derivative, CUV 0, if and only if their flows commute.
Lemma 2.4.2
Two vector
=
Proof Denote the flows of U and V by Ft and G,. The equation Ft o G, G, is the flow of V. Hence we have G, o Ft implies that F-t o G, o Ft o G, o Ft) V Ft*V and therefore T(Ft)-l (jd,,G,) o Ft jd-,,(F-t ds ds =
=
=
d
_pUV
( dt ) t=0
=
Conversely, *
Since Fo V that
s 1--4
integral
F-t
o
curves
Ft V
=
=
(-.4-) dt t=0
V
that X UV
assume
=
0.
0 which is
=
equivalent
* to - !- Ft V
dt
=
0.
*
V for all t. This implies integration yields (Ft) V G, o Ft is an integral curve of V. From the uniqueness of I we get F-t o Gs o Ft G, for all t, s.
V
=
*
an
=
=
2.4.1. Let M be
a n-dimensional manifold and f U1,...' U"T pointwise linearly independent, pairwise commuting vector fields defined on an open neighbourhood of x E M. Then there exists a coordinate chart (V, p) centred at x whose Gauflian basis vector fields Ui. satisfy Oxi
Corollary be
a
collection
=
Proof.
Denote the flow of
k Uk by Ft and let
O(X',...,x')
=
F.1,
o
...
o
F.1, (x)
sufficiently small (xl,...,x') E K'. Since the vector fields Uj are pairwise commuting so are their flows F j (cf. Lemma 2.4-2). Hence we for
have for every i E
O(X 11
...
fl,...'nj
Ix n)
=
This
implies V).(Ei)
basis
JE1,...' Ej
Fx i Fx1, o
d =
(Xi
-1
o
,
...
0
_P i_l
0
P+1 Xi_1
p(X1'...'xn))
of K'. Since the vector fields
=
0
...
o
F; , (x).
Uj for the standard
f U1,...' U,,j
are
linear
independent the map 0 has maximal rank and is therefore a local diffeomorphism. Let VV C K' be an open neighbourhood of 0 such that O(z) is well defined for all z E W and one-to-one on W. We can now define
A W)
=
(V)()/V)l 0-1).
1
Corollary 2.4.2. Let M be a 2-dimensional manifold and U, V be vector fields which are at each point linearly independent. Then M admits local coordinates (XI, X2) such that a,,i 11 U anda,,2 11 V.
92
2.
Analysis
on
2.4.1
Proof. By Corollary f, h with [f U, hV] =
[f U, hV]
we.
have to show that there exist functions
only
0. We calculate
=
0
manifolds
Vf U(hV)
=
=
f h17UV
=
f h[U, V]
=
f h QU, V]
-
+ -
'7hVf U
f dh(U)
-
hdf (V)U
+ d
f h17VU
hdf (V)U
-
f dh(U)V
+
ln(h)(U)V
d
-
ln(f)(U)V).
wu, wV be the 1-forms which are dual to U, V. Then any solution (f, h) of the uncoupled system of linear ordinary differential equations
Let
0
=
d
ln(h) (U)
-
wV QU, V]),
0
=
d ln(f) (V) +
wu ([U, V])
[f U, hV] 0. This system of differential equations can be solved by the fundamental theorem for ordinary differential equations 2.4. 1. 1
satisfies
In the
=
following
sections
we
will encounter various kinds of tensor deriva-
tiv,es. It is therefore practical Definition 2.4.3. Let D be it
fields. If
to formalise their a
common
map which maps tensor
properties.
fields
into tensor
satisfies
(i) D(T,(M)) C T,'(M), (ii) D(W 0 0) D(O) 0 W + V) 0 D(W) ("product rule"), (iii) D commutes with contractions, =
then D is called
derivation.
a
2.4.3. Let M be
Corollary
The Lie-derivative
Proof.
Ft*Cr^o
Xv
We have to =
S
CrFt*o
is
a
manifold and
d
dt
be
can
a
Proof. Writing
an
field
on
M.
Ft*o
CfFt*0
-4
is linear
(so
the
interchanged with this operation).
Proposition 2.4.2. Two derivations coincide if they fields and junctions.
we
vector
the third property. This follows from
verify only
and the fact that
8
derivative
V be
derivation.
a
arbitrary
tensor field
V)
in
coincide
on
vector
a
coordinate representation
(9
dxj,
obtain
Do
=
D(0'1`,9j, ii j.,
0
...
(9
aj,
0
dxjl
(9
...
...
r
+
E'O"ii
-
-
-3,",9j,
+ 3,
t=1
aj,
0... &
D(ai,)
&
aj,
...
&
(2)
0... 0
dxjl
(3
aj,
...
0
0
dxjl
(9
D(dx3")
...
(9
0
...
dxjl
(9
dxjl.
2.4 Vector fields and
Hence
we
only have
differential equations
ordinary
to show that D is
uniquely determined
93
for tensor
But this follows from D(w(V)) & V)) EO(M). D(C'(w I I Cll(D(w&V)) Cj1(DwOV+wODV)) Dw(V)+w(DV) for arbitrary I vector fields V and tensor fields w E 710 (M)
fields
E
w
=
=
=
-
Recall that vector fields tions. We show
alised to
now
can
be considered
b,
D,
can
on
func-
be gener-
be derivations. Then the commutator
[D, D] a
acting
arbitrary derivations.
Lemma 2.4.3. Let
is also
derivations
as
that the commutator of vector fields
derivation.
:=
D
D
o
-
D
o
D
Moreover, the Jacobi identity
[D, [D, b]]
+
[D, [D, D]]
[D, [D, D]]
+
=
0
holds.
Proof. For the first
assertion
we
to check that the
only need
product
rule
is satisfied. This follows from
D(DWOV)+W&DO)
DoD(WOO)
.6 o
and the fact that the term
respect to D and
The second assertion is
+
Do
D o
+
0
special incident
a
commutators of the form AB
[D, [D, DI I
0
J5V)
symmetric with
is
A -
of
a
general property of
BA: The summands in
[D, [D, Dfl, [D, [D, DI I D
o
b
o
b
-
D
ob
+
b
o
D
o
f)
-
b
o
J5
o
b
b
-(f) b o
b
o
D
-
o
D
o
D
-
b
o
D
-(D
o
b
o
o
5
o
D
-(D
o
D
2.4.4. For any vector
IU, IV, W]
+
fields U, V,
1W, [U, V11
+
W
we
IV, [W, U11
o
D)
D
o
b)
o
D)
2
o
D
-
pairwise.
Corollary
b
6
0
cancel
o
1
6
3
+
o
4
5
4
3
2
1
have
=
0.
D
o
D
94
2.
Analysis
Proof. Clear on
on
manifolds
since vector fields
be considered
can
as
functions.
Proposition
2.4.3. For
both X
Proof. Clearly, tion 2.4.2
we
only
vectorfields U, V,
[U, V]
and
need to show that
vector fields. For any function
-PV.CU*f
UoVof
=
[.C U,
-
Vo
formula holds for functions.
have
we
X V]
are
they
coincide
=
=
2.4.4
X[U,V]
=
[XU, _PV1
derivations. By
f we have [.CU, XV] Uof [U,V] of
Corollary
derivations acting I
on o
Proposi-
functions and
f
=
CUXV
C[U,Vjf.
implies for
9
f
on -
Hence the
any vector field
W
[.CU,.CV]W
XU(.CVW) [[U, V], W]
CV(XUW)
-
=
X
[U, V]
if
one
to
diffeomorphism but there
considers smooth maps which
[U, [V, W]],- [VI A W11
W.
The Lie bracket of vector fields does not
respect
=
only
is also are
not
transform naturally with especially simple relation necessarily diffeomorphisms. an
N be a smooth Proposition 2.4.4. Let f : M tor fields on M. If f', T7V are vector fields on N -+
and
T'f (W ,)
[V, W] f (x) Proof.
=
fVf ( )
for
all
c
x
M,
then the
map and
with
V, W be Tf (V,)
formula Txf QV, W])
holds.
Let
V (W (V
o: N o
f
R be
a
smooth function. The assertion follows from
V ((Tf (W) (
(V(W(w)))
o))
0
o
f
(Tx f (V) (Tf (W) ( o)))
o
f
f
2.5 Differential forms While it is
possible to avoid the usage of differential forms, they are important tool in analysis and mathematical physics that I have chosen to include them in this book. Differential forms will be used occasionally in the book, for instance in the treatment of electromagnetism. The reader can skip this section on first reading but she or he is such
vec-
an
advised to read the motivation below. This section builds on the theory of anti-symmetric tensors which is presented in Sect. 2.3.1 starting at page 77.
2.5 Differential forms
Differential forms areas
are
totally anti-symmetric covariant tensors. physics where this anti-symmetry
95
There
in mathematics and
are
proves to
be of great importance.
(i) Systems of partial differential equations: derivatives of
of Schwarz the
Recall that
by the
lemma
C' function commute. If
one a higher solution satmust differential has a equations, any partial isfy this "integrability condition". For the existence of a solution it is often sufficient to ensure that this integrability condition holds. Since anything symmetric applied to something anti-symmetric vanishes, such conditions can be naturally expressed by the requirement
system of
that certain differential forms vanish.
(ii) Integration: Recall from linear algebra that the volume spanned by n vectors Jbi,..., b,j in K' is given by the determinant I det(B)l where B is the linear map given by Bei bi and enj is the =
standard basis of K'. As the determinant is differential forms car6
are
totally anti-symmetric, generalisation. The lemma of Poin-
its natural
(Theorem 2.5.5)
and the theorem of Stokes
(Theorem 2.5.2)
integral theorems of Gau.B and Stokes are the for superiority of,using differential forms. good examples (iii) Physical applications: There are also direct physical applications of differential forms. They are a prerequisitive for understanding gauge theories (cf. (Bleecker 1981)) of elementary particles and in particular the theory of electromagnetism (cf. Sect. 5.2.3). which unifies the classical
Recall from Definition 2.3.8 that the set APM
p-forms
is
a
vector subbundle of
Definition 2.5.1. p
by
OP (M)
If M is
(cf.
a
To (M)
over
of all differential forms of degree w I (cf. Definition 2.3.13).
=
will sometimes denote OP (M)
we
of all
by
S?P (M,
R)
below)
The definitions and to
o w
P
real manifold
Remark 2.5.2
alt
:
U,,CM AP(T,,M)
P
We denote the set E
=
T'M.
properties of p-forms given a pointwise manner.
in Sect. 2.3.1 carry
differential forms in
Lemma 2.5.1. Let w,,q be
differential forms
and V be
a
vector
field.
N the exterior product satisfies (i) For any smooth map 0: M A O*W O*n. O*(W Aq) N is a local diffeomorphism then 0* (v I w) (0* v) _j (ii) If 0: M holds. (O*w) (iii) The differential form w can uniquely written as --->
=
=
--+
U)
(X)
1:
=:
1
where
(x ...... xn)
is
a
...
Wi'...j,
(x) dx"
coordinate
system.
A
...
A
dx p,
96
2.
Analysis
manifolds
on
Proof. The proof follows directly There is
a
simple formula
from the definitions.
which relates the Lie-bracket and the interior
product. Lemma 2.5.2. Let the
formula [U, V]
w
_j
w
be
a
differential form and U,
C U(V
=
w)
I
V
-
-i.CUw
Proof. If w is a 0-form then we have [U, V] _j w right hand side vanishes for the same reason. If have by the derivative property of CU
[U, V]
J
CV
W
=
(.C UV)
=
CU(V W) J
=
X
U (W (V))
V
-
V vector fields. Then
holds. =
0
w
by definition. The
is
a
1-form then
we
(.C UW) (V)
-
J.CUW.
Assume that the assertion of the lemma holds for 1-forms and p-forms. For any (p + 1)-form w we find p-forms w' and 1-forms W with w
I:nj=1 q'
A w'. Hence
[U, V]
get
we
[U, V]
J
J
A
n
n
QU, V] J 77')
A
w'
[U, V] J w'
A
-
n
(-CU(V J 77') A
VJ
-
(.CU(V J w')
-
V
-CU?7')
A
w'
JXUw )
n
(.CU(V Jq')) A
A
w'
-
77'
(_C U (V J w'))
+
((V J 77)
w')
(V J -CU?7')
A
A
w'
(V J _C Uw')
n
(C VJ
U
w') (V J w'))
Un)
CU
A
VJ
A
A
A
(X Uw'))
-
(V J 77')
A _C
Uw'
(-C U77i) A VJwi + (.C Uq') A V J w'
-
-
(V J 77)
A C
Uw'
n
(CU (V CU(VjW)
J
-
Hence the assertion follows for
(71'
A
w'))
-
VJ
(-CU(77'
A
w'))
VjxUW.
arbitrary degree by induction.
2.5 Differential forms
By
97
far the most important construction for differential forms is the
terior derivative which will be introduced in the
following
Theorem 2.5.1. For each p E N U
a
d: such that the
-->
unique
map
S?,'+'(M)
hold:
following properties
(i) d is K-bilinear, (ii) d d 0 (iii) d(w Aq) dw o
QP(M)
there is
f 01
ex-
theorem.
=:
=
A 71 +
(_I)qW
d77 for
A
all
q-forms
and
w
r-forms
77Y
(iv)
For
f
S?'(M) (i.e., functions f:
c
the usual
M
K) df
---->
coincides with
differential. The operator d
Definition 2.5.2.
of Theorem
2.5. 1 is called the exterior
derivative. Observe that for the definition of the exterior derivative
we
any additional structure. This fact indicates that in many
it will
play
fundamental r6le. In comparison, the Lie derivative only defined with respect to a given vector field.
a
tensor field is
2.5. 1. First
Proof of Theorem is
an
(dw)jU
h: M
--+
show that d is
a
local operator, i.e., if U
a
compact closure and w,,q c QP(M) satisfy WjU =,qlu (dq)ju. To see this let V be an open set with f) C U and
=
R be
h(w -,q) d(w
This
we
of
open set with
then
Since
do not need
applications
-
a =
smooth open function with -,q we obtain from (iii)
77)
implies (d(w
=
-
d(h(w -,q))
77))1V
the arbitrariness of V Since d is
a
hjV
=
0 and
hlm\a
=
1.
w
we
=
=
dh A
(w -,q)
+
hd(w
-
0 since both dh and h vanish
have therefore
local operator
we can
proved
(dw),U
==
restrict to chart
77)).
on
this set.
By
(d?7)lu.
neighbourhoods.
We will prove the theorem by showing that for each chart (U, W) there is a unique operator d which satisfies properties (i)-(iv) above. Let W (E
QP(M) and write (i)-(iv) imply
dw
w
=
El
...
< i,
d
Wil 1
...
1
ip
dx"
A
(d(wil jyj
A
dx"
...
+ Wj'... j,, d
...
A
A
...
A
dx'T,.
dx',
E ...
...
wi,...j, dx"
(dx"
A... A
dx'T,))
A
...
A
dx',,
Properties
98
2.
Analysis
manifolds
on
Y 1
...
d(wi,...i,,)
dx"
A
A
dx"'
A
...
E(-l)j-ldx"
+ I
...
A
dx'.i+'
E I
Thus this dw
El
...
A
...
d(wi,...i,)
dx'T'
dx"
A
A
uniquely defined
also guarantees existence
Properties (i) and 77 E
A
A
...
dx'P
have shown that dw is
we
explicit formula :-=
...
dx'.i -I
A
...
ddx'j
A
A
d(wj,...jP) (iv)
are
dx"
A
A
...
(iii)
clear. For
if it exists. Furthermore
once we
have shown that
dx',, satisfies (i)-(iv).
A
calculate for
we
w
E
S?P(M)
f2q(M)
d(w Aq) j,
A
dx"
dwi,...i,,,qj,...jq
A
dx"
d
Wil I
...
ip%,
A
...
dx'p
A
A
dxjl
A
dxj'
A
...
A
dxjq
-; jj <... <j,
E 1
...
A... A
dxi,
A... A
dxjl
1: jj <... <j,: n
Y-
+ I
<...
Wj,...jPdqj,...j,
A
dx"
A
...
A
dx'p
A
dxjl
A
...
A
dxjq
1: ji <... <j,: n
E I
...
dwi,...i,
A
dx"
A
...
A
dx'P
A 77j,
...
j,
dxjl
A... A
1: j, <... <jq
(-I)P
A
...
1:5 h<
...
dqj,...jq
dw A 77 +
Property (ii)
dx"
A 1
A
A
...
A
dx'p
dxjl
(-I)Pw
A
A
...
A
dxjq
dq.
is follows from
ddw
E
=
1
...
dd(wil ip)
and the fact that for any function P
ddf
=
d
A
dx"
A
...
A
dx',,.
...
'of axi
f
Cz
P
dx
i =
Y
C' (M,
R)
we
a2f
-dx3
axiaxj
A
have
dx'
=
0,
dxjq
2.5 Differential forms
where
we
a2f
have used that
.9Xj'aX.j
is
in i and
symmetric
99
j by the lemma I
of Schwarz.
Remark 2.5. 1. Notice that this definition of d coincides with Definition 2.2.7 in the
case
If N
2.5.1.
Corollary map, then
of 0-forms.
we
have
is
second manifold and 0: M --+ N d(O*w) for all p-forms w on N.
a
0*(dw)
=
Proof. This follows since properties (i)-(iv) are obviously O*dw and since the exterior derivative is unique.
c
satisfied for
I
2.5.2. Lie derivative and exterior derivative commute: Let
Corollary w
smooth
a
701 (M).
S?P (M) and V E
Then the
XVdw
=
equation
dXVw
holds.
Proof. Denote the flow of V by Ft. Then
have
we
d(Ft)*w
=
(Ft)*dw
and therefore d
d.CVw
=
((dt
d
(Ft)*w
)
d
it- d(Ft)*w
It=0
It=0
d
(Ft)*dw
=
XVdw. It=0
We wish to
give
a
formula for dw which does not
depend
on a
chosen
coordinate system. The idea is to link the exterior derivative to the Lie
derivative and the interior Lemma 2.5.3. Let
w
product.
QP(M)
E
XVco Proof.
=
and V
C-
V idw +
70'(M).
Then
we
have
d(V Jw)
We prove the lemma by induction. If p 0, then we obviously V dw. V 9 w J Assume now that 0 and CVw dw(V) =
have V J w
the 'assertion has been
proved for
Since the formula which coordinate
(A)
=
=
=
we
neighbourhood
E
==
ii<
...
all q
=
E
f 0,
.
.
.
,
pj
and let
want to prove is local
E
w
we can
S?P+ 1 (M).
restrict to
U and write
Wil
...
ip+,dx"
A
...
A
dx'P+'
=
wi A
dx'
a
100
Analysis
2.
on
manifolds
where wi G S?P(U) are suitably chosen differential forms. Recall that a derivation and that therefore
XV
is
XV(wi
A
dx')
(
-CV
1
,
,
CVwi holds. On the other hand, +
VJ
=
A
dx')
sign(up+l)uv+l (.CVwi
dx'
(9
CVdx i)) dx'
A
+ wi A
CVdx'
have
we
d(V J (wi (dwi
(9
p+lESp+l
+ wi 0
d(wiAdx')
sign(up+l)o-p+l (wi
7,+iESp+l
E
p!I!
VJ
E
PM
dx'))
A
dx')
((V J wi)
+ d
A
dx'
+
(- 1)Pwi
A
(V
dx'))
induction
(V J dwi)
=
(- 1)P+l dwi
A
(V J dx')
dx'+(-I)Pdwi
A
(VJdx')
A dx' +
induction
+
d(V Iwi)
A
I)P (- I)Pwi
+
d(V i dxI)
A
induction
=
where
XVwi
A
have used
we
dx'+ wi
d(V
J
XVdx'
A
dx')
d(dx'(V))
=
d(.CVx')
CVdx' (cf.
Corollary 2.5.2). We
can
now
use
1 the
preceding lemma
in order to prove
invariant
an
formula for the exterior derivative.
Proposition
2.5. 1. Let
w
Then the exterior derivative
E
QP (M) and
of w
is
V0,
be vector
Vp
fields.
given by
P
dw (Vo,
.
.
.
,
VP)
Pi,
E (- i)icvi (W (VO,
Vp))
i=O
E (-I)i+jW(XVVj'V0.... Vi
+
I
i
I
...
1-17j
I
...
I
VP)
O
)7 (-I)ivi Mvo'...'
P))
V
0
i=O
+ .....
O
where
means
that the
corresponding
vector
field
is
left
out.
Vp)
2.5 Differential forms
101
equality follows trivially from the first equality. We will Voew proposition by induction. If w E S?O(M) then dw(Vo) XV0 (w) which implies the equality in the case p 0. Assume now that
Proof
The second
prove the
=
=
,
the assertion has been proven for q E Lemma 2.5.3 implies
V 1)
dw(Vo,
=
(Vo J dw) (Vi,
=
CVOw(Vj,..., Vp+l)
and let
w
E
S?v+' (M).
VP+ 1)
indi tion
,-=
f 0, -pj
d(Vo Jw)(Vl,..., V,+I)
-
xv0 (W(Vi'..., V
W(Vi,
-
Vi-1,
-CVOVi, Vi+1,
VP+1)
i=1 induction
E(-l)j-,-Cvj((VO JW)(V1,
VV+ 0)
.,Vj- I, Vj+1
j=1
inL tion
(_l)j-l+k-l(V0 Jw)(.Cvj Vk, VI I
I<j
-^
Vj, -C
"I
VP+1))
Vk,
-
- -----
VO (W (VI,
VP+
P+1 -
E(-1)i-1W(.CVOVj, V1,
Vi-1, Vi+1
VP+1)
....
i=1
P+1
+
E
-C
Vj (W (VO, V,
I ....
vj-17vj+17
...
I
VP+1))
j=1
(_I)j+kW(XVjVki, VO,V1'.. Jj, --,: ki -Vp+l))-
+ 1<j
P+1
E(-1)j.CVW(VO,...,Vj j
.....
VP)
j=0
+
E O<j
(_l)j+kW(
jVk1,V(0'.
V
....
;Vki -Vp) -
102
2.
Analysis
on
manifolds
2.5.1 The lemma of Poincar,6 The lemma of Poincare is the generalisation of the following two facts familiar to physicists. (i) Every curl-free vector field has a local scalar potential and (ii) every divergence-free vector field has a local vector potential. The lemma of Poincar6 is a good example for the elegance of differential forms.
give the reader a better idea how the lemma of Poincar6 arises we will briefly recall the introduction of scalar and vector potentials and then translate this discussion into the language of differential forms. To
Let F: U C R 3
(i)
Then for every f : V --> R with is
applied
--+
x
R3 be G
F1 V
a
smooth map which satisfies rot(F) 0. a neighbourhood V and a function =
U there is =
grad(f).
In Mechanics this mathematical fact
to conservative force fields F. The function
is called the
f
associated scalar potential. Consider
(ii)
now a
aV3 aV1 aV2 + 5x3 axr + =a x
3 map V: U C R
R 3 whose
-+
divergence vanishes,
0. Then for every x E U there is a V and another map W: V --> R' such that rot(W) =
neighbourhood =
W is called the vector
V. The map arise for
potential of V, Vector potentials the elementary theory of electromagnetism.
instance in
In the
language of differential forms, these (and similar) results can be a simple, unified manner. To see this, we will briefly sketch classes of isomorphisms which will be studied in more generality and
treated in two
more
detail in Sect. 4.2.
The standard scalar
product (")R3
of R 3 induces
which maps vector fields to the 1-forms, V map is denoted by 0: w --+ wO, where
-->
0
(W ' )R3
are
will defined since the scalar
product
to the standard orthonormal basis this
column and
row
The scalar
::::::::
an
isomorphism
(V7')R3.
The inverse
These
w. isomorphisms non-degenerate. With respect isomorphism just interchanges =
is
vectors.
product also induces an isomorphism between O-forms an isomorphism between 1-forms and 2-forms. Let
and 3-forms and 1 (X ,X2,X3)
be the standard coordinate system of R3 and set
This defines linear
*1
dxl
A
dX2
*dxl
dX2
A
dX3'
*dx
2
dX3
A
dx',
*dx
3
dxl
A
dX2.
isomorphisms
*:
A dx
A'(R3)
--,
3,
A3(R3)
A 2(R 3) They can be extended to 2- and 3-forms by for all forms in R 3 .
-
and
*:
A'(R 3)
demanding
**w
=
w
2.5 Differential forms
df 0, rot(F) grad(f) above relating F (ii)
Observe that Hence the
=
(i)
(*dP) , div(V)
=
and
and
=
103
*(d * 0).
V and W translate into
f,
the assertions
P satisfies dP P f with df (ii) If the 2-form *0 satisfies d *0. I-form W with dW If the I-form
(i)
0, then locally there
exists
0-form
a
=
I
0
=
0, then locally there exists
a
=
The lemma of Poincar6
generalises these
(Lemma
Theorem 2.5.2
QP(M) with p > 1. If dw bourhood U of x and a (p The
proof
facts.
Poincar4). Let M be a manifold and w 0, then for every x E M there is a neighdO. I) -form 0 E S?P-'(U) such that wjU
of
=
-
=
of the theorem will be
Lemma 2.5.4. Let M be
a
corollary
a
to the
smooth manifold and
following lemma.
define for
any t E
[0, 1]
(q
with
the map
it: M There exists d
K + K
o
Proof.
For
o
a
d
w
G
(il)*
-
Qq+ 1 ([0,
Vl,..., Vq+j be
Let
M,
X
([0, 1]
X
X
linear map K: Qq+ =:
1
--+
(t, X).
M)
_+
Qq (M)
>
0)
(io)*. 1]
Kw,,(vl,...,Vq)
we
[0, 1]
--+
M)
x
=
and V1,
J0 i(it* (at
vector fields
on
J
...
i
Vq
E
TxM
w(t,x))) (V1
M.
I
...
we
define
Vq) dt.
I
Then, using Proposition 2.5.1,
obtain
dKw(Vj,..., Vq+l) q+1
1
E(_l)a-IV
t
a
J
Z
W(t, X)) (VI,
.....
Vq+l) dt
0
a=1
1
(_I)(a-1)+(b-1)
+
fo
it* (at
J
W(t,x)) ([Va Vb] V1.... 7
,
I
va
ai
...
i
For any vector field V R x M, given by (t, x) In the
following
it is
these notations and
(p
+
I)-form
dKw is
on -
...
M
i
Vq+l) dt. we
(07 V)
will denote the canonical lift of V to E
TtR (D TxM, by the
same
symbol
Vo at. notationally advantageous using [VO, V,,] [at, Va] [(,9t, 0), (0, Va)] given by
dKw(VI.... Vq+l) 7
bi
to write
=
=
for
=
V.
With 0 the
104
2.
Analysis
on
manifolds
q+I
E(-l)a-IVa*j W(t,,)(V6,V1,---,Va,--.,Vq+j)dt n 0
a=1
1
(-I)a+b
+
W(t,x) (VO,
[Va, Vb], V1
I
0
1
Va,
Vq+l) dt
Vb,
q+1
fo
-
Va
0(W(t,x)(VO) V1,
-
-
-,
Va, a
-
-
-,
Vq+l)) dt
a=1 1
E
-
(_l)a+b
fo
W(t,x) ([Va,
Vb] VO, VI.... i
O
Analogously,
Kdw is given
Va,
Vb,
Vq+l) dt.
by
1
JO
Kdw(Vl,..., Vq+l)
it*(at Jdw(t,x))(VI,
dw(t,x) (Vo, Vl,.
Vq+i) dt
Vq+l) dt
q+1
1
1
E(_I)aV
x) (V(
a
0
....
Vq+l))dt
a=O I
7-
+
fo
(_l)a+b
W(t,x) ([Va
i
Vb] V1.... 7
O
Vaa ...... b.... Vq+l) dt, 7
Taking
the sum, d
o
K+K
d,
o
all terms with
> I
a
cancel and
we
arrive
at
(d
o
K + K
o
d)w(VI,
.
.
.
,
Vq+l)
=
fl
19t
0
(W(t,,) (VI,
Vq+l) dt
0
(iI)*W(V1'
Proof of Theorem
f(Xl,...,Xn)
:
2.5.2. Let
I:n J(Xa)2
(V, o)
<
be
Vq+l)
-
(iO)*U)(Vjj
chart with
a
o(x)
....
Vq+I).
0 and
W(V)
Setting
a=
n
U
:=
W- 1(1 (x
...,x n)
:
E(Xa)2
<
a=1
and U
F(t, y)
-
:==
0-1(t2W(Y))
U which satisfies
we
F(O, y)
have constructed =
x
and
F(I, y)
a
=
smooth map F: y for all y E U.
[0, 1] x
21.5 Differential forms
Since F o i 1 is the
identity on U and
(F il)*w
have for any p-form w, follows now from
d
and dF*w
KF*w
o
F*dw
=
o
=
0
proof KF*w explicitly.
if there
dF*w
=
F o io the constant map y --+ x, we (F o io)*w 0. The assertion
and
=
i*,F*w
-
io*F*w
=
w
of the lemma of Poincar6 allows
us
to calculate
differential form w E Qv(M) is closed if dw 1 a differential from CU E S2P- (M) with W
Definition 2.5.3. A
and is exact
o
w
0.
Observe that the =
K
-
=
105
exists
=
0
=
d(D.
terminology the lemma of Poincar6 simply states that every closed locally exact. In general, this is not true globally. below 2.5.3 is a global version of the lemma of Poincar6. Corollary
In this
differential form is
Definition 2.5.4. Let M, N be manifolds and f , f : M --4 N be smooth maps. f, f are homotopic if there is a differentiable map F: [0, 11 x M --4
f(x) for all x E M. The map f (x) and F(1, x) homotopy. A connected manifold M is called contractible if the maps id: M M, x --> x and the constant map c.,(): M --> M, x --> xo are homotopic. F(O, x)
N such that
F is called
If x0, x,
=
=
an
are
in M and -y:
[0, 1]
i-->
M is
a
curve
from x0 to x, then
F(t, x) 7(t) homotopy between the two constant maps cx,, and c,,, Hence the following lemma implies that contractibility does not depend is
=
on
.
the choice of x0.
Lemma 2.5.5.
Homotopy
is
an
equivalence
relation which
we
denote
by
x for all t. Proof The relation f -- f is clear since we can choose F(t, x) P between F and then and F is : a f homotopy (t, x) (I t, x) f g g is a homotopy between g and f whence g -- f. Assume that f -- g and g -- h and let F, (respectively, P) be homotopies between f and g (respectively, g and h). Let 0: [0, 1/2] ---> [0, 1] be a smooth map which
If
-
-
satisfies
W 0(0) 01 1, (ii) 0(1/2) =
=
(iii) and
d
5(1/2) dF-
0(t)
=
I
0 for all k > I
=
-
0(1
-
t).
Then the map
A
[0, 1] is
a
homotopy
M,
between
(t, X) f and h,
,
f F(O(t), x)
if t
E
I P(O(t), x)
if t
E
[0, 1/2) [1/2, 1] I
106
Analysis
2.
manifolds
on
N be smooth homotopic maps f, f: M differential form of degree p > 1. Then f *w
Lemma 2.5.6. Let
QP(N)
be
closed
a
and
-4
-
w
E
I*w
is
exact.
Proof
Let F be
.
S?P(M)
dKF*w
=
Corollary
S?P(M) Proof. id*w id*w
-
homotopy
a
between
f
f and
be
dKF*w + KdF*w
2.5.3. Let M be a
closed
a
(i')*F*w
=
contractible
differential form.
Then
manifold,
have
we
(i')*F*w
-
f2P+'(M)
and K:
be the operator defined in Lemma 2.5.4. Then
=
p
f*w
-
f *w.
! 1, and
c-
w
is exact.
w
Let cx,,: M -- M the constant map x -4 xo. The differential form (cx,,)*w is exact by Lemma 2.5.6 and the assertion follows from
=
w,
(cx.)*w
=
0.
1
2.5.2 The theorem of Frobenius
The
guiding idea
linearisation of is often much
a
in
analysis
is that it is much
simpler
to work with
a
function than with the function itself.
Analogously,
it
simpler
to
specify properties
of the tangent bundle of
manifold than to describe the manifold itself. We in the
are
a
therefore interested
following problem.
Let E be
a
vector subbundle of TM. What
sufficient conditions for the N c M with TN
If N is
a
=
(local)
are
the necessary and a submanifold
existence of
EN?
submanifold of M then TN is
a
subbundle of TM which has
the property that for any two sections U, V of TN (i.e. any two vector fields along N which are at each point tangent) the commutator [U, V] is
again
a
section of N. This follows since N is
On the other
hand,
instance, take M this vector bundle cannot exist
a
a
manifold in its
not every subbundle E of TM has this
=
R 3 and E
we
have
f a(z,9,, +,9,,) -i% [z,9,, +,9y, a,] =
=
submanifold N of M with TN
=
+
b(9,
:
own
a, b E
E(,,,y,z).
right.
property, for
R1.
For
Hence there
EN. This motivates the
following definition. Definition 2.5.5. A vector subbundle E
for any of E. An An
two sections
of TM is called integrable if U, V of E the Lie derivative [U, V] is also a section
integral manifold of E is integral manifold N is called
a
submanifold N of M with TN c E. Ex for all x E N. if TxN
maximal
=
2.5 Differential forms
The theorem of Frobenius
(cf.
Theorem 2.5.3
below)
107
asserts that inte-
is also sufficient for the local existence
grability integral manifolds. This the order In to justifies terminology. verify integrability it is sufficient to consider a single frame for E, i-e, k linearly independent locally defined vector fields V1, Vk which at each point x span E,,. .
.
.
,
Lemma 2.5.7. A k-vector subbundle E
if
there exists
commutator
Proof
local
a
[Vi, Vj]
frame f V1, a section of E. .
is
of TM is integrable if and only Vk I of E such that for all i, j the
.
.
,
It is clear that the condition is necessary. Let are functions a', 0' with U
of E. Then there
Ek 0% i=1
=
Theorem 2.5.3
and V
=
(U
V
(Theorem
Vi
+
a'0i [Vi, Vj ]
C-
E.
of
Frobenius, contravariant form). of TM. Then through every x E M there a locally unique maximal integral manifold N,, of M if and only if E integrable. Moreover, Nx depends smoothly on x.
Let E be is
any sections
Eik=1 a'Vi
and
[U, V]
is
U, V be
smooth subbundle
a
The basic idea of
proof is as follows. Let V,.... Vk be a frame of E and N' be the submanifold swept out by the integral curve of V, through x. At each y G N' we can consider the integral curve of V2 through y. All these any
z
curves
E
integral
N2
form
we
together form integral curve
curves
take the
subset N3 of M and
a
a
set subset
of
N2 of M. Now for
V3 through
so on.
z.
These
integral
One then has to check that
Nk really
is a submanifold. This is not entirely straightforward because Nk-' are not submanifolds. It general, the intermediate sets N2, Nk, are submanifolds can be shown, however, that the subsets N2'. of M if the frame I V1, Vk I is carefully chosen. We will prove the theorem for an equivalent covariant form (cf. Theorem 2.5.4 below) using an analogous strategy. in
..
.'
..'
.
.
.
,
A k-dimensional vector subbundle of M
sectionn,i-k kern(w), where w' linearly independent E and
fV1,
is the dual
...
)
are some
can
be described
suitably chosen,
1-forms. In
i ....
Vn- k I be
basis, then
fact, let f Vn-k completion to a frame
a
E=
V,,J
be
of TM. If
as
the inter-
at each a
fw 1
point
frame for n
n-k
n i=1 kern(w').
Definition 2.5.6. A collection of finitely many pointwise linearly independent 1-forms 1wil is called a Pfaffian system. The Pfaffian system is integrable if the vector subbundle E= ni=1 kern(w') is integrable.
108
If
2.
a
Analysis
on
manifolds
1-form wi vanishes
then
N, i.e., wi (v)
submanifold
on a
should also vanish:
dw, being N, 1 u, v E T, N. This is equivalent to (dw' A w the same integrability condition as before: intrinsic to
A
dw(u, v)
wn-k)
A
...
wn-k be 1-forms which
Lemma 2.5.8. Let dent at each point.
wn-k I Proof. We extend f w 1, be the let f Vl,..., Vnj dual basis.
...
Awn-k
to
0
=
=
E
v
TN,
0 for all
0 which is
X
linearly indepen-
are
n-k
:=
coframe
a
=
n i=1 kern(w ) is n- kj. for all i E
Then the vector subbundle E
integrable if and only if dw'Aw 'A
0 for all
=
wnj
of T*M and
E is then
spanned by jVn-k+l
Wi(Vk)
Wi([Vk, V11)
Vnl-
We calculate
dw'(Vki V1)
=
Vk
0
WiMl)
V1
-
0
-
=
-Wi([Vk, V11)-
n immediately that dw'(Vk, Vj) (i E 11, kj) vanishes Vk, V1 E E if and only if [Vk, Vjj E E for all Vk, V1 E E.
It follows
all
-
-
-
-,
for
(Theorem of R-obenius, covariant form). wn-k be a Pfaffian system which satisfies dw' A wl A Let wl,... A n-k all k 1. Then there exist coordinates (x 1 0 i E 11, n w for Xn ) and functions w iwith Theorem 2.5.4
...
,
=
-
a
W"
=
W 3
(xi
Xn )dxj
(j=1,...'n-k).
ThemanifoldN=fyEM:xj(y)=xj(X)VjEfl,...,n-kllisamaxn-k integral manifold of E n kern(w'). In particular, Theorems
imal
2.5.3 and 2.5.4
are
equivalent.
Proof of Theorem 2.5.4. As by induction over k. Let k
=
1. Then E
tor subbundle
outlined
nn-1 j=1 kern(w')
:==
spanned by
above,
wi
=
w'(V)
0
(i
W (X11... En-1 j=1 3
We
assume now
1,
.
.
.
,
a
one
n
-
1),
(xl,... Xn) I
dimensional
vec-
vector field V.
such that V
there must exist functions
By
=
with
wj'
Xn )dxj.
that the theorem has been proven for k k
==
1,
wn are linearly indepenf be any function such that df w 1, Clearly, this system of differential forms satisfies the integrability -
Let
dent.
=
is
will prove the theorem
single, non-vanishing
a
Theorem 2.4.3 there exist coordinates Since
we
.
,
.
.
,
conditions
By
our
induction
ddf
A
df Awl
A
...
A
wn-k
=
0,
dw'
A
df
w'
A
...
A
wn-k
=
0.
A
assumption there
exist coordinates
(': 1...... ,n)
with
2.5 Differential forms
fi(: )d; '
df
We will
not
show that there is
now
defines the
depend on: n-k+2 ......
as
rn
Equation (2.5-3)implies that f
.
&n-( -i) only.
function of
coordinate functions by of I-forms
Q'-kl
Pfafflan system 101, which the original Pfaffian system but does
a
vector bundle
same
J&P,
-
-
f
,
say
f
We
can
&n- (k
=
dp-k, df I
-,
(2-5.3)
fn-(k
+
+
109
and
-
therefore substitute
1). At each
n-kl
df,
wise linearly independent and span at each T*M. Hence there are functions M with w
M the
x EE
are
a
of the
same
each pointsubspace of
df where
Mdx" +h"
=
j
one
M the two sets
E
x
must be
,
n-(k-1) (h3%J=i_.,n-k is an invertible matrix at each point. This implies that d&i + hi (&) df there are functions h' such that the differential forms Qi 3
=
span the
same
n-i-k kern(wi)
as
space
Wn-kj.
f wl,
nn-k j=1 kern([?)
=
=1
is
Since the vector bundle E
integrable
we
obtain from Lemma
2.5.8 0
=
dQ'A Q1
A
...
S?1
=
dh'A df
=
dh'A df Ad&'
A
A
A A
Qn-k ...
...
Qn-k
A A
d:p-k
It follows that the functions h'
pend only
(and therefore also the I-forms Q ) &n-k+i (where we have used f =&n-k+i).
on
Qn-k
Since the 1-forms
On-kj
the Pfaffian system
do not can
depend
:p-k+2,
-on
also be considered
as a
forms define
a
the
have
proof
we
are new
such that Qi
7
&n
Pfaffian
Mn-k+l parameterised by : ' ......p-(k-i) These I-vector subbundle of TM'- +I. At the beginning of
system of the space
there
...
de-
-
already established the
assertion in this case, hence
(x1,...,Xn-k+i) of Mn-k+l and functions S?jl These equalities also hold for the II:n-k j=1 S?, (x)dxj.
coordinates
forms 0' considered
on
The assertion follows
M since
now
they only depend
&I
on
......
from the fact that the 1-forms w'
n-(k-i).
are
linear
combinations of the 1-forms Qi.
2.5.3 Orientable real manifolds This section is
We
were
global
a
prerequisitive for the following
led to the definition of
concept of
a
a
manifold
vector space and have
section
on
integration.
by the localisation of
seen
that
a
manifold
can
the
be
110
2.
Analysis
thought of as The example
on
manifolds
collection of local R's which have been
patched together. sphere shows that, globally, a manifold may be very different from R'. In this section we give a (very primitive) global classification of manifolds by dividing the collection of all manifolds into two classes. More sophisticated global classifications can be found in books on differential topology such as (Guillemin and Pollack 1974) or (Bott and 'Ri
a
of
a
1982).
Definition 2.5.7. A real
manifold M is orientable if there exists a vanishing n-form v on M. An orientation of M is the choice of one of the two equivalence classes ffv : f E C'(M,R+ \ fOj)j, f -f v : f E C' (M, R+ \ 101) 1. An oriented manifold is an orientable manifold together with an orientation. nowhere
Proposition
2.5.2. A real
has
an
the
differential
f (Uk Wk) I kEN
atlas
,
D
(Wa (Wb)-I)x 0
Assume that
Proof. "= ":
manifold M is orientable if and only if it for all a, b E N and all x E Ua n Ub
such that
v
RI
:
is
a
---+
RI has positive determinant.
nowhere
vanishing
n-form and let
f (Vi Oj) IjEN 7
be
a
countable atlas. In order to
simplify
charts such that for each k > 2 there is
notation
we
renumber these
j < k with Vj n Vk =A 0. We set (Ul, Wj) (Vi, 01) thereby trivially defining an atlas for V1 which satisfies the positive determinant condition This atlas can be extended by an induction argument. Assume that we have defined an atlas a
=
f (Ul (PI) for the set V1 U
and let
j
A
...
U
Vk which satisfies the positive determinant condition be an index with Vk+j n Vj =h 0.
a
A
nowhere
dxn and
satisfying (Ok+l)*l/ vanish
on
fk+,(y)
<
7
kj
E
There is
fjdxl
...
A Wk) I
vanishing
a
=::
nowhere
fk+ldx'
function fj: W(Uj) vanishing function
A
...
A
--
R with
(Wj).v
fk+j: W(Vk+l)
dxn. Since neither
fj
nor
wj(Uj) n'Ok+l(Vk+l), we have either fj fk+,(y) > 0 0 for all y G pj(Uj) n Ok+I(Vk+l). In the first case we -
(Uk+1 (Pk+l)
=
7
whereas in in the second
(Uk+li Ok+l)
=
(Vk+l; Ok+l)
case we
(Vk+l
7
1
2
,Y ,Y
3
fk+l
=
fk+l
set
X1-2
where Xl-->2 is the reflection defined
XI-2(Y
and
Yn)
0
V)k+l)
and
fk+l
=
-fk+l
by =
(Y2' Y1, Y3'...' Yn).
--
R
fk+l
or
set
fj
2.5 Differential forms
In either 0. This
case we
have then
(Wk+l)*I'
fk+ldx'A
=
Adx"' and
...
ill
fj-fk+l
>
implies
det(D
(Wj (Wk+l)-') dx1A o
...
A
dx"
(Wk+l)-l)* dx1
0
A
fj fk+1
det(D (Wj det(D (Wi
and therefore
0
0
(Wk+l)-l) (Wk+l)-l)
(CPj
A dx"
...
dx1
A
...
A
dXn
(Wk+l)-')
0
> 0 in
either
case.
We still have
0 for any i E let fj be defined
kj with by (Wl)*v f1,...'k+lj A dXn (1 E fp, qj). Since all fj (1 E 1, fldx' A kj have the same sign and also fk+l and fj have the same sign it follows that sign(fi) sign(fj) sign(fk+,). Hence det(D (Wi (Wk+l)-l) > 0 Let jgk: M [0) 1] JkEN be a partition of unity subordinate to fUkjkEN and let
to show that
>
0. For 1 (=-
4
UinUk+l
o
=
...
-
-
-,
0
=
n
V
T_gk(Wk)*(dxl
...
dXn).
k=1
This is
gk(x) x (E
a
smooth, well defined n-form
since at each
x
only finitely
many
and the support of each A is contained in Uk. Let all indices with gi., (x) -4 0. Then
are non-zero
Uj and il.... iP
P
( oj)
*
v.,
=
E gi., (x) D (Wi.,
o
( oj)
-
dxIA
1)
...
A
dXn
(x)
j=1
does not vanish since all gi., D
have the
same
are
( oj,.,
strictly positive o
( oj)
-
1) 1
wi,
and all
(x)
I
sign.
Definition 2.5.8. Let M be
E on (M) an oriented manifold and v representative of the orientation. An oriented atlas is an atlas I Pa (Pa) JaEA such that for each a E A there exists a strictly positive function Vaj n: Ua -- R+ with
be
a
i
...
( Oa)*V where
(xl
....
ented chart is
I
xn) a
are
::::::
Vaj
...
ndX1
...
dx',
the standard coordinates
chart which
belongs
to
an
of Rn. A positively
oriented atlas.
ori-
112
Analysis
2.
2.5. 1
Example
on
manifolds
(M&bius band,
continued
from page 53). 2.1.2 is not orientable.
band M defined in notation
VV+
Example definition,
in its
as
7r-1 ((a, 2a)
=:
map D
(WI
Using
==
has positive determinant for all
.
the
the set U, intersects U2 in two and and W_ 7r-'((O, a) x (-b,
(-b, b))
x
(W2) 1) -
0
The M6bius
E
x
same
subsets',
b)).
The
W+
and
negative determinant for all x E W_ Assume now that M is orientable, i.e.-, that there is a nowhere vanishing 2-form v on M. Then there are .
nowhere
vanishing functions fl,
:
V1
-4
R,
f2: V2
R
-4
with
fldxl
v
((W2) 1)
*
A
f2dx'
dX2 and dX2
(( P2)-1)*
P
==
f2dxl
A
dX2.
(((Pl ((P2)-l)* (fldx' A dX2) we f2 (x) (W1 (W2) 1) h for all c )IV+ U W_ Since the determinant det (D (W 1 ((P2) 1) changes sign get contradiction. From
-
V
obtain that
det (D
A
0
=
0
x
-
0
we
a
Thus the M6bius band is not orientable. It is
a
simple but good exercise to actually build a M6bius band from verify using this model that there are closed curves along
paper and to
which there does not exist any continuous frame.
2.5.4
Integration
on
real manifolds
In this section, we restrict to K R. This is necessary since we need to employ partitions of unity. See Remark 2.5.2 below for the integration of complex valued functions. =
One
usually introduces integration as a method to determine the volume open, bounded region B C R n- The main idea is as follows. We divide B into small parallel epipeds Bi. Each Bi carries a number f(Bi) representing the volume of Bi. Summing up all these numbers gives an approximation for the volume of B. Clearly, the function f which maps parallel epipeds into real numbers must satisfy certain properties. The most obvious property is that if we divide Bi into two disjunct parallel Ai U Ci, we have f(Bi) ,:z f(Ai) + ffi), epipeds Ai, Ci with Bi at least if Bi is sufficiently small. Then, choosing an infinite sequence of
an
=
fJBi,,,1iE1(a)1a (a E N) ofsuch divisionswe obtain a sequence of numbers lEiEI(a) f(Bi,a)JaEN which in most cases of interest has a well defined limit, the choice for
volume
vol(B)
of B. Linear
algebra indicates the following
Let
e,,J the standard basis of R', 101.... 0- 1 its dual basis, and bij,..., bi,, those vectors which span the parallel epiped 01 A Aon (bi, I,. Bi. The number f(Bi) bi,,,,) is then the Euclidian f.
)
=
...
-
-,
volume of Bi with respect to the standard Euclidean scalar product. This function clearly satisfies the additivity condition above. If one knows
2.5 Differential forms
how to determine volumes
0: B 01 A
...
(for
R
-->
113
also integrate continuous functions differential form
one can
densities) by replacing the differential form 001 A A on.
instance
A on with the
mass
...
turn to manifolds. The main
problem here is that we do not have a linear space in which to embed the cubes. However, by now we are familiar with the idea of translating concepts to their infinitesiLet
us now
mal counterparts. Since the tangent space was introduced as the linear approximation of the manifold, it is natural to place our parallelepipeds which
use
the linear structure of Rn into the tangent spaces rather than words, we divide M into small sets Vi
into the manifold itself. In other
such that each of these sets
corresponds
to
a
parallel epiped
in
T,,.M,
where xi is a point in Vi. We cannot define a canonical volume because E,, 1. This a general manifold does not have a preferred frame I El, , .
indicates that it is 'more natural to define the volume of
special To
(cf.
case
are
set first. We will later
of
of all
We
clearly have
recover
.
.
than to
directly
the volume
as
a
our
discussions
we
will define integration for
n-
necessarily smooth.
not
Definition 2.5.9. Let M be set
n-forms
4.2.1).
Definition
simplify part
forms which
a
integrate
continuous
a
real
manifold.
We denote
On (M) the
by
n-forms. Q (M). Let (U, o) be
f2n (M) c
a
chart and
w
cz
f2cn (M)
an n-form with compact supp(w) C U. Writing o (xl,..., x') there A A dxn. We is a unique smooth function wi n with w ndxl w,
be
=
...
...
...
define
W
Wi
...
ndxl
A
...
/\ dXn
where the last expression is the usual
U)i
integration
...
n
o
cp-ldxl
...
dXn,
in Rn. We still have to
show
(i) (ii)
fu
that
w
does not
' O
depend
on
the chosen
chart,
how to extend this local definition to manifolds which may not be covered by a single chart, how to extend this local definition to n-forms which do not have
(iii)
compact support.
(i):
Let
(U, 0)
be another
chart, denote by (y 1,
yn)
the
corresponding
coordinate functions and set axa 77
Then
we
=
sign
(det(I
ayb
1))
have gxa
W
:::=
Wi
...
ndxl
A
...
A dXn
=
Wi
...
n
det(f
ftb
1)
o
Odyl
A
...
A
dyn
114
Analysis
2.
on
manifolds
qxa 77wi
...
det(f
n
J)
o0dy'
A
...
dyn
A
and therefore
JOM
W1
oO-1dy'
n
...
10 =
dyn
...
Wi
...
0'0-1 det(I
n
(U)
77
=
Wi
W1
7
...
n
...
n
-0-1
O-ldxl
o
Hence the definition is indeed coordinate orientation in advance and restricts to In order to address
employ
a
(ii)
E
w
Qn(M)
be
a
...
(p
o
dyn
0)-ldxl
...
dXn
dXn.
independent
if
fixes
one
an
integration globally
we
will
C
an
oriented n-dimensional real
I (Ub) Ob)
closure, fa index with
GB
SUPP(fa)
Since each
is
an
partition of
a
E
x
C
b(a),IP6(a)
oriented atlas such that each Ub has compact unity subordinate to fUbjbEB and b(a) is an
Ub(a)
M has
SUPP(fa)
many
manifold
n-form with compact support. Then
aEA
finitely
o
...
oriented atlas.
and to define
IM where
1) dy'
qyb
partition of unity.
Definition 2.5.10. Let M be and
an
qXa
a
and
-
neighbourhood
supp(w)
which is intersected
is compact the
sum
by only
in the definition
above is finite. Let and
19c1cEc
6(c)
finite
be
an
we can
be another partition of unity subordinate to f(UbIbEB Since all sums involved are supp(g,) C
index with
Ub(c).
calculate
aEA
J(U
faw b(a) Wb(a)
aEA
9c
fa U)
9C
f" W
9c
fa W
cEC
I EEI I: E
aEA cEC
(Ub (a)
,
W b (a)
)
cEC aEA
E cEC
f(U6(WP6(c))
gC
1: aEA
f.
2.5 Differential forms
115
grCEC
implies that the definition is independent of the chosen partition of unity. Since by coordinate invariance it is also independent of the chosen oriented atlas our definition of integration over n-forms with compact This
support is well defined. We will now address point (iii). If w G Qr' (M) does not have compact support, its integral may not exist. This is completely analogous to the integration of functions f : R ---> R. For our purposes the following extension is sufficient.
Definition 2.5.11. Let M be and let
w
E
w(a) : Ua 1...n
modulus
oriented n-dimensional real
an
manifold
A, (Pa) I aE A be an oriented atlas and for each a A dx'. The R be defined by (Wa)*W W (a)ndxl A I... the continuous n-form jwj locally defined by ((Pa)*lwl
S?n Let C (M).
of
p(a)" jdx1 A
is
A
...
I...
Let M be
":
--+
w
...
dxn.
oriented n-dimensional real manifold and
an
(Ua, W,,)aEN
be
a countable oriented atlas such that each Ua has compact closure. As a preparation to the following definition we first need to give a meaning to
arbitrary w E S?n(M) which do not 11 E kj let Ij necessarily have compact support. For each j E f 1, fj+l,...,kl : UjnUj = _ 01. ifq E Qn(U,U UUk) has compact support
the
expression f fulu
...
JWjjkEN
uU,
for
C
=
.
.
.
,
...
G
then WE
we
fu,u
have
Qn(M)
is
...
UUk
general
a
C
Ekj=
1
(f(Z i,Wj)
differential form
we
f(i jnuwj)
if
define
k
Jul
U
...
UUk
jm'wj)
j=1
Definition 2.5.12. Let M be and
be
A, Wa)aEN
a
oriented n-dimensional real
w
E
Qn(M)
is
c
manifold com-
integrable if the (monotonically
f fak: U1 b(a) is an
...
UUk
U
and
index with
w
E
...
U
Q (M)
independent
aEAk
Ukj
where
any
I
jWjjkEN
UI U
nition is
n u, oj
sequence
11 Clearly,
J(41
countable oriented atlas such that each Ua has
pact closure. The n-form
increasing)
an
1E j
a
fakW7
Pb(a),Vb(a))
partition of unity subordinate
supp(fa)
C
Ub(a)
-
to
f(UbIbEB
is bounded.
with compact support is integrable. The defisee this let (Vb, V)b)bEN be
of the chosen atlas. To
second countable oriented atlas such that each Vb has compact closure. Then for each k E N there is a j(k) E N such that U, U Uk C a
...
116
VI
2.
U
Analysis
on
Vj(k). Hence JU)JJkEN is UUk
...
IfUJU
...
We
can now
manifolds
fu,u
...
I (A) I
UUk
bounded if
fV, U-I ,(,) 1wJ which implies ffv,u U-I j IWIIjEM is bounded. <
U
that
...
...
define
for differential forms which may not
integration
have compact support. Definition 2.5.13. Let M be
Pa, (Pa)aEN
and
be
a
pact closure. For any integrable
J
W
W
-
"l U-UUk
for k
fUIU-UUk+I
f?n(Rn)
(E
lim k
This limit is well defined since
J
w
=
M
I
oriented n-dimensional real
an
countable oriented atlas such that each
00
fulu
...
we
define
I
:_
integral by
UU,
integrability implies W
its
manifold Ua has com-
that
JUkU"*UUk+l IWI fU <
00 J=k
f fu IU-UUk WJkEN is a Cauchy sequence. integrate any integrable n-form over any subset open subset U and any (not necessarily integrable) n-form over any open subset V with compact closure since these forms are clearly integrable oo
and therefore that
Observe that
we can
with respect to the (open) submanifolds U and V. The following two lemmas are direct consequences of Lemma 2.5.9
and w, cD
for
(Linearity
on C (M)
of
integration).
differential forms which
are
our
definitions.
Let M be
a real manifold integrable. Then we have
any real numbers a, b
IM (aw
+
bCD)
=
a
IM
w
+ b
IM
0.
Lemma 2.5.10. Let
M, N be oriented real manifolds and f : M --> N diffeomorphism such that f,.v is a representative of the orientation of N for each representative v of the orientation of M. Then we have for any w E w E Q"(M) be
a
C
fm f*W JN =:
W.
Complex valued functions play a very important r6le in functional analysis and quantum mechanics and vector valued differential forms are used in both physical and mathematical gauge 'theory. (For instance, the analogue of the electromagnetic field strength is described by a Lie algebra valued differential form.) Readers interested in integration over such objects can find some elementary definitions in the remark below. In this book, however, we will use these concepts only in order to motivate the definition of mean curvature vector field (cf. Definition 4.4.2).
2.D Differential forms
Remark 2.5.2. We have restricted to K
=
R since
117
partitions of unity do
general over complex manifolds. It is possible however to integrate complex valued differential forms over a real manifold M. Let M be a real manifold. A complex valued differential form is a map w: x ---> Wx not exist in
p entries
where w,,: TXM X x TM -4 C is multi-linear and anti-symmetric. Denote by OP (M, C) the space of all complex valued differential form ...
of
degree
p. For each
U) C
on (M,
S?P (M,
C) QP(M, R)
w
iin
differential forms Wre,
C) integrable
if both Wre and
integrable complex valued
over an
fm We wish to show that
C
W
re
+
over
is clear. With
wirn
form
n
fm
=
integration
Additivity
is C-linear.
there exist two such that
w
uniquely defined re
+
=
iWirn. We call
integrable. defined by
are
is
w
The
integral
ifm Wiin
complex valued differential
w
E
on (M,
C)
and
a
=
forms
are +
ia"n
fm
re))
have
we
fm =I (a fm ((a (a JM re
aw
ia'm) pre
+
+
iirn)
A M
airnwirn)
rere
=
-
re
(a a
=
re
K
=
iiii)
i(a re.irn
IMA Jm)
fm pre
+
+ i
(a
a'mw re))
+
re
fmWim
+
a"n
i.irn)
W.
This definition or
-
+ ia
fm
ain'
re
=
+
can
C and 93 be
differential form of
be further extended to vector spaces. Let K R over K. A vector valued =
k-dimensional vector space degree p is a smooth map w:
a
x
-*
w,,
where
p entries
T,,M
u)x:
is multi-linear and
x
...
x
TxM
--+
93
by f2P(M, 93) the space of degree p. If w E S?P(M, 93) and 93, there exist k uniquely defined differential Denote
anti-symmetric.
all vector valued differential form of
JUI,
-
-
-
)
Uk
forms w' form
w
a
G
QP(M)
with
w
is called
definition is
integral
fixed basis of
of
a
=
I:k
w'u, A
if all w'
vector valued differential
integrable integrable. It is clear that independent of the chosen basis. We can now define are
vector valued differential form
w
via
k
(
J M
W
=
E a=1
(IM W-)
V,
the the
Analysis
2.
118
This definition is
6k
basis
Y:k
manifolds
on
independent of the chosen basis
A060
with
we
have
w
c'
since for any other
_Tk= c7),86,3
=
where CDO
A)3w' and therefore
(im
J C,)3 ) 6, M
,8=1
A'3w'
bc,
a
E
=
a=1,0=1
A13 -
( fm
6,
a=1,3=1
k
(JM
E a1pha=1
We turn
to the theorem of Stokes which
now
integration formulas of GauB and Stokes which
generalises the classical generalise funda-
in turn
b
mental theorem of First
f' f '(x) dx
calculus,
f (b)
=
need to define the concept of
we
a
f (a).
-
manifold with
boundary.
The pair (M,,)M) is an n-dimensional oriented boundary if there exists an n-dimensional oriented manifold 1 1 and an embedding t: M --4R such that
Definition 2.5.14.
manifold with
(i)
topological boundary &(M) is an (n l)-dimensional sub1 1 is which to diffeomorphic manifold of aM, (ii) for each x c &(M) there is a positively oriented chart (U, p) of 1 1 centered at x with W(U n t(M)) ly E W(U) : yl < 01. the
-
=
An oriented is
manifold
with
boundary
is
compact if
t(M)
U
at(M)
c
compact.
We will Let
M
\
usually identify M with t(All) and W with&(M). ICI be a manifold and N a hypersurface in 1 1 such that
N is connected. Then
boundary,
even
if
k
is oriented and
Stokes would not hold for Theorem 2.5.5
(M, N)
is not
w
are
(Theorem
integrable
M
oriented manifold with
compact. The following theorem of
(M, N). of
Stokes).
compact real manifold with boundary and dw and
an
M and
over
(M,W)
Let w
E
be
on-I(M).
an
oriented
Assume that
W, respectively. Then Stokes'
formula
Ja
W
M
=
JM
dw
holds.
Proof are
Let
jUj, ojjjErj
intervals
an
oriented atlas of 1
(a , b ).... (a ', bjn)
The definition of assume
be
a
with Wj (Uj) manifold with boundary
that for each
Uj
1 =
such that for each
(a , b )
x
...
x
j there
(a' , b ').
implies that we can also equality Wj (Uj n M)
which intersects OM the
=
2.5 Differential forms
ly
E
W(U)
:
subordinate to
y' < 01 holds. IUj Wi IjEN
Let
Jfjjj
N be
c
119
partition of unity
a
7
We show first that
d(fjw) holds for each j E N. We may write
dx'+'A
Adx' which
...
(2.5.4)
fj W fjw
=
implies d(fjw)
A E'i=, wkdx' A ,(-I)'+laiwk'dxA ...
dx'-'
...
A
Adx'.
We set A
[aj, bj]'
=
(aj', bjl)
x
...
(a. , b. )
...
(aj, bjn)
x
C R
n-1
A
(a3i, b3 )
where
indicates that the ith interval is omitted. Since
compact support in 2.5.4 is
(aj', bj') x
(aj, bj)
x
...
the left hand side of
w3k
has
Equation
given by
d(fjw)
_1)i+IC9xikdx'A =
=
f(al,b!)x f(a ,bl)x
Ad xn
...
i+ 1 ...
x
k dxl
...
d xn
axi
W", V, 3
kdxl ...
x
(& ",V")
...
dxn
axi
n
WiJ (X1,
(-I)i+l
+
.
.
.
,
b3 ,.
..
Xn
i=2 =0
-
A
Wi (XII..., a...... Xn
dxl
...
dx'
dXn
=0
f[a,j,b.jj1
i
(W1 (0, X2,..
Xn)
1
_W
(aj, x
2 ....
if
;'0 w3j (bj, X2.
Uj
) dx 2.
..
intersects
d Xn
9M,
-
.1-7
f[ag,bj 1,
Xn)
1
Xn) -w3,(aj,x
2
Xn; )dX2
..
A Xn
otherwise. There
are
three possible
cases.
120
Analysis
2.
manifolds
on
0 then aM n Uj 0 and both integrals vanish. (i) If M n Uj If M n but OM 0 n 0 the right hand side vanishes (ii) Uj 4 Uj =
=
because
we
integrate
empty
over an
set. The left hand side vanishes
by
calculation
our
(iii)
Assume that M n
Jmn4j d(fjw) f(.?,
Uj : 0
and OM n
Wi (0, X2,.
=
.7
b?) x
...
x
.7
f
=
Uj = 0. ..,
x') dX2
...
dxn
,
(a!, b `) .7
.7
Wj(o,X2,...,Xn )d X2 I
,9 amnz
Then
A
...
A
dxn
j
famnz j fj since the
pull back
of
dx1
to OM vanishes and therefore also the
n
-
1-
form
W (0, X2
...,x')dxl
We conclude the
A
...
A
j
A
...
proof by summing
A dxn
over
all local
:,Q
fm
to
aM(i
>
2).
integrations:
00
fm
dw
pulled back
E
d(fjw) j=1
j=1
fmnL j d(fjw)
00
Y j=1
Corollary
2.5-4. Let M be I E S?n-
mani
Id and
Proof.
Since M is
fam
=
As
W
fo
Jamnul. fjW Ja
W
w
=
a
an
(M).
W.
M
n-dimensional, oriented, compact,
Then
fm
manifold without
dw
=
real
0.
boundary,
OM
0 and
fm dw
0.
1
application of Stokes' theorem we prove that for every evensphere every vector field must have a zero.' This theois also known as the "theorem of the hedgehog" since it shows that impossible to perfectly comb an "ideal" hedgehog.
an
dimensional unit rem
it is
Lemma 2.5.11. Let M be
manifold and f,
f:
M
--+
M
an n-dimensional, oriented, compact, homotopic maps. Then
f
f
m
for all 7
w
E
M
Qn(M).
Recall that
we
only consider smooth
vector fields
real
2.6 Connections and
on
-
lary
1
implies that there
Lemma 2.5.6
Proof.
(M)
with
f *w
-
f*w
=
exists
a
121
structures
differential form Co
1
2.5.4.
Sn.
on
Proof.
If
E
dCv. Hence the assertion follows from Corol-
sphere in R'+1 and V be a a point x0 E Sn with V(xo)
Theorem 2.5.6. Let Sn be the unit
field
projective
is
If n
even
then there is
0 for all
V(x) :
x
Sn
(E
we
can
vector =
0.
normalise V with respect
product -, -)R11+1 of R'+', i.e., we can assume 1. For each x E S' we without loss of generality that T) V)R11+1 can identify V(x) with a tangent vector of R'+1 and, since T.,Rn+l and R1+1 are canonically isomorphic further with a point in R'+'. Our S'. normalisation implies then that we have defined a map V: S' 1 Let 7 be an integral curve of V with -y(O) x. Then ('Y(t))'YW)R-+1 2 (X7 V(X))R",+I. Hence V(x) is implies 0 I ('Y(t)1'Y(t))R-+1)Jt=o Ut to the Euclidean scalar
=
--->
=
perpendicular
to
and the homotopy
x
[0, 1]
F:
x
is well defined. It satisfies
Since -id is
=
=
=
homotopic
orientation of Sn
(for
n
S'
--+
X
_4
S'
cos(7rt)x
F(O, x)
=
x
and
to id and the
even)
+
sin(7rt)V(X)
F(1, x)
-x
=
for all
x
E
Sn.
diffeomorphism -id changes the 2.5.11 implies that for every W E
Lemma
on (sn)
is holds. This in turn not true. Hence
implies
our
is",
is" fs,,. w
=
0 for all n-forms
initial assumption
which is
w
V(x) 7
0 for all
x
E
certainly
S' must be
wrong.
2.6 Connections and There
is
one
projective
feature of A' which
we
have
structures
ignored
far in
so
our
efforts
p. 89
spacetime. Given any two different points in An there is a [I unique line passing through them. This global structure has an infinitesimal counterpart given by the directional derivative of vector fields. In fact, these lines are exactly those curves -y: [a, b] --> A' which satisfy 0. (Observe that this expression is well defined, i.e. for any Dff- y)
to localise
=
vector field V with
Recall that in
V(-y(t))
Chap.
I
=
we
(t)
we
have
have relied
to introduce inertial observers. Here
we
on
DV( ) this
=
0.)
affine
will introduce
structure in order a
generalisation
of
p.
1
125]
122
2.
Analysis
manifolds
on
general manifolds. This will be done by generalising
it to
the directional
derivative D. In the affine space An the difference of a derivative of a map 0: An a vector field V: An --> K' is blurred. In fact, let e 1, en
A' and
-
be the standard basis of Kn. Since -
the derivative of
in direction
w
at
x
is
given by
-
the derivative of V in direction
w
at
x
is
given by
it is difficult to
0:
map
commonly both
are
M and
-->
x
+
2-24wie.
and
wi ei av" axi
the difference between both kinds of derivatives.
see
Accordingly, they M
0
a
vector field U
denoted
on
by
D. Consider
now
0 at x in analogous
direction wx is now given by Txo(w.,) E To(x)M whereas the derivative of V is given by TxV(wx) E Tv(x)TM. We do not obtain vector but
a
M. The derivative of
a
element in the tangent bundle of the tangent bundle. It follows that this derivative cannot be used for defining straight lines. an
In order to obtain in TM
ues
we
ing definition
an
will need is
an
same as
the usual derivative D.
Definition 2.6.1. A covariant derivative
Tol(M)
V:
-4
711(M),
V
--+
VV,
or
VV(W)
connection V is
VWV
:--:
such that the
with val-
additional structure, a connection. The followexpression of the idea that infinitesimally a connection
an
17 should be the
analogue of the directional derivative
for all vector fields U, V, W, all functions f, h, following holds.
'--:
a
map
Va VbWa OXb
and all a,
0
E
R
(i) '7f V + hWU f VVU + h'7WU, (ii) VW(au + OV) aVWU + 017WV, NO 17wf U (W f ) U + f VWU. =
=
-
=
The torsion
of V
is the tensor
(U, V)
-->
field
Tor(U, V)
=
VUV
VVU
-
A covariant derivative 17 is called torsion-free
if
-
[U, V].
in addition to
(i)-(iii)
the equation
(iv)
Tor
=
0.
holds. This definition is
justified by
the
following
Theorem 2.6.1. A map V: Tol(M) ant derivative if and only if for each
(xO
I
.
.
.
,
xn- 1) centered at
x
such that
theorem.
T11(M) x
E
is
a
torsion-free
covari-
M there exist coordinates
2.6 Connections and
projective
123
structures
,9Va
( ) '9Xb
holds at
x
for all
Proof. Assume
first that there exist coordinates such that the
properties (i)-(iv)
2.6.5 is satisfied. Then it is clear that covariant derivative
Let V be
centered at
x.
of
Equation
torsion-free
a
satisfied.
are
(U,
covariant derivative and
a
3)
be two charts
We denote the coordinates with respect to these two charts
W (y), -P(y) (D( o W- 1))ab igXb
by x (y)
(2.6.5)
V.
fields
vector
(7b Va
=
Ix
3'(y).
=
=
Observe that
(D(W
,-1))ab
0
ax' =
ayr
and
o
Setting
,VWV
Va
V,95 19: b
fVa
fVa -9.:i'Vb, .97
a
f'C 19X,::
-95c'Wb
=
have
we
lacblqX ,WaVbg
Wa,9X", Vc,9X; +
=
=
where
and
F'b,9x.
V,9X.19Xb
X
f;,cb,9, ,::fVafrbaxc,
+
a
' Xb ab. Hence
and
a
obtain
we
C
(VWV)
fVaa:i", frc a,,ta
Wd
axd
PcbWaVb
+
'a
aXe 9.,
a
ax
a;,-a CqXe a;,c
,ox d
9;-,a
&,-a +
aj b
Xd Xf
, 'c af
-
a: c
(_)c VWV
a., , ax,
0_ _c
_Vf
+
axf
P
a. a
ax
d
a, b
Vf
9xf
a2;i c
a, ,a CqXe +
Wd
c
ab
WdVf
_
Xd a.: a aXf aXe
.cb WdVf a
+
WdaxdVe
aXe
(
Wdax Vf
WdaXd, Vf
9xf
From
5X7
"
a.:; a 19: b
a2,, c
(
+
9Xf,9X d aXe
+
92j h
(5
-
a., h
(,VWV)e
=
cb)
WdVf
Pa c
19x d 9Xf
XfgX d
a;,-a +
aj b
9x d gXf
2LWd aXd Ve ax,
b)
fahI
WdVf
yWdVf
+ re d
get
we
therefore
(9Xe
Fdef Now
we can
show that
9j a
h
92,: h
9Xe 9. a
9., b
,h
-jX-f_9Xd +_;X h_ -Xd 5Xf ,
covariant derivative is torsion-free if and
if there exist coordinates such that the
equation
only
in the theorem holds.
Expressing the condition of being torsion-free in coordinates we see that it is equivalent to Fab Fbca in any coordinate system (xo.... Xn-1 If the covariant derivative is induced by coordinates in which the Facb =
I
vanish at x, then in any other any coordinate system have red f
a2: l which is _5 r D_X_ -Tax 'T aX.
clearly symmetric
in
(xo.... Xn-1) I
e
and
f.
we
For the
124
Analysis
2.
assume
converse,
generality)
that
manifolds
on
that the
z-a(x)
Facb
transformation of the form :F in b and
c.
At.;p
0
=
we
5-Xb
A ac ab
=
Pacb
=
(Pacb)
-
+
a, b and
consider
a
(without
loss of
quadratic coordinate
xa + .1 Aa XbXc, where A' is symmetric bc 2 bc
=:
2;,- c
Aacb
=
5X _b at JF
=
ja b
-
19Xa,9Xb
Acab,
0. Our assertion follows
by choosing I
Ix
For later reference
proof
now
aXa
ja b i
=
Facb
symmetric in
have then
a,, a
whence
are
0. We will
=
collect the coordinate expressions derived in the
we
of Theorem 2.6.1 in the
following corollary.
Corollary 2.6. 1. Let (xo, Xn-') be a local coordinate system on a manifold with connection (M, V). Then there exist functions FbI, ra (bc) such that for each vector field V the covariant derivative VV is given by .
.
.
V b Va and the
I
==
9b Va + _pa buc
function Fbac transform under
(x0 according
xn-1) aj a
92Xh
19X h
7Xbqj c
a C
Lemma 2.6.1. Let extension
unique
(M, V)
of V
to
transformation
(: o ....... n-l)
VV: T,,r(M)
--
19. a 19Xd
19Xeh d 7Xh Yb_ a. c Fre r
+
be a manifold with connection. There general tensor fields,
V: Tr (M) such that
coordinate
to
A a
_4
a
-
T'r+ , (M),
Tr(M)
is
a
0
-
exists
V V)
derivation for every tensor
field
V.
Proof. It is clear that we can extend VV to tensor fields Uniqueness follows directly from Proposition 2.4.2.
as a
derivation. 1
In
special relativity, timelike straight lines represent freely falling partilightlike straight lines represent light signals. In Euclidean space, straight lines are the shortest curves between any two points. One way to define a straight line -y: (a, b) F--> A' is to require that the acceleracles and
tion
=
Dffy)
connections
Let t
as
vanishes. This definition carries
over
to manifolds with
follows.
V(t)
be
vector field
along a curve t 1-4 -y(t). Then we can ' which is defined in a neighbourhood of the path of -y. It follows from Corollary 2.6.1 that the covariant derivative of '1 restricted to -y in direction is independent of the extension ' . Hence the following definition makes sense. 1--4
extend V to
a
a
vector field
2.6 Connections and
Definition 2.6.2. Let V be and t
-->
V(t)
vector
1 (t)
Then
(i)
a
covariant
a
field along
V (t)V(t)
=
covariant derivative of V
projective
derivative,
125
structures
t
-y(t)
-4
a
curve,
9a
is the
-y.
(AVa(t) +.pbayc(t) b(t)) dt
=
along
-y.
(ii) A pregeodesic is a curve -y satisfying V 0. (iii) A pregeodesic -y is a geodesic if V (iv) A geodesic is complete if it is defined for all t G K. (v) A projective structure is a maximal equivalence class of =
tions which all have the
The notation field with
V (t)V(t)
W(-y(t))
=
can
V(t).
(V (t)W)
0
same
be
justified
Then
we
as
follows. Let W be any vector
have
(( babWa
Y
connec-
pregeodesics.
rba,_ b Wc)Oa)
+
0
C
-Y
d dt
d dt Hence the notation
comes
from
(W 0,Y)a
+
Fba_ bWc
0
'Y) 19a
Va + ra bW Vc)Oa-
identifying
V and W which
are
different
maps but assign to each point -y(t) the same vector. We will now introduce the notion of parallel transport. The follow-
ing
motivation may
seem
to be
mathematically imprecise. Nevertheless,
mathematicians used such arguments (before the advent of the French Bourbaki school which introduced a new level of precision in mathemat-
ics)
in order to introduce connections. These
and
are
with
arguments capture very well
of the intuition which leads to the notion of
some
a
therefore worth
knowing,
even
of salt: One may view
grain
a
a
covariant derivative
though they
have to be taken
covariant derivative
as a
connec-
tion between
infinitesimally neighbouring tangent spaces. Let V be a x E M and E TxM. The idea that T,,M is the infinitesi-
field, approximation of the manifold M near x is sometimes expressed by stating that x + is a point infinitesimally close to x. (Strictly speaking, this "addition" of points and vectors in general manifolds does not make vector
mal
Assume that M is
submanifold of Kn+k
Then we can identify subspace of K k+m and x + does make sense. This point will in general not lie on M but still be close to M if is small.) Let v,, E TxM which we want to "parallelly" translate to the point x + . sense.
T,M with
a
a
.
linear
We will then have vx+C v., + Jv where Jv is small. Since this is an infinitesimal process Jv should depend linearly on v and the difference =
. This bilinear map mally neighbouring points.
vector
sider the
now
curve
connects the
In
tangent
spaces of
our
infinitesi-
17bcVb c. Co n coordinates, we have jva a curve -y between two distant points x, y E M. We can divide into infinitesimal segments and parallelly translate a vector a
=
_
[p. 1211] I
p. 132
126
2.
manifolds
on
TXM successively along these infinitesimal
E
vx
Analysis
curve
segments. The
concatenation of these infinitesimal map P
:
ly
TxM
which in
TyM
-+
parallel translations gives a linear general will depend on the interme-
diate tangent spaces and therefore on the curve -y. These ideas will be made precise in Definition 2.6.3 and Proposition 2.6.1 below.
Definition 2.6.3. Let
(M, 17)
be
-/:
be
a
manifold
(a, b)
-+
with connection and
M
A vector
field V along -y is called parallelly transported along -/ if 17 V(t) for all t E (a, b). A parallelly transported vector field is often simply called parallel.' curve.
a
=
0
In
particular, a geodesic is a curve whose tangent vector is parallelly transported. For this reason, geodesics are sometimes called auto-Parallel curves.
2.6. 1. Let
Proposition
[a, b]
-y:
--+
M be
a
curve
(M, V)
be
which
can
a
manifold with connection and smoothly extended into both
be
let di-
rections. The map
P^y: T-Y(a)M where V:
V(a)
=
Proof.
[a, b]
v, is
Since
a
-4
T-y(b)M7
TM is the unique
--*
linear
-y([a, b])
v
parallel
-4
V(b),
vector
field along
-Y with
isomorphism. compact, there exist finitely
is
many charts which
the
curve -y. Without loss of generality we can assume that -Y cover is contained in a single chart. (Otherwise we could divide -Y in segments which are contained in single charts. A successive application
of the
the a
proposition to these segments would imply Vt 0 general case.) The equation 17 V(t) =
the G
proposition
in
reduces to
(a, b)
system of first order differential equations for the coefficients Va(t):
1 a + rba . bV, 0. It follows from the fundamental theorem for ordinary differential equations (cf. Theorem 2.4.1) that the map P is well dely C
fined.
(t)
Letting
-4
-y (a + b
-
t)
and
f/ (t)
=
V (a + b
-
t)
we
clearly have -
17,:YV(t) and
a
=
0 if and
only
fortiori that P
is ^/
It is
possible
to
if 17
recover
an
V(t)
=
0. This
implies that
isomorphism.
the covariant derivative from
(P
,
P I
parallel transport.
The relation between these concepts is similar to the relation between the Lie-derivative and the Lie-transport. But note that there is no Lie-
transport along
a
single
curve.
2.6 Connections and
Proposition -y:
[a, b]
rections. For Then
for
2.6.2. Let
(M, 17)
be
which
can
curve
a
Proof. Let W(a)
=
S-0
(P t, (V(t
8
+
-
s))
FbacWb au ( t,,)c. Using
a
t
V(t))
-
.
Then 17W 0 implies -!LWa da 1419 expansion it is easy to see that there
P t (V(t + s)).
d
a
be
--
lim
V (t)V(t)
exists
a
-
any
127
structures
manifold with connection and let smoothly extended into both dilet each s E [0, b M, a -4 -y(t + s t) a). t,,: [0, s] vector field V(t) along -y we have
M be
-
projective
=
Taylor along t,, with
smooth vector field U
C
W07) Setting the
a
=
W(O)
s we
=
d
((au t") lc=O)
o'Fb(c Wb(0) a
-
obtain from W (0)
=
V (t + s) and
aa
+
U(9)9 2.
(jddo, , t,,)1,=0
=
- (t)
equation
p t'sV(t + 8) which
V(t)
_
implies the
Corollary
=
V(t
+
8)
_
V(t)
+
SrbacVb(t)
c(t)a
U( 8 )82
a
assertion.
2.6.2. A connection is
uniquely
determined
by
its
parallel
transport. Since inertial observers played an important r6le in Chap. 1, a thorough understanding of geodesics and projective classes should be important for the globalisation of the results in Chap. I (cf. Chap. 3). But they are also of independent geometric interest. In fact, the classical development of Euclidean geometry is built on the concept of straight lines (and therefore on the concept of projective classes). Theorem 2.6.1 implies that torsion-free connections are a straightforward globalisation of the usual derivative of vector fields in vector spaces. Proposition 2.6.3 shows that also from the viewpoint of geodesics it is sufficient to consider only torsion-free connections. Lemma 2.6.2. Let
difference
V
-
t
Proof. We have
is
V, t be a
tensor
connections
S(U, f V)
only
a
manifold
M. Then their
field.
to show that
function-linear. The
on
(U, V)
S(U, V)
,
=
VUV
-
7_ UV
is
non-trivial part of this assertion follows from
=
Vuy V)
=
(U
=
f VUV
0
f )V -
-7uy V)
-
+
f VUV
ft
uv
::--
-
(U
-
f )V
f S(U, V).
-
f tiuv
p. 159
[I
p.
1
129]
128
Analysis
2.
on
manifolds
Proposition 2.6.3. Let t be a connection. Then there exists torsion-free connection 17 which has the same geodesics as Proof
We define
.
VVW
-
VWV
17VW [V, W] is
=
-
t VW
1
Tor(V, W),
a
unique
where
Tor(V, W)
the torsion tensor of V. Since
Tor(W, W)
-
2
0 for all
W, both connections have the same geodesics. Further, V is torsion-free by construction. For uniqueness note that if we add any additional, non-vanishing, in the covariant entries skew symmetric tensor field S
T21 (M)
(E
for any
hand,
field S E
loose the property Tor 0. On the other in the covariant entries symmetric tensor there exists a vector v., with S(v.,, v,,) :h 0. This implies to V
we
=
non-vanishing,
T21 (M)
that the
geodesics with respect to the connections 17 and V have initial velocity v., do not coincide.
+ S which
I
However, there exists infinitely many torsion-free connections with the same pregeodesics. This means that each projective class contains infinitely many torsion-free connections. Lemma 2.6.3. Let V and 17 be torsion-free connections and 93 be the projective structure generated by V. Then 7 cz T if and only if there exists a one-form 0 such that t 0 (& id + id & 0 (or, in coordinates, V =
-
1'6'r
-
F1 be
2ja 0 C) ). (b
.
Proof. Let Zbac Pba,. 1-b`e. Then their pregeodesics can coincide o n ly d VbVc Za for all _Tdbe VbVcVa va vectors v'. This implies _Ta Vb v'V 11 be be =
-
=
-
for all vectors
symmetric Sbea C,d
in
which is
equivalent to 6e(d Zbae the lower indices b, c we get v
6e _Ta (d be)
:=
1 =
3
(d
_
cd
-
dc
6a_Tb d be
that Za
.
cd
For the
=
(n
a
6a c
-deb)
+
(6eZa d be
_
ja_Te be ) d
)
-
e
=
-
I
.
26a Od) (C converse
notice that same
2.6.3. Let
connection such that
VVV
=
VVV
+
O(V)V,
whence both
pregeodesics.
will be used in Sect. 3.2
(M, V)
be
a
manifold with M there is
connection. Let
be M "
for every pregeodesics with initial velocity v., (E itx with respect to V t coincide. Then V and t generate the same projective structure
such that the and
_
c
-
following corollary
Corollary
(6e_radb
b a a and b gives Sbbcad nZCd 'Ec'd + rdac jac Zd0b + 26a where It follows + 1)_ra Od (n + 1)'-IZbbd ed (C Od) ,
connections have the
The
0
0. Since Z is
be)
6a_re b cd ) +
Contractil-ig the indices Za
6a Zbee (d
ja Ze
( (je_ra b
-
if
x
(E
an
open set
it ,
C
2.6 Connections and projective structures
We
Proof.
the
use
in the proof of Lemma 2.6.3. It is Zd VbVeVa 0 for all vectors v E i1x be
notation
same
as
VbV,Vd sufficient to note that Z, be a (I
already implies
6(,dZbc)
-
129
6a Ze (d be)
0.
[p. I
Written down in reduces to it
natively,
0 coordinates, the geodesic equation a + Fbac b c system of second order differential equations on M. Alter-
a
be considered
can
tangent bundle TM. In is given by d
system of first order equations on the coordinates, this system of differential equations d a
_1Y
=
dt
The
corresponding
701(TM),
and
can
as a
vector field
be
Va
Va,
=
dt on
_Fa bcVb VC. ,
-
TM is called the
geodesic
spray F E
invariantly defined by r (V.)
TO '. (at)
where -y,.,,. is the unique maximal geodesic with 7,.,, (0) v,,. The following proposition justifies the definition.
=
x
and
(0)
Proposition 2.6.4. Let (M, V) be a manifold with connection and V be the geodesic spray. If A: (a, b) -- TM is an integral curve of V, then 7rTM o A is a geodesic. Conversely, for every geodesic -y there exists a unique integral curve A of r with ?rTM o A y. =
Proof
Let vx E TxM and -yv,, be
defines d
Av, (t)
=
translation at
a geodesic with (to) vx. This geodesic v. (t) in TM. Clearly, 7rTM o Avx -yvx and Tt V :(a) T0 11_ (-Yvx o -rt) (at), where Tt is the t dt
Avx (t) TtAvx (,9t)
a curve
t, Avx is
s
-4
Conversely, 7,\(t,,) with
t +
=
=
s.
-
A(-yv,,, oTt)(0) dt
curve
of _V.
let A be
(to)
I
Since
integral
an
=
=
an
integral
A(to). By
curve
is the
velocity
vector of -Yv.
of r and consider the
the construction
geodesic above, ,\(t(') is an
of V. Since its initial
point in TM is A(to), it must coinuniqueness part of the fundamental theorem for 1 ordinary differential equations (Theorem 2.4.1).
integral
curve
cide with
We have
A(to) by
that
the
manifold is
locally isomorphic to A' (considered as a It is not true, however, that a manifold with connection is isomorphic to A' with its affine structure. In fact, for a manifold with connection there do not generally exist charts which map geodesics into straight lines. For this to be the case a necessary seen
a
set with differentiable
structure).
condition would be that the connection is in the as
the canonical connection of A'. We will
identify
the
lines in A'
2.6.4).
now
same
projective class
show that
one can
still
geodesics which pass through a given point with all straight which, pass through a given intersection point (cf. Lemma
127
1]
p. 159
2.
130
As
a
Analysis
on
manifolds
consequence of
Proposition
2.6.4 and the fundamental theorem
for
ordinary differential equations, each x E M has a neighbourhood W of 0 in TxM and there is a J > 0 such that for all v c )/V the
M with (O) v is defined. By choosing W geodesic 7v: (-J, J) small enough we can normalise the interval [-J, J]. In fact, observe that ,,v (t) a , (at) for any a E K. Hence for every x E M the zero vector 0 E TxM has a neighbourhood U C TxM such that for all v E U the geodesics t 1-4 -yv(t) with initial velocity v is defined on the interval -
=
=
[-1, 1]. Definition 2.6.4. Let
exp:
jv
E
TM:
(M, V)
-y,(1)
be
a
manifold
definedf
is
--+
v 1-4
where
X
(V)
7rTM
=
is called the
,
M
exp(v)
exponential
It follows from the fundamental theorem for tions that the set of all
with
A(O)
=
v
v
TM,
E
is defined up to
with connection. The map
:=
map
expx(v)
:=
-yv(l),
of V.
ordinary differential equaintegral curve A of F
for which the
(including) parameter
value 1, is open.
Hence the domain of exp is open in TM. This also implies that for any x the intersection of this domain with T,,M is open in TxM. This set is also
all
a
star-shaped E
as a
consequence of the
equation ,,v(t)
a ,(at)
for
R.
Proposition 2.6.5. For each point x E M there exists a neighbourhood U of 0 E TM such that expx is a diffeomorphism from U onto a neighbourhood U of x E M.
Proof.
Let
V
E
TxM and
f)
d T expx (f))
dt
expx (tv)
)
dt
(tv)) It=0
E
ToTxM. Then
d
dt
It=0
-YtV (1)
)
d dt
It=0
-Y' M
)
V
lt=O
which
implies that T exp,, is an isomorphism. Now the assertion follows from the inverse function theorem.
Corollary
2.6.4.
There is
Exp: 1 is
a
diffeomorphism
--
a
M
onto its
neighbourhood x
M,
vy
F--->
U
of Ox
E TM
(y, exp(vy))
image
Proof. The corollary follows from the fact that TExp non-singular whenever T exp is non-singular.
is
such that
=
T7rTM T T exp I
2.6 Connections and
e,,J
be
a
V"E7,'1(va)2
<
rJ
is
B,(x)
:=
exp(f3,(Ox))
T,M. The
phism
Jv,,
=
:
map exp.,:
&(0)
-->
sufficiently small
for any
r
> 0.
Hence
a
we can
is
a
diffeomor-
a
coordinate
These coordinates
coordinates and the
corresponding
chart is called
Lemma 2.6.4. Let
(M, 17)
(U, p)
be
be
of 0,, c
define
(expx-1 (y)) a.
a
M.
c
neighbourhood
:=
xa(y)
system by
lei,
Let
ei for each vector v,,
=
f3,(O.,)
Then
i
basis of T,,M and write v,,
v
131
structures
provides especially practical coordinates.
2.6.5
Proposition
projective
are
called normal
normal chart.
with connection, x E M, and Then W maps the x. lines through 0 E K'.
manifold
a
normal coordinate chart centered at
a
x onto the straight Furthermore, the Christoffel symbols with respect 0. satisfy _ra (b c) (X)
geodesics through
to the chart
(U,
=
Proof. The
first assertion follows
immediately from the
construction of
normal coordinates.
vb vc vanTo prove the second assertion we must only show that ra bc ishes for all vectors v E TxM. Let y be the geodesic through x with (O) v. Since its coordinate expression is a straight line the coordinate =
components satisfy a
rbac(,y(t)) b(t) c(t)
=
At t
In Euclidean space,
0 and therefore 0
=
b
a (t) +rba
C
=
0 this
a convex
equation reduces
to
U set is characterised
(t)
rba,(x)vbv
C
C
(t) 0.
by the requirement
that any two points x, y E U can be joined by a straight line which is contained in U. For a manifold with connection we call a set U convex if any two points x, y E U can be joined by a unique geodesic which is contained in U. We will now show that each point has a convex neigh-
bourhood. Theorem 2.6.2. For each
bourhoods U,, with
x
nn" , Un
=
E
M there is
a
sequence
of
convex
neigh-
JxJ.
Proof. Let (xl,...,Xn) be a normal coordinate system and (U,W) the corresponding chart. For y E Image(expx) n U we define the distance function
d(y)
:
=
[(Xa (y)
_
Let
B, (x)
=
ly
c
Image(expx) n U
:
d(y)
<
enough, there exists a neighbourhood T7V(r) of Ox E Exp maps W_(r) diffeomorphically onto B,(x) x B,(x). 0 the neighbourhood W(f) shrinks to the set JOxJ C TXM. of the continuity of exp and exp(Ox) x, there is an f E (0, r)
Choosing
r
small
TM such that For f
--
Because
=
exp(twy) Bx(r) for all t E [0, 1] and wy E T7V(f). We will show that (for f sufficiently small), any two points y, z E Bf;(x) are joined by a geodesic which does not leave Bf;(x). Let -y be the 1 geodesic starting at y with velocity vector Exp- (y, z). This geodesic
such that
joins
y with
E
z
by
the definition of
Exp.
132
2.
Anady,!_Ar
manifolds
on
We have to show that for f small
Bf:(x).
The idea of the
enough
the
curve
7 cannot leave
follows. Since this
proof geodesic would have to re-enter Bf (x), in the given coordinates it would have to be curved at least of order 1 If somewhere outside Bf; (x). On the other hand, geodesics are generalisations of straight lines. This should give a contradiction for f small enough. While this is the geometrical idea of proof, it may not be entirely clear from the analytical implementation which we will present is
as
now.
If -y would leave maximum which is
and 11 d dt2
o
equality
choose the
a
L .) (X) a b F(abc
we
curve
(Jab 7arba ) a b d(-y(t)) C
geodesic equation. Since
and
small that
The
1
a
have used the
r so
B,(x).
constructions of f. Hence
gives
we
=
original
definite for all y G
2.7
j:n
.
_p,a'(X), a, b be
which
=
.
directly compute
can
Ena=1 (( a)2 + a a) d(-y(t))
do -y
where in the last
we can
then the map t --+ d(7(t)) would have a than f At this maximum we have Ad o 0 7 dt
bigger
-Y :5 0. We
d2
a_t2
Bi;(x),
F(ab,) (x)
Jab
Xa (y)
=
0,
bc F(a,_)(y)
-y does not leave
must have
-Ld dt2
o
is
positive
Bx(r) by
the
> 0 at the maximum
contradiction.
Examples of connections
p. 125
[I
p.
In this section, we will introduce two examples will both become important in Chap. 3.
137]
of
connections which
2.7.1 The Levi-Civita connection
The
following
definition
(An, ("'61)
is
a
and Minkowski space not necessarily affine spaces.
generalisation of Euclidean space (A n' ) to real manifolds which are
Definition 2.7.1. A
pseudo-Riemannian manifold (M, g) is a real man-tensor field g which is everywhere ifold M together with a symmetric (0) 2 non-degenerate. We will often simply write (u,v) instead of g(u,v).
The
of a vector u with respect to g is defined by 1jull V-lg(u, u) 1, general it is not a norm in the sense of linear algebra (the triangle inequality only holds if g is positive or negative definite). Let i, j, v E norm
=
but in
11'...'nj
and define
(770 ij
If
(M, g)
is
2.7
Examples
-1
if i
=j
< V,
1
if i
=j
> V,
0
otherwise.
pseudo-Riemannian manifold then there
a
such that for each point which satisfies
E
x
M there is
g(ei, ej) v
We say that g has signature dex of g. A basis jej,...,enj
orthonormal basis and
orthonormal
an
=
(,q,)ij
a v
I enl
nj
G
Of T'M
(2.7.6)
.
(n-v)
times
is
basis
a
133
times
)
+
and call
satisfying Equation (2.7.6)
v
the in-
is called
an
local frame
a
JE,.... En I such that for each El I (x).... En(x) I is an orthonormal basis
in its domain of definition
called
of connections
x
is
frame.
Definition 2.7.2. A
pseudo-Riemannian manifold is called a Riemanif g is positive definite and is called a Lorentzian manifold has signature (-, +,... +).
nian manifold
if g
I
Lemma 2.7. 1. Let
each mal
x
E
M has
(M, g) be a pseudo-Riemannian manifold. Then neighbourhood U which is the domain of an orthonor-
a
frame. Let U be
Proof.
coordinate
neighbourhood of x and denote the inby 1 91.... an I. We apply a variant of the Schmidt orthogonalisation procedure to this frame. There exist functions A, on a
duced GauBian frame
'
U such that U1(X) U. We will
x
El,
.
.
.
,
:=
Enj=1 Aj(x)aj 1
En I
vector fields. Let
tj
U111JU111.
=
pointwise linearly independent g (Ek,
Ej)
=
0 for k
U such that
satisfies
g(Ul, Uj) =h
0 at all
points
induction argument in order to define a frame which is orthonormal up to a permutation of the frame use an
Now
1 and g (Ek 7
Ek) Ej..=, A'j (x)aj does
Ui(x)
that
assume
vector fields
we
have constructed
jEj,...,Ej_jj
such t*hat
1. There exist functions not lie in
Al
on
spanf Ej,...' Ej_jj
and
ji
=
g (Uj,
Ui k=1
satisfies
g(&j, 0j) =7 (tj &
satisfies g
fE17
....
En}
0. The vector field
==
I
be
a
11 7 9
(& & 7
=
Ek)
E-i
=
Minkowski spacetime has
D,
a
i.e.
U_j1jj U-ill
0 for k E
suitable permutation of
the usual derivative
Ek
9(tki kk)
11,
It,
.
....
.
.
7
is well defined and ,
i
-
11. Finally,
t" 1.
metric q which is constant with
Dq
0. This is
let
1
respect
to
equivalent to the fact that parallel transport of vectors is an isometry. Furthermore, D is the only torsion-free connection which satisfies this requirement. For a general =
134
2.
Analysis
on
manifolds
pseudo-Riemannian manifold, there an analogous statement holds. Theorem 2.7. 1. Let Then there exists
Vg
a
(M, g)
exists
a
unique
connection for which
be
a pseudo-Riemannian real manifold. unique torsion-free connection V which satisfies
0.
=
This connection is called the Levi-Civita connection.
Proof of Theorem 2.7. 1. Recall that the connection V is torsion-free only if VUV [U, VI for all vector fields U, V. A short VVU calculation shows that the condition Vg 0 is equivalent to U (V, W) VUV, W) + V, VUW). The strategy for the proof is to use these two conditions in order to calculate the only possible candidate for the LeviCivita connection. It is then easy to verify that this candidate satisfies all the relevant equations. To exploit that V is assumed to be torsion-free consider the difference if and
=
-
=
U
(V, W)
-
W
(U, V)
=
(VUV, W) + V' VUW) ('7WU' V) (U'VWV) (VUV, W) (U'VWV) + (IU' W1, V)
=
-
=
-
-
(VVU, W) + U, VVW) to this equation the right (17UV, W) + 17VU, W) + (U, IV, W]) + [U, W], V). eliminate Here can (VVU, W) using that V is assumed to be torsionfree: (VVU, W) (IV, U], W) + (VUV, W). Putting everything together
If
we
add V
U, W)
=
hand side becomes we
=
we
finally
obtain the Koszul Formula
17UVI W)
2 +
(U
V' W)
W, [W, Ul)
+ V
w, W)
-
W
(u, V)
-
(u, IV, WI)
(w, IV, Ul)
-
(2.7.7) 1
The
following proposition gives
slightly
a
geometric characterisa-
more
tion of the Levi-Civita connection.
Proposition 2.7. 1. Let (M, g) V be a torsion-free connection. transport is an isometry. Proof. Assume a curve we
and t
calculate
be
a
pseudo-Riemannian manifold and 0 if and only if parallel Vg
Then
=
first that V is the Levi-Civita connection. Let t
i--4U(t), d dt
(U(t), V (t)V)
t
-->
V(t)
(U(t), V(t)) =
0 which
=
be two
V (t) (U(t), V(t))
implies
sequently, parallel transport
is
an
i-->
- (t)
parallel vector fields along -Y.
that
(U, V)
isometry.
is
V (t)U, V(t)
independent
be
Then +
of t. Con-
2.7
Conversely, u, v,
vector fields
transported 0
parallel transport
U, V
a curve
with
(t) A V) ='('Nt) g) A V) + (t)g) (U, V). At t 0 this implies =
of connections
is
an
-y with
U(O)
V
=
(V
that
assume
be vectors. We choose
w
Examples
=
u,
(O) V(O)
=
=
isometry and let w and parallelly
v.
Then
we
V (t)u' V(t))+(U(t), V 0
=
(Vwg) (u, v)
135
obtain
(t)v
=
which proves the
assertion.
2.7.2 The
Let M be is g
Weyl
a
conformal
02j.
=
connection
real manifold and g be a metric on M. A second metric to g is there is a positive function Q: M ---> R+ \ f 0 1 with
A
conformal structure Q is an equivalence class of conformal chapter we will see that the Michelson Morley experdirectly leads to a conformal structure rather than a Lorentzian
metrics. In the next iment
metric.
Given This
a
conformal structure (t there is
generalises
a
class of
adapted
connections.
the Levi-Civita connection of the previous section.
Definition 2.7.3. A
triple (M, Q , V), where M is a n-dimensional manconformal structure on M, and V a torsion-free connection is called a Weyl structure if for every g C- (t there exists a one-form p such that Vg W 0 g. The connection V is called a Weyl connection
ifold,
C-'
a
,
=
In the w
E
following
TjO(M) (cf. the lemma
use
will
we
use
the exterior derivative dw of
Theorem
2.5.1). In Theorem of Poincar6 (Theorem 2.5.5).
2.7.2 below
we
a
p-form
will also
Readers who have omitted Sect. 2.5 can replace dWab by 2!a[aWb] Using thi equality the lemma of Poincar6 can be understood in our
special
case.
Lemma 2.7.2. Let
(M, Q , V)
AdW
is
Proof.
Let g E (t and
independent of
=
Q2g.
2S?dS2 0 g +
Vj Hence F
j
=
2
dW
=
-
-1 2
be
a
Weyl
structure. Then the
2-form
F
g E it.
d o
The 2-form F is called the
Then
Q2Vg
=
does not
length
we
(p
have +
2dlnf?)
depend
on
0
the choice of g
curvature of the
Weyl
structure. We
will motivate this term in Sect. 2.8.1 below. Theorem 2.7.2. Let x
E M.
structure
(i)
F
(M, (t, V)
be a manifold with Weyl structure and neighbourhood U such that for the induced Weyl (U, (t, V) the following statements are equivalent.
Then
=
0,
x
has
a
Aijalysis
2.
136
(ii)
c,.rL
There exists
masafolds
e
a
(t which has Levi-Civita connection V.
Proof. We first show that (ii) implies (i). Let g 0. Then we have Vg 02g such that Vj j Q3dQ (D g. Hence W QMS? and dW 2SMS? (& j
(t be any metric and (p (9 g
=
=
=
For
"(i)
=:
=
note that F
(R)"
=
ld o 2
0. Hence
=
(Theorem 2.5.2) implies x and of a function f : U ---> R -e-f df 0 g + e-f df (3 g 0.
lemma of Poincar6 hood U of
V(e-f g)
=
Corollary there exists
=
0.
an
application of the a neighbourW. Consequently,
the existence of
with
df
=
1
=
(M, E', V) be a Weyl parallel, non-vanishing n-form
2.7. 1. Let a
.
assume
y. Then there is
an
that
(up
to
(t such that V is the Levi- Civita connection
sign) unique metric g En) g and I S?(E,,. .
structure and
I
,
for
every
g-orthonormal basis f El,
.
.
..,
of En 1.
Proof. For any metric g (=- Q we define an n-form A as follows. We let f Ej,..., Enj be an orthonormal basis with dual basis fOl, O'l and 01 A denote g (Ei, Ej) E I 1, 11 by ej. Then A is defined by A A on. Since An(TM) is I-dimensional there is a unique g E Q such that -
=
-
p. For this metric and any vector
0
=
VV (y (El,
.
.
.
,
v we
-
-,
...
calculate
E,,)) n
=
(17vp) (E,,.
.
.,
En)
+
1: I_t(Ej,..., Ei-1, 17VEi, Ei+l,..., E,.)
n
=
1: I_t(Ej,..., Ei-1, g(17VEi,, Ej)EjEj, Ei+,,..., En) i=1 n
=
E g(VvEj, Ei)ei n
=
Y- -2 Ej (Vv (g(Ei, Ej))
-
(Vvg) (Ei, Ej))
i=1 n
=.=
-E 2 cj o(v)g(Ej, Ej)
n =:
-
2
P(V).
i=1
Definition 2.7.4. Let
timelike
(or spacelike)
(M, ( , V) curve.
be
a
V (t) (t) for
all t in the domain
Weyl structure and -y be a smooth affinely parameterised if
We call -y
I
of definition of -y.
(t)
137
2.8 Curvature
It is clear that timelike
only
if
they
pregeodesics
if and
affinely parameterised
are
geodesics.
are
(M, (t, V) be a Weyl structure and t -y be a smooth (t. and there exists a all Then all t 0 E g g( (t), (t)) :h for that such is affinely parameterised. 7(t(s)) reparameterisation %s) 7(t(s)) is affinely param-y(t) is affinely parameterised then s If t
Lemma 2.7.3. Let
F-->
with
curve
=
t-4
-4
eterised
and
if
only if
Proof. We denote -y (t (s)).
there exist a, b E R such that
- 4dt
by
(),
dot,
a
by
ds
a
=
at + b.
%s)
Let
prime,
Then
Xv (t) W' W) X'71Y(sm) 7
ly (s (M)
(0)
ds
g
(7Y(S) Is t
1(8)
) (g (dsTt )2 (dt_ds is
dt
ds
d
(s))
(s) Tt
(V '
dt
that
ds
( ' (S) Tt
2
ds
implies
d
and
s
ds
2Sg(
(s), ' (s))
dt 2
affinely parameterised
(s),
+Tt g if and
d's
ds
9('V (t) W,
dt2
dt
gMt)
only
(s))
if
W)
W)
I
immediately from the fundamental thealready affinely parameterised, the differ-
holds. The first assertion follows for
orem
ODES.
If t
-+
'y(t)
ential equation reduces to
is d2s ft_2
=
0 and the second assertion follows.
1 13
2.8 Curvature 179
In Sect. 2.6 of
we
have
seen
that the covariant derivative defines Given
parallel transport along -y(l) x, this parallel transport
-y(O)
curves.
=
=
R,y: TX'M
ly
=:
vx, in
TX'M'
small
defines
VX
-4
loop a
-y:
[0, 1]
i-->
notion
M with
map
P vX ly
spacetime and in Euclidean space we always general the vector P ly vx depends on the loop -y.
While in Minkowski P vx
-->
a
a
have The
p.
141]
138
Analysis
2.
manifolds
on
'Ya3, b
f (a, b)
V 2
4
la,b
f (a, 0) a,b
theorem shows that if
following
there exists
a
Theorem 2.8. 1. Let U, V,
w
E
(M, V)
TxM. Then there R: TM
with the
be
a
exists
a
TM
x
rectangular loops then
restricts to
one
well defined limit where -y
--->
jxj.
manifold with connection and well defined tensor field TM
-->
TM
(u, v, w)
t--4
R(u, v)w
x
x
E
M,
following property.
Let U C R 2 be
an open neighbourhood of (0, 0) and f : U -+ M be an v. For any a > 0, b > 0 2-surface such that f,,al u, f,,a2
immersed with
=
[0, a]
7a,,b:
x
[0, b]
[0, 2a
+
2b]
=
let N,b be the closed
C U
--+
f (U)
curve
C M
f (t, 0) f (a, t
t
f (2a
f (0,
-
a)
+ b
-
t, b)
2a + 2b
-
t)
fo r
0 < t <
for
a
for
a
fo r
0 < t <
a
+ b < t < 2a + b a.
Then I lim ab (a,b) (o,o)
P
7abW
-
W)
-R(u, v)w
=
holds.
If U, V,
R(u, v)w
W
is
are
R(u, v)w The
vector
fields with U.,
=
u,
V
v, and
W,
w
then
given by
following
=
(VUVVW
-
VVVUW
section will be of great
this point in time it is better first to
-
V[U, V]W) Ix
importance in Chap. skip forward.
5.
However,
at
2.8 Curvature
139
It is easy to check that
Proof.
R(u, v)w
(VUVVW
=
-
VVVUW
-
V[U' V]W) Ix
does not
depend on the extensions U, V, W and is therefore a tensor field. 4 We denote the four segments of the curve 'Ya, b by b 'Ya, b) reP 0 and spectively-. Let W (a, b) observe that W (W) depends
7al,
=
smoothly
(a, b).
on
7a2,
P'Yal,
b
Vf*,92W(a, b) This and
W(0, 0)
,ya4' b
=
W(a, b)
we
have
0.
=
b
(V192V191W)(0,0)'
-
is
(P'd,b)
^YO, b which implies
P
Wb-
a
segment
curve
4
=
V 2,
=
parallel transport
immediately imply
w
=
R(u, v)w Observe that the
b
Rom the definition of
w
independent =
W(O, b).
of
a.
Hence
In
particular, obtain
we
W-W
'/a b
I
( 2, 'Yal, (W) b- (PN4,b (P'/a3,b (P^/a2,b P'Ya, (W) (PNI,b)-I) (W)) P^ya4,b (P 3, (W (a, b)) W(0, b)
b
P
OP
4
'Ya ,b
OP
3
OP
N
'Ya,b
b
b
1
0
0
I
0
b
-
b If b
0 then W is
=
'Ya
a
b
simply the parallel transport
of
w
along
al,b
which
implies
Vf
*
91
W (a,
0)
==
V al,
W(a, 0)
=
0,
(2.8.8)
b
and therefore 1
(P^YabW W) I -'::
-
'
ab
b
(P74 a
I +
-
b -
Since
'ya4,b
does not
does not
depend
summand is therefore
(Vf*a,W(O, b))
-
Vf*
a,W(0,0))
I
Pld,b (- (Py3a,b (W(a, b)) a
-
W(O, b))
Vf*a,W(O, b)).
depend
on a
b
on a
either.
the
parallel transport along
(2.8.9)
'd,b) P'd, b'
By Proposition 2.6.2 the limit of the first given by
140
Analysis
2.
on
manifolds
I
lim
P
(Vf
4
a,b-0 b
b
a,
W (0,
-
V
f* a,
W (0,
0)
(PN,b (Vf*a,W(O, b))
lim
=
b)) 4
b-0
-
Vf*a,W(O, 0)
Vf*a2Vf*Ca1W(O' 0), In order to summand in
complete the proof Equation (2.8.9),
we
need
only
to show that the second
W (0,
b))
1
bp'Ya,b(a (P'/a3,b (W (a, b))
3(a, b)
4
has the limit 0 for
(a, b)
-
-
V
f*,91
b)
P'ya4,
=
depends smoothly
b
( (P
on
where
0, 0,
implies
are
P703,
Vf*a,
(W(a, b)) an
b)
-
W(O, b))
(a)
p(O, b)
(a, b).
'Yal,O,
curve
I
W (0,
Since
b)
Taylor formula yields
curve
3
'YO,b
is constant which
0 for all b. The
=
curve
b
0)
)
ab (a, b),
+ 0 (b) +
smooth functions. The
aVf
-
of the
application
id and therefore
=
is inverse to the
W (0,
3
'Ya,b
(a, b),
o (a,
b)
0.
-->
We will first show that this summand is continuous in
p (a,
W (0,
In
analogy
to
Equation (2.8.8)
we
3
_Ya,O
obtain
0 and therefore
=
V(a, 0)
=
P, 3'OW(a, 0) P, 3'OW(a, 0)
-
W(O, 0)
-
W(O, 0)
=
a
=
-
a17f a,W(O, 0) 0.
a
Rom
o(O, b)
=
W(a, 0)
stant and that their
ab p(a, b)
we
conclude that both 0 and 0 we have W(a, b) ab
vanishes. Hence
==
are
con-
(a, b)
and
b_ (a, b)
(a, b)
I
b is smooth. In now
0
=
sum
I
( (P,,.3 a
particular, (
(W(a, b))
-
W(O,
,b
is continuous. The
b))
-
Vf*a,W(O,
equation 3(0, 0)
=
b)) 0 follows
from
lim a-0 a
for any b
> 0.
(p7a3,b (W (a, b))
-
W (0,
b)
Vf*,g,W(0, b), 1
2.8 Curvature
Definition 2.8. 1.
Riemann
-tensor R (1) 3
The
(Bianchi identities).
connection. Then R
satisfies
(VuR) (v, w) is
is called the curvature tensor
(or
tensor) of (M, V).
Lemma 2.8.1
If V
141
(VvR) (w, u)
+
torsion-free, then
R also
R(u, v)w
Let
+
(M, V)
the second Bianchi
+
manifold
a
with
identity,
(VWR) (u, v)
+
R(w, u)v
=
0.
=
the first Bianchi
satisfies
R(v, w)u
be
identity,
0.
Proof. Consider a normal coordinate system (xl,...,Xn) centered at E M and let U, V, W be extensions of u, v, w E TxM such that these
X
vector fields have constant
system. We have then coordinate
components with respect
[U, V]y
neighbourhood
and
=
we
=
=
=
X
vector field X
to our coordinate
0 for all y in this [V, W]y [U, W]y V W VU VV VW U Ox. For any
=
X
=
X
get
(VUxR) (V, W)X
VUx (R(V, W)X)
=
-
R(V, VUXW)X
VUX(R(V, W)X)
=
VUX (VVVWX
=
+
-
-
-
-
R(VUXVW)X R(V, W)VUXX R(V, W)VUXX
VWVVX)
-
VV VWVUX
VW VVVUX X
=
The second Bianchi
QVU' 1'7V"7WD. X.
identity follows now immediately from Lemma 2.4.3. identify we assume in addition that 17 is torsion
For the first Bianchi
free. We calculate
R(u, v)w
+
R(v, w)u
+ =
=
R(w, u)v
VuVVW -VVVUW +VVVWU -VWVVU +VWVUV -VUVWV VU ('7VW
VWV) + '7V ('7WU +'7W (VUV -'7VU) 0 + 0 + 0
=
-
-
VUW)
0
Definition 2.8.2. Let sor
Ric(u,v)
F E
02(M)
=
the
(M, V) be a manifold with connection. The tentr(R(.,u)v) is called the Ricci tensor. We denote by 2/n-fold multiple of the anti-symmetric part of Ric, 1
F(u, v) =n (Ric (u, v)
-
Ric (v,
u)).
-
Fp. 137 __T]] I I
p. 184
142
2.
Analysis
manifolds
on
Lemma 2.8.2. Let
Then F
1 ==
-
n
Proof 7',,M.
f E1,
Let
.
.
R(u, v)Ea
.
En I be
,
torsion-free
connection.
n
a
-R(E,,, u)v
=
a
nF(u, v)
Applications
to
+
R(Ea, v)u
(R (Ea, u) v
-w'
2.8.1
with
manifold
a
basis, f w 1, I be dual basis and u, v c the first Bianchi identity and the antisymmetry of R(., -) we
Rom
obtain
(M, V) be tr(R(., -)) holds.
-
(R(u, v)E,,)
Weyl
and therefore
R (Ea, =
v) u)
-tr(R(u, v)).
structures
The differential form F has is
a particularly geometric interpretation Weyl connection, justifying the following definition
a
Definition 2.8.3. Let Then
we
K(u, v)w
call F the
F(u, v)w
+
(M, (t, V)
length
be
if V
manifo Id with
a
curvature and
Weyl structure. K, defined by R(u, v)w
the directional curvature.
To motivate these terms coined
by Weyl9
we
will need the
following
lemma. Lemma 2.8.3. Let F
=
-Id p 2
and
(M, Q-', V) g(K(u, v)w, w)
Proof. Let U, V, W be We
can
also
be
assume
manifold with Weyl structure. Then for all vectors u, v, w and all g (E C
a
0
=
tensor fields with
that
[U, V]
=
U,,
=
u,
V
v, and
W.
W.
0.
g(VUVVW, W)
(17ug) (17VW1 W) O(UMVVW, W)
-
-
9(VVW' VUW) +VU (g(VVW, W))
gTVW, VUW)
+
VU
(2
V
-
g(w, W)
2('7Vg) (W, W)) -W(U) +
2U 2
0
V
0
( O(U)v
In the German
kriimmung-
(2IV
0
g(W, W)
g(W, W) -
2
are
1(17vg)(WW)) -g(VVW'VUW)
2
-
g(w, W)
original, they
-
+
vu(,P(V)g(w, W)) W(V)U g(w, W))
called
-
Streckenkriimmung
and
Richtungs-
2.9 Variation of
+
- O(U) P(VWW' W)
-
2
143
geodesics
g(VVW' VUW)
1(V UW) M 9 (WI W) IU*V*g(WW) 2 2
+
-
2 O(VUV)g(w, W). This
implies (using [U, V]
g(R(U, V)W, W)
0)
==
g(VUVVW -2
VVVUW, W)
-
((7UW) (V)
-
07V O) M) 9(W, W)
I -
-dw(U, V)g(W, W). 2
Since W is
arbitrary,
we
obtain
tr(R(U, V))
the first claim. The second claim follows
g(K(U, V)W, W)
=
=
g(R(U, V)W, W)
w
G
Ply
P
w;zz
loop w
is
-
g(- IdW(U, V) W, W)
=
0.
2
of the type
given
in Theorem 2.8.1.
approximately
w
+
abR(u, v)w
w
+
abF(u, v)w
ly
which proves
g(F(U, V)W, W)
-IdW(U, V)g(W, W) 2
TM and -y be a small Then the parallel transport Let
-
nd o(U, V)
from2
+
abK(u, v)w. EW-L
abF(u, v) is the factor by which the parallel transport (relative) length of w and that abK(u, v)w is the change of
It follows that I +
stretches the
parallel transport. Since the parallel transport length invariant it is impossible to compare lengths at different points. There is an important consequence to this fact, the so-called clock paradox of second kind (cf. Sect. 3.3). direction of
of
a
w
due to the
vector does not leave its relative
2.9 Variation of
geodesics
In this section we investigate the infinitesimal analogue of I-parameter families of geodesics. This section is technical and can be omitted on first reading. It is a prerequisitive for Sects. 4.5 and 4.6 in the chapter on pseudoRiemannian manifolds. Section 4.5 is concerned with metric preserving diffeomorphisms and used in the discussion of cosmological models (Chap. 6). Sect. 4.6 is necessary for understanding the complete proofs of the singularity theorems which are presented in Chap, 9.
144
In
2.
Analysis
manifolds
on
will see that freely falling particles can mathematically by (certain) geodesics. A cloud of such particles moving in spacetime corresponds then to a smooth 3-parameter family of geodesics. It is therefore of physical interest to study families of geodesics. Here we will study the slightly simpler sub-case of I-parameter families and its infinitesimal analogue. Let f : Z M be a differentiable map and recall that a (smooth) vector field along f is a smooth map X: Z ---> TM with IrTM 0 X(X) f (x) for all x G Z. We denote the space of vector fields along f by E' Q Any smooth vector field X on M induces a vector field k: x t--> X.,
Chap.
5
we
be described
--+
A vector field U
Xf (.,). f
via
on Z also induces a natural vector field. along f,,Ux. Important examples of this construction are given by fields along curves and by vector fields along canonical immersions
x
vector
-4
of submanifolds
(cf.
Sect.
Lemma 2.9. 1. Let
4.4).
Z
f:
M, U, V
-->
E
To'(Z),
and
X, Y
E
To' (f
f
Then V
UX
(U,60OXa
:=
Fbac(f*U)bXc),qa
+
is
a
well
defined
vector
field along f and satisfies the following properties. f
(i)
UX
V
is
function-linear
in U and R-linear in
f
(ii)
V U (WX)
=
f
(iii)
+ W
Vf UX for all functions W E C"O (Z);
f
V
Uf* V
Proof. We have
-
-
f* [U, V]
=
Tor (f* U,
f* V).
to show that the definition is invariant under coordinate
of M. Then f
Vf* U
V
transformations. Let
phism
dW (U) X
X;
0 be
we
a
diffeornorphism
of Z and
0 be
V)*U(x
V
'Y
+
O,3U-y(ajXaq0(0-1)5)00-1 (ayfba,3(0 -1)-Y)(0jO0UJ)XC 0-1
a
rb
0
C
((UOa Xa M
-->
M be
Of
UO*X
=
a
iC(OOfbUO)Xc) Oa) 00-1.
depend on the coordinates chosen diffeomeorphism. We obtain
( U,3 a
0
+
pdoa)
Fba (O0(0
0
e
0
0
for Z.
f Xd)
f)bUO) (ador
(U,3 p_adoa)
aa
+ ra b
Hence the formula does not
0:
diffeomor-
f-'r,
UX)
(,q
Let
a
obtain
fa
0
f xd
)aa
3feXd + (adoa)
0
f aoXd)
2.9 Variation of
b
rb,(ae (C -0 ) of 19Of 'U'O(Od0c) a
+
((adoa)
UO
0
Fba_('9e_0b)
+
f xd
e
0
f 19'8Xd
((,g-ad 0a)
+
f (adoc)
0
0
e
145
geodesics
)19.
f
f)aefeXd ) 0a
0
The GauBian basis vector field with
respect to the coordinates induced by 0 are given by 0.0a. Taking this into account we see that the Christoffel symbols transform as given in Corollary 2.6.1. This implies that our coordinate formula defines a well defined vector field along f. Equations I (i)-(iii) follow directly from our coordinate expression. If
f
is
lated
immersion then the covariant derivative
an
entirely
Lemma 2.9.2. Let
X, Y
E
along f
can
be calcu-
in M.
To' (f).
Let
f:
Z
0,
M be
--->
be vector
Tol (Z),
and
M which coincide with
f. U
immersion, U, V
an
fields
on
E
f
and X at all Points y X
f (x). Then
we
have V
UX
o
f
at all
E Z.
This lemma
justifies writing
this notation
extensively
instead of
Vf* UX
(VCj) of.
We will
use
in Sect. 4.4.
Proof of Lemma 2.9.2. Let x E Z. Since f is an immersion there exists n-dim(Z) of a neighbourhood of U of x, a neighbourhood V C K 0, and a local diffeomorphism F: U x V --4M with F(x, 0) f (x) J6r all x c U. X o F-'. We may extend U, X to U x V such that FU and X =
=
Then
we
obtain
('7 CJ .' )
-
,
o
F
=
-b
Ua,9a X
F
o
(1,b,&akc)
_
=OiF allia
a
(Xb F-1) o
o
i9iF alliajXbaa (F-1)i
Uja.Xb J Restricting the
for all
x
E
last
_
(_Vb
o
ac
o
-
F
(Tab, F) (F U)aXc (1-abc F)* (F U)aXc o
*
o
-
F) (F U)aXc.
expression
to U
(VCj9)
F(x, 0)
o
F
x
101 gives ==
(Vf UX)
x
I
U C Z.
Lemma 2.9.3. Let
the
F
o
'a
f:
Z
4
M, U, V
E
7-01 (Z),
and X
E
To' (f ).
equation
R(f*U, f*V)X holds
=Vf U Vf VX- Vf V Vf UX- Vf
[U, V]X
Then
146
2.
Analysis
The
Proof.
manifolds
on
equation follows directly from the definition of the Riemann
tensor.
I
Observe that
f
may not be
immersed submanifold. This is important for the
an
does not need to be
an
immersion and therefore
f (z) following
application. Definition 2.9. 1. A
geodesic
M, (s, t) geodesic.
M such that
f (s, t)
variation is
for
each
a
the
6, 6)
f
map
(a, b)
x
-4
f (s, t) is a We denote the velocity of the geodesics by ft : T(,,t) f (19t) and the deviation vector field along the geodesic f (s, -) by f, : T(,,t) f (a,) --+
E
s
curve
t
F-->
=:
=
.
Proposition
2.9. 1. Let
f : (s, t)
-4
f (s, t)
M be
a
geodesic
variation.
f
Then vector
ft satisfies the geodesic equation V field f, satisfies the equation
f v
f
at V 9t fs
+ R (f,
ft) ft
-
0 and the deviation
tft
f
(Vf at Tor) (ft f ,)
-
Tor (ft, V
at fs)
=
0.
Proof. The geodesic equation follows directly from the definition of geodesic variation. Observe that [f, ft] f. [,9s, at] [f. (0,), f. (at)] =
a
=
0 and that therefore =0
f
V
f
f
atV ath
=V
f
9, ft + V at [ft, f,] +
atV
=
R(ft, fs)ft+ f + Tor (ft, V
t7at (Tor(ft, f,))
0
Vf 0,Vf 9tft + (Vf ajor) (ft, f,)
at fs)
It is often sufficient to consider the infinitesimal variations. This
justifies
the
Definition 2.9.2. A Jacobi field is which
satisfies
the Jacobi
V V J + R(J,
)
analogue
of
geodesic
following definition. a
vector
J
along
Tor( ,
V J)
field
a
geodesic
-y
equation -
(V Tor)( , J)
-
=
0.
Proposition 2.9.2. Let -y: [a, b) ---> M be a geodesic. The Jacobi fields along -y span a 2n-dimensional linear space and any Jacobi field J along 7 is uniquely determined by J(a), V (a)J-
Proof. Without loss of generality we may consider a single chart which geodesic. The Jacobi equation reduces then to a system of
contains the
2.9 Variation of
n
second order differential
ordinary
equations,
or,
geodesics
equivalently,
147
to
a
sys-
first order differential equations. Hence the assertion follows from the fundamental theorem for ordinary differential equations
tem of 2n
(cf.
ordinary
2.4-1).
Theorem
1
Corollary 2.9. 1. Let -y: [a, b] --> M, t F-4 exp,, ((t a) u.,) be a geodesic. A vector field J along -y is a Jacobi field which vanishes at x -/(a) if and only if there is a vector vx E TM with J a) (ux + svx)).,9,. exp ((t -
=
=
It is clear
Proof.
that, given such a x. Proposition
vector vx, J is
-y which vanishes at
along
-y which vanish at
characterised
x
by their velocity vector
'7 ('a) exp ((t for all vx
TxM and TxM is
E
Definition 2.9.3.
geodesic
-y
joining
vanishes both at
Proposition there exists and
T,
2.9.2
points
and y and and y.
2.9.3.
an
expx:
x
Two
ux E
+
svx)). a,
=
vx
n-dimensional vector space.
an
x
along
the Jacobi fields
'7 (a)J.
a) (u.,
-
Jacobi field
a
implies that
n-dimensional vector space and are The assertion follows since
an
span
-
E
x, y
M
a non-zero
are
conjugate if there is a field J along -Y which
Jacobi
Two points x, y G M are conjugate if and only if TxM in the domain of exp such that exp(ux) y =
T,,.TxM
---
TyM fails
to have maximal rank.
M joining these Proof If x, y are conjugate, there is a geodesic -/: [0, 1] points and a Jacobi field J along -y which is non-zero but vanishes at x and y. Let ux be the uniquely determined vector which satisfies exp(tux) -y(t). By Corollary 2.9.1 there exists a vector vx E TxM \ 01 --
two
=
such that
J(t)
=
T(O,t) exp(t(ux
+
swx))(,9,).
The assumption
J(1)
0
=
that the linear map T(o,l) exp(t(ux + swx)): K2 --* TM does not have maximal rank which in turn implies that T". expx: T,,.TxM TyM does not have maximal rank.
implies
To prove the
by
the
converse
requirement
assertion,
that expx (ux)
0. The vector field
J(t)
expx (tux) is then
non-zero
a
=
=
we
just choose the
0 and
T.,,,,,
T(O,t) exp(t(ux
exp
+
((A
ds I ,=o
(ux
+
0
:
vectors ux, vx
svx))
swx))(i9,) along -y(t)
Jacobi field which vanishes at
x
and y. I
2.9.4. Let y: [a, bJ -- M be a geodesic without conjugate points. For every pair of vectors W-Y(a) E T-Y(a)M, 17v-y(b) IE T- (b)M there is a unique Jacobi field J along -y with J(a) ?-D-y(b) W-Y(a) and J(b)
Proposition
=
=
-
148
There
Proof. and
Analysis
2.
ii-y(b)
vectors
are
Ty(a)M with -y(t)
E
U-Y(a)
with
T-y(b)M
E
manifolds
on
-y(t)
exp.,(b)((b
=
-
=
expy(a) ((t
Oa-y(b))-
-
a)u_ (a))
Since -y does not
have conjugate points the linear maps
T(b-a),,,(, )
:
T(b-a)u,(,,.)T-y(a)M
---
T-y(b)M
expY(b):
T(b-a)i -,(b)T-y(b)M
-4
T-y(a)M
expY(a)
and
T(b-a)i,-,(,,) are
both
isomorphisms.
Hence there
vectors
are
and, -y(b)
V-y(a)
d
170-y(b)
T(b-a)u,(.)
exp_Y(a)
(b
W-y(a)
T(b-a)fi,(b)
eXPy(b)
(b
a)
-
ds ls=o
(U-y(a)
such that
SV-y(a)))
+
d
Let J,
Then
,
-
a)
ds I s=o
S' -y(b))
+
J2 be the Jacobi fields defined by J,
=
J2
=
JI vanishes
T(0,t) exp((t T(o, t) exp, ( (b -/(a)
at
-
-
a) (U-y(a)
+
8V7(a)))(a,9)7
t) Oi-y (b)
+
8 -y (b) ) ) (Os )
-
and has the value
has the value W-y(a) at -/(a) and vanishes at J, (t) + J2 (t) is well defined since 7rTm o J, (t)
'CV-y(b) at y(b) whereas J2 -y(b). Observe that the sum 7rTM 0 J2 (t) -y (t). By the
=
=
linearity of the a
(ft-y(b)
Jacobi equation the vector field J J, + J2 along -Y is also Jacobi field which has correct values at both 7(a) and -y(b). This proves
existence.
=
is clear since the space of solutions has dimension
Uniqueness
2n which is
just the dimension of the vector space
Jacobi fields
can
Proposition 2.9.5. field along t -4 -y(t)
=
Let =
E
x
M,
exp(tux)
Since
Let
f
is
(f,),,,=o
is
f: (s, t)
ft
=
f,
=
a
a
-4
expx (t (u +
T(,,t)f(,9t) T(,,t)f(a,)
geodesic
J(O)
=
=
TxM, and J be the Jacobi
=
0 and V
map in direction
expx(v[u])
Tu;,:
E
ux,vx
with
differential of the exponential
Proof.
I
also be used to calculate the differential of the exponenan n-dimensional "vertical" subspace of Tu.TM - !+ tvx) I t=0 E Tu. TM, where vx E Tx M. v[,,] dt (ux
tial map restricted to which consists of all
the
T-y(a)M (D T-y(b)M-
sv))
=
E
v[u)
J(O) is
==
v,
Then
given by
J(I).
M and
Tt(u+,,) expx((u + sv)[t(u+sv)])7 Tt(u+sv) expx((tv)[t(,,,+,,)I).
variation with
Jacobi vector field
f(s, 0)
along
=
x
Vs the vector field
-y which vanishes at 0. Rom
2.9 Variation of
f
[f,, ft] we
=
have
0
f,
we
obtain
V J
=
0 and
ft
Fas) Is=O To exp., ((u as
+
field J also satisfies
=V
=
geodesics
f
atf,
To exp. ((u
Tor(f,, ft). At t 0, sv) [o])), whence V 1 (0)
asft
=V +
+
v. The Jacobi To exp., ((v) [o]) sv) [o]) J(I) (,9,) T(0,1)f T(u) exp.,(vfu]). The ==
=
149
=
=
=
vector asser-
tion follows since initial value and derivative characterise Jacobi fields
uniquely.
I
Space and time from a global point of 3.
view
p. 137
The content
of this chapter is mainly physical. In Sect. 3. 1 we show experiment of Michelson and Morley indicates that spacetime admits a conformal structure. A conformal structure is not sufficient to describe spacetime adequately. In Sect. 3.2 we generalise the notion of inertial observers which leads to the existence of a projective structure. (One of Einstein's key observations was that this projective structure is closely linked to the phenomenon of gravity. This will be pursued in Chap. 5). In Sect. 3.3 we use our physical postulates in order to show that the conformal and the projective structures of spacetime form a Weyl structure. Here we closely follow (Ehlers, Pirani, and Schild 1972). The proofs in this section are technical and can be omitted without loss of continuity. In Sect. 2.8 we introduce a further physical postulate which restricts the Weyl structure to a Lorentzian manifold.
[I
that the
3.1
rays: the conformal structure
Light
In Sect. 1.4
field of
we
light
have
(cf.
cones
that spacetime is endowed with
seen
Postulate
1.4.1).
an
invariant
In
analogy to the discussion in light cones infinitesimally, i.e.,
the previous chapter we will define these in the tangent spaces rather than in spacetime itself. The discussion in
Chap.
I may
seem
to indicate that for each
x
E
M
the
tangential space TxM can be identified with (R', TI). Since for each non-degenerate bilinear form g., of signature (-, + +) there exist .
linear coordinates
(xo, x',
(i, j
11),
f 1,
E
.
.
.
,
Xn- 1) such that gx
may be
tempted
=
-
d
.
.
,
(x 0)2 + Jij dx'dxj
simply replace Minkowski spacetime (An, 71) by a general Lorentzian manifold (M, g) However, the Michelson-Morley experiment only determines the paths of light rays.1 In other words, from the Michelson-Morley experiment alone one can only infer the existence of a conformal structure (An', E,,)' where (t17 .
.
.
,
n
-
one
to
-
=
ff?277
:
0 E
COO(A
n ,
R+ \ 10 1) 1.
For the definition of space. This
wave
(In
length
to indicate that
Sect. 1. 4 we
we
used the affine structure
needed the Euclidean structure of
implicitly we used a Lorentzian metric rather than a conformal structure in order to interpret the Michelson-Morley experiment. However, the outcome of this experiment is a null-effect, i.e., , AZ ,z 0 which is independent of the Euclidean structure chosen. seems
M. Kriele: LNPm 59, pp. 151 - 169, 1999 © Springer-Verlag Berlin Heidelberg 1999
p.
1
156]
152
Space and
3.
time from
global point
a
of view
of A' to
single out a constant representative n E (1_1'77 This is not possible general manifolds.) Hence in a global setting the Michelson-Morley experiment leads to the following postulate. .
for
(Existence of a conformal structure). (Conformal) spacetime is a pair (M, Q ), where M is a n-dimensional manifo Id and (t a conformal structure of signature (-, + This
Postulate 3.1.1
We will call We will to
recover
to
it)
given by the paths of light
structure is
conformal
a
conformal structure of signature
rays.
(-,
+,
.
.
.
,
+)
Lorentzian.
that, given a Lorentzian conformal structure, it is possible light rays (cf. Postulate 3.1.2 below and the discussion leading see
-
(M, (t) be a manifold with Lorentzian conformal (or lightlike) hypersurface N is a hypersurface such (t) the induced metric on N is positive semi-definite
Definition 3.1.1. Let structure. A null
that
(for
any g E
but not positive Let
(M, ( )
be
definite. manifold with
a
a
Lorentzian conformal structure and
hypersurface. At each point x E N there exists a unique 1-dimensional subspace 1., C TM which is tangent to N and 0 for all v E lx, g E C For, if there where two such satisfies g(v, v) N C M be
null
a
=
vectors V1 i V2 E a
vector
E
w
impossible
TxN which
spanf vi V21 ,
Ux
E
TxN lx \ Ox I Vx I
of V and U in N
curves
Definition 3.1.2. Let
(M, (_")
structure and N C M
null
satisfies g( , geodesic
vector
null
not collinear then there would exist
g(w, w)
since g restricted to
vector fields with
integral
were
with
a
='O
are
the
be a conformal differential equation
a Proof.
First
respect
we
+
gad '9bgdc
null
geodesic
b c
C9dgbc 2
will show that the coordinate
representative
E
t But this is
and g c Q . Then -y
11 a.
expression
to coordinate transformations and with
g G
w
be a manifold with Lorentzian conformal hypersurface. A curve -/ in N whose velocity for every metric g E Q is called a conformal
Lemma 3.1.1. Let
satisfies
for all
< 0
positive semidefinite. If U, V are E lx \ jOx I for all x c N then the reparameterisations of each other. is
respect
is invariant with to
changes
C If V is the Levi-Civita connection of 9 then
7a
+
gad 19bgdc
'9dgbc 2
of the
we
have
3.1
whence
have
we
2f9ad
rays: the conformal structure
to show that for any other metric
only
Levi-Civita connection
e-
Light
t'
we
have
11
V
1Od(e 2f gbc)
Ob (e 2fgdc)
b c
2
e
2f
153
g E (E"with
But this follows from
==
149dgbc
gad abgdc
Y
2 0
+ 2abf
Let N be
dinates
a
null
a
ad
9
_
ad f gbcyb
hypersurface containing -y. We may chose for N such that 91 spans 1,, for each x.
c
coor-
(xl,...,x n-1 )
Since
the bilinear form gJspanJa2-.'a,,_11 is positive definite and 91TN does not have full rank, we have gii OVi E f 1, I I We may exn , =
-
.
.
.
-
tend the coordinate system to (xO.... I Xn- ') such that g (,9o, at N. In these coordinates we have (after normalising ) we
gad (abgdc
obtain
ordinates
we
tion follows from
proved
is
equivalently, adgll
or,
9igll
!gad (_adg1 1). 2
=
0, the lemma
9ad (_,g011) 11 (al)a
=
0 Vi G
f 1,
n
Since in
once we
a gad
have
-I
our co-
seen
-J1. This d
=
=
a, and that
equa-
I
-
0. Then \ fOJ be a vector with g(v, v) an (up reparameterisation) unique conformal null geodesic through x -y(O) with (O) v.
Corollary
3. 1. 1. Let
there exists -y
b c "adgbc) 2
have
91) =
v
E T,,M
=
to
=
=
Proof. By the fundamental
theorem for ODEs,
given
any function
A(t),
the differential equation
a has
a
+
gad '9bgdc
2
'9dgbc
b c
-
A a
=
0
unique solution for any v E TxM. It is easy to see that for any two A, the solution curves are identical up to a reparameterisa-
functions tion.
I
The
preceding corollary shows that at each point there is a unique con0 for all 9 E C geodesic in any direction Rv where g(v, v) This implies that there are exactly as many conformal null geodesics as there are light rays. In addition, it is easy to see that in the case of the Lorentzian conformal structure induced by Minkowski spacetime the light rays defined in Sect. 1.4 and the conformal null geodesics coincide. Hence we feel justified to link our infinitesimal Lorentzian conformal structure to light rays in spacetime by identifying them with conformal null geodesics. formal null
=
Postulate 3.1.2
formal
null
(Light rays).
geodesics of
The
its Lorentzian
light rays of spacetime conformal structure.
are
the
con-
154
3.
Space
and time from
a
global point
A Lorentzian conformal structure is all
causality. The following definition
is
of view
we
need in order to
straight
a
forward
investigate generalisation of
Definition 1.4.6. Definition 3.1.3. Let T be ture
(-, +,..., +),
a
Lorentzian conformal structure of signa-
, and A, U
g E
c
M.
A vector w is called spacelike, if g (w, w) > 0, timelike if (w, w) < g 0. The vector w is called 0, and lightlike (or null) if g(w, w) causal if it is timelike or lightlike. A vector field V is timelike (respectively, lightlike or null, causal, spacelike) if for each x E M the vector Vx is timelike (respectively,
(i)
=
lightlike
(ii) is
null, causal, spacelike). conformal structure global timelike vector field V. or
The Lorentzian a
(t is time orientable
Assume that (E is time orientable. A time orientation is lence class
if there equiva-
an
W if g(V, W, ) < 0 of timelike vector fields V where V at some point x E M. Let [V] be a time orientation of ( . A causal vector u is called future directed (respectively, past directed) if g(u, V) < 0 (respectively, -
g(u, V) > 0). (iii) A curve -y
is called spacelike (resp., timelike, lightlike, causal, directed, past directed) if all its velocity vectors are spacelike (resp., timelike, lightlike, causal, future directed, past directed). A timelike curve is often called a world line when one whishes to emphasise that it can represent the history of a (small) material object. (iv) The chronological future of a set A relative to U is
future
I+ (A, U)
x
The causal future
J+ (A, U)
There
are
Ix
=
M
cz
of
a
E M
I
a
future directed,
timelike
curve
-y C U
from A
causal
curve
-y C U
from A
to
xJ
to
xJ
set A relative to U is
3
a
future directed,
analogous definitions for
the
chronological past
l-
(A, U)
and the causal past J- (A, U) of A relative to U. If U M we omit the term "relative to M" and write T+(A), etc. If A fxJ is a =
==
single point
we
Definition 3.1.3 is
independent
Lemma 3.1.2. Let
I+ (x, U) is open.
write I+ (x,
x
E
U)
etc.
of the chosen
M and U be
an
representative
open
g E t
neighbourhood of x. Then
3.1
Let y E I+ (x,
Proof. a
(xo,
.
pact neighbourhood of line f from
straight
such that any
> 0
an a
supf0'(t)'Nt)) Hence these lines
.
G
VI
V n -/. Then the (coordinate) of V there exists
E
z
By compactness
straight line '
f'(t)
:
.
y, and let
to y is timelike.
z
from
z
with
Z(f, f') < a satisfies : f(t) G VI < 0.
lsupfg( (t)'i(t)) 2
<
all timelike in V. Since
are
155
and -y C be a timelike curve from x to y. Choose Xn- 1), let V C U be a (small enough) com,
U)
coordinate system
rays: the conformal structure
Light
fill
they
a
whole
neighbour-
hood of y the assertion is proven.
J+(x,U)
The set
will
we
I
does not need to be closed relative to U.
3.1.2. Let U be open and A be any subset
Corollary I+ (A U)
=
1+ (1+ (A U), U)
C
J+ (A U)
=
=
I+ (J+ (A U), U)
J+ (J+ (A U), U)
Proof. The only inclusions which
A:
a
causal
[1/2, 1]
and A is Fix
a
as
are
curve
-->
M from
metric g e
to
z
not obvious
are
a
I+ (J+ (A, U), U)
c
A and
E
y
--
x.
E a
timelike
[0, 1]
--
a
z
curve
M of 4
from y to x which is timelike near x. ,' and let U be a timelike vector field along -y which
causal
(piecewise)
a
J+ (1+ (A U), U)
=
M from y to z and The concatenation - :
[0, 1/2]
M:
of U.
-
I+ (A, U) and J+ (1+ (A, U), U) C I+ (A, U). Let x E I+ (J+ (A, U), U). Then there exist
M,
However,
later, J+ (x, U) is closed if U is chosen small enough.
see
curve
(1). There is an 6 > 0 such that g( (t), (t)) < -,E for R+ be a smooth function which satisfies (I c, 1). Let 0: [0, 11 all t C- [0, 1 for 0 and (t) > 0 6]. For any (small enough) 0(1) 0(0) V The curves t s > 0 let f (s, t) f (s, t) all connect expy (t) (s 0 (t) (t)). We denote derivatives with respect to t by -1(t). y with x and f (0, t) satisfies all t E
U(1)
=
-4
-
=
=
-
--+
=
=
a
dot and with respect to
(g(f (s, t), f (s, t)))'
=
=
s
by
a
prime. We calculate
Vfas(g(f (s, t), f (s, t))) 2g(Vf if, f) 2 g(V, 2g(Vf f', f) 2g(V (OV), =
=
(g(j(s, t), j(s, t)))'(O,t)
for all t
c] and the F-] for s > 0 small enough. Since we have g( (t), (t)) < -E < 0 for all t E (I -,E, 1] the curves also satisfy 6 / 2 < 0 for sufficiently small I s 1. This proves that g (j (s, t), f (s, t)) < we have obtained a timelike curve t f (so, t) from y to x. This curve may have a kink at the parameter value 1/2 where the original curve -Y Hence
we
curves
t
have
-4f (s, t)
are
timelike
on
[0,
1
< 0
E
[0,
1
-
-
-
1-4
passes
through
z.
Using
difficult to smooth out t this
curve
-4f (so, t)
near
t
a =
coordinate chart it is not
1/2
while preserving that
is timelike.
The inclusion
gously.
Lemma 2.1.7 and
J+(I+(A,U),U)
C
1+(A,U)
can
be shown analo-
I
156
Space and
3.
time from
Lemma. 3.1.3. 1+ (A) Lemma
Proof. By
I+(A)
is open, there is
1+(J+(A)) Let
an
I+(A). J+(A), -y
bourhood of in
endpoint
I+(A)
y
be
a
1-(x)
J+ (A)
C I+
U.CAI+(x)
=
is clear. Let E
of view
is open. The inclusion
int(J+(A)). Since int(J+(A)) int(J+(A)). Hence x E I+(y) C c-
x
n
(A).
=
E
x
int (J+ (A)),
=
3.1.2,
int(J+(A))
C
global point
a
causal
curve
from A to x, and.U be
a
neigh-
Since any small enough deformation of -y has future we can deform -y thereby obtaining a timelike curve from
x.
U,
A to u.
I
Definition 3.1.4. Let
M, and
U be
C:(U)
an
fy
=
E
open
(M, Q )
be
a
Lorentzian
neighbourhood of x.
M:
3
a
conformal structure,
x
E
We call
future directed conformal null geodesic -/ C U
from
x
to
yj
the
integrated future light cone of x relative to U. There are analogous definitions for the integrated past light cone and the integrated light cone. If U M, we omit the term "relative to M and write Cx+, Cx-, and Q, "
=
(M, Q )
3. 1. 1. Let
Proposition
mal structure. Then each
x
(i) C (U) \ jxj sn-2
manifold with Lorentzian conforan open neighbourhood U diffeo-
is
a
smooth
hypersurface which
is
diffeomorphic
to
R,
x
(ii) C: (U) n I+ (x, U) i) CX+ (U) C I+ (X, U)
=
-'2
a
has
to Rn such that
morphic
(P.
be
E M
0,
=
J+ (X, U).
P. 158
3.1.1 will be
Proposition
a
corollary
to Lemma 3.1.4 below which is
result from Lorentzian geometry. Choose a representative g E V_' and denote the associated Levi-Civita connection by V. The exponential map a
allows x
us
to
E M with
space at
identify
the causal structure of
the causal structure of
an
open,
a
convex
convex
neighbourhood
set of the
of
tangent
x.
Lemma 3.1.4. Let
(M, (t)
be
Manifold with Lorentzian conformal strucneighbourhood of x E M with respect to
ture, g (=- ( , and U be a convex the Levi-Civita connection of g.
(i)
y E
where
I+ (x, U) v
a
(ii) J+ (X, U) The
(respectively J+ (x, U)) if and only if y timelike (resp., causal) vector. I+ (X, U),
=
expx (v)
future pointing =
complete proof of Proposition 3.1.1 requires the material from page immediately after the proof of Proposition 2.6.4 up to the end of Sect.
129 2.6.
3.1
g(u, u)
decomposed
vOu +
and
vo
scalar
as v
vi- here vi- E
T,,M. For
on
h(v, w)
=
>
E
every
u'
v
g(v, w) 0 let
TM
E
+
B,(O.,)
be
be
is
fv
=
01
=
Euclidean
a
E
"
M
obviously a neighbourhood of 0., in T.,M. similar argument as in the proof of Theorem 2.6.2 shows that U, expo.. (B, (0,,)) is a convex neighbourhood of x for small enough E.
h(v, v)
Ej.
<
This set is
The set S,
fv
=
T., M
G
submanifold of TXM which is
g (v,
:
v)
0, g (u, v)
=
diffeomorphic
to the
(n
=
-
-
-,,/-E/-21
I
a
uniquely
g(u, v)
:
2vowo
157
E T,,M
U
can
=
bilinear form
E R. The
product
=
Choose any g E Q and let -1. Each vector V E TM
3. 1. 1.
Proof of Proposition timelike vector with
rays: the conformal structure
Light
A
is
a
2)-dimensional
sphere Sn-2 fz c Rn-1: rn-l(Zi)2 I I and which lies in the boundary of B, (Ox). The map (0, 1) x S, --+ M, (t, v) i- expx (tv) is a parameterisation of Cx+(U,) \ jxj which proves the first claim (i). Since the map exp,,: B,(Ox) --> U, is a diffeomorphism assertion (ii)
round
=
follows from Lemma 3.1.4 and the fact that every causal vector W E B,(Ox)q is either timelike or null. Assertion (iii) is a trivial consequence I of Lemma 3.1.4 (ii).
Proof of Lemma 3.1-4. (i): We prove the statement for 1+ (x, U) (the proof for J+ (x, U) is analogous). The exponential map is a diffeomorphism of a neighbourhood 0 of Ox E TxM onto U. For any geodesic -y we obtain V (g( , ) 2g(V , ) 0 whence the velocity vectors of =
=
their causal class. It follows that exp, maps geodesics timelike vectors into T+(x,U). We have to show that for each point
do not
y E 1+ (x,
U)
the vector
The double
cone
into 3 connected vectors
the U
change
3V E
Ox
=
v :
E T.,M is
g(v, v)
necessarily timelike. 01 divides 0 \ Ox
=
components: the future and past full
(Ox,+, Ox',-)
diffeomorphism
:
(exp,,) (y) fv E 1
with
and the set of expx: z
=
0
--+
U
exp., (v)
spacelike
we
see
vectors
cones
of timelike
(C-.',').
that the set
Applying
Cx(U)
divides U into the sets
CxO,
jz
E
CxO,
-T+ (x, U) there is a timelike curve co's, respectively. For every y which connects 1--4 t x and y. From g( (a), (a)) < 0 U, -y -y: [a, b] we know that -y must initially enter Cx,+. If the assertion does not hold X
then y E Cx+ (U) U Cx,'. Since 1+ (x, U) is open we can assume without loss of generality that y E Cx,'. Hence -y must intersect Cx at some point
7(to)
where -y leaves
Cxo,+.
Since
(to)
is timelike and future directed it is
and points into C',+. But this is to the construction of the point -y(to).
transverse to
For
(ii)
Cx
at
-y(to)
a
contradiction
it is sufficient to note that the set of causal vectors in
is the closure of the set of timelike vectors in
TxM.
TxM I
158
p. 3.56 p.
1
159]
Space and
3.
Since
a
relative one
time from
a
global point
Lorentzian conformal structure allows to
lengths (at
given point),
a
point. We just loose
So far
we
only
absolute calibration.
an
have considered
structure further
we
angles and spatial geometry at
measure
it is sufficient for
3.2 Inertial observers: the
our
of view
projective
light propagation.
structure In order to
specialise
must take into account other fundamental prop-
erties of nature. In
Chap. I inertial observers have played an important they are not subject to any physical forces one can physically implement inertial observers (or particles) by freely falling observers (or particles). We will use freely falling or inertial observers as the other input into our theory besides the Lorentzian conformal structure induced by light rays. The following postulate reflects that the movement of an inertial observer depends on his/her initial velocity and initial position. This is
theoretical r6le. Since
the main content of Galilei's law of inertia.
(Existence
Postulate 3.2.1
of inertial
observers). Through
any
for any timelike direction Ru there exists (up to point (M, g) and extension) exactly one inertial observer -Y: R -- M parameterisation x
and
E
which passes
through
Postulate 3.2.1
with
x
singles out
a
11
velocity
Ru.
collection of paths in spacetime. In Minkows-
ki space, inertial observers move along straight, timelike lines. This is again a global characterisation which we need to overcome by formuit
lating
i.e. in the
infinitesimally,
tangent bundle rather than in
space-
time itself. An infinitesimal
description of (unparameterised) inertial observers in Minkowski space is that their spatial acceleration vanishes. This property can be generalised as follows.
(Law
Postulate 3.2.2
(UxWx)
chart
of
coordinate system
we
inertia).
dt2 The chart maps W,,: Ux
x.
x
E
M there exists
a
x
d 27 a
at
For each
such that with respect to the corresponding have for all inertial observers passing through x
centered at
-->
R'
11 a. depend smoothly
on
the parameter
X.
its very formulation Postulate 3.2.2 is independent of the chocharts, we need to express it in an arbitrary coordinate system. In
While sen
by
Sect. 2.6
we
dependent ever,
have
seen
that the derivative of vector fields is coordinate
and therefore not well defined in
we can use
Theorem 2.6.1 and
a
Corollary
general
manifold. How-
2.6.1 in order to define
a
3.2 Inertial observers: the
projective
connection with respect to which Postulate 3.2.2
manifestly
coordinate free
can
159
structure
be formulated in
a
manner.
Let (U, ( ) be any chart and x E U. We define the Christoffel Fb', with respect to this chart at x by
symbols
192Xh
ap a
Fbc where
(x
x')
are
19xhq; bCq;; C'
the coordinates with respect to the chart
(Ux, ( x)
provided by Postulate 3.2.2. We will now prove that this construction gives a well defined connection 17. Let (1 ,O) be a second chart with x E
1.
Then
definitions
have
we
the
give
Pba,
same
a-: '
We need to show that these two
-
5X IT a93 a.-F
connection.
Indeed,
we
calculate
a2Xh
9. a a
Fb
C
axh q.,t-b,% c aj a a, k
axh 9.: l
a
TX-1 5
&7& axh qj b Cqj a
lq': k
2X h
9; k
9, l 0, m
5Xh
9., b
a., l 9._ m
a, _k
X,' 5y -57X-b k
which is
exactly
Thus
have
can
we
Given
a
q.;
,
9, ,- b
c
_5X__1,qj ca; b
^k
rim,
,
the transformation formula
provided by Corollary
2.6. 1.
as
follows. There is
a
connection V such that inertial
pregeodesics.
are our
3.2.2 holds. In
I(Vxi Ox)JxEM
of
compatible fact, Corollary
observers, 1A, Mj.,EM
is not the P,
charts such that the formula in Postulate 2.6.3
implies
that
exactly
those collections
which induce torsion-free connections that have the
as
V
are
also
possible choices.
This
implies
the
same
following
Corollary 3.2.1. Postulates 3.2.1, 3.2.2 determine a,projective structure 113 such that each particle is a pregeodesics with respect to any 17 E 93. Weyl characterised the connection as a field which forces a particle to be transported parallelly with itself in space and time. Thus we have arrived at a geometrical explanation for the law of inertia postulated by Galilei. 3
In order to understand the about
projective
PI
collection of inertial
only collection
pregeodesics corollary.
a,7 m -
X, k
92,;J
well defined torsion-free connection r and Postulate 3.2.2
be restated
observers
iqj a +
Yxh
+
a2; l
-5X---1 O' ca;-b
a2Xh
qj a
-57X_1 5_X _O b
axh +
corollary below
structures.
we
need to know
a
little bit
more
[I
1- 5,811
p.-127 129 P.
1
161]
160
Sp- ce ;---und
3.
3.3
O'nie firoin
a
global point
Compatibility: Weyl
We have obtained
of vif,- w
structure
Lorentzian conformal and
a projective structure but geometrical structures is still unspecified. We will now introduce a further postulate which links observers and light rays and therefore these two structures. It is an experimental fact that one can chase light rays with material observers arbitrarily closely, provided one uses enough energy. The following is a formalisation of this
the
relationship
a
of these two
idea. Postulate 3.3.1 Each
x
(Compatibility
M has
E
neighbourhood
a
with the causal U such that
for
structure). \ JxJ
all y E U
we
have y
=
As
-y(t) for
a
rays
an
inertial observer -y
through
--* y G
x
1+ (x, U) U I- (x, U).
first consequence of this compatibility axiom we can determine light connection instead of the Lorentzian conformal structure.
using the
Lemma 3.3.1.
geodesics
which
Let
The are
conformal null geodesics coincide with those lightlike.
pre-
somewhere
E M and let U be the intersection of the neighbourhoods of provided by Proposition 3.1.1 and Postulate 3.3.1. Let p be a conformal null geodesic from x to some fixed point y E U. Proposition 3.1.1 implies that y lies in the boundary of 1+(x,U). Hence there is a sequence of points yi E 1+ (x, U) which converges to y. For each i let 7i: [0, 1] F--4 U be the pregeodesic which corresponds to the inertial
Proof
x
which
x
are
observer which
moves
from
x
to yi
(cf.
Postulate
3.3.1).
Let
v
\ fOJ
be
accumulation point of the (bounded) sequence i(O) E T,,M and let v. By the continuous dependence of -y be the pregeodesic with (O) solutions of differential equations on initial conditions and parameters an
(cf. Theorem 2.4.1) there are for each point -y(s) (s E [0, 1]) and each neighbourhood V of -y(s) infinitely many pregeodesics -yi which intersect V and whose velocity vectors at s converge to :y(s). This implies that -y is causal and that -y C I+ (x,U). Since yi y and T! is compact the pregeodesic -y reaches y. The inclusion J+(I+(.T,U),U) T+(x,U) and y c -y n C: (U) imply -y c 1+ (x, U) \ 1+ (x, U) Cx+ (U). Since Cx+ is a null hypersurface and -y is lightlike (causal but not timelike) -Y must be a conformal null geodesic. That -y coincides with M follows now from the uniqueness of conformal null geodesics. For the converse we simply need to note that both pregeodesics which have an initial lightlike velocity vector and conformal null geodesics are uniquely determined by initial point x and initial velocity direction -*
=
=
R (O).
I
In the rest of this section existence of
a
natural
we
Weyl
will show that
structure.
our
postulates imply
the
3.3
Compatibility: Weyl
161
structure
a Lorentzian conformal structure and 93 be projective structure such that the Postulates 3.1.1, 3.2.1, 3.2.2, and 3.3.1 are satisfied. Then there exists a unique V E 93 such that for all I-P o (9 g. g E Q-" there is a one-form o with Vg
Theorem 3.3.1. Let (t be a
=
The
into several lemmas.
proof of this theorem will be split
Lemma 3.3.2. Let g be a Lorentzian metric and ' abc ': (ab,) be a 0 for all null vectors v. totally symmetric tensor such that zA (v, v, v) 7--
=
Then there exists
one-form V such that A(abc)
a
79(agbe)
':--
-
Proof. sym(d 0 g) clearly satisfies the condition of the lemma. We will verify that this is the only possible choice. 1. Then -1 and e E t--L with g(e, e) Let t be a vector with g(t, t)
now
=
=
t
e are
, Ab (t we
null vectors and from
e, t
e)
e, t
=
3tatb e'
A(abc) (tatbtc
+ 3te bec
e
aeb e')
=
0
obtain 0
' A(abc) (tatbtc
0
' A(abc) (3 tatb ec
3taebec), b + ea C).
(3.3.1)
+
e
(3.3.2)
e
Setting
(3,A(abc) tatb + 2A(t, t7 t)tdgdc)
19C
Equation (3.3.2) A (e, e,
e)
is
=
equivalent
-
3 GA (abc) tatb
Analogously, Equation (3.3.1) A
Finally,
(abc)
to
ea,btc
=
t9(e)
=
equivalent
g(e, e)?9(e)
Ve E
3
t-L.
to
1
A(abc) tatbtc 3
79(t)
=
9(ab?9c)
a
ebtc.
implies 1
1 =
=
_I
the definition of 79
A(abc) tatbec
is
c)
_
3
'0 (e)
=
3
9(t) 019(6)
=
9(abl9c) tatbec.
and
IA(abc) tatbtc
=
g(t, t)79(t).
By the polarisation identity for symmetric 3-tensors, we know that A(ab,) 1 coincides with 9(a0c) on a basis and our claim is proved.
4
The
proof of
Theorem 3.3.1 is rather technical. It
loss of continuity.
can
be omitted without
164
Proof.
3.
Space
Let
f Eo,
and time from
.
.
.
E,,- 1 1 be
,
be the dual basis. We
can
global point
a
an
orthonormal basis and
write L
La bc &
=:
La for all indices a, b, c. We consider .b COS (0) EA + sin (0) EB , where A -7 + Eo
2
L (N,
N)
=
=
-1(1 2
cos(20)),
-
L (Eo,
Eo)
c
(9
lightlike
(9
f wo,
Ea,
vectors
n-
1
.
where La bc
given by
B and 0 E
N
=
In order to as a
-1 2
Fourier
sin(20),
calculate
we
COS2 (0) L (EA EA)
+
Wb
[0, 21r]. c(N)N, we expand L(N, N) -1 + cos(20)), sin 0 COS 0 2 (1
exploit the condition L(N, N) polynomial in 0. Using COS2 0 and sin 0
of view
,
+ 2
cos(O)L(Eo, EA)
+ 2
sin(O) cos(O)L(EA, EB)
+
sin2(0) L (EB EB) I
sin(O)L(Eo, EB)
+ 2
I =
L(Eo, Eo)
+ 2
(L(EA, EA)
2L(Eo, EA) cos(O)
+
I(L(EA, EA)
+
-
2 =
for
some
c(O) (Eo
function
order 2 which
+
right
> 3.
cos(O)EA
we can
c
i3ABEA
+ 2
+
write
L(EA, EB) sin(20)
sin(O)EB)
side
c(O)
+
Fourier
polynomial of polynomial of order < 1 would be a Fourier polynomial of aAB + )3AB COS 0 + 'YAB sin 0. The a
a
Fourier
=
given by
1
aABEO
+
must be
implies right hand
hand side is then
2L(Eo, EB) sin(O)
The left hand side is
c(O).
that
Hence
L(EB, EB))
L(EB, EB)) cos(20)
since otherwise the
order
+
+
+
1^YABEB + (OABEO + aABEA)
COS
0
2
('YAB EO
+ 01AB EB) sin 0 +
cos(20) (2IOABEA (17ABEA 113ABEB) sin(20). 7ABEB
2
+
+
2
2
A comparison of coefficients gives
L(Eo, Eo)
+
I(L(EA, EA) + L(EB, EB))
2
aABEO
+
-
3ABEA
-7ABEB,
+
2
EA)
=
)3AB E0
+ CVAB EA ,
2L(E0, EB)
=
'YABEO
+
L(EB, EB)
=
L(EA, EB)
=
2L (E0,
L(EA, EA)
-
2
CVABEB
,
3ABEA -YABEB7
I'YABEA + 2I 3ABEB-
2
(3.3.4)
(3.3.5) (3.3.6)
(3.3.7) (3.3.8)
3.3
We obtain
Compatibility: Weyl
165
structure
linear system of equations for the components L' (b < c) bc c (g Wb (9 0 Ea. From Equation 3.3.5 we obtain immediately
a
-
of L
L'(bc)
=
LB OA =Of6rBVfO,Aj,
LOOA
A
L OA
OA B,
2
=
(3.3.9)
-aAB.
2
Further, neither OAB nor aAB can depend on B since the left hand sides Equations (3.3.9) are independent of EB. Equation 3.3.6 implies in addition that LB OB - aAB. Hence aAB cannot depend on A either. We will therefore write Ao := aAB and AA 13AB. Equation 3.3.8 implies LoAB 0 for A 4 B and LC for 0 pairwise different A, B, C. We AB of
=
=
=
also obtain
I
A
LAB
=
B
'YAB and
LAB
'AA.
=
These two equations are be used to eliminate
consistent if 'YAB AB. Equation 3.3.7 L (EB I EB) in Equation 3.3.4 resulting in
only
=
L(Eo, Eo)
+
L(EA, EA)
CeABEO
=
IOABEA + 1^YABEB + 1OABEA 2 2
2
2
implies LOOO
This equation
+
can
7ABEB
==
AOEO
+
AAEA.
Ao which is independent of A. Hence B yo := -LoAA, We also have Loo B A A -L AA for B =h A and L A If +LA set L all we coefficients AA. /.t 00 AA 00 are determined. It is now straightforward to check that La ja A ) + (b (be) /_,a gbc. Conversely, this tensor indeed has all the properties listed in the LoAA
LoBB for A =h B
=
+
LAA
=
set
we can
=
=
,
-
c
lemma.
Corollary sor
for all
L(., .): T,,M x .,M --4T.,M be a symmetric L(N, N) 11 N for all null vectors N and g(L(v, v), v)
3.3.1. Let
such that vectors
v.
Then
for
each g E Q
there exists
a
1-form
ten=
0
A such that
I
L(v, w) where
Proof.
/V
is
=
2
defined by A(v)
This follows
condition
(/\(v)w
+
/\(w)v)
=
=
g(v, w)1V
g(A0, v) for
immediately
g(L(v, v)v)
-
all
v
E
Vv, w
E
TM,
T ,M.
from Lemma 3.3.6 and the additional
0 for all vectors
v.
We are now ready to prove the main result of this section. Proof of Theorem 3.3.1. We will first show that there exists a unique V E T such that for each representative g E (t there is a one-form W
with
(VVW)
a =
VbabWa
+
gad(' (abgdc 2
+
acgbd
-
lydgbc)
166
Space and
3.
time from
a
global point
I
+
Wd9bc
2
-
of view
gd(00c) vbwc.
(3.3.10)
Recall the formula for A the
assumptions
La(bc)
ga(bk)
::::
(n + 1) Ab we
provided by Lemma 3.3-5. Since L'bc satisfies Corollary 3.3.1 there exists a one-form A such that 0 implies 0 /Nagbc. The property Laba ga'La(bc)
of
=
-1
Ab
-
(n
2
1) Ab
-
whence A
,
0 and therefore
=
0. Hence
have I -
1
-
abc
2
(19b9ac
19c9ba
+
Aa
aagbc)
-
bc
I 2
An
application
such that 0 From
=
-
gd(b(Pc) +
of Lemma 2.6.3
implies that there Equation (3.3.10).
0. This proves
Equation (3.3.10)
Vagbc
Odgbc
aagbc
a
unique V
G
q3
obtain
we
d c
d
1 abgdc
-
is
ga(bOc)
_
racgdb
I
19agbc
Pagcb
-
2
19agbc
2
+
+
19c9ab
19b9ac -
-
ac9ab)
ab9ac 2
Ocgab + gc(a(Pb)
2
(Pb9ac +
gb(a(Pc)
Wagbc-
Hence
(M, (t, V)
is
Weyl
a
structure and the theorem is
1
proved.
3.4 Reduction to the Lorentzian structure P. 161
[I
P.
1
169] There
are
experimental
facts which indicate that
ture has features which have
therefore necessary to
seems
no
specify
the
a
general Weyl
struc-
our
actual universe. It
geometrical
structure of space-
counterpart in
time further.
It is
plausible
of
2.7.4)
an
to
identify
The freedom t
server. zero on
an
affine parameterisation (cf. Definition a standard clock carried by the ob-
inertial observer with --*
at + b
corresponds
to the freedom to choose the
the time axis and to choose the unit in which time is measured.
An atomic clock
roughly
be found in textbooks
on
works at follows
(detail
of this mechanism
Quantum mechanics). Each
atom has
a
can
charac-
teristic minimal energy E which it can absorb. A very short while after the absorption of such a package of energy the atom will emit a photon
whose
by
E
can
frequency
=
v
with respect to the rest frame of the atom is given frequency is characteristic for each sort of atom it
hv. Since this
be used to build
a
clock.
3.4 Reduction to the Lorentzian structure
In Sect. 1.4.3
we
have
167
how to calculate E from the world lines
seen
of the atom and the
photon in the context of special relativity. Unfortunately, we cannot simply apply this calculation here since the number E did depend on the Minkowski metric n and not merely on its conformal class. But it is very suggestive to identify this atomic clock with the standard clock t given by the affine parameter of the world line of the atom.
We will now see that this identification gives rise to a global effect which has not been observed. Let x, y E M, y E 1+ (x) be two events and consider two atoms of the same kind which are moving from x to y
along different paths
(cf. Fig. 3.4.1). Y
'Y1
(al)
'Y2
-yi:
We will
[0) 011
assume
M in spacetime Mi 'Y2: [07 Ce2l they move initially along the same --+
that
^/2 (Ce2)
'Y1
Fig.
3.4-1. The world lines from
atoms which X
path
in
there is assume
'Y1
(0)
^f2 (0)
aA
x
initially and finally
to y of two at rest with
to each other
spacetime and that their clocks such that
1 (t) that just before reaching an e
> 0
For these observers
10
respect
are
we
V":YA (t) 9 ( A (t)
y
are initially calibrated at x, i.e., 2 (t) for all t E [0, 6). We will also they are again moving side by side.
obtain
7
A (t))
aA -,
9(
=
A (t)) A (t))
A
(t)9) (:YA (t)
A (t))
i
-dt
;-
JO
9( aA
+2
fo
g
A (t)
(V' /A (t) 9(
A (t)) A (t)
7
A (t))
A (t)) A (t))
aA
( A (t)) dt. 0
7
dt
168
Space- and
3.
Since the In
time from
frequencies
(g( , (a,),
were
, (a,)))
-
-
global point
a
initially equal
of
view
obtain
we
In
(9( 2 (a2)
in
(g ( l (al), 1 (a,)))
+In
,
2 (a2)))
(g( 2(0), 2(0)))
-
In
-In
(g ffi (0), 1 (0)))
(9( 2(Ce2)i 2(Ce2ffl
Let
Q be any 2-surface which is bounded by the curves -yl, plication of the theorem of Stokes (Theorem 2.5.5)5 gives
(g ( , (a 1),
In
, (a 1)))
In
(9 ( 2 (a2),
-y2. An ap-
2 (a2)))
0"
f ( i (t)) J dW w
==
=:
J
Hence the parameterisation of both the
curves
of both atoms
_
f012
W(
2 (t)) dt
0
F
-2
C Q
tification
dt
Q
and therefore
-
by
our
iden-
different at y, even though they were the same at x. As a consequence, the frequency of an atom clock would depend of the history of the atoms constituting it. This does not
-
frequencies
to be the
seem
Moreover, the spectrum of far away stars is aphistory of the atoms which constitute these 0. conclude that dW case.
parently independent stars.' Hence
we
Postulate 3.4.1 F
are
of the
=:
(No
second clock
effect).
The
length
curvature
1
=
-
of spacetime vanishes identically. 2 dW
Notice that the tation than the
justification of justification of
Postulate 3.4.1
requires
more
interpre-
other axioms. One may argue that it is the weakest link in the chain of arguments which leadsto general our
relativity. Corollary 3.4. 1. Assume that Postulates 3.1.1, 3.2.1, 3.2.2, 3.4.1 hold. Then spacetime is a Lorentzian manifold (M, g). Proof. This follows immediately We have
3.3. 1, and
from Theorem 2.7.2.
arrived at a geometrical structure which gives the framedescription of space and time. The arguments which lead to Corollary 3.4.1 may seem so compelling that the reader could ask herself or himself why we started with Newton's theory of spacetime work for
The
a
gist of our argument
is that the difference of the
frequencies at -Y1 (al) paths -yi, 'Y2 unless W can be chosen to vanish should be plausible even without appealing to the theorem of Stokes. It is needed for a strict proof though. This argument does not depend on the identification of atomic clocks with
'Y2(Ce2) 6
now
is
non-zero.
That this is the
case
the affine parameter of their world lines.
for suitable
=
3.4 Reduction to the Lorentzian structure
169
instead of
motivating our postulates directly. In fact, from a purely conceptional point of view it is, advantageous to analyze the measurement of space and time relations and to use this analysis in order to arrive at a Lorentz structure. This program has been carried out by Ehlers, Pirani, and Schild (1972) who also arrive at a Weyl structure and reduce it to
a
Lorentz structure via Postulate 3.4.1. With
our
preparation this
highly readable and certainly recommended to physicists who are interested in the operational approach. We have used a more historic approach for two reasons. Firstly, most readers are familiar with the classical description of spacetime (albeit less formalised, perhaps). Secondly, Newton's theory is also very compelling on first sight. So are Galilei's theory and the special theory of relativity. What is more, when these theories where still young and generally accepted it was very difficult to see how to improve them. In fact, most physicists and philosophers would have claimed that these theories are correct in an absolute way. There is no doubt that the Lorentzian description of spacetime will not be the last improvement either. It is even a prominent topic of current research to try to incorporate general relativity into a new general quantum theory of spacetime. It is generally believed that this new theory will be qualitatively very different from the geometrical theory we have presented here. The reader should recall that we started with macroscopic properties of rays of light which do not take into account the quantum nature of light. Also, we have always assumed that space and time are continuous rather than discrete. Hence there are several points where our theory of spacetime may prove inadequate. It must also be said, however, that a conceptionally satisfying theory of quantum gravity article is
does not yet exist. Our description of
and to date there is
no
of spacetime betten
spacetime is much better than previous theories theory which describes the global properties
other
[p. I
166
1]
P. 171
4. Pseudo-Riemannian manifolds
P.
We have learned that spacetime can be described by a Lorentzian manifold. In this section we will investigate the slightly more general case
of pseudo-Riemannian manifolds in detail. The development of the theory of spacetime will be continued in Chap. 5 where we motivate Einstein's equation, the central equation in general relativity which links matter to gravitation. Readers who wish to get to Einstein's equation quickly may skip most of Chap. 4. They only need to read Definition 4.2.2 and Sect. 4.3 up to and including corollary 4.3. 1. For mathematicians, this chapter contains the essentials of (pseudo)-Riemannian differential geometry. Almost everything we present here will be used in the following physically motivated sections. Prerequisitives of this chapter: Sect. 2.7.1 and Sect. 2.8 (up to but not including Lemma 2.8.2). Recall that
pseudo-Riemannian manifold (M, g) consists of an n-dimena symmetric, everywhere non-degenerate 20)tensor field g. We will often denote g by (., .). A pseudo-Riemannian (M, g) is called a Riemannian manifold (respectively, Lorentzian manifold) if g has signature is (+, +) (respectively, (-, +, +)). The simplest example of a Riemannian manifold is Euclidean Space, (Rn, d(xl )2 + .+ d(xn)2) and the simplest example of a Lorentzian manifold is Minkowski spacetime, (R', -d(x 0)2 + d(xl )2 + + d(xn-1)2). In this book, we are especially interested in Lorentzian manifolds a
sional manifold M and
(
.
.
.
,
-
.
.
,
.
.
,
...
mathematical models of spacetime. Riemannian submanifolds (cf. 4.4) of codimension I can be thought of as instants of time. They
as
Sect. will
play
an
important r6le when
we
discuss the initial value
in Sect. 5.4. Pseudo-Riemannian manifolds which nor
Riemannian
are
rarely applied
in
are
physics. However,
at any additional cost to widen the discussion to this
Unless
it does not
come
general case. geometrical objects are undermore
explicitly stated otherwise all from the metric and the Levi-Civita
stood to be derived
Remark 4. 0. 1. The
problem
neither Lorentzian
investigation of hypersurfaces
connection.
in Euclidean space has
tradition in mathematics and it has led to many important (mathematical) developments. (Pseudo)-Riemannian manifolds are the
a
very
long
We
only collect those facts which are essential to an understanding of Einstein's equation which will be presented in the next chapter.
M. Kriele: LNPm 59, pp. 171 - 254, 1999 © Springer-Verlag Berlin Heidelberg 1999
Pseudo-Riemannian manifolds
4.
172
of these
hypersurfaces and therefore of indepenwe will not push this angle, readinterested in differential ers primarily learning geometry should keep in mind the following example of a Riemannian manifold. Let M C R' be a hypersurface and consider for each x E M the R' tangent space TxM as a subspace of R'. To be concrete, let t: M be the natural inclusion and identify T,,M with t.TxM C T,(x)R' Pz R'. and define the We denote the standard scalar product of R' by Riemannian metric g of M by
natural
generalisation
dent interest to mathematicians. While
--->
g(V7 W)
:`
(t*Vl "W)R11
for all vectors v, w E TM. While this class of examples is rather
sualising
simple, it is sufficient for viimportant features of Riemannian manifolds. Whereas
most
Euclidean space is trivial in the sense that d(xl)2 + + d(Xn)2 is a constant tensor field with respect to appropriate coordinates, g is non-
constant in
-
-
and the curvature of its Levi-Civita connection does
general
not vanish.
given by the sphere by t: S2 -4 R 3, x f--> x the canonical inclusion. In this case we have T,,M ly E Rn 01. We can parameterise (a dense open subset of) the sphere (XI Y)R3
Example 4. 0. 1.
S2
=
fx
E
R3
The
simplest, non-trivial example
(Xl)2 + (X2)2 + (X3)2
:
11
=:
C R
3
.
is
Denote
=
=
using the chart (U, W) where
Cos 0 1
W_ (01 0)
sin
=
Cos
0
0 cos 0
.
sin 0
Let
f El, E21
be the standard orthonormal basis of R 2. From
-
-1
'9
( o-').Ej
490
-
=
-
cos
sin 0
sin
sin 0
ao Cos
-
=
0
0 cos 0 0 Cos 0
sin
Cos
0
we
obtain goo
=
COS2 0. We could now use 0, goo 1, goW g(ao, ao) the Levi-Civita concalculate to (Equation (2.7.7)) =
=
=
the Koszul formula nection and
given
we
through the formula study general submanifolds calculating these quantities.
could determine the Riemann tensor
in Theorem 2.8.1. In Sect. 4.4
and present better
techniques
for
we
will
A submanifold of Minkowski space does not necessarily inherit a Lorentzin fact, the example C,,+ (U) \ jxj shows that a hypersurface ian metric -
4. Pseudo-Riemannian manifolds
173
in Minkowski space may not inherit any pseudo-Riemannian metric. In contrast to the Riemannian case, it is probably not wise to try gain-
ing intuition for Lorentzian manifolds from studying submanifolds of Minkowski space. We should instead use the intuition we gained in the previous chapters. Lorentzian manifolds serve as models for space and time
as a
unit
and in this
-
physical
way their
geometry
can
be under-
stood best. a
have
a
pseudo-Riemannian manifold is everywhere nonisomorphism of vectors and one-forms.
Since the metric of
degenerate
we
canonical
(M, g)
Lemma 4.0.1. Let
be
pseudo-Riemannian manifold. The metv , where v (w) T ,M --* T,,*M, v
a
isomorphism all W E T,,M. for w) g., (v, ric,induces
-->
an
Proof. This follows immediately
from the fact that g is
isomorphism by (.)0: T*M naturally extended to tensor fields.
We denote the inverse
phism
can
be
Definition 4.0.1.
-
V)
.,v,+,)
-
completely
and the
Let
E
Then
T,,r(T--M).
we
T.,M. This isomor
define the completely
V) by
covariant tensor
V),(Vl,
.
-*
non-degenerate.
:-::::!
0(vi,
-,Vs,
contravariant tensor
OW.... Wr+S)
= -
I
-
-,
(V,+i) ...... (v,+,)')
00 by
(W,)O, w",
-
-
-,
r+sl)
components of V) are often simply denoted by V)j,...jjj i,jl j". components of 00 by 0" The
The
(.)0, (.)'5
isomorphisms
of indices". This
j,
and the
and
lowering
...
...
...
are
often referred to
terminology
is motivated
abstract index notation. We write
(90)ab
=
as
by
"raising
their expression in the
gab, (V )a
=
gabVb
=:
Va i
symbols " " and 5" should be easy to remember since there is an analogous notation in music. One of the most important inequalities in linear algebra is the CauchySchwarz inequality for positive definite scalar products (*7')R",' It states that for every pair of vectors v, w the inequality and
(.0)a
=
gab Wb
a
=:
.
The
(V W) Rn I
us,
7
(W W) R'11, I
inequality clearly generalises to Riemannian manifolds. To analogous inequality for Lorentzian manifolds is much more im-
holds. This an
(V V) Rn
portant.
4. Pseudo-Riemannian manifolds
174
(Inverse triangle inequality).
Lemma 4.0.2 ian
and let v,
manifold
u)
Let
be causal vectors. Then the
(M, g)
be
a
Lorentz-
inequality
VF-(VV)I VR-W*
(v, W holds.
Proof. The inequality holds trivially for null that both
sume
number
a
w
and
such that
w
v =
av
vectors. Hence
timelike. There is
are
+
This
e.
a
vector
implies (e, e)
>
we can as-
I
e
0, 0
<
and
v
a
(w, w)
a2 (V7 V) + (e, e) and therefore
(V, W) 2=
2 a
2
(V, V)
((W, W) '(e, e)) (V, V)
==
>
(W, W) (V, V)
.
P.
I
p.
176
In the rest
of this
may be omitted
For any
T,,.TM d
Tt (U +
section
on
we
prove
a
somewhat technical lemma which
first reading.
E T,,M, the n-dimensional "vertical" subspace T,,., T,,M given by the image of the map ,,M --+ T,,TM, v F-+ v-[,l
u
is
tV)lt=o.
We
can
equip T,,;,,.TM with
f)[u],
a
of
pseudo-scalar product by
general, the push forward of the expodefining (v, w). nential map fails to be a linear isometry of the spaces Tu.,,TM and TM. However, if one of the vectors ij[,,], fv[,,j is aligned with u[,,], we have the following
In
invariance.
Lemma 4.0.3
Tu.T
+
tv) I t=0
,Cu[uj d Tt (u
(GauJ3 lemma).
M.
E
If there
and
Proof. The and
v
Consider the
Cz
+
tw) I t=0,
T,xM \ f 01, and [u], E T, M with,5[,,]
w
then
[u] ), Tx exp (fv [u]
(v, w)
formula which is linear in v[u] and w[,,]. Since therefore replace v by u in this formula.
parallel we can map f : (s, t)
expx (t(u +
sw))
E
M and let
;sw + sW)[t(U+SW)j)
T( ,,t)f (,9t)
Tt(,,+,,v) expx((u
f,
T(,, t) f (a,)
Tt (u+,w) exp., (i_w [t (u+,,,])).
f
curve
u
ft
Observe that The
M,
(u
dt
a
G
Ru C T,M and
fv[u)
assertion is
are
x
E
(Tx exp (, u
Let
are v
t
we -*
f
have V
expx(t(u
for all s since V is torsion free. =V c,) is a + sw)) geodesic, which implies
tf,
9,ft
=0 .
f V
f
at Ut, M
17 at ft,
f"
)
+
( f"Vf
a, f"
f,
If7 a, f
t
Existence of Lorentzian and Riemannian manifolds
4.1
V a, 2 This equation
be
can
I
f
-
V
f
1
the assertion follows
Ut, ft)
=
9, (U
2
+ 8W,
U
sw
+
easily integrated yielding (ft, f,) by setting (s, t) (0, 1).
t
175
(u, W)
=
(u, w).
.
Hence
=
4.1 Existence of Lorentzian
and Riemannian manifolds This section is included on
for
its theoretical interest and
can
be omitted
first reading.
Every manifold
Theorem 4.1.1.
M carries
positive definite
a
metric g.
Proof. Let J(Ua, 0101,,EAJ be a collection of charts which covers M and If,,,I,EA be a partition of unity subordinate to JU,,I,EA. The bilinear form
(dxl
*
f, (x)
9X:=
1
0 dx
+
-
-
n + dx" (D dx
-
aEA
well defined and
evidently
is
and
supp(f,.,)
E
v
TxM be
Let
symmetric. non-zero
any
j be
fo, (X)
(v V)
Y. (( (Pc,)*V)j)
A line
erates
a
is
it admits
a
line field 1
Manifold
(non-oriented) a
M carries
line
> 0
vector field U gen-
field
a
Lorentzian metric
point
X
E
are
not
if and only
1.
Riemannian metric of M and M. Each
on
2
1-vector subbundle with base manifold
a
possible to have line bundles which (cf. the example of a M6bius band).
vector fields
Proof. Let h be a
(((WUj)*V)i)
fai (X)
line bundle but it is
Theorem 4.1.2. A
if
>
tangent bundle TM. A nowhere vanishing
generated by
G
i=1
field (or line bundle)
M of the
x
positive definite.
then that gx is
implies
2
i=1
aEA
index with
n
n
9
an
vector. The estimate
M has
assume
that there exists
neighbourhood
a
U such
that for all y E U the line field 1 can be represented by ly RUy, where U is a vector field on U. We can assume without loss of generality =
that
h(U, U)
(o)-tensor 2
now
=
1. Then U is determined up to
field g
h
:=
attention to U
-
2U5
0
again. Let
W
factor 1. Hence the
a
globally well defined. We restrict JU2,..., Unj be a completion of U to a is
local orthonormal frame with respect to h. That the tensor field g has and is therefore a Lorentzian metric follows from signature (-, +, , +) .
g(Ui, Uj)
=
.
.
h(Uj, Uj)
=
6ij, g(U, Uj)
=
0, g(U, U)
=
I
-
2
=
-1,
176
4. Pseudo-Mernannian manifolds
Conversely, be
that M admits
a Lorentzian metric g and let h M. Then at each point x E M we have a TxM defined by A =9 ikh3*k 1 where we use the
assume
Riemannian metric
a
linear map A: TxM
--->
on
-
I
Lorentzian metric g to raise with respect to h since
h(v, Aw)
hkiv kA.i
=
for all vectors v,
wj
or
=
lower indices. The tensor A is symmetric
hkig"hj,Vkwj
Hence there is
w.
hjlAlkvkwj
=
h(w, Av)
=
h-orthonormal basis of eigenvectors
an
of A. Since g has signature (- 1, 1, the linear map has one negative , 1) and n I positive eigenvalues. It follows that a each point x there is a .
.
.
-
uniquely determined 1-dimensional subspace 1,, of T,,M which is spanned by the eigenvectors corresponding to the negative eigenvalue. Clearly, 1 defines
a
line field of M.
Corollary
Proof
.
field 1 E
X
now
V
Sphere S2
The
4.1.1.
does not admit
a
If there is
a
S2 exactly
two vectors
non-vanishing
cover
R'
:
(Xl)2+(X2)2+(X3)2
a
on
S2 then there is also
Riemannian metric. Then there
construct
=
E
Lorentzian metric
a
S2. Let h be
on
fx
=
Lorentzian metric.
Ux
1x with h(Ux, Ux) on S2. At
E
1. We will
=
vector field V
line
a
at each
are
x
we
choose
U,. and consider all great circles through x. These great circles all of S2 and each point but f xj is intersected by exactly one
such great circle. The points x and -x are intersected by all great circles. We define now V on S2 \ f -xj as follows. Along each great circle -Y we
t
let
1-4
E f U, (t), Uy (t) I be uniquely determined vector such that is smooth. These vector fields along great circles have each a
V, (t)
Uy(t)
-
limit vector in
I U_
two
from
great
arcs
the other and V
limits limt-,
-
x,
U_ x 1. Let
to
x
-x.
now
Since
depends smoothly
Vy,(t)
fU-,,, -U_xJ
G
must coincide. But this
and that therefore
we
implies
- j
and
a
_4
S2,
smoothly
[0, 7r]
_/2:
rotate
_-
S2 be
one arc
into
parameters it is clear that both
on
that all
have defined
[0, 7r]
:
we can
limt.., V.,,(t)
arcs
have the
E
same
fU__ X' -U_Xj I-xl
limit at
continuous, non-vanishing
vector
field V. This contradicts Theorem 2.5.6 and therefore there cannot be
Lorentzian metric We will
see
in
on
S2.
Proposition
may want to discard
8.1.1 that there
P.
other
reasons
why
Hodge
star
operator
1
In Sect.
U
are
compact models of spacetime.
4.2 The volume form and the p. 174
3-791
have
seen that for a general manifold without addipossible it to define an integration of n-forms integration offunctions. In an oriented pseudo-Riemannian
2.5.4
we
tional structures it is but not
an
a
I
one
4.2 The volume form and the
Hodge
star
177
operator
manifold there is a canonical n-form, the volume form. This allows us to define the integral over a function (cf. Definition 4.2-2). The Hodge star operator is a canonical isomorphism of p-forms and n p forms. It can be used to put Maxwell's Equations which describe electromagnetism and ligh't (cf. Sect. 5.2.3) into an especially simple form. A further motivation is given in the introduction -
to Sect. 2.5.1.
The discussion
reading and will
of
the
This section draws
uct
on
on
star
operator can be omitted Of this book.
on
first
Sect. 2.5
induce
4.0.1)
Definition
(cf.
isomorphisms
The
Hodge
not be needed in the rest
a
pseudo-scalar prod-
T,M.
Lemma 4.2. 1.
The bilinear
g[r]: Tr(TxM)
X
S
form
T,(TxM)
R,
C11
crr+s +'0
non-degenerate, symmetric bilinear form. The pseudo-scalar product 2 g[r] is positive definite if g is positive definite. is
a
S
Proof. For 0, 0
0 )ai b,
(V
G TI
a
calculate
we
d
d,
a,,+ic.,+, V)C1.+1 a,,_+'. Obl' '+ j.'.b,,+,.g
+ ""'
...
a
= ...
(Tx M)
b '+,,
..
...
...
X
=
V)al
gbid, ...
...
ga,,+,.c,,+,.
-
9 b,, d
...
a,a,,+i
...
,
a,,+,.Odl
d,, d,, +
...
gbid,
...
gb ,+,-d ,+,--
It follows that the total contraction of this tensor is symmetric in its constituents Oal a,, a,, + i... a,,+,. and Od, d.,d,+, ds+,.. Let
fel.... be
a
and
...
...
...
W1....
en,
i
I
W
nj
ej pair of dual orthonormal bases and 0 c T,,r(TxM). Since (w )O w' (the sign depending on the signature and i) we have (ej) =
=
g[r8] (0, eai
0 ea., (9
W
bi
implies that g[r]
which in turn
is
s
(6al,
n,.)
&
ea,,,
non-degenerate. The
W
bl'..., W n,.)
assertion for Rie-
mannian metrics follows from
g[r 8
V*ai
V)) al
...
b
....
7
W
ea,,,
bl'..., W n,,)
a,,
b_
CI'--CS'
di
'...
X
2
The
converse
'd,
Cec
......
e,"
W
di
is not true since 9
vanishing bilinear
form
on a
'.
[0 ] n
..7
is
W
d,
)6alel
always
...
6a,, c,, 41 di
...
definite since it is
one-dimensional space.
6b,d, a
-
non-
Pseudo-Riemannian manifolds
4.
178
The metric
measures
also the volume of
independent
a
vectors,
(M, g)
Lemma 4.2.2. Let M. Then
a
orthonormal
Proof The
be
a
neighbourhood
jg[O](yjt)j
p, -/-t with
forms an
has
x
not only the length and the angle of vectors but parallel-epiped spanned by a collection of linearly
n
-pseudo-Riemannian manifold and x E U such that there exist exactly two nn!. Furthermore, if fEj,...,Ej is =
then M., (El,
frame
En)
c=
first claim follows from the fact that An
,,M)
is
nal vector space and g[(]x a non-vanishing bilinear form n Let E, I be an orthonormal frame and jw
dual frame. Then the tensor fields
o
=
wl
19 In'] ( O, 0) I Hence M
El, by Ej .
.
.
,
=
det(A3 ).
wl
=
En I AEj
A
A Wn
...
I
=
0 Wn
n
I
The
or
Tl'(U) a
always 1
preceding lemma implies (by continuity)
that
n!.
=
Evidently,
n
if there exists
n-form
non-vanishing
a
connected pseudo/,t which satisfy the
on a
at most 2 n-forms
are
Ig [0 ] (p, ) I
normalisation condition
only
=
linear map
-1.
Riemannian manifold there
if and
n!. If
is defined
det(A. )/_t (E,,..., E,)
=
The second claim follows since the determinant of
to either 1
=
n
which transforms orthonormal bases into orthonormal bases is
equal
be the
satisfy
Ig [0 ] (/,t, /-t) I
is any other orthonormal frame and A E A'j Ej then M (E,,..., En)
=
it.
-
n! alt (W) satisfies
=
1-dimensio-
a
on
this is the
M, i.e.,
on
case
if M is
orientable.
If (M, g)
Definition 4.2.1.
is
oriented
an
pseudo-Riemannian manifold unique n-form pm c 0
with orientation 0 then the volume form is the with
Ig [0 ] (ym, ym) I
define
The
1. If U C M is of U by vol(U)
=
n
we
the volume
following definition reduces
open set with
an
=
compact closure
f?U AM.3
to the volume
integral
in Euclidean ge-
ometry Definition 4.2.2. Let
fold
with volume
set U is 3
(M, g)
be
form defined by fjU f/-t.4
Readers
who
an
p. Then the
have
oriented
Integral
omitted
Sect.
dXn'
where
fl(U) V/_j_det((gij)jj)dx'
...
2.5 W
pseudo-Riemannian Mania function f over an open
of
is
can
chart
replace map
fu pm
whose
by
domain
contains U. 4
As in the
preceding footnote,
-((gij) ij) dx' (p-' Vj-det
.
.
.
f, f p
dXn.
may be
replaced by
f, pm
by
fW(U) f
0
4.2 The volurne form and the
Lemma 4.2.3. Let
and
(x',
dinate
an
star
179
operator
oriented pseudo-Riemannian manifold coordinate system. Then the coor-
positively oriented expression of pm is given by .
.
.
,
xn)
a
I-tm
Proof.
be
(M, g)
Hodge
If
Jdet ((gij)1
=
(yl,...,yn)
is another
det((ax, '97) b)
then the number
positively
is
positive.
A
...
A dxn.
oriented coordinate system We denote the metric
com-
a,
ponents with respect
to
(y 1,
yn) by jab
.
gxc gab
which in turn
J
and obtain
yXd b
a
gcd
implies 2
9xa
det ((jab)l
det
ay b
,
-
9xa
ay b
)
1
)
det
((jab)l
d_et ((Jab)1:5ab
I
_((gii)1
given by
Hence the n-form
((
det
)
A
...
A
dx' for any
positively oriented coordinate system is globally well defined. At x we can Jab- Calculating choose a positively coordinate system with gab(X) =
det J_((gij), in this coordinate
<
ij:5,,) I dx IA
...
A
dXn
system implies that this n-form coincides with
Definition 4.2.3. Let W E divergence of o, div ( p), is
divW(A',..., Ar-1, V,
....
T,'(M) a
I
be
tensor
a
tensor
field
in
field
on
1
T,- (M)
Then the
(M, g). and
pm. I
given by
VS) n
E (VEaW)
(0a, Al'... Ar-1, vi, ,
...,
V,
a=1
where
JE1,..., En; 01
This definition is
which
can
be
....
Onj
independent
seen
is
(any) pair of dual,
orthonormal bases.
of the chosen orthonormal basis
best in coordinates: the equation
-
a
fact
div( o)', ii....i. -l(x)
=
Vk(P (X) is clearly invariant and agrees with the definition above if the GauBian frame is chosen to be orthonormal at x.
[p. I
176
1]
p. 137
Pseudo- Riemannian manifolds
4.
180
Lemma 4.2.4. Let
and U be
(M, g)
field
vector
a
on
be
an
oriented
pseudo-Riemannian manifold formula
M. Then the
_CUpm
(div(U))[zm
=
holds.
Proof. Let
x E
Christoffel
M and choose normal coordinates centred at
symbols Va
=
of V vanish at
Hence using Lemma 2.5.3
x we
(
aa and 0,,
)
obtain at
we
(rdet ( Fgab)l
=
A
...
dx IA
*
+
det ((jab),: ab: ,,)XU (dx' det ((jab)
1 < a, b<
n) 1: dx
Since the
=
0.
x
XUpM =.CU
=
x.
have
1
A
A dxn
...
...
A
...
Edxl A
...
A
dXn
A dXn
A.CUdXa
A
...
A dXn
a=1 n
A
d(U J dXa)
A
...
A
dXn
a=1 n
Edxl A
A dUa A
...
...
A dXn
a=1 n
7
qUa dx
1
A
...
Ad Xn
qxa
a=1 n
1:
IqUa qxa
a=1
Since
x was
Corollary
arbitrary 4.2.1
)
4m.
we
=
have
(Theorem
XUpm of
U be
do-Riemannian
(div(U)),4m. =
GauB).
I
(div(U))p everywhere. Let
field
(M, g)
be
an
M, and V
oriented pseuM an open
manifold, of M with smooth boundary W Assume that 9V is a smooth sub0 for mani Id and that there is a vector field n along aV with g(v, n) have Then all v E TaV and g (n, n) we E 1, 11. 77 a
vector
on
C
subset
=
=
JV div(U)pm
=,q
JaV(n, U)
pav.
4.2 The volume form and the
Proof
Lemmas 4.2.4 and 2.5.3
d(U
pm)
J
=:
d(U J pm).
inclusion. Then
we
Hodge
imply (div(U))I-tm by t: OV ---> M,
star
XUpm
=:
Denote
x
F-4
=
IV div(U)[tm fV d(U Jpm) fav (n, U) We will
now
introduce
canonical
a
q
=
=
U-Jdl-tm +
=:
the canonical
x
pm) 77 U, n) n J Mm (Theorem 2.5.5) implies
have t* (U J
Hence the Theorem of Stokes
181
operator
q
=
(U, n) pav.
pav.
isomorphism between
the space of p-
and the space of (n-p)-forms An -P(TxM). See Sect. 2.5.1 for the motivation of this isomorphism. forms
AP(TxM)
Proposition 4.2.1. Let (M, g) be an oriented'pseudo-Riemannian mannj there is a unique linear ifold with volume form p. For each p E f 0, isomorphism *: AP (7:.,M) -- A'-P(TxM), 0 i--> *0 .
gI 0 P] (**, 0) An-p(TXM). such that
g
=
n-
The operator * is called the induced by * the Hodge star
Proof.
Let
[' 1 (0
A
n
Wnj
jej,...' enI,
A
where
we
wa,,-p, U)bi
A
,
all
V9
AP (Tx M), 0
E
E
Hodge star operator and the isomorphism isomorphism. be
=
n
.
0, p) for
and qj (ei, ej). Then the set jWai A thonormal basis of An-p(TxM) with
g[O _P] (Wai
.
A
...
have assumed a, <
a
...
Wb,,-p )
...
dual pair of orthonormal bases A wa,,-p Tai< is an or-
...
=
Jai bi
N,
an-P, bi <
<
...
...
?7a,,
-
<
,
Ja,,
-
bn-p*
1,
b,,,
-
We
1, ;
can
therefore write
E ai<
...
ai<
...
g
X
e-a,,-,, )Wal
*O(eai,
[0 _Pj (*0, Wal
A
...
n
Awan-7,
...
g[O] (V) A Wal
A
A
...
n
X
...
Awan-p)
77al '*'77a,,-,, Ujai A
ai<
A... A Wa,,-,,
wa,,-,,,
77a i...
?7a,-,, Wal
A
...
A wan-,,.
The last formula proves that *?p is uniquely defined by the defining property of *. Further, our explicit basis representation shows that *0 exists. Since 9 [0 representation of
n-P] *0
g[O] (0
A
n
does not
depend
0, /_t) on
is
an
invariant
the chosen basis.
equation,
our
I
182
4. Pseudo-Riemannian manifolds
Corollary 4.2.2. Let 0 E AP (T,,M) and pair of orthonormal dual bases. Writing 77i
g[O ] (OAWai A
Au)a,,-,, [077ai
...
n
a,<
...
en 1, (ei, ej) we
:=
...
Wnj
?7a,,, _'Wai A
...
,,
Proof. Let f ej,...' enj, assume
a, <
bn, f a,.... anI *U)ai A
a
Awa,,-,.
Proposition 4.2.2. The operator *: AP(T, M) -- An -P(T M) isometry for even index v and an anti-isometry for odd index v.
and
be
have
<
ap, ap+l
Jbi,...' bn1
=
A Wa,
...
Wnj
<
...
11,
=
be
-
a
-
-,
...
...
nj. By Corollary
sign(al,..., an)Tla,+l
=
an
pair of orthonormal dual bases < bp, bp+l < bi < <
< an,
...
is
,
...
4.2.2
na,, wa,+, A
have
we
A wa,,,
...
and therefore
g[o_p](*Wai n
A
...
A
wa,,, *w
b1
A
Aw
...
sign(al,..., an)sign(bl,..., bn)?7a,+, 0
X
gIn _P] (Wap+1
=,q[O_P](Wap+l
A
A
A
...
n
where
we
O_P](WaT,+,
g[OI(Wal P and
(by
our
A
...
A
bp+ 1,
=
Waj,' Wbi
selection and
we
.
.
A
,
an
A
...
ordering
6ap+lbp+l holds,
.
*'*
bn
=
A
A
A
Wb,,
Wb,,)
Wa,,, Wbp+l
A
...
...
A
...
A
Wb,,,
Since
-
Wbp)
of
A
bp+ 1 A
U)a,,,
have used that 9 In
vanishes unless ap+ 1
wa,,,, Wbp+l
A
...
=
qaj
77ap Jai bi
...
6apbp
indices)
6a,,bn= Jaibj
Japb,,
have
gInO_pl(*Wal
A
...
A
Wa,,,, *,b
1
?7ap+l
A 0
9
...
A
w
Wal A
b,,) ...
A
Wap, U)b'
A
...
A
wbp
Tlai
(_j)vg[Oj(Waj P
Lemma 4.2.5. Let
of
index
N
v
*1
(M, g)
with volume =
it,
*A
=
be
form
an
A
...
Wa,,, Wbi
oriented
y. Then
(-')';
A
A
...
A
Wbp
pseudo-Riemannian manifold
4.2 The volume form and the
(n-P)-g [0] (0, *V)) for
[on-,,] (*0, 0)
9
Hodge
An-p(T,,M); (iii) 0 A *0 g [0P ] (0, 0) y for 0 (_1 )v+p(n-p)O. (iv)
all
all
AV (T
0, ?p
E
have 9 [on] (*I,
A)
=
star
0
183
operator
AV (T., M), 0 E
C-
M);
=
Proof (i):
definition
By
we
9
=
[on] (1 Ay, /-t)
=
9
[on]
is non-degenerate and An (T,,M) is a and the first claim follows since g [I] n one dimensional vector space. The other claim can be proved analogously
but it also follows from
Using Lemma
(ii):
g[O_P](*O,,O)
=
(iv). 2.3.8
directly
we can
g[OI(O AO,tL)
(_I)p(n-p)g[O](,o A 0,/,t)
=
n
n
=
calculate
n
(_j)P(n-P)g[O_ v] (*0, 0) n
(iii):
From
4.2.2
Proposition
get
we
0 gln'](O A *040 =9[n-p1(*O,*V))
The assertion follows
now
(iv):
9
from 9
(-1)"401(0,0-
n
AP(TM)
For every
=
we
have
=
9
[on] (*0 A 0, A)
=
g
[0 1 ((_ I)P(n-P) 0 A *0, /_t)
101 (* * 0, 0) P
n
n
P
(_l)p(n-p) (_I)vg[Ol (0, 0), P
whence the claim follows from the
Lemma 4.2.6. Let
and
E
w
Proof. at
x
be w
Let
=
fdx"
*W
...
=
A
oriented
an
div(w)
have
and choose
is linear in A
we
=
pseudo-Riemannian manifold
(- 1)P-
I
*
d
* w.
normal coordinate system centered x. Since the equation to
a
is orthonormal at
jai ......Onj
such that
proved
M
E
x
be
(M, g)
S?P (M). Then
non-degeneracy of g[PO]
w we can
dx'p where ii
assume
<
...
fsign(il,..., in) Wp+i
<
...
without loss of
ip.
We obtain at
?7i,,dx"'+'
A......
Since the first derivative of the metric components at in the calculation of d * w and get
x
generality that x
dx -. vanish
we can
ignore them
n
(d
*
w)x
=
1: ajfsign(ii, j=1
.
in)?7i,+l
...
?li,, dxj A
dx ,,+ I
A
...
A
dx ,,
184
4. Pseudo-Riemannian manifolds
Let
E
be the number which satisfies dxj A dx -+ I A
I 1, 11 -
A dx3
where
...
A
dx -
j, and let u[j) be the permutajp which tion of i 1, into the last entry and which leaves the moves j iP relative order of all other entries fixed. Then we get sign(jj, Jn) sign(ij,...'ip) sign(u[j]). We get then
dx-'
71
A
...
.
.
.
-
<
<
...
,
.
*(dxl
dxl,'+, A
A
sign(ii,
=
=
=
sign(ii
Oj
we
.,qj,-, dxjI
in)'qj,
_j
i
(dxj
dx"
A
aj
_j
(dxj
-
dx"
A
A
...
A
A
...
A
dxjP-1
......
A......
dxjp-l
77jp-,
dxj
in)77jl
-,
A
.,qjp-, dxjI
-
(-I)P-1sign(ij'. x
Hence
,
-
dx"')
A
in)sign(a[j]),qj, sign(il,..., i,,)sign(a[j]),qj, x
=
...
.
...
A
A
...
T/j,
dx"'
I
-
dx',,
get n
(*d * w)x
E aj f sign(i
=
in) ?7i,+
1,
i...
(- I)P-
77i"'
1
j=1 x
sign(ij.... in)77ji )
...
77jp
-
1
19i
J
(dxj
A
dx"
A
...
A
dx'P
n
E,9j f 77i (- 1)P_
=
1
ai
J
(dxj
A
dx"
A
...
A
dx'P
j=1
(-1)P-1div(w),,.
4.3 Curvature of The material p.
189]
pseudo-Riemannian manifolds
presented
in this section up to and
including Corollary
4.3.1 is fundamental for Sect, 5 on relativity. The rest of this section can be omitted on first reading. The Riemann
V
[U, W]
W of
(or curvature) a
tensor
R(U, V)W
=
connection has been introduced
Since it arises in such
VUVVW-17VVUW-
geometrically in
Sect. 2.8.
natural way, one would suspect that the Riemarm tensor of the Levi-Civita connection plays also an important r6le in the a
physical theory of spacetime. This is indeed so as we will see in Chap. 5. Here we will merely investigate some of its mathematical properties. The Riemann tensor satisfies a collection of useful algebraic identities.
4.3 Curvature of
Proposition 4.3. 1. Let (M, g) M, and t,u,v,w G TxM. Then
R(u, v)w
+
be
pseudo-Riemannian manifolds
a
pseudo-Riemannian manifold,
R(u, v)
=
-R(v, u),
R(w, u)v
=
0
(R(u, v)w, t)
=
(R(u, v)w, t)
=
R(v, w)u
+
185
(First
-
Bianchi
(R(u, v)t, w)
(R(w, t)u, v)
x
E
Identity),
,
.
Proof. The first identity is clear from the definition of the curvature identity is just the first Bianchi identity which holds
tensor. The second
for any torsion-free connection Since
(R(u, v) (w =
the third
+
t), w
+
+
(R(u, v)w, t)
proved
is
Lemma
2.8.1).
t)
(R(u, v)w, w) identity
(cf.
once we
+
(R(u, v)t, w)
have shown that
+
(R(u, v)t, t)
(R(u, v)w, w)
,
=
0
for all vectors u, v, w Cz T,,M. We can assume that there are vector fields U, V, W which are linear combinations of Gaufflan vector fields
with constant coefficients and have
[U, V]
=
satisfy U.,
=
u, V.
V, W
w.
Then
we
0 and calculate
(R(u, v)w, w)
=
=
(VUVVW, W) VVVW, W) VVW'Vuw) VU ('7VW' W) VV (VUW, W) + VUW'Vvw) VU ('7VW' W) '7V (VUW, W) -
-
-
=
-
I
I
U
0
2 =
where in the last
V
0
(W, W)
V
-
-
2
U
-
(W, W)
0,
equality
we
have used that the vector fields have
stant coefficients.
The first Bianchi
0
=
0 + 0 + 0 + 0
identity implies
=
1
H
(R (u, v) w, t)
2
+
4(-)
R(t, u)v, w)
+
H
R(u, v)t, w)
+
(R(t, u)w, v)
(R(w, u) v, t) 5
+
4
6
(R(w, t)u, v)
(R (v, w) u, t) 1
+
3
(R(v, t)u, w) 3
+
(R(u, w)t, v)
con-
186
4.
Pseudo-Riemannian manifolds
2
6
(R(v, w)t, u)
+
5
(R(w, t)v, u)
+
=
Here
we
have used the first and third
cancel and the
Recall that
(+)
terms
by taking the
are
+
(R(t, v)w, u)
(R(w, u)v, t)
identity
+ 2
(R(v, t)u, w)
to show that the
(-)
terms
equal.
trace
the curvature tensor. In the
2
I
we can
construct
pseudo-Riemannian
Ric with the metric to obtain
a
a
(') 2
tensor Ric from
case we
can
contract
function.
Definition 4.3.1. Let the
function Scal
:=
(M, g) be a pseudo-Riemannian manifold. Then gab Ricab is called the scalar curvature of tr(Ric) ==
9.
The second Bianchi quence which will be of
Lemma 4.3.1. Let
identity (Lemma 2.8.1) has importance in Sect. 5.3.
(M, g)
be
a
the
following
conse-
pseudo-Riemannian manifold. Then I
div(Ric
-
-Scalg)
Proof. Rom the second Bianchi identity 0
=
0.
2
we
infer
=
n
E ,a
((VEaR)(u,v)w+(V
V
R)(E,,,u)w+(V U R)(v,Ea)w
a=1
div(R) (u, v)w where
+
VvRic(u, w)
-
VuRic(v, w),
have used the property Vg 0 of the Levi-Civita connection. and with contraction v w a Replacing gives 0 2div(Ric)(u) VuScal. we
=
=
-
I
Proposition 4.3-2. Let (M, g) be a pseudo-Riemannian manifold and let (X11 Ixn ) be normal coordinates centred at xO E M. In these coordinates, the metric tensor g satisfies ...
gab(x)
=
gab(XO)
-
IRacbd(XO)XcX
3
d
+00 Xc 12)
Proof Since (xl.... Ixn ) are normal coordinates centred at xo The Chris0. The Koszul formula (2.7.7) implies symbols satisfy r. (xo)
toffel
=
that 1,3
29" Oigij +,9jgil
-
algij)
4.3 Curvature of
Hence
we
2(gklF
gkiFiki)
aiglj
: :
3
ajgil
+
019ij
-
+
1919ij
partial derivatives
first
implies that the
which
187
obtain
+
U
pseudo-Riemannian manifolds
+
19jgil
-
19iglj
=
2ajgil
of the metric components bc 9,,g vanish at
vanish at x0. It follows immediately that then also all x0. At x0 we can calculate for the Riemann tensor Ra bcd
(R(a, 19d) 19b)
a
(Vjqc(F dai) a
a
lubcd 1 2
1 2
We will
VIOPr c19') )a
-
-
19dFb(c
(49cabged gae (acl9bged gae
+
-
show that
now
19cad9be acaegbd
-
c9c,9egbd
-
-
adabgec
+
490bgcd
==
acadgab
190bgec
-
'9d'9cgb,-
+
adaegbc)
adl9egbc
(4.3.1)
at xo. Recall that in normal
coordinates centred at x0 all rays t --> (txl.... tx') which pass through a 0 for all x (xl.... x n) x0 are geodesics. Hence we get rb(c (tX)Xb x' =
=
small
enough.
This
implies
a
b
e
0
ax d
(29aeFbc(tX)X XC)
ad ((ab9ac(tX)
+
(adabgac(ft)
+
19dacgba(ft)
(19dgac(ft)
+
19cgda(ft)
,9x t
+ 2
Observe that
19dgac(O)
lim t_0
Hence
=
acgba(ft)
-
19agbc(ft)) X bxC)
-
adl9agbc(ft)) XbXc
-
aagdc(ft)) Xc-
0 and that therefore
119dgac(ft) t
=
d('9d9ac(0))(X)
dividing the equation above by
=
t and
19e'9d9ac(0)Xe-
taking
the limit t
0
we
get 0
=
(190bgac(0) + 2
=
+
Xb xc(4,9bC9d9ar_(0) this
(xl.... IXn) that
Gabcd
adaagbc(O)) X bXc
-
1909cgda(0) -
d x
we
=
(2190dgac(0)
is
arbitrary (for
:=
-
adaagbc(O)
equation with 0
Since
19d19c9ba(0)
(490dgac(0)
Contracting
implies
+
-
219(d19bgJa1c)(0)
a0agdc(0)) XcXe. +
2abacgda(O)
-
2abaa9dc(0))-
obtain
adaa9bc(0))X bxcxd small -
enough values)
a(dalal9bc)(0)
==
this equation
0. We obtain
188
4. Pseudo-Riemannian, rnanifolds
Gabcd
0
Gbcad
+
+
Gcabd
I
(2adabgac(O)
3
+
-
209c'Odgab(O)
adl9agbc(0) -
2abacgad(O)
+
-
abaagcd(O)
acaagdb(O))
1 +
-
3 +
(2adacgba (0) 2a,,adgbc(O)
-
-
adabgca(O)
+
2acaagbd(O)
+
2a,,abgcd (0)
-
C9cabgad (0)
lyaabgdc(O))
I
+
-
3 +
(2190agcb (0) 2ab19dgca(0)
(3adabgac(O)
3
adaagbc(O) This last equation 0
=
+
+
-
-
ad'Ocgab (0)
aaacgbd (0)
abacgda(O))
3ad19agbc(0)
adabgca(O)
implies
-
+
+
3acadgab(O))
19dacgab(0)-
(4.3.2)
now
Gdbea
1
3
(2aaabgdc(O) +
=
We
2acaagdb(O)
aa'9dgbc(O) -
+
=
2abacgda(O)
-
Nadgca(0)
acadgab(O))
+ 19bacgda(0) 2(OcAgde(O) 3
interchange
19caagbd(0) ad'90ca(0)
-
+
acaagdb(O))
-
b and d in this
equation and get Oa adgbc (0) + adacgba (0) + comparison with Equation (4.3.2) gives 19caagbd(0)
0. A
which is
It follows that
equivalent to our assertion 190bgcd Equation (4.3.1) implies
a Rbcd (0)
==
gae (0)
=
aclodg,,b-
(acabged(O) acaegbd(O)) -
and therefore
Racbd(O)
where
follows
+
we
Radbc(0)
=
190cgad(0)
-
=
a0cgad(0)
+
=
-3,9baagcd(O)i
a0agcd(0) 190dgac(0)
+ -
190dgac(0) -,90agdc(0) 19b0agcd(0)
have used
now
from
a
Equation (4.3.2). The assertion of the proposition Taylor expansion Of gab(x) around the point xo. I
Corollary 4.3. 1. Let A be a tensor field which is pointwise defined an algebraic expression of g and its first two derivatives. Then A is algebraic expression of g and the Riemann tensor R.
as
an
4.3 Curvature of pseudo-Riemannian manifolds
189
Proof. Proposition 4.3.2 implies that any tensor field A which is a pointwise invariant function of g and its first two derivatives only depends on g and R.
I
F[5pTT1W47j] I I
For 2-dimensional
manifolds, the Riemann tensor reduces to a function, a much simpler geometric quantity. To see this, let I e- I, e2 I be an orthonormal basis of T'M. Since (R(., .)., -) is anti-symmetric in the first two and in the second two entries the expression (R(ei, e2)e-2; ei) does not depend on
the chosen orthonormal basis and defines
is easy to
(For explicit
a
function K: M
that this function describes the curvature tensor
see
formulas cf. Sect.
R. It
-4
uniquely.
4.3.1)
higher dimensional pseudo-Riemannian manifolds such a simple relationship does not exist. However, it is possible to define a function K which maps the space of non-degenerate 2-dimensional subspaces of For
T'M into the real numbers. Let G nondeg (TM) 2
fspanfu,,,v,,l
=
:
Ux
be the set of all two-dimensional
spacelike
or
Vvx,9jsPan u_v,,,j subspaces
is
non-degeneratel
in TM which
are
either
timelike. The
Definition 4.3.2.
function -
K: G nondeg (TM)i--->R, 2
spanjux,vxjt-->
(R(ux, vx)ux, vx)
UX, UX) (VX' VX)
is
(Ux, Vx)
2
of M.
is called the sectional curvature
Observe that this expression
_
well defined since the denominator does
nondeg not vanish for any 17 C G 2 (TM) and both the numerator and the
denominator transform
Al'ux 1
+ A 2Vx, 1
fj.X
=
by
a
factor
det(A)2
pseudo-Riemannian manifolds
Those
ture reduces to
a
under
a
change
of basis fix
A2'ux +A 2VX. 2
function
on
for which the sectional
M should be
curva-
especially interesting.
Proposition 4.3.3. Let (M, g) be a pseudo-Riemannian manifold and M. If the sectional curvature satisfies K(Hx) x c K(flx) for all I-T, fIx E TxM then there exists a number c C- R with Rx (u, v)w c ((v, w) u (u, w) v) for all u, v, w E TxM. =:
-
Proof. Assume that K(Hx) does
not
depend
TxM. Then the definition of the sectional an
orthonormal basis
fel,
-
-
(R(ei, ej)ej, ej) for
some
constant
clude that
c.
-,
=
enf c
we
on
the choice of
curvature
implies
plane in given
that
have
( ej, ej) (ej, ej)
_
(ei,
From the tensorial property of
ej)2) R(-,
we
con-
p.
2091
4. Pseudo-Riemannian manifolds
190
(R(u, v)v, u) for
allu,v ET,,M. For
0
=
(R(u
=
(R(u, v)v, u)
+ w,
v)v, u
+
w)
E
w
every
-
((Uj U) (V, V)
C
=
T.,M,
(u
c:
(R(w, v)v, u)
+
u
(U, V)2
obtain
we
+ w,
_
w) (v, v)
+
(R(u, v)v, w)
+
U
-
V)2
+ W,
(R(w, v)v, w)
+
(u, w) (v, v) (w, w) v, v) c( (u, u) (v, v) 2 (U, V)2 (u, v) (w, v) (V7W)2) + 2
+
-
=
2 -
+
(R(u, v)v, w) (u, u) (v, v) (U, V)2
R(u, v)v
2
_
c( (u, u) (v, v)
+
(u, w) (v, v)
2
-
(u, v) (w, v)
+ 2
c((v, v)
-
which
R(u, v
=
R(u, v)v
w)(v
+
w)
c((v, v)
-
+
R(u, w)v
-
C((w, W)
R(u, v)w
+
-
U
-
u
(u, v) v)
-
c((w, v)
-
u
u,
(W, V)2
(w, w) (v, v)
(u, v) v)
-
c((v + w,
-
_
(V, W)2
+
=
=
(W, W) (V, V)
+
(u, v) v), w)
-
implies R(u, v)v c((v, v) polarise again and get
now
0
u
(U, V)2
_
-
+
v
u
u,v
-
R(u, v)w
+
(v, u) w)
-
w)
for all U,
-
E T.,M.
v
w) (v
+
c((v, w)
u
+
We
can
w))
(u, w) v)
-
R(u, w)w
+
(W, U) W)
c((v, w)
This
u
-
(u, w) v)
+
R(u, w)v
-
c((w, v)
u,
(v, u) w).
-
implies R(u, v)w c( v, w) u (u, w) v) R(w, u)v c( u, v) w (w, v) u) for all u, v, W G T,,M,i.e the expression R(u, v)w c((v, w) u (u, w) v) is invariant with respect to cyclic permutation. Since the cyclic R(u, v)w + R(w, u)v + R(v, w),, vanishes (cf. Proposition 4.3.1) and -
=
-
-
-
-
(u, W) the
cyclic
C((.V, W)
u
v
-
(v, W)
U
+
V, U)
R(u, v)w (u, W) v))
sum -
of
-
W
-
(u, V)
c((v, w)
u
-
W
(W, V)
+
U
(W, U)
-
(u, w) v) gives
-
0
=
3
v
=
0,
(R(u, v)w
Proposition
4.3.4
tional curvature K:
-
I
-
(Lemma of Schur). If dim(M) ! 3 and the secGnondeg R reduces to a function on M then (TM) 2 -*
it is constant.
Proof. Proposition
4.3.3
with R,, (u,
c(x) ((v, w)
v) w
implies that there
exists
a
function
x
--+
c(x)
u, w) v)
for all u, v, w E T,, M. Let W be vector fields whose covariant derivatives vanish at x and which
U, V, satisfy U.,
=
=
u, V
v,
Wx
=
(VtR) (u, v)w
u
w.
=
-
Then
we
obtain for any vector t E TM
dc(t) ( v, w)
u
-
(u, w) v).
4.3 Curvature of
dc(t) ((v, w)
=
be
Let
u
We
can
we can
u
u, w) v)
-
dc(u) ((t, w)
+
191
identity (Lemma 2.8.1) implies
Hence the second Bianchi 0
pseudo-Riemannian manifolds
v
+
(v, w) t)
-
dc(v) ((u, w) t
(t, w) u)
-
.
vector which is not
a
choose t also
assume
lightlike and w =h 0 be orthogonal to u. orthogonal to w. Since dim(M) > 3 u, v, w are linearly independent. The equation
and
w
=
that
v
to be
above reduces then to
dc(v) u
0
not
dc(u) v) (w, w)
.
0. Hence dc vanishes on implies dc(u) lightlike and therefore must vanish identically.
which in turn are
+
=
Observe that the
preceding proposition
all vectors which
I
is false for two-dimensional pseu-
do-Riemannian manifolds.
Proposition
4.3.4 motivates the
Definition 4.3.3. A
if
curvature
In
following
definition.
pseudo-Riemannian manifold (M, g)
nondeg the sectional curvature K: G 2 (TM)
Proposition
4.5.2 below
we
will
locally classify
all
-4
has constant
R is constant.
pseudo-Riemannian
manifolds of constant curvature.
4.3.1 2-dimensional
pseudo-Riemannian manifolds
formulas for the Levi-Civita connection and the curvature of 2-dimensional pseudo-Riemannian manifolds. These formulas will be used in Sect. 7.4. We collect
tensor
pseudo-Riemannian manifolds are the lowest dimensional pseudo-Riemannian manifolds which are not trivial. Let 71 772 E f 1, 11 and let f El, E21 be an orthonormal frame with 0, g(El, Ej) n2. By Corollary 2.4.2 g(El, Ej) qj, g(El, E2) 2-dimensional
-
1
=
=
=
there are coordinates (t, q) and functions A v e- (t, q) 19q. e- (t, q) 0t and E2
A(t, q), v(t, q)
such that El
=
Proposition 4.3.5. Let (M, g) be a 2-dimensional pseudo-Riemannian dv(E2)EI dA(Ej)E2 and the Levi-Civita manifold. Then [El,E2] connection is given by =
'7E,El VE2 El The curvature is
=
=
-
- 71772dv(E2)E2, dA(El)E27
'7E1E2
VE2E2
=
completely determined by
-
=
dv(E2)EI,
771772dA(El)El.
the scalar curvature
Seal,
192
4. Pseudo-Riemannian manifolds
R(Ej,E2)Ej=-22ScalE2, R(Ej,E2)E2=2IScalEj, 2 2 1
Ric 2
K(TM)
Scalg,
2
Scal,
where
Scal
=
-2
(77, (El
9
E,
*
A +
(dA(Ej ))2)
+ 772
(E2
*
E2v
+
(dv(E2 ))2)
)
-
Pro0f. We calculate first the commutator of the vector field El, E2, Using ordinary derivative of R 2with respect to the coordinates (t, q).
the
[El, E2]
D(e-A,9q)(e-'at) D(e-'at)(e-Ai9q) -e-'e-, dA(at)aq + e-e-ADaq(,9t) + e-'e-Adv(aq)c)t e-'e-ADat(aq)
=
-
-
-
=0
-dA(EI)E2
+
dv(E2)El
-
e-'e- A
[0t, Oq]
Our formulas for the Levi-Civita connection follow
'7E, El, El ) '7E2 El, El ) '7E, E2, E2 ) '7E2E"2) '7EjE2 E, )
=
El
9,qj
=
0,
E2
0
771
=
01
El
*
772
=
07
E2
'6
772
2 =
2 =
2 =
2 =
,
=
'7E1 E2 (dv(E2)El
'7E,Ej, E2)
from
0)
17E2El, E, -
dA(Ej)E2, Ej
=
?jjdv(E2)i
Ej,7E,E2) -Tlldv(E2)) '7E, E2 '7E2 El, E2 ) =
VE2 El, E2
-
(dv(E2)El
'7E2 E2, Ej) The curvature
directly
-
dA(Ej)E2, E2)
(E2, VE2Ej)
can now
be calculated
=
=
772dA(El)E2
-772dA(El)
straightforwardly.
R(El,E2)El =7EjVE2Ej -7E27E,El
='7Ej(dA(Ej)E2) dv(E2 )7E,El -
El
*
El
*
+ +
-7[E,,E2]El 771772VE2(dv(E2)E2) dA(El)7E2 El
A E2 + dA (El) dv (E2) E, + 771772 E2
9
E2
e
1/
E2
4.4 Subinanifolds
-
dv(E2)dA(E1)E1
(El
El
*
+ 771772
Since
=
=
Ric(El, E2)
Ric(E2, E2)
E2
(dv(E2
+
e v
obtain Ric
tr(R(., El)El) 772
=
))2))E2-
02 (R(E2, El)El)
(R(E2, EI)E1, E2)
Ric(E2, EI)
=
02 (R(E2, E2) Ei)
+
=
tr(R(., E2)E2)
ni
1
2
-77,
=
(R(El, E2)El, E2))
=
'
01 (R(El, E2) Ei)
=
0,
(R(El, E2)E2, EI)
(R(El, E2)El E2) 7
Scal g, where
Scal
-271,772 R(El, E2)El, E2)
=
-2(,ql (El
-
+ 772
(E2
*
the sectional curvature is
K(T M)
-772
=
tr(R(., E2)El)
=
=
Finally,
*
(dA(El))2 E2
(dA(El ))2
A +
(E2
+
have
we
Ric(Ei, Ei)
we
9
?11772(dv(E2))2 E2
+
193
*
El
E2
9
* v
A + +
(dA(El ))2)
(dv(E2 ))2)
given by
(R(El, E2)El, E2) =
Scal.
-
(El, Ei) (E2, E2)
-
(El, E2)
2
2
4.4 Submanifolds
pseudo-Riemannian manifold (M,g) and an (immersed) subM it is of interest which geometrical structure Z f: Z inherits from (M, g). Examples for physically especially interesting submanifolds are spacelike hypersurfaces (describing an instant of time, cf. Sect. 5.4) or the integrated light cone. In Chap. 9 closed trapped surfaces, a class of submanifolds of codimension 2, will play a central Given
a
manifold
-+
r6le. In this section
will
Remark
4.4-1.
f. Very
often the immersed submanifold is
we
explicitly a
refer to the immersion
subset of M and
f
is the
canonical injection. In this case it is more convenient to omit any references to f. We will often speak of tensor fields along Z instead of tensor fields
along f.
194
4.
Pseudo-Riemannian manifolds
We denote the space of vector fields along Z by To' (f) and will use the notation introduced ft-Alowing Lemma 2.9.2. In this notation, Lemma 2.9.1 is fundamental for the following. Lemma 4.4.1. Let
(M,g) be a pseudo-Riemannian manifold and f : Z ---> M be an immersed submanifold of M. Then the following holds for all vector fields U, V E To'(Z), X, Y Cz 'To'(f
0)
Vf" Uf. V
(ii)
d
Vf' Vf- U
-
X, Y) (U)
f. A V],
=
Vf.UX, Y)
=
X, Vf.UY).
+
Proof These properties follow immediately
(cf.
Lemma
2.9.1)
and the fact that V is
Definition 4.4. 1. Let
Z
f:
M be
--
an
a
from the definition of
Vf* UX
Levi-Civita connection.
immersed
I
pseudo-Riemannian
manifold and g be the metric of M. The map f (or the immersed submanifold
Z) is called non-degenerate non-degenerate (0) -tensorfield. A non-degenerate submanifold 2 is also called a pseudo-Riemannian submanifold, and a non-degenerate hypersurface is also called a pseudo- Riemannian hypersurface. Let Z be a non-degenerate immersed submanifold of (M, g) and denote by (T--Z)-L the set of all v E Tf(.,)M with g(v,w) 0 for all The decomposition Tf(.,)M w E f* 7:,,Z. induces f*T.,Z 0 (T if f *g
is
a
=
=
projections v
such that
T v
=
v
__>
+
vTE f*TZ v' for
Lemma 4.4.2. Let Z be
(M, g)
and U, V E
every a
To' (Z).
and v
is the Levi- Civita connection
Proof Observe that VUV
(i)-(iv)
2.7.1)
is
_L
Tf (,,) M.
non-degenerate immersed submanifold of VUV defined by
('7 f*Uf* V)
of (Z, f *g).
imply that
f
is
the Koszul
an
immersion.
Proper-
equation (cf. Equation
a
I
non-degenerate
immersed
submanifold of
The map 1:
symmetric
tensor.
(T. Z)
G
is satisfied for V.
Lemma 4.4.3. Let Z be
(M, g).
v_L
is well defined since
in Lemma 2.9.1
2.7.7 in Theorem
-4
Then
f*vuv:=
ties
E
v
IV (Z)
X
I
Tol (Z)
-
(TO, M)
(U, V)
1-4
I(U, V):=
in U and V and
(,7f*uf*v)
Junction- linear.
I is called the
shape
4.4 Submanifolds
Proof.
The map _Y is
clearly function-linear
in its first
195
entry. Recall that
the Lie bracket of any two vector fields tangent to Z is itself tangent to Z. Hence Y (U,
0, which
V)
-
_Y (V,
U)
(V f.
=
implies that
in turn
U f,, V
Y is
U)
V
f. V f,, symmetric. But then -
(f,, [U, V])
=
-L =
ff must also be
I
function-linear in its second entry.
have considered vector fields tangent to Z. We obtain similar relationships for vector fields normal to Z. So far
we
a non-degenerate immersed submanifold of field on Z and N be a vector field along Z (Tf(x)f,,Z)J- for all x E Z. Then Vf.VN is well
Lemma 4.4.4. Let Z be
(M, g).
Let U be
such that
defined
N(x)
and
a
E
vector
have
we
Vf,, VN where
0
denotes the
(Vf*VN)
=
(I(V, .), N)
lift of indices with respect
to the induced metric
f *g.
immediately from the coordinate expression that Vf VN does not depend on the extensions of V, N off Z. Hence it is well defined. Let X be a vector field along Z and XT, X-L its tangent and normal Proof
It follows
*
part. Then
we
have
(Vf*VN, X) (Vf*VN, X-L)
+
=
Vf*VN, XT) =0
((Vf* VN) _L,
=
-
=
11
X
+
Vf*V N, XT)
N, V f* VXT )
((Vf*VN)-J-,X-L) N, (V VXT) (Vf*VN) X--L) N, y(V,XT)) -
f*
-
Lemma 4.4.5. Let Z be
a
non-degenerate
immersed
Z be a smooth curve. For (M, g) and let -y: [a, b] -L (T_ (a) Z) there is a unique vector field N along f -Y --*
o
(i) N (a) (ii) N(t) (iii) and
=
E
n,
(f*Ty(t)Z)-L
(Vf* N)
1 =
for
0.
all t
c
[a, b],
submanifold of
every vector
with
n
196
4. Pseudo-Riemannian manifolds
We will write
transport of Let
Proof.
which span N
N(t)
P-L
=
Ej,..
-,
to these
-L
at every
Fj: [a, b]
in
point. We
(TE)
-L
(Vf* N)J-
=
can
N(t)
as
R such that
--+
decompositions we obtain
the equation
to it
as
the normal
be orthonormal vector fields
En-dim(_v)
(T,,(t) Z)
functions
refer
parallel
n.
along f o-y with values
are
and
n
decompose
(Vf N)Ja
o
-y
any vector field
Ni (t)Ej (t). Similarly, there
=
1
(Vf* Ej)
0 reduces to
along f
=
TjEj.
==
With respect
(- -N'+NjFj)Ej. Hence dt
first order system of
ordinary uniquely determined by its
differential equations and each solution is initial values N'(a) n'.
I
=
Proposition 4.4.1 (GauB equation). Let Z be an immersed non-degenerate submanifold of (M, g). Denote by ER the Riemann tensor of (Z, g) and let U, V, W, X E 701 (Z). Then we have f
*
g
(-'-R (U, V) W, X) =
Proof.
(R(U, V)W, X)
Since this is
a
+
tensor
(I(U, X), I(V, W)) equation
-
(I(U, W), I(V, X))
we can assume
that
[U, V]
=
0. We
calculate
(R(f* U, f* V) f* W, f* X)
'7f*U'7f*Vf*W' f*X) (,7f*V,7f*uf*W, f*X) Vf*uf* (VVW), f*X) ( ,7f* Uz (V, W), f* X) (vf*vf* (VUW), f*X) (Vf*V1(U' W), f*X)
=
-
+
=
-
-
f* (VUVVW), f*X)
=
f* (VVVUW), f*X)
-
=0
Vf*u (_Y(V1 W), f*X)
+
--;,0V
-
=
-
,
f*V (_Y(U1 W), f*X)
+
(f* (OR(U, V)W) f*X) ,
(YA W) I(VI X))
+
Since I is neither
I
a
tensor field
derivative of I is not defined
on a
(.Y(V, W), Vf*uf*x)
Z
-
(Y(U' W), Vf*Vf*X) (I(V, W), 1(U, X))
-
nor a
tensor field of
priori. However, it is
for any vectors u, v, w c T., Z and any vector fields w the expression u, V. v, W,, =
=
M, the
covariant
easy to check that
U, V, W with U.,
4.4 Submanifolds
(vf.wl) A V) depends only
on
Vf.W(I(U, V))
=
I(VWU, V)
-
the values of u, v, w. This of 1.
justifies
-
197
I(U, VWV)
to call V_Y
defined
as
above the covariant derivative
Proposition 4.4.2 (Codazzi Equation). Let Z be a non-degenerate immersed submanifold of (M, g) and let U, V, W E To' (Z). Then we have
(R(f,,U,f.V)f,,W)-L N be
Proof. Let
=
(Vf.U-ff) I(V, W) (Vf.V_iT) I(U, W). -
vector field
a
Z which is
along
orthogonal
to TZ.
(R(f, U, f,,V)f,,W, N)
Vf.UVf,,Vf.W, N) (Vf.VVf.Uf*.W, N) Vf*[U,V] f* W, N) Vf*Uf* (VVW) N) (Vf*UI(V, W), N) Vf*Vf* (VUW) N) Vf*V.Y(U, W), N) -
-
+
,
-
,
(1([U, V], W), N)
I(U, VVW), N) -
-
+
I(V, VUW), N)
I(VUV, W), N)
-
+
Vf*UI(V, W), N) Vf*VI(U, W), N) (I(VVU, W), N)
(Vf*UI) (V, W), N) The assertion follows since N in
was an
-
(Vf*VI) (U, W), N)
arbitrary
vector field with values
(TZ) 1.
For
some
1
purposes the
shape
tensor is too
complex and also too rich in simpler geometrical quantity one can the shape tensor over all direction in the
information. In order to obtain average at
a
given point
x
C-
Z
a
submanifold. Consider can
then
Sk-1
=
a
k-dimensional Riemannian submanifold Z
IV
G
Tx Z
*
:
(f g),, (v, v)
=
I J, This set is
a
submanifold of the Euclidean space
(TxZ, (f *g).,).
the volume form of Sk- Iconsidered
as
good
M. We
C
the set of directions in T,,Z with the unit
identify
definition for the average of I
Average
VOI(Sk- 1)
We denote
submanifold of
over
sphere
compact Riemannian
by *
psk-,
(Tx Z, (f g),,).
A
the directions in T.,Z is then
fS2
')I-tS2,
198
Pseudo-Riemannian manifolds
4.
where the vector space valued integral is defined as in Remark 2.5.2. This method would not work for pseudo-Riemannian manifolds which are not Riemannian since in this to
We will to
case
the set of directions does not
correspond
compact pseudo-Riemannian submanifold of T,,Z.
a
a
now
calculate the average in the Riemannian case. This leads can be straightforwardly generalised to arbitrary
formula which
pseudo-Riemannian submanifolds. For simplicity we only consider a 3dimensional submanifold Z C M. The general case is analogous but calculationally more elaborate. Recall from Example 4.0.1 that we can parameterise a dense open subset of S2 using the chart map o given by
Cos 0 I
(-7r, 7r)
(0, 27r)
x
S2,
_4
(0, 0)
sin
-4
Cos
0
0 Cos 0
sin 0
where T.,Z is identified with R 3 via the induced metric
noting 1, (gS2)OW
on
2
0, (gS2)00
:--
orthonormal basis
I Cos 01dO do. Let e4, .... e,, be are bilinear forms P with A
we
have
(gS2)00
:_-
0. Hence Lemma 4.2.3
COS
:--
an
S2 by gS2
orthonormal basis of
an
je-1, e2, e3j. De(gS2) (0o, ao) implies
(T,,Z)-L.
tLS2
There
n
Y Ii (V, W) ej
1" (V, W)
i=4
for all v, w E T,,M. Since the volume of S2 is obtain for the average of I.,
given by
fS2 ItS2
:--
47r 3
we
ej
47r
S2
i=4
Cos
7r/2
n
3
j j27r
4,7r
/2
i=4 n
+ 2 sin
0
Cos
0
Cos
0
cos(O)dodO ej
+ 2
Cos
2 0 sin 0 Cos offi12
0
0 Cos 0 sin 01i23
2
0 COS2 Oyi22
2
offi
+ sin
+ 19""
33) cos(O)dodO
ej
7r/2
3
Ef 4,7r i=4
( Cos 3 07r_yiI,
+
7r/2
n
(Yi1I
+
'ri22
+
-R'i33))
i=4 3
Ef j=1
,
sin
sin 0
0 COS2 0.yi1 I
0 Cos 0 sin 0 lil 3
Cos
Cos 0
2w
2
-7r/2
0 Cos 0 0 Cos 0
sin 0
Cos
i=4
n
sin
0
w/2
1: 47r + 2
fl,
*
g (ej,
ej).B' (ej, ej).
e-i
3 Cos
0,7r_yi22
+ 2cosO sin
2
0,r yi
33)
Cos
OdO ej
4.4 Submanifolds
This
199
justifies the following definition.
Definition 4.4.2. Let
f: Z -+ M be a non-degenerate immersed subpseudo-Riernannian manifold (M,g). The mean curvafield H is defined by
manifold of ture vector
the
dim(Z)
I
E
H.,
dim(Z) where
f el
7
edim(Z) I is
f *g(ei, ei)-Y(ei, ei),
i=1
orthonormal basis
an
of TM.
It is easy to see that H,, does not depend on the choice of orthonormal basis. The mean curvature vector field plays a prominent role in the
investigation of black holes (cf. Chap. 9) and is closely linked to the theory of minimal surfaces (cf. Lemma 4.4.8 below). The normalisation 1
factor more
is
dim(Z)
more common
in the mathematical literature than the
alternative 1. Since unusual normalisations
logical
then bad normalisations
we
are even worse
have retained this factor.
Definition 4.4.3. Let Z be
non-degenerate immersed submanifold of Z ---+ TZ--L along Z with (n, n) 1 is called a normal vector field or simply a normal The shape operator S,,: T, Z -4T Z of Z associated with n is defined by S,,u, v) .Y(u, v), n). Rn the orthogonal projection. Then the secDenote by Irn: (T Z) ond fundamental form k. corresponding to n is defined by ir,, (1 (u, v))
(M, g).
A
(local)
vector
a
field
n:
=
.,
=
,,
k (u,
v) n.
Definition 4.4.4. Let Z be
shape of (M, g). degenerate hypersurface The term second
non-degenerate immersed hypersurface fundamental form of a nonsimply denoted by S and k. a
operator and second
The
are
fundamental form
from the
theory of hypersurpredates Riemannian gea non-degenerate hypersurface Z is Idimensional. Therefore there exists an (up to sign) unique normal vector field n: Z --+ TM along Z. Hence even without choosing a normal, k and S are uniquely determined up to sign. In Euclidean geometry, the induced metric g is called first fundamental form since it f allows to determine fundamental geometrical quantities such as angles and lengths. The second fundamental form k is another (o)-tensor field. 2 Since the GauB equation reduces to
faces in Euclidean space (R3, ometry. The normal space of
comes
which
=
(ZR(u, v)w, t)
=
k(u, t)k(v, w)
the curvature of the surface Z is k also determines the
shape
-
k(u, w)k(v, t),
completely determined by k. Moreover, a hypersurface is
tensor and therefore how
200
4. Pseudo-Riemannian manifolds
curved in space. Hence k is indeed a geometrically fundamental quantity justifies the name second fundamental form.
which
Lemma 4.4.6. Let Z be Then the
shape operator
damental
form by k(u, v)
S is
a
is
non-degenerate hypersurface with normal n. given by Su -Vf*un and the second fun=
=
(Vf*un, f*v) (n, n).
-
The
self-adjoint and the second fundamental form k
Proof. Let V be
(ff (u, v), n)
vector field with V,
a
Vf*uf*V, n)
=
v.
=
Then
(f*v, Vf*un).
is
we
shape operator
symmetric. have
(Su, f*v)
The normalisation,
(n, n)
implies( Vf*un, n) fundamental form
k(u, v)
=
we
=
self-adjointness
ff (v,
U).
analogy
to the
average of the
.
of S and the symmetry of k follow from
curvature vector field
mean
one
can
_Y(u, v)
introduce the
shape operator.
Definition 4.4.5. Let
surface.
('7f*un, f*v) (n, n)
(ff (u, v), n) (n, n)
The
In
0 and therefore the first assertion. For the second
calculate
Then the
f:
mean
Z
--
M be
curvature is
non-degenerate immersed hypergiven by H(x) n'Itr(k,,). a
=
The most important mathematical application of this concept arises in theory of minimal submanifolds.
the
Lemma 4.4.7. Let
Then the
(M, g)
be
an
a
yz
(which implicitly defines Proof. -n
oriented
pseudo-Riemannian manifold pseudo-Riemannian hypersurface in M with normal n. volume form of Z is given by
and Z c M
It follows
an
directly
=
n-lym
orientation
of Z).
from Definition 4.2.1 that either nJ pm
J /,tm is the volume form of Z.
The rest vature
of this section is an illustration of the concept of primarily directed to mathematicians.
mean cur-
or
4.4 Submanifolds
(M, g)
Let
be
oriented, Riemannian manifold and
an
consider
a
201
compact
submanifold B C M of codimension 2. We denote the space of all smooth hypersurfaces Z C M whose boundary is B by 9X(B). Then the question
naturally arises which Z E TZ(B) has minimal volume vol(Z) f, Mz. In general, there may not be a minimising hypersurface or it may not be unique. Nevertheless, it is relatively easy to derive a necessary condition any minimising hypersurface must satisfy. Assume that Z is a minimising hypersurface and let U be a vector =
field such that for all
U(x)
h(x)n(x)
=
B then the flow
for
function h: Z
some
Ft of U generates
-4
R. If
h(x)
=
0
smooth
I-parameter family Ft (Z) of hypersurfaces in 9X(B). Hence a necessary condition that Z has minimal volume is given by (IT ) 0 /-t Ft, (Z) t 0 fFt, (Z)
Zt
x E
a
=
=
-
=
(M, g)
Lemma 4.4.8. Let
and Z be be
a
an
vector
oriented
field
in
a
be
an
oriented
pseudo-Riemannian manifold
non-degenerate hypersurface with normal neighbourhood of Z which satisfies
(i) n(x) fi(x) for 1 (ii) (ii, fi) =
all
E
x
n.
Let ii
Z and
=
and h be
a
function and d
)t=o
atwhere H is the
Proof.
mean
Since all
U
=
Mi. For the
(n
flow Ft of
U
we
have
qzhHpz,
-
Ft (_1)
J z
curvature
of Z and
77z
=
(n, n)
E
objects are smooth we can interchange differentiation integration over Z and obtain, using Lemma 2.5.2,
with respect to t and d dt
d
)t=o J
M Ft (_)
=
Ft, (Z
Ft
=
=
=
f E,.... Enj
be
an
=
-
J pm +
U
dt
+
we
t=O
J
n
J pm + div
n
J
J pm
=
=
ii and
get
n
=
X
=
orthonormal frame with E,,
be the dual frame. Then
div(ii)
"
,
=
Let
d
(p -) ( ( ) Ft"pfz -CU(Az) fz (n AM) JXUttm fz [U, n] (hii) ym) fz Qhfi, n] fz (-dh(n) div(hii))n fz hdiv(ii)pz.
n-1
E Oa(V Ea ii)
E Oa(V Ea ii) + on (,V EnEn
a=1
a=1
01,..., on
4. Pseudo-Riemannian manifolds
202
=0
n-1
(5Ea)
0
+
En, VEnE.,
a=1
(n
-
1) (n, n)
Since for each function h with
H.
obtain
I-parameter family of hypersurface can only 0. For if there would be a point x E Z \ JBI extremise volume if H with H(x) =A 0 then there would exist a neighbourhood Ux of x in Z and
hypersurfaces Et
E
9R(B)
hIB
0
=
with Zo
=
we
Z
a
a
smooth
=
a
function h with
(i) h(y) 0 for all y (ii) h(y)H(y) > 0 for (iii) h(x)H(x) > 0. =
This would
imply
E
B U
(Z \ U ,),
all y E B U
f,,qEhH1,tE 4
Z,
0 in contradiction to the
extremality
of
vol(Zo). The equation H 0 can be understood as a differential equation for a function f which describes the hypersurface Z. It is a classical =
problem in differential geometry to monograph on this subject (for M
solve this equation. A ==
R')
is
comprehensive
(Nitsche 1975).
Hyperquadrics
4.4.1
we study the simplest non-trivial class of hypersurfaces of Rn. These examples will be used in Chaps. 7 and 6.
In this section
Let V
,q,: Rn
Rn
x
--
n
E Xi Yi
(X) Y)
R,
pseudo-scalar product
Xk yk.
k=v+l
j=1
be the standard
E
+
of index
v.
Proposition 4.4.3. The pseudo-Riemannian manifold (R' \ 01, 71v) is I x E Rn \ 101 : q,(X, X) C1, foliated by hypersurfaces Quadv- (c) where c E R. The hypersurface Quadn,-'(c) is non-degenerate if and only if C =h 0. n
=
These
Proof.
hypersurfaces Quadvn-i(c) It is clear that every
Quadvn-i(c). c.
Since
df
x
are
G
called
Rn
hyperquadrics.
\ fOJ
lies in
ThesetQuadn-l(c) Ej=, xjdxj + Enk=,+, xkdxk
isthezerosetofx
V
exactly -4
f,(x)
=
0 the map
2.1.1
n
subset
77,(x,x)-
does not vanish unless
f has constant rank in Ix E Rn : x 7 1 implies that Quadv (C) is a hyp ersurface Proposition The tangent space at x is given by Jv G Rn : 71, (X, V) 01.
x
one =
01.
-
=
of Rn
Hence 01
4.4 Submanifolds
Assume that
c
0
=
Then
-
have 77, (x,
we
x)
T, Quae,
(c). Since for any tangent vector induced metric is degenerate. Assume
that
now
with 77, (w, v) 0, this would
=
0.
Tx Quad'v
c: -4
(c).
a
0 and therefore
have q, (x,
vector
v)
=
G
x
0 the
.,Quad'-'(c)
w
V
Since
0 for all y E
have also 71, (x, w) R'. This is impossible since we
0. The map
Quad'v-1(c)
-->
(xi I...Ixv, xV+1 I...IXn)
__,
t:
an
=
I
Lemma 4.4.9. Let
v we
Suppose there was
v E
imply qv (W Y) non-degenerate.
qv is
is
=h
c
0 for all
=
203
Quad,,:vl(-c), (Xv+1.... x-, xi n-v
I
I
...
Ix V)
anti-isometry.
Proof. The hypersurfaces QuadL.n-1(c) and 77v (X,
for all
x
X)
Rn. The map
E
vectors u,v
t
n-v
-?7n-v (44
:--
is an
TxQuad'-'(c)
E
Quad'-'(-c) 4x))
since for each
anti-isometry
TxQuadn:,'(-c)
=
coincide since
have
we
pair of
Tlv(v,w)
-?7n-v(t*V, L*W)Lemma 4.4.9
implies
hypersurfaces
Quad',,-'(c)
Definition 4.4.6.
that it is with
Possible
to restrict attention to those
0.
c
The
Pseudo-sphere of dimension n, index V, and rahyperquadric Sn-i(r) Quadn-l(r'). If v 0 we write Sn-i(r) instead of Son-'(r) and Sn-1 instead of Sn-1(1). dius
r
is the
=
=
V
V
Lemma 4.4.10.
Sn-i-v,
where
The pseudo-sphere Sn,-(r) Sk denotes the k-dimensional
is
diffeomorphic sphere.
to R'
x
unit
Mr: Rn --> Rn, x _, -lx maps Sn,-'(r) onto It is therefore sufficient to prove the lemma for r 1. ConS,"(1). sider the map f : R' X Sn I v -4 Rn' (y, Z) _4 (y, + Iy z). The
Proof. The diffeomorphism
=
-
equation 77,, (f (y, z), f (y, z)) set of Sn-
1
(1).
I
implies
that the
f has also
a
image of f smooth inverse map, f
hence it is in fact
a
diffeomorphism
The map
(y,,\/ll_+Jy _li),
V1
-
Lemma 4.4.11. Let
c
of
11c.
constant curvature
=k
=
0. The
-
hyperquadric
onto
Quadn-l(c) V
is 1
a
sub-
(y, i)
Sn-1(1).
is
a
=
I
manifold
Pseudo-Riemannian manifolds
4.
204
Proof.
The
is
shape operator
given by
( V_ICI x
Sv
whence the
shape
=
-D
v
M
V_ICI
tensor reads
("/-ICI ) u
1(u, V) and the GauB equation
=
-77V
,
c
x
Ici
V-jCj
v
(Proposition 4.4.1)
reduces to
C
77, (R(u,
v)v, u)
4.4.2 Umbilic and
=
ICII
( V (U, U),qv (V, V)
totally geodesic
_
nV (U,
V) 2)
submanifolds,
The investigation of submanifolds is a classical field in differential geometry. Naturally, one concentrates on submanifolds whose shape tensor is of especially simple form since only for these classes one has
of at least a partial classification. In this section we collect elementary definitions and results which should be covered in a course of differential geometry. Readers who are primarily interested in physical aspects can skip
a
chance
those
this section.
M be an immersed pseudo-Riemannian f: Z pseudo-Riemannian manifold (M, g). A point x E Z is called umbilic if there exists a vector n E T,,M with f *g(u, v)n for all U, v E T,,Z. The submanifold Z is called _Y(u, v) (totally) umbilic if all points in M are umbilic. 0. The submanifold Z is called totally geodesic if I
Definition 4.4.7. Let
submanifold of
--
a
=
=
M be an immersed non-degenerate subf: Z manifold of the pseudo-Riemannian manifold (M, g). Then the following statements are equivalent.
Lemma 4.4.12. Let
-+
(i) Z is totally geodesic; (ii) For every curve -y C and
only if f
o
Z
we
=
(iv)
For every
The
a
geodesic of (Z, f *g) if
a
For every v with (O) f,,v then there is
(iii)
have: -y is
geodesic with respect to (M, g). E T,,Z we have: If -y is the maximal
-y is
curve
-y:
[a, b]
a
6
--->
parallel transport of v along
> 0
Z and every -y
geodesic
in M
7Q-J,6]) C f (z) v E Ty[al Z we have:
such that
satisfies
f.P,,(v)
=
Pf ., (fv).
4.4 Submanifolds
Proof. (i)
<-*
Recall that V
(ii):
implies that the curve Conversely, from the symmetry I
0
=
I
=
0
T,Z there
a
of
v
Hence the
=
(O)
_Y( , ). Clearly,
+
only
if
f
o
-y is.
infer that
we
ff (V,
,y
if and
V)
=
0 VV. since for each
v
f.V
and
curve
P'Y(V)
E T,,Z.
17f. f.P'Y(V) (iv)"
equivalence "(i)
v
-
The
parallel transport
1( ' PY(V)).
follows in the
same
way
as
"(i)
4=>
1
00". The
E
v.
=
Trivial.
Let -y be a along 7 satisfies 0
f,, V
decomposition above
-y with
geodesic
(iii): (iv):
(ii) (i)
P,(v)
is
=
geodesic
a
of I
' *
The assertion follows from the
f,,
f.
-y is
205
following lemma shows that totally geodesic submanifolds by the infinitesimal neighbourhood of a single point.
are
char-
acterised
M be two totally -M and f: Proposition 4.4.4. Let f: Z geodesic submanifolds of the pseudo-Riemannian manifold (M, g). If there with f,,T,.,Z f,,T t then there are neighbourare points x G Z, -, E hoods U of x and 1 of _;i- with f (U) f(Z ). -->
=
=
Since expx is
Proof.
a
local
diffeomorphism
near
Ox
E
TxZ there
are
neighbourhoods )IV, V_
of x, i which are swept out by geodesics through of M by x, Jc. The image of these geodesics under f, f are geodesics of two sets of these Lemma 4.4.12. Since the tangent vectors geodesics at
is
f (x) a
each form
neighbourhood
4.4.3
U of
a
neighbourhood of 0
f (x)
=
f(x)
with U n
E
f,,TxZ
f (W)
=
=
fj;, , there f( V_). 1
U n
Warped products
Many standard spacetimes have a generalised product structure, the warped product structure. To write down curvature expressions for this class in general will save work in Chaps. 6 and 8. Definition 4.4.8. Let
(Z
x
F,,7rz*gz
+
(r
manifolds function. Then
be pseudo-Riemannian
(Z, gz) (F7 9F)
of dimensions nz, nF and r: Z -4 R+ the pseudo-Riemannian manifold
\ 101
0
be
a
smooth
7rZ)2 7rF*gZ)
,
where 7rE: Z x F --> Z and 7rF: Z x F -- F) are the canonical projections, is called the warped product of Z and F with warping function r
206
4. Pseudo-Riemannian manifolds
We
identify vector fields X an Z (respectively, V on F) with the k -7'(T,,,_k 0 (respectively with f7 satisfying We Cali X G' satisfying 7rz,,V V). (respectively, V) the lift of X (respectively, V). In the following, we will make use of this identification and denote both vector fields by the same letter. Further, for on Z x F there are unique vector fields X on Z and any vector field can
vector field
=
V
by
F with
on
=
X + V.
For every x E M we denote the submanifold Z Z., and the submanifold 17rz(x) I x F by F ,.
X, Y be vector fields on Z x F which U, V vector fields on Z x F which
Lemma 4.4.13. Let
fields vector fields
vector
f7rF(X)j
x
are
Z and
on
of Z
are
x
F
lifts of s of
li
F. Then
on
0) 7rz*vyx::= v7rz*y7rz*xl (ii) IrF*'7yX O (iii) VXU VUX d(Inr)(X)U, (U, V) grad(ln r), (iv) 7rz* VUV (V) 7rF*VUV VITF*U7rF*V. =:
=
=
=
-
=
Proof. These equations
(2.7.7)). (i), (ii): Since
be verified
can
using
the Koszul formula
(Equa-
tion
VXY, V)
that
F. From
=
therefore to 2 function
on
Z is
--+
2.4.4
get (7r-,) [X, V]
we
(VXY, V)
=
-V
=
=
=
*
X, Y)
-
van-
-
tangent
to the fibres
for all vector fields Y which
2
0 and
to show
to the fibre
Z the first summand vanishes. The second summand
=
formula
tangent
0. The Koszul formula
[X, Y] is tangent to Zx (iii): From [X, U] 0 we get VUX is
only have
we
are
(7F) [X)'VI 0 (2.7.7) reduces V, [X, Y]). Since (X, Y) is a
*
(7rF)*X
0 and
ishes since
VXU
isometry,
an
0 for all vector fields V which
Proposition
(7rz)*V
since
Z ,
-/rz:
=
=
are
The covariant derivative
VXU.
Fx since
Y, VXU)
=
lifts of vector fields
VXY, U)
-
on
=
0
Z. The Koszul
implies
(VXU, V)
==
X
(U, V)
=
X(r2 9F (Ui V))
2rdr (X)gF (Ui
=
V)
2 =
-dr(X) (U, V). r
(iv):
equation follows from
The
V'Vux)
(VUV, X)
=
-
(
1
dr(X)U
V, r
-1 (U, V) (grad(r), X)
.
r
(v):
For each
metric 91F is
a
x
E
Fx is a submanifold of M whose induced 2 r (X)gFmultiple of the metric on F, gjF,,'
M the fibre
constant
=
4.4 Submanifolds
207
Hence their Levi-Civita connections coincide and the assertion follows. I
Corollary 4.4.1. Let -y a geodesic if and only if
(i) V (ii) V
z F
Proof. We
F
can
at all t. Then
write
(t)
X(7(t)
is
we
have
assertion follows if
of
==
be
a curve
in Z
x
F. A
X (-y (t)) + V (-y (t)), where X, V
curve
-y is
V,;/
==
are
vector
and
V(-y(t)) is tangent to F,,(t) Zy(t) + VXX VXV + VVX + VVV and the
tangent
to
project this
we
Lemma 4.4.14. Let vector
(,yz, IF)
( F) F) gr ad (In r), -2d(lnr)( -v) F.
Z
fields such that
=
vector to
I
TZ,, and TF,
X, Y, Z be vector fields on Z x F which are lifts of U, V, W be vector fields on Z x F which are lifts
Z and
fields on fields on
vector
F. Then
(i) irz,R(X, Y)Z R-r(7-r.X, -7rz.Y)7rz,,Z, (ii) 7rF,, R(X, Y) Z 0, (iii) R(X, Y) U 0, (iv) R(X, U)Y -IVVr(X, Y)U, (U, V) VXgrad(r), (v) R(X, U)V (vi) R(U, V)X 0, (vii) 7rz,,R(U, V)W 0, (Viii) 7rF,, R(U, V) W RF (U) V) W +-' rT (grad(r), grad(r)) ((U, W) =
=
=
r
-
=
=
=
V
Proof. Assertions (i) and (ii) follow directly from Lemma
(ii).
-
(V, W) U).
4.4.13
(i)
and
-
(iii):
We may choose
R(X, Y) U
=
=
=
X,
Y such that
[X, YJ
=
0. Then
VXVyU VyVXU VX (d In r(Y) U) Vy (d In r(X) U) (VV ln(r)(X, Y) dlnr(VXY) + dln(r)(Y)d ln(r)(X)) -
-
-
(VV In(r) (Y, X) d In r(VyX) + d In (r) (X) d In (r) (Y)) U -
-
=
(iv):
dlnr([X,YI)U
Since
[X, U]
R(X, U)Y
=
=
=
0
we
=
0.
have
VXVUY VUVXY VX (dlnr(Y)U) dlnr(VXY)U -
-
U
208
4.
VV In
=
dlnr(VXY)U dlnr(VXY)U In d (VV r(X, Y) + In r(X)d In r(Y)) +
dlnr(Y)VXU
+ =
r(X,Y)U
-
U
I
vvr(X, Y)U. r
(vi):
We
R(U, V)X
directly calculate
can
=
=
VUVVX
-
VVVUX
VU(d
In r(X)V)
-
-
V[U,V]X
VV(d In r(X)U)
-
d In r(X) [U,
V]
2
=
VV(In r(X)) (U, V)
+ d In
2
-
r(X)VUV
-
VV(In r(X)) (V, U)
2
d In r(X)VVU
-
d In r(X) [U,
V]
0
where
we
(v):
have used that
Since
[U, V] (R(X, U)V, W)
=
VUV VVU. (R(V, W)X, U) -
=
0, the
vector
R(X, U)V must be
tangent
Z,,. The
to
(R(X, U)V, Y)
assertion follows from
-IVVr(X, Y) (U, V).
(R(X, U)Y, V)
r
(vii): (viii)
This follows from
R(U, V)W, X)
-
R(U, V)X, W)
Observe first that the Levi-Civita connection induced
=
on
0.
the fi-
bre F,
equals the Levi-Civita connection Of 9F since both metrics differ only by a constant factor r2 (x). The result follows from the GauB equation (Proposition 4.4.1) since by Lemma 4.4.13 (iv) the shape tensor is I given by 1(U, V) (U, V) grad(In r). =
-
Lemma 4.4.15. Let
fields vector fields vector
Z and
on
F. Then
(i) Ric(X, Y) (ii) Ric(X, U) (iii)
X, Y be vector fields on Z x U, V be vector fields on Z x
on
Ric (U,
V)
=
Ricz(irz,,X, 7rz,,Y)
=
are
F which
are
lifts of lifts of
'F'7'7r(X, Y), r
0,
=
x (U, V), (iv) Scal Scalz =
-
F which
Ar
RiCF (Ui V) + 1ScalF r2
+
r
2njr
zAr
(nr-1) r2
(grad(r), grad(r))
nl,,(nF-1) -2
(grad(r), grad(r)).
4.5 Isometries and
Killing
vector fields
209
trz(R(., X)Y)+trF(R(X, .)Y), (i), (iv) while assertion (ii) is a (i) Formula (iii) is implied by and 4.4.14 Lemma of (v). (iii) consequence Lemma 4.4.14 (v) and (viii) and assertion (iv) is just the metric trace of (i) and (iii). Since
Proof
follows
Ric(X, Y)
=
tr(R(., X)Y)
=
from Lemma 4.4.14
directly
4.5 Isometries and
Killing
vector fields p. 189
diffeomorphism which preserves the metric. pseudo Riemannian manifolds with many isometries are especially simple. The relevance to the theory of space and time comes from the fact that observations indicate that our universe is well approximated by Lorentzian manifolds with many isometries (cf. Chap. 6).
An isometry is
a
Definition 4.5. 1. Let
and
(M, g)
(1 1, j)
I [I
be ps eu do- Riemannian mani-
folds. An isometry is a diffeomorphism 0: M -- 1 1 which preserves the such that metric, (O*j) g. A local isometry is a local diffeomorphism 0 =
=
g., at all
points
Lemma 4.5. 1. Let
and U C then
0
=
11
be
0 if
x
(M, g)
M.
and
(1 1, )
be
pseudo-Riemannian manifolds
connected open set. If 0, 0: U --+ 1 1 are local isometries, and only if there is a point x G U with TO T,,V). a
=
The two isometries
Proof.
E
clearly coincide
on
the closed set V
ly
=
G
Ty,01. Since V is non-empty and U is connected, we only TyO need to show that V is open. Let y e V and W be a normal neighbour-
U
:
=
hood of y. Then for every z E W there is a vector expy(v[z]). But this implies O(z) O(expy(v[zl))
=
=
exp,b(Y)(TyO(y[z]))
=
V)(expy(v[z]))
=
O(z).
v[z]
E
TyM
with
expO(Y)(TyO(y[zJ))
Hence
01W
=
01-,V
=
=
and I
therefore W C V.
Definition 4.5.2. A
defines
z
Killing
vector field is
a
vector
field
whose
flow
local isometries.
manifold (M, g) is stationary in a region U c M if there Killing vector field in U. It is static in U if this Killing field is orthogonal to spacelike hypersurfaces.
Lorentzian is
a
timelike
vector
Clearly, only very special pseudo-Riemannian manifolds can have nonKilling vector fields. A simple example is given by a metric which does not depend on one of the coordinates. Then the corresponding Gauffian vector field is a Killing vector field.
zero
Lemma 4.5.2.
only if V '
is
A
vector
field
is
if.C g 0 Ve - ' de.
Killing if and only
antisymmetric. In this
case we
have
=
=
if and
p.
11
210]1
210
4.
Proof.
The first
Pseudo-Riemannian manifolds
is clear since the Lie derivative
equivalence
derivative
along the integral equivalence we calculate
Gc g)(U' V)
we a *
V
add to
-
of
.
(qu' V)
V
V 5 (V) U).
(u, V)
zero.
+
+
(vu ' V)
U' V V)
-
Levi-Civita connection the terms marked
a
It follows that
anti-symmetric.
U' X V)
-
V U' V)
-
is the
X
In order to prove the second
V (U, V) + (U1 VVO
have used that for
Here
with
is
-C A V)
=
=
curves
is
a
Killing
vector field if and
Now the assertion follows from
(de)ab
=
only
if
2V,,, bj.
[Fp.-209--q] p. 255
be Killing vector fields. Then [6, 61 is Proposition 4.5. 1. Let also a Killing vector field, i.e., the Killing vector field on a pseudoRiemannian manifold form a Lie algebra.
X
only to show that the commutator of two Killing vecKilling vector field. From Proposition 2.4.3 we know that , X, ] V) for any tensor 0. In particular we obtain I [ ' Mg
We have
Proof.
tor fields is
0
X 'C?7g
a
[X X77'CO
=
-
=
0
be
Lemma 4.5.3. Let
1,y
is
Proof
a
,
Jacobi
7(s) (s,t)
vector
field,
s
and
a
Killing
is
1--4
-4
geodesic,
Ft(^ (s))
-dt-Ft-y(s)
note that '7
)
a
+
is
vector
is
field
by Ft. Since Ft the a
curve s
F-->
variation of is
=
a
geodesic. Then
a
Ve ( ' )
is
an
isometry for each
Ft(-y(s))
is also
geodesics and
a
t
geodesic.
its deviation
Jacobi field. For the second property
anti-symmetric by Lemma
'V )
and -y be
is constant.
s F-+
Denote the flow of
Hence
and
field
0'
0
-
=
4.5.2. Hence
V ,-y) 1
0.
of this section we will investigate highly symmetric pseudomanifolds. These results are of independent mathematical interest and will be used in Chaps. 7, 6.
In the rest
Riemannian
pseudo-Riemannian manifold (M, g) 0. cally symmetric if VR Definition 4.5.3. A
is called lo-
=
implies that the components of R with respect allelly propagated frame are constant functions.
This definition
to
a
par-
4.5 Isometries and
Lemma 4.5.4. Let
be
(M, g)
Killing
vector fields
211
pseudo-Riemannian manifold. It is loonly if for every curve -y and all vector fields cally symmetric if wich W are parallelly propagated along -y the vector field R(U, V)W U, V, is also paralley propagated along -y. a
and
The
Proof. fields
equation VR
0
=
implies
for
parallelly transported
vector
U, V, W =0
V (R(U, V)W)
(V R) (U, V)W
=
+
R(U,V V)W
+
+
R(V U ,V)W R(U, V)
V W
0
along -y. Hence R(U, V)W is also parallel along -y. Let Conversely, let , u, v, w G T,,M and -y be a curve with (O) and that of W be the assume w along -y parallel propagation u, v, U, V, the vector field R(U, V)W along -y is also parallel. Then the assertion follows from =0
(17 R) (u, v)w '1'7 (0)(R(U, V)W) -R(V =
-
=
R(u, V
(O)V)w
-
(O)U, v)w
R(u, v) V,
(Off
0.
I
(M, g)
Theorem 4.5. 1. Let
and with
x
E
M,
,
E
AR(u, v)w
1 1. If =
and
(1 1, j)
there exists
f?(Au, Av)Aw
neighbourhoods U, 1 of x,.:
and
a
a
be
locally symmetric manifolds
linear isometry A: TxM -- T=1 all u, v, w c TM, then there are
for unique isometry 0:
U
with T
A.
Proof.
We
only need
to prove existence since
uniqueness follows from
neighbourhood U of x exp.,-' is a local isometry. First sufficiently small. By Proposition U for 2.6.5 there is a unique w., c T,,M with exp(w.,) every y E for every uy E TyM there is a unique iiw. G Tw.,,,. (TxM) y. Further, with Tw. exp,,(iiw.) Twx T,,M C Tw.TM there is a uy. Since U, Lemma 4.5.1. We will show that for
some
the map 0: U --* 1 1, y F--* exp, oA note that 0 is well defined if U is
normal
o
=
vector Ux such that
i1w.,,
Proposition 2.9.5 that
dt
uy,uy)
It=0 =
Ux). It follows therefore from J(1),J(1)), where J is the unique
(wx
+
4. Pseudo-Riemamnian.
212
Jacobi vector field
V J(O) 0. (uy)
=
=
along
the
-y: t
curve
ii,,. From the definition of 0 T expj
TATexp.,
-
1
(uy)
=
f (0, t) with J(O)
-4
we
0 and
=
get
T exp;z, TA
=
T exp,;,- (AUAw.,j
and
by the same argument as before it follows that there is a Jacobi j along the geodesic : t i--> exp(tAw,,) which satisfies J(O) 0, and j (0. (uy), 0,, (uy)) V J (0) Au, (J(I), j(1)). j ly
field
=
=
Let
=
f El, f kj,
E,, I be
parallelly propagated frame along -y with El orthonormal, parallelly propagated with Ei(O) frame along AEj (i E fnj). With respect to these frames the Jacobi equations for J and J are given by and let
.
.
.
.
,
.
.
,
&I
a
=
be the unique =
n
d2 ji +
dt2
n
d2 ji
Y R'IkI jk
and
-Tt-2
+
k=1
Y_ -I ilkl jk. k=1
is the unique geodesic with (O) we have used that A (O) which The functions R'1kj and R1kj are each constant implies that ti by Lemma 4.5.4. Since we assume AR(u,v)w R(Au,Av)Aw for.all R'1kI u,v,w E T,,M, the definition of our parallel frames implies R'1kI jk (0) for all i, k. Further, the functions ji, jk Satisfy jk (0) 0 and
Here
=
=
=
=
(by
the definition of
our
frames) Ajk(O) dt
=
_dt _jk(O).
mental theorem for differential equations 2.4.1 all k and we get
=
Hence the funda-
implies jk (t)
=
jk (t)
n
(j(j)
j(O.U Y, 0. UY)
,
f(11: ji(j)jk(j)j(t,(j)j kk (1)) i,k=l
n
=
E ji(j)jk(j)j(ki(0), kk (0)) i,k=l n
ji (1) jk (1)j (AEj (0), AEk (0)) i,k=l n
=
1: ji(j)jk (1) g (Ei (0), Ek (0))
=
9 V(1) 1 J (1)
i,k=l
g(UY' UY). and the assertion follows from the
g(u, V)
1(g(U + 2
V,
U
polarisation identity +
V)
-
g(u, U)
-
g(V, v)).
for
4.6
Length
and energy functionals
213
Proposition 4.5.2. Let (M, g) and (1 1, j) be pseudo-Riemannian manifolds with constant curvature c and a. They are locally isometric if and a. only if they have the same dimension and signature and satisfy c =
Proof. Observe first that the conditions are necessary. We show now that a pseudo-Riemannian manifold with constant curvature is necessarily locally symmetric. Let t, u, v, w E T'M and U, V, W vector fields which satisfy U.., w and whose covariant u, V v, W derivatives vanish at x. From Proposition 4.3.3 we get =
=
,,
(VtR) (u, v)w
=
Vt(R(U, V)W) R(VtU, v)w R(u, VtV)w R(u, v)VtW -
-
=
If
(M, g) x
any
If
(k, )
and
Tx M, 5
G
(M, g)
E
cvt ((V, W)
have the
same
T k there exists
(1 1,
and
-
have the
a
U
(U, W) V)
-
Corollary non-zero
constant curvature then this isom-
same
k(Au,
A
global
classification of
dicates that
Hyperquadrics have
orthonormal bases
TyQuadn-l(c)
:
c
Let
:
f,,.
0
4.6
0
to
Rn It is
--+
an
Quad,
an
(Wolf 1977). a
very
manifolds with constant The
following Lemma large isometry group.
in-
Quad,n-1(c). For any pair of TxQuadvn-1(c) and ffj,...'fn} C
and x, y E C
isometry 0:
Quadn-l(c)
--->
V
Quad'-'(c)
with
V
Rn be the linear map which maps x, ej,...' en to isometry of (R',,q,) onto itself and therefore also
In this section
functional mannian many
itself. Since
is the desired
(c)
Length and
for
1
fi (i Ell,..., nj).
onto maps-Quadvn-1(c) n-1
of
0.
fel,...,enj
there is
V
Proof. Y, fl,
TxM and the
(M, g).
pseudo-Riemannian
difficult
more
Lemma 4.5.5. Let
0,(ei)
E
pseudo-Riemannian manifold with a hyperquadric Quad, n-1(c)
a
isometric to
locally
curvature is much
=
be
w
constant curvature. Then there is
which is
(c: 0)
(M,g)
4.5.1. Let
0.
dimension and signature then for linear isometry A: Tx M -- T;-
Av)Aw for all u, v, etry satisfies AR(u, v)w assertion follows from Theorem 4.5.1. =
=
0*(ei)
=
0(ej)
=
fi, the
restriction
isometry.
energy functionals will
study the problem of extremising the length sufficient conditions in the Rieand the Lorentzian case. Here we will lay the foundation surprisingly strong global theorems in differential geometry and
we
find
necessary and
1
214
4. Pseudo- Riemannian manifolds
(e.g. the Theorem of Myers,cf. (ONeill 1983, theorem 10.24)) and for the singularity theorems in general relativity (cf. Chap. 9). This section is
mathematically more involved than the other chapter and can be skipped on first reading. section uses material from Sect. 2. 9.
sec-
tions in this This
In Riemannian
geometry the length of
would need to model the
one
curve
a curve measures
in space. It is
how much wire
fundamental geo-
a
metrical experience in Euclidean geometry that for any given (not too distant) pair of points there is a curve of shortest length which connects them. In Lorentzian geometry the
length of a causal curve can be interthe time observer needs in order to traverse this an preted proper world line. Since in special relativity moving clocks are slower (twin paraas
doxon)
expects that for any
one
point there is
curve longest For other signatures the problem
to non-trivial results
(cf.
(not
two
causal
a
Lemma
of
curves.
The
distant), causally
related
extremising length does
not lead
4.6.9)
For the discussion in this section it is
widen the class of admissible
too
which connects them.
curves
to the
technically advantageous continuous, piecewise
to
smooth
advantage lies in the fact that in many situations it is much a continuous, piecewise smooth curve with certain
easier to construct
properties than
smooth
a
Definition 4.6.1. Let -y:
[a, b]
--
M be
Then the
curve.
(M,g)
be
a pseudo-Riemannian manifold and continuous, piecewise smooth curve in M.
a
length of 7
This definition makes defined derivative
is
b
defined by L(-y) since
sense
everywhere but
a
piecewise smooth
on
a
set of measure
of the chosen parameterisation. In the
pendent
V/j__ g ( (t), (t)) I dt.
case
has
curve
a
well
It is inde-
zero.
of Euclidean space
it coincides with the
length one would define through the approximation of -y by polygons. The following lemma guarantees that there are no repercussions in considering piecewise smooth curves instead of smooth curves.
Lemma 4.6.1. Let -y: [a, c] --+ M be a piecewise smooth there is a sequence of smoothly immersed curves -yi: [a, c] converge
pointwise
Proof Assume that M and \:
[b, c]
(Xl,...,Xn) that b
-
2-'11
2.1.7 there
---
to -y and
are
a
Then
M which
limi-,, L(-Yi)
=
L(-Y).
-y is the concatenation of two smooth
M where
such that A is >
satisfy j(t)
curve. -->
and let tj
ft(b)
=
A(b).
given by =
b
-
smooth functions
t
curves M: [a, b] ---> coordinate system (t,0,...'0). Let io (E N such
We choose
2-' where i
Wj,,0j: [a, c]
>
a
io and i
-+
[0, 11
E
N.
By Lemma
such that
4.6
Oj (t)
>
Oi (t)
We define the
I
for all t E
1
for all t <
2
=
0
for all t
=
0
for allt E
> -1
for all t >
1
for all t E
2
pi with
curve
and energy functionals
Length
respect to
215
[a, ti- I], ti+j], [b, cl,
E
[a,ti-1], ti], [b, c], coordinate system
our
by
t
/-t'(ti-i)
+
I
(W,(S) (Ak(s)
ti
cik
where the constants Notice that
1,t(t)
smooth there is
jy"(t) t E
=
Iti+14
-
c
c
for
-
by
C O,(S))
+
jkO, (s)) ds,
the condition
for t < ti-I and
number
a
/-t(b)l < c1lb tj [ti, ti+,] we obtain -
determined
are
-yi(t)
+
I
-
A(b)
-yi(t) A(t) for t > b. Since /-t is > 0 such that I Ak (t) I < c and I yk (t) -A (b) I all t E [a, b]. Hence using W(t)oi(t) > 1/4 for =
=
b
ti
c Wi(s)Oi(s)ds
<
Ak (b)
Ak (ti- I I
+i.-
b
W,(S)Ak (s)ds
ti-I
5,koi(s)ds
+
<
b
From ti
uniformly
Corollary
[a, c]
2-1
we
get
IcikI
-
lengths of
M is
+
1).
Since
i, the
curves
-yj converge to the
length
4.6.1. Assume that
-
tj+j).
2(2c
:5
bounded with respect to
to -y and the
-y:
-
(2c + 1) (b
(M,g)
is
a
cik, oj(t), Oi(t)
-yj converge of -y.
pointwise I
Lorentzian metric and that
piecewise smooth future directed causal
curve.
there is sequence of smoothly immersed timelike curves 7j: [a, which satisfies j(t) -4 (t), -yi(t) --> -y(t), and L(-yi) --- L(7). We
Proof.
can assume
nation of two smooth
[t(b) cone.
=
A(b)
There
such that both are
to p and A and
Lemma 4.6.1
without loss of curves
are
generality
cl
Then -->
M
that -y is the concate-
M and A: [b, c] M with [a, b]. A(b) and A(b) point into the same future mu:
--
sequences of timelike
curves
--->
pi and
satisfy Mi(b) Ai(b) p(b). For provides a sequence -yi,j of curves ==
=
Ai which pair
each such
converge of curves
such that -yij converges
4.
216
,nanifolclms,
to the concatenation of Mi and
future directed
so
is 7ij for
Ai. Since both, Ai and
j enough.
We
can
pi
assume
are
timelike and
without loss of
generality that
all - ij are timelike. It follows that the sequence consists of timelike curves and converges to -y.
4.6.1 Variation of
length
f-Yi,ijiEN
and energy
In Euclidean space, the shortest curve between two points is the straight line connecting them. In Minkowski space, the longest causal curve between two
points
x, y E
I+ (x), is also the straight line connecting them.
Fig.
4.6.1. A broken
smoothed out
In
a
by
a curve
lightlike geodesic can be of arbitrarily small length
Riemannian manifold "without holes" it is
general
intuitively
clear that any two points can be joined by at least one shortest curve. In a Lorentzian manifold, the infimum over the length of all curves which connect
x and y is always zero since we can join any two points lightlike geodesic which then can be smoothed out to give a smooth curve of arbitrarily small length (cf. Fig. 4.6.1 and Corollary 4.6.1). It is also clear that there does not exist a curve of maximal length connecting x and y since we can always choose a spiralling spacelike curve of arbitrarily large length (cf. Fig. 4.6. 1). However, we will see below that in many situations there exist curves connecting causally related curves
by
x
a
broken
and y which maximise L in the class of all causal curves. 5 pseudo-Riemannian manifolds which are neither Riemannian
rentzian do not admit any non-trivial solutions to the
problem,
even
if
one
restricts to
spacelike
or
nor
Lo-
length extremising
timelike
curves.
These
ar-
guments will be made precise in Lemma 4.6.9 below. 5
Our a
examples
solution if
also
one
imply
that the
restricts to
length extremising problem does not spacelike curves instead of causal curves.
have
Length and
4.6
Fig.
energy functionals
4.6.2. A
curve
distance between two
Z1
217
minimising the spacelike sub-
manifolds Z, and Z2
actually investigate the slightly more general problem where endpoints x, y are replaced by submanifolds without boundary Z1, Z2 (cf. Fig. 4.6.2). In order to solve the length extremising problem in the Riemannian and in the Lorentzian case we will study 1-parameter families of curves f : [a, b] x (-E, E) --+ M, (s, t) -4 f (s, t) such that E Z, and f (s, b) (E Z2 for all s. If 7 extremises the length f (s, a) 6 functional L for all smooth curves which connect Z, with Z2, then we have A, L(f (s, .)) for all such I-parameter families with - (t) f (0, t). ds We will
the
=
=
ls=o
Through
the
of
investigation
d
2
dsyls=o L(f (s,
.))
we
will arrive at sufficient
conditions. Definition 4.6.2. Let be
a curve
Z1, Z2 be submanifolds of M and
which connects Z, with
A continuous variation
-y:
[a, b]
--
M
Z2-
f: (-E, E)
x
[a, b]
--+
M, (s, t)
called piecewise smooth if there are numbers tj.... ) tk E is smooth, where to := a, tk+1 := b and i x [t,,t,+,] f,
-->
f (s, t) of -y
(a, b) f 0,.
is
such that
Q. (continuous, piecewise smooth) variation f of -y connects Zi with Z2 if f (s, a) E Z, and f (b, s) G Z2 for all s c (-E, e). We denote the vector field T(,,t)f (as) along f by fs, the vector field T(s,t) f (,Yt) along f by ft (where defined), and call the (piecewise smooth) vector field (t) : (f,)) I s=o along 7 the variation vector field. C=
..
,
A
=
Lemma 4.6.2. Let -y: [a, b] ---+ M be a smooth curve which connects two submanifolds Z1, Z2. For any vector field along -y with (a) E Ty(a)ZI, (b) E Ty (b) Z2 there exists a variation f of -/ which connects Z, with Z2 and which has variation vector field .
Proof.
A2(0) a
Let M, C Z, and /LL2 C Z2 be smooth curves with pi(O) -y(a), (a), and A2(0) (b). We can now extend to -y(b), Al(O) =
=
=
vector field -";' such that yj
denotes the flow of E 6
In the Lorentzian
we
case:
set
all
(i
E
f 1, 2
f (s, t)
=
are
integral
F, (-y (t)).
smooth, causal
curves
curves
of
If F
218
4.
If -y:
Pseudo-Riemannian manifolds
[a, b]
M is
--
piecewise smooth then
is discontinuous at those
points where
-y fails to be smooth. We will therefore need the technical definition.
following
Definition 4.6.3. Let -y be a continuous, piecewise smooth curve and V be a piecewise smooth vector field along -y. For each to E [a, b] we set
,,AV(to)
lim
V(t)
V(t).
t_tO't
It is clear that
zAV(to)
0 if and
Lemma 4.6.3
(First
variation of
spacelike
lim
-
t_tO't>tO
only
if V is continuous at to.
arc
length).
Let y:
[a, b]
-4
M be
timelike, continuous, piecewise smooth curve, let 77 and f : [-E, E] x [a, b] sign(( , M, (s, t) 1-4 f (s, t) be a continuwith variation vector field . Deous, piecewise smooth variation of note by tj.... tk E (a, b) the points where -y fails to be smooth. Then the derivative of L with respect to s is given by a
or
=
I
d
(is L(f (s,
1
-77
dt
ls=o k
VI-1_Mti)Mti)) I
00
b. a
Proof. If 6 is small enough, all curves f (s, -) are either timelike or spacelike, and the integrand \/I_((ftjt))I is differentiable for all (8, t) where f is differentiable. We can therefore exchange on every smooth piece of 7 the differentiation and the integration. Now the assertion follows from d
ft
-
ds-
ft' ft)
f =)
_Utl ft) ft
-1f V17-W t)
Corollary
V
4.6.2. Let
(M, g)
be
,
Oft
)
ft =--
f,
a
'R
at fS
V _(f-1ft) t
't at
Riemannian
or
Z1 Z2 be two submanifolds without boundary, and ing Z, with Z2
f,
77M--t,
Lorentzian -y be
manifold,
a curve
connect-
-
(i) If (M, g) is Riemannian and -y is shorter than any other (neighbouring) curve connecting ZI and Z2 then -y is a pregeodesic which intersects both
Z1, and Z2 orthogonally.
4.6
Assume that Z1, Z2
(ii)
are
Riemannian
point. If (M, g) longer than any other causal
to
Proof. We We
pregeodesic which
prove
assume
only (i)
V(t)
a
V
:=
degenerate which is
Z2 then -y is
connecting Z, Z1, and Z2 orthogonally. is
completely analogous. pregeodesic. Observe that it is case
( 1A )
0 for all t. Hence
=
-1^1y) 0 I
assumption there would be
be
or
curve
and
since the other
if
causal
a
219
intersects both
first that -y is not
pregeodesic if and only our
curve
energy functionals
submanifolds
is Lorentzian and -y
a
causal
a
Length and
a
point to
(a, b)
c
with
V(to) =h
a
by
0. Let
vector field
, along -y such that (V (to), , (to)) < 0. BY continuity there is a neighbourhood (t-,t+) of to such that (V(t),61(t)) < 0 for all t c (t-, t+). Let now W be a smooth, positive function with support a
(to, t+) and W(to) =h 0. (Such a (V(t), p(t) i (t)) > 0 for all t
in
Then
function exists C-
[a, b]
by
Lemma
2.1.7).
and does not vanish in
a
neighbourhood of to. Taking a variation with variation vector field < 0. This implies that there o i we obtain therefore (Tdds L(f (s, -))) ,=o =
shorter
are
We
orthogonally. Let
be
6(a)
is
than 7 in contradiction to our assumption. that -y is a pregeodesic but does not intersect Z, Then there is a vector v E Ty(a)Zi with ( (a),v) < 0.
curves
assume
a
now
variation vector field with
tangential such that
field
(jdds L(f (s, -)))
to
Zi there is
f (s, -)
connects
< 0 in
a
(a)
variation
Zi with Z2
contradiction to
,=o
=
f
our
v
and
6(b)
=
0. Since
of -y with variation vector for all s. Again we obtain
assumption that
-Y is
It follows that -y intersects Zi orthogonally. observe that the same argument holds equally well for
length
minimising.
Finally
The discussion above does not square root of
This
apply to null curves problem can be avoided
Z2. I
since L involves the
if
one
considers the
b
1 dt. 7 energy of the curve -y, E(-y) := f' E ( (t) (t)) a Unlike the length functional this integral expression does
the parameterisation of the curve. While it is not true that timelike curves which extremise L also extremise the energy will
see
below that this property almost holds.
Lemma 4.6.4
(First
variation of
continuous, piecewise smooth
f (s, t) field .
depend on spacelike or integral, we
be
curve
M be energy). Let -y: [a, b] x : f [-(E, 6] M, (8, t) [a, b] --4
and
---*
a F-->
continuous, piecewise smooth variation with variation vector by t,.... tk E (a, b) the points where -y fails to be smooth. Then the derivative of E with respect to s is given by
The
name
"energy"
I
comes
from the fact that in the Riemannian
just the kinetic energy of a mass point of mass the integrand has nothing to do with energy.
tegrand case
a
Denote
is
case
the in-
1. In the Lorentzian
220
4. Pseudo-Riemannian manifolds
k
d
(ds
)
E(f (s, .))
Proof. Consider
fb
=
1'=O
"
E (J (ti),
(ti))
+
i=1
piece of
a
)
dt +
-y where it is smooth. The assertion follows
from I d
(f" f')
2 ds
ft
It follows that
merely
a
,
If7 a ' ft
a curve
ft,
Vf a, f,
which extremises energy is
a
geodesic (and
not
pregeodesic).
We will now, derive sufficient conditions for
between submanifolds without
length
If7 at ft, f '
(ft, f ')
curves
to extremise the
But first
boundary.
need
we
a
tech-
nical lemma. Lemma 4.6.5. Let 7 be a spacelike or timelike pregeodesic and denote the orthogonal projection to the orthogonal complement of -Y by (.)-L.
Then
for
every vector
V
field
along
the
-
formula
(V V)'
=
V (V)
holds.
Proof. The vector field V can be decomposed into its part orthogonal to , W, and its part tangent to , W , where W is a smooth function. From V =W +W and ( ,W) =Oweget -L
(V V)-L where
we
=
dW( )( )
have used that
The assertion follows
1
+ W
(VA
a curve
now
is
a
+
W' V ) implies In the
(V W)
following we
Lemma 4.6.6
like
V W
will
=
only
if
V
=
0
-
0
=
0
V (V').
freely interchange
(second
if and
pregeodesic
since
variation of
timelike
I and
arc
V
when 7 is
length).
a
Let 7 be
geodesic. a
space-
G 1-1,11 and let f: [-E,E] x geodesic with 77 [a, b] -- M, (s, t) --+ f (s, t) be a continuous, piecewise smooth variation of -y. If we denote by ti, tk E (a, b) the points where f (s, -) fails to be smooth then the second derivative of L o f with respect to s is given by or
=
.
(
..
d2 ds 2
L (f (s,
77 15=0
7
jb
+
(R( ,
dt
4.6
(
+
Length
and energy functionals
221
f
fs
V, V,541
+
1
R(
dt
k
as denotes the variation vector
where
jection
()
to
Proof. Using 4.6.3
b
Is=O, field
a
-J-
and
the
orthogonal
pro-
' .
the formula for the first derivative in the
proof
of Lemma
obtain
we
d2
-
V77 (ft ft) d82 I
d
( V7-77ft,
f
ft
ds
,V
ft)
ash f
f
77
/?__3 (ft ft)
V I
ft,
V
I
f
77 +
Vf a, f
a, ft,
V
-If (Y t)
aft) ) ( Vf
ft, V
a,ft
t
ft,
+
'9,
If7 9, f
t
(4.6.3) f
f
From V
a ,ft f v
=V at f, and f
f
f we
V a, ft,
get
f
f
f
a., V 9, ft =V as V at f , =V at V 9, f,
Vf as ft
f
f
ft, V a,
V
f
f
V
atf,,V
ft,
V
f
a"ft
Since for every vector field V
f
atfs) )
9t V 95 f,
along
we
-y
+
R(f, ft) f,
and
+
have V
(ft, R (fs, ft) f,) =
V-L
(V'f") + 77 (ft, ft)
ft
and
I
1
(V, V) V VJ-) Equation (4.6.3) simplify to
therefore
=
,
the second and third summand in
+
f
Vfn- (f
t
I
ft)
atfs)
f
atf,)
222
4.
Using
Pseudo-Riemannian manifolds
the
product formula for the
f
ft, Vf
term
a,,f,)
V at V
we
finally
obtain b
d2
(
L(f(s, dS2
-)))
=
1 '=O
I
and the first with
(, )
equality in the
=
,
t
f
+
( (V aif') (
*' :7:_-If=t):: ( ft, V a,
+
'
f
77
f
fS)
(ft, R (f, ft) f,)
)
f
V at ft,
-
f
V
f'
dt
assertion follows since
f (0,
-Y is
a
geodesic
q.
Since for every smooth piece of the variation f the equation
( V4-L "74 ) 1
holds, the
There is
second
an
equality
-L
V -L'
=
follows from
)' ( V 74-L' -L ) -
an
integration by parts using
formula for the second variation of the energy
analogous
integral. Lemma 4.6.7 and
f : [-,E, E]
x
(second variation M, (s, t) [a, b] --
smooth variation
f (s, -) fails to be to s is given by
of
7.
Denote
energy) Let -y be a geodesic f (s, t) be a continuous, piecewise by tj tk c (a, b) the points where -
...
7
smooth. Then the second derivative
d2
d82
of
-*
E(f (s, Is=0
fb ( V4, V4) f.')
+
f
+
V
a,
I
,=,,
f
(R( ,
)
+
'a'.
k
f
+ 11=0
of E with respect
)
dt
4.6
Proof. The first equation
(ft (S' 2 j_S2
and energy functionals
ft (S'
f
f
V
Oft' V Oft
f
f
f
V
atf" V atf,
f
f
+
f
atf
V
"
f
equality follows
from
V
+
V '9 ' V at fS,
ft
f
f
+
V at
V'9'fS'
ft)
ft
f
an
) V' atft) (R(f, ft)f,, ft)
atf. f
V 7 a
proof
V 0, V C'), ft I
(R(f, ft) f, ft) f
the
f
+
f
V at fS,V at f,9
The second
223
follows from
d2
1
Length
f, V
a 'f
+
"
ft
+
integration by parts exactly
as
in
1
of Lemma 4.6.6.
M be Z1, Z2 be submanifolds of M and -y: [a, b] which both intersects E2 submanifolds orthogonally. from Zj Assume that f is a (continuous, piecewise smooth) variation of 7 which connects Z, with Z2. Then
Lemma 4.6.8. Let
--->
to
a curve
( (V a'f') 1'=O' ) f
holds, where lz,,
=
(.Yz2( (b), (b)), (b))
denotes the
Proof The
assertion follows our
(
tive
form
shape
tensor
of Zi.
immediately from the definition I Z1, -y(b) 1 Z2-
implies
.))
)
that for
a
geodesic
(respective 1 y 1,9=0
tively, a
LEY
I
d2E(f (s, .)) )
-d-s-7
the space of all variation vector fields
-y).
It is
an
shape
variation the second deriva-
) is a quadratic jS=O the central geodesic
along form
-y. The associated bilinear form is called the index
of
of the
assumption -y(a)
d2 a_S2 L(f(s,
on
.Yz, ( (a), (a)), (a))
a
tensor and
Lemma 4.6.8
-
infinite dimensional
analogue
1,L
of -Y
(respec-
to the Hessian of
function.
Definition 4.6.4. Let
Z1, Z2 be submanifolds of M or points in M and which connects Z, and Z2 and intersects both geodesic [a, b] I submanifolds orthogonally. Denote by TZ'l,Z2 -y be the space of piecewise smooth vector fields along -y which are tangent to El at a and to Z2 at --
M be
a
b.
The energy index form is the bilinear
form
224
Pseudo-")
4.
E, -Y
-nnamfolds
I
Ti
R,
'T ,,.27
X
( (V 6 V 6 ) i
either
(R( l
('Z2( l (b), 2 (b)), (b))
+
If -y is length
+
spacelike or timelike defined by
and
dt
(1z, ( l (a), 6 (a)), (a))
-
satisfies
1, 11, the
77 E
index form is
IL11'' Z2 -1 11' Z2 'Y
X
-
/
1 -Y -r,'l,-2
V41j-, V 62-L )
77
+ 77 -
R,
-->
Tj
(R( l, )6,
+
dt
(Iz2( l (b), 6 (b)), (b)) (1zj (61 (a), 6 (a)), (a))
Corollary 4.6.3. Let 7 be a geodesic from Zi to Z2 which intersects these submanifolds orthogonally. The index form IL,-y is positive semiZ1, Z2 definite if -y minimises length and negative negative semi-definite if -Y maximises length. ,
Proof.
For L the assertion follows from the
d
L(f (s, -) The
L(-I)
=
proof
+
s( is-
for the energy
Taylor expansion
d
8
L (f (s,
-)
integral
)'=0+2 is
exactly
dS2 the
L(f(s,
.))
+O(S3). 8=0
I
same.
following lemma summarises in which cases there is a non-trivial extremising problem for E and L. In particular, it implies that the extremising problem has only in the Riemannian and Lorentzian case nonThe
trivial solutions. Lemma 4.6.9. Let
(M, g)
be
a pseudo-Riemannian manifold, 'Ply Z2 M be a geodesic boundary, and -y: [a, b] which connects Zl with Z2 and intersects both submanifolds orthogonally. If -y is a null geodesic assume in addition that,,7y(a) Z Ty(a)ZI and (b) V T-y(b)Z2-
be
submanifolds of M
without
--
E'-Y
is positive (respectively, negative) semi-definite then signature (+ +) (respectively,
N if 1 , Z' has
(ii)
Let
E,-y, -L
1 1,Z2
be the
bilinearform
E'-Y
1 1, Z2
positive (respectively, negative) definite +) (respectively, g has signature (+
is -
restricted to then either
or -
g has
signature (-
+
+) (respectively,
-L. If
g
TE,'Y,-L L
Z, Z2
4.6
7 is causal
-
53b 52
-
-y(b) (iii) If
are
(respectively, -y is spacelike spacelike (respectively, 5'b
the index
signature (+ g
form IL,-y ZI,Z2 ,
+)
energy functionais
or
null)
32
timelike)
are
-Y(a),
at
positive semi-definite then
is
225
g has
or
L,-y
index
(iv) If the
Length and
form has signature (-
is
negative semi-definite then
either
+)
+
7 is timelike
Z1, Z2
are
spacelike
-y(a), -y(b)
at
or
g has
+)
signature spacelike Z1, Z2 are timelike
-
-y is
-
-
Proof. Let J
E
every k E N
we
R
and
\ 101
v
be
vector with
a
parallel translation
vector field vanishes at the
The
TI(a)M
E
consider the variation vector field
where V is the
a) b-a
-y(a), 7(b)
at
of
v
Wk (t)
along
J. For (v, v) 1 V (t) sin ((t k =
=
-
-y. This variation
endpoints
of -y and is therefore in El "'Z' 'Y
-
equation
jb V Wk, V Wk (R(Wk, ) a) b-a) jb b-a) ) sin2((t -a) (R(V, ) )))
E'-Y
1 1, Z2 (Wk, Wk)
Wk,
+
a
2
Ir
J
k7r
Cos2
a,
k7r
I
V,
+
V
implies
that for
sufficiently large k the integrand
sign(IE"y,Z2 (Wk, Wk)) Assertion
(i)
b
=
follows
has the
-
a
sign
as
6,
("Y(a), v)
=
0.
same
sign(6). immediately
from 6
=
(v, v).
For the rest of the prove we will assume in addition that 0 for all t E [a, b]. Consequently, we have (Wk (t), (t)) =
(ii):
Assume that
IE,'_I,Z2 'J-
is
positive semi-definite and suppos6e either
that -
-
2-dimensional
there is
a
negative
definite
there is
a
or
I-dimensional
definite and that -y is In both v
I
subspace of T,,M restricted
cases
there is
subspace restricted spacelike.
a
vector
v
E
Ty(a)M
to which g is
with J
=
that either g is Riemannian
or
negative
(v, v) < 0 and TE,-,, Icannot be positive :=
-I and ,,. Hence sign (I E"'Y'-L ,Z2 (Wk, Wk)) ZI,Z2 our assumption, and therefore
semi-definite. This contradicts
-
to which g is
that
it follows
226
-
4. Pseudo-Riemannian manifolds
g is Lorentzian and -y is causal.
Since
(a)
TI(a) -"I ( (b)
the submanifold must be semi-definite _T E,-y,l
sics. For
1 1,Z2
large enough
at
completely analogous.
is
Z1,Z2
The index form
and Zi (Zb) is orthogonal to -y(a) (-y(b)). The proof for negative
Ty(b)Z2)
spacelike is
defined for spacelike
only
k the relation
L,-y IXE (Wk; Wk)
Wk
,
L,-y
Sign( 1 1, -, (Wk, Wk))
=
2
or
timelike
geode-
implies
771EEl"Y
2
and therefore
J_
Z2
(Wk7 Wk)
sign(,qJ).
Assume that g is not definite. Then v can be chosen such that L,-y This implies that 1 1, _,2 is not positive semi definite either. -q. There is to (iv): nothing prove if (M, g) is a 2-dimensional Lorentzian
(iii): 6
=
manifold. -
-
Suppose
g has not
signature (-
dim(M)
Then
v
+
-
-
-
+) (respectively (respectively, timelike).
or
that
: 3 and -y is spacelike
can
sign(,q6)
that either
be chosen such that
I for
=
Tj and
large enough k implies
L,-y
sign( 1 1, Z2 (Wk) Wk))
that I L,-y is not Z1,Z2
negative semi
definite.
I
4.6.4. Let
Corollary
(M, g)
be
a
Riemannian
or a
Lorentzian
manifold
and 7: [a, b] -- M be a spacelike or timelike geodesic which connects Z, with Z2 and intersects both submanifolds orthogonally.
(j) If with
L, -y
is definite then for all variations f: [-e, E] 2 non-vanishing variation vector field there is a 6 r
x
E
[a, b] (0, c)
-->
M
such
that
L(-y) L(7)
-
-
< >
L(f(s, L(f(s,
Vs E
Vs E
[-6,6] [-6,6]
in the Riemannian case, and in the Lorentzian case.
L,,y is not Z1,Z2
ii) Ifi
[a, b] L(7)
>
L(-y)
<
-->
-
-
Proof.
M
L(f(s, L(f(s,
To prove
vector field
.
semi-definite, there of -y such that
(i)
let
Since
Vs
f be
and
1 15 2( _' _)
< 0.
variation vector fields
L(-y)
+
jS2 1 L'I,Z2
2
In Lemma 4.6.15 below
f: [-E, E]
x
in the Riemannian case, and in the Lorentzian case.
variation of -y with
L(-y)
L
7
proof of (ii) let
L
a
[-E, El [-c, c]
L(f(s,
assertion follows from
For the
E
Vs E
exist variations
+
non-vanishing variation IS2,L,-y + O(S2) The 2 Z15Z2 ( ' ) .
0 and Lemma 4.6.9.
variation fields with
"Y IL Zl,--2(
+' +)
> 0
By Lemma 4.6.2 there are variations f of -y with . Now the assertion follows from L(f(s,
+0(82). we
will extend this result to null
geodesics.
4.6
Conjugate and
4.6.2
focal
Length
and energy functionals
227
points
points is closely linked to semi- definiteness of that Z1, Z2 are points. Before investigating this relationship we will need to collect a few important facts about Jacobi vector fields in pseudo- Riemannian manifolds. The existence of conjugate
the index form in the
case
Lemma 4.6.10. Let -y:
fields which
(V J' j
vanish at
[a, b]
--->
M be
point to
some
a
geodesic
[a, b].
E
Then
and
have
we
J be Jacobi
J,
(J, V j)
-
It follows from
Proof.
(R(J, (R(j,
V J,
that
J,
V j)
V J' i
-
diate consequence of
)J,
+
) J,
+
V J, V J) V j, V J)
J,
V j)'
is constant. Hence the assertion is
J(to)
=
i(to)
=
0
an
imme-
I
.
that it is possible to split Jacobi fields into a tangential orthogonal part unless -y is a null geodesic. Moreover, the tangential part is always trivial. We show
and into
now
an
[a, b] all for 11 (t) (t)
Lemma 4.6. 11. Let -y: -y with
along
(i)
The vector
along
field
-->
M be
t E
[a, b],
-y is
a
a
be
geodesic,
and J be
Jacobi
a
Jacobi
vector field field along -Y.
a
field if and only if there
are
numbers a,,3 with (t) (at + 0) (t). (ii) The following statements are equivalent. =
(a) (J(t), (t)) 0 for all t E [a, b], (b) there are two different numbers c, d E [a, b] with (J(c), (c)) and (J(d), (d)) 0, (c) there is a number c G [a, b] with (J(c), (c)) 0 and 0. (c) J, (c) =
=
0
=
=
'
)
(V
Proof. (i):
If
11
=
we can
write
(t)
=
O(t) (t).
equivalent
(ii):
to
(t)
=
R(O(t) ,
0.
The assertion follows W (t)
satisfies
Since
0
0 which in turn is
the Jacobi equation reduces to
once we
=
have shown that
(J(t), (0)
0. But this follows from
4. Pseudo-Riernannlaji mauiiold,,
228
-0
(J,
V V
V
(V V J, )
=
7
=
)
J,
(-R(J, ) , )
J,
+
=
V )
0.
I
Corollary 4.6.5. spacelike and J be jT
and J
to
1
Let y: [a, b] --> M be a geodesic which is timelike or Jacobi field along -y. Then the orthogonal projections
a
J-
to
Proof. Without loss jT is given by
of
Then
,V
are
q
generality
(J, ) .
(V J) T.
J)
also Jacobi
fields along
assume
Since 7 is
In the
same
a
way
that
-y.
(, )
geodesic we
=
1, If.
77 c-
obtain V jT
we
1:Y
get T
V ,7 jT From
jT
we
obtain
=
(V VA
R( jT, )
=
0 and therefore T
,V V jT + R( (jT,
V V jT
=:
(V V J) +
=
where j
we
have used
jT and
_
R(.,
V V jT
assertion follows
now
=
from
=
0
we
0
get
R(J-L, )
(R(J, ) )T
=
0,
(cf. Proposition 4.3.1).
From J-L
V V J`
The second
=
=
R(J, ) .
V V J.
I
Let -y: [a, b] --> M be a geodesic and J be a Jacobi field which vanishes at a and c E (a, b). There is a geodesic variation f : (s, t) F-4 f (s, t) of -y with variation vector field J
(cf. Proposition 2.9.1 and Corollary 2.9.1). geodesics pass through -y(a) and "intersect -y(c) to first order" though they may not actually meet this point. Up to second order the geodesics segments f(s,')[a,,] have all the same length as 71[a,c]- It is therefore plausible to expect that 7 will not extremise length beyond These
7(c). A
typical example where the geodesics meet in both points y(a) and given by those great circles of the unit sphere sphere S2 which intersect both the south pole at -y(a) and the north pole at -y(c). Since in this example all neighbouring curves intersect -y(c) at an angle different from zero, they can be deformed so that they meet 7(b) and have length shorter than -y (cf. Fig. 4.6.3). This is also true in the general case, even if the neighbouring geodesics do not actually intersect 7(c)
-y(c)
(cf.
is
Theorem
4.6.1).
4.6
Length
and
energy'functionals
229
For
arbitrary submanifolds Z1, Z2 an analogous statement cannot simply no canonical way to compare different submanifolds along a curve. Hence we cannot speak of a "first point where a given normal geodesic fails to minimise length between Z, and Z2". However, we can ask at which point a geodesic -y orthogonal to be formulated. There is
a
submanifold Z fails to minimise distance from Z. In order to
this question
points
to
we
will need to
pair which
a
consists of
a
Definition 4.6.5. Let Z C M be a
with
geodesic
called
(i)
E
V J(a), v)
(ii)
and y: [a, b] --> M be Ty(a)Z. The point -y(c) is
submanifold
E
Ty (a) -", J(C)
J (a) E
a
Z, (a) (Ty(a)Z)-L \ of Z along -y if there is a Jacobi field
7(a)
focal point
a
+
Observe that in the
J
along
-y with
0,
: --
(I(J(a), v), (a))
case
answer
the concept of a pair of conjugate submanifold and a point.
generalise
that Z is just
=
a
0
for all
v
E
single point,
T- (a) -"condition
(ii)
is
empty and the definition reduces to the definition of a pair of conjugate points. The following lemma explains why we also demand condition (ii). Lemma 4.6.12. Let Z be
a
submanifold of
M and -y:
[a, b]
--+
M be
-y(a) Z, (a) (T-y(a)Z)-L \ T-y(a)ZM is a variation of -y through geodesics If f : (- E, E) x [a, b]
geodesic with
E
-
to Z then the variation vector
thogonal
Jacobi
(i) (ii)
(a)
(iii)
V, 4(a), v)
is
a
E
a
thogonal
is
or-
satisfies
field,
be
+
(.Y( (a), v), (a))
=
0
for all
v
E
Ty(a)Z-
vector field along -y which satisfies (i)-(iii). Then f : (- c, E) x [a, b] -- M of -y through geodesics orwhich has variation vector field . a
variation
to Z
Proof. "=*": Assume that f That
field
T-y (a) Z
let
Conversely, there is
a
E
a
is
follows from
f (s, a)
c
Z for all
s.
V, 4
V
through normal geodesics. Proposition 2.9.1 and property (ii)
variation
a
Jacobi field follows from
The equation
f
implies that
=
f
atfs
for every vector field V
=
V
C%ft
tangent
to Z
=0
V)
=
f
f
ft,
V
=V
S
(ft, 1 (fl, V)) holds. This proves
(iii).
(ft, V) S'-'
a
f
-
ft, 9SV)
230
4. Pseudo-Riemannian manifolds
Let y: a
vector field
fine
(-c, e)
along
--->
Z be
a
V(s)
I
p with
with
curve
T,(,)Z
A(O) V(O)
=:
and
=
(a) and V be (O). We de-
f(s,t) exp(tV(s)) thereby obtaining a variation of -y through geodesics normal to Z. This variation has variation vector field if and onlyif f,(O,a) (a) and7 (a)fs '7 (a) - We clearly have f,(O, a) =
=
A(O)
(a)
=
=:
have to choose V such that
possible,
let
Then
have
we
A(O)v
=
for any choice of V. Since
V(s)
=
=
V(O)
17A(O)V
PjYJ[O'S] =
=
'7 (a)fs Vfft '7A(O)V V (a) . To that this is always =
see
(7 (a) )
(a)+sPj-
pqo,s]
(a) and, using
'7A(O)P1-Ltj[O'S]
(a)
+
we
=
(cf.
Lemma
4.4.5).
Lemmas 4.4.4 and 4.4.5,
P/_'-tl[0,0]
( V (a) )
O
('7A(O)Pit1[O'Sj (-Y(f,, '), ft)'
where the last
inequality follows
(1 Yll W), ft)
=
=
for all vector fields W which
(a)
(7 (a) )
+
from f
-
(7
+
V
-
f
W
OSW' ft
(W, Vf atfs) tangent
are
V'%ft
W,7 (a) )
to Z.
If -y is not a null geodesic then the neighbouring geodesics provided by Lemma 4.6.12 are of the same causal type as -y. However, if -y is null then this property is not guaranteed. The following lemma clarifies the situation for null
geodesics.
a submanifold of M and -y: [a, b] -4 M be a I Z, (a) E (T-Y(a) Z) \ Ty(a) ZLet be a vector field along -y which satisfies properties (i)-(iii) in M of -y Lemma 4.6.12. Then there is a variation f : (- c, c) x [a, b] through null geodesics orthogonal to Z and with variation vector field 0 for all t E [a, b]. if and only if ( (t), (t))
Lemma 4.6.13. Let Z be
null
geodesic
with
-y(a)
E
-+
=
Proof. Assume that f
is such
a
variation
through
f
equation (ft, ft)
0
therefore
(a)
imply
0 for all t E
implies
)
=
0
V
alft,
0. Lemma 4.6.12
[a, b]
-
ft) (ii)
null
geodesics.
The
Vf atf, ft )
and
and Lemma 4.6.11
(ii)
4.6
Conversely,
that
assume
is
(iii)
of Lemma 4.6.12 and is
f
in the
as
proof of Lemma
Length
and energy functionals
vector field
a
orthogonal 4.6.12 but
along
-y which satisfies
231
(i)-
to -y. We construct the variation
will choose V
now we
so
that it is
null at every point of /,t. To do so, let s -4 W(s) E Ty(,,)M be a curve with 0 for all s. As in the proof of Lemma W(O) (a) and (W(s), W(s)) =
=
4.6.12 let M: (a). We set
and
-4
s
V(s)
M(s)
Z be
with
tt(O) -y(a) and A(O) P'AJ[O'S] W(s). Clearly, V(s) is normal to Z at all
=
VA (V (s), V (s))
E
=
2
a curve
(V AV (s), V (s)
=
2
( (V V(s))
=
s
V (s)
0 for all s. Hence f (s, t) implies (V(s), V(s)) exp(W(s)) is a null through geodesics normal to Z. We have to choose W such that the variation vector field of f coincides with . Since we have f,(0, a) #(0) (a) we only have to arrange W such that 0
=
=
variation of -y
=
=
f
at f, (0, a)
17
=
174.
,
From
(I (f, .), ft)o
-
(V*) )
=
(cf. proof
of Lemma
4.6.12)
we
obtain f
atfs)
f
=
(0,a) =
( '9"ft)
(0,a)
(0,a)
(VAV(S))-L(O,a)
(I (A (0), .), V (a))
-
T
(VAV('5))-L(O,a) + (7 (a) )
(VA(O) (P J[O'S] W(S) I
(V
A( PI/-'1 [0, S] W(O))
+
+
(V (a) d
P-L AJ[O'S]
18=0
( Fs W(S)) JS=O)
T
+
(a)
)
is- W(8)
)
(7 d
(V4,
We have
(a)
Since
I
:
)
=
0 since
is
Ty(a) I' we get therefore
space of the null
T-y(a)
(a)
(7
+
( (a), X)
C-y(a)
cone
=
01.
(a)
)T
JS=O
C
Hence
orthogonal
to -y
( (V4) (a))
Ty(a)M
(V4)
,
at the
is
a
point
tangent
(Lemma 4.6.11). =
0. The
(a)
tangent
just fX
is
vector to
C-y(a)
E
at
I
(a)
and
we can
choose W such that
d ds
W) I 8=0
f
implies
tfs (0, a)
V q
=
V4
and
we are
done.
V4)
.
But this
4. Pseudo-Riemannian manifolds
232
Fig. 4.6.3. Conjugate points sphere
We need another technical lemma Lemma 4.6.14. Let Z be
[a, b]
-y:
J1, J2
--+
are
M be
Jacobi
==
0
for
J1 (t) t E
Proof.
Rom the
,
the
to Lemma 4.6.10
pseudo-Riemannian submanifold of M and orthogonally at -/(a). If
which intersects Z
fields which satisfy Ji (a)
(I(Ji (a), v), (a))
for all
a
geodesic
a
analogous
on
all
v
E
C-
Ty(a),E,
V (t) J2
Ty(a) Z
then
and
(17 Ji (a), v)
+
J1, J2 satisfy
V (t) JI
i
J2 (t)
[a, b]. proof v
of Lemma 4.6. 10
J1) J2
)
-
JI
we
i
V
know
J2
))
0
-
Hence the assertion follows from
(V (a)jl, J2(a)
-
Ji(a),7 (a)h
(I(JI (a), J2 (a)), (a))
+
I(J2 (a), J1 (a)), (a))
=
0.
I We
can now
present
a
theorem which links focal
points
to
length
extrem-
ising geodesics.
(M, g) be a Riemannian or a Lorentzian manifold, M be a geodesic which submanifold, and -y: [a, b] orthogonally at -y(a). If (M, g) is Lorentzian we also assume
Theorem 4.6.1. Let Z be
a
Riemannian
intersects Z
that -y is timelike
--*
4.6
(i)
The
submanifold
Length
Z does not have
and energy functionals
focal points along
-y
if
233
and
only
if the index
form
L,-y
positive semi-definite in the Riemannegative semi-definite in the Lorentzian case. If L, -y 1 ,J-y(b)} (VI V) 0, then there exists a function vo with V vo , i.e., V corresponds to a reparameterisation of . (ii) The point -y(b) is the only focal point of Z along -y if and only if the index form is positive semi-definite in the Riemannian and negative semi-definite in the Lorentzian case. there is a non-vanishing vector field V: [a, b] ---+ M along 7 which L,-y 0 and has values in satisfies 1 , ,y (b) II (V V) is
nian and
=
-
-
=
(iii) if
There is
the index
focal point -y(c) of Z along form is not semi-definite. a
-y with
c
b
<
if
and
only
Proof. Observe that the cases (i)-(iii) are mutually exclusive and comprise all possibilities. Hence it is sufficient to prove the "if"-parts only. (i): Since there are no focal points of Z along -y there are n linearly independent Jacobi fields Jj: [a, b] -4 TM which -
(a)
everywhere perpendicular
are
(b) satisfy Ji(a) for all form
(c)
v G
a
basis of
Let V be
a
-y(t)'
=
V
0,
.
.
.
V--L, V
,
n
-
V-L
V
, Ji (a), v
)
1)
for every parameter value t E
(c)
there
with Vvo
+
R(V-L,
YVI
(.ff (Ji (a), v), (a))
+
variation vector field which is
vanishes at b. Because of
(i
and
T,,(,,)Z T, (a)ZE
to
V V
V
I ,
are
+
tangent
F_'
0
(a, b]. to Z at
smooth functions v':
v'Ji.
==
We will
a
(a, b]
now
and -4
R
calculate
1
j)k jk
,
+
vk
V j,
(V Vl, VkV jk) (i)i Ji vi V Ji) bk A (V V VkV jk) ViV Ji,' kJk) (biJi, IkJk) VV-L,VkV,jk ) ViJi,i)kV Jk) +
+
I
+
,
+
Lemma4.6.14
+
+ =
(I)i Ji
i)k A)
7
V V_L,VkV jk) Vi Ji, V (VkV Jk) (vi Ji V V Jk ) (i)i Ji, i)k A) V V_L, VkV jk ) (V' R(V +
-
7
=
+
Vk
+
+
(70i, i)kJk)
,
,
234
4. Pseudo-Riemannian manifolds
where in the last we
equation
we
have used that Jk
Jacobi fields. Hence
get
( V, V--, 17
V
(R(V--, ) V-L,
+
Then
a.
(b), V(b)
=
0,
V-L(a)
v' have bounded extensions
to
Ji, i)i ii)
(j,
k
Assume for the moment that the functions to
are
V(a)
=
v
have continuous extensions
Ty(a)Z,
e
VJ_' Vk V jk
+
and the fact that all
imply
a
b
fa
(V_L' Vk V
dt
Jk
=
_
I(V(a), k(a) Jk), (a) v
=
Writing
obtain
we
77
(VI, VkV jk )t=a
b
IL,-t
(( V
Zj-y(b)j (V V) I
V' ,
V V' ) + (R(V-L,
)V-L,
)) dt
(ff (V (a), V (a)), (a)) 6 =
(b'Ji, b'Jj) dt + 77 ff (V (a), Vk (a) Jk), (a))
77 'a,
-
=
77
77
(1 (V (a), V (a)), (a))
f
(b'Jj,,b'Jj) dt.
'a',
Since -/ is timelike in the Lorentzian case the vector field V is always L,-y is semi-definite. Furand we can conclude that 771 Zj-y(b)l (V, V)
spacelike
L",'fy- (b) I (VI V)
77 1
thermore,
that
implies
in turn
0 if and
V-L vanishes
only
a.
There
basis of
are
Ty(a)Z.
dim(Z)
geodesic
also
can
constant, which
are
k v
f Jil (a),..
have continuous extensions
(a)j
.,
which form
a
This follows from Lemma 4.6.12 since it is
construct normal to Z. We
vectors
vi
since it vanishes at b.
We still have to show that the functions to
if all
because otherwise
assume
we
variations of
that
Jk(a)
could subtract
a
=
possible to direction tangential
in any 0 for all k
till
...
1
idim(z) I
suitable linear combination of the
Ji, from Jk. For each t we decompose V(t) into a part U tangential to ji diiii(E) (t) I and a part & tangential to spanjjk(t) : k V spant Ji, (t), fill idim(Z) I It is clear that all v" have smooth extensions to a since f ji, (t), Jj,1,1.(z) (t) I are linearly independent for all t (E [a, b]. Since & =V-v"Ji, andjk (k V fill idim(Z) 1) are smooth and vanish at a there are vector fields W, Kk such that & (t a) Kk. (t a) W and Jk and Kk(a) These vector fields satisfy W(a) V jk(a). From V U(a) .
...
.
.
-
1
.
.
.
,
...
1
=
Cj-L
=
Vkjk
we
get W-L
=
=
-
=
Vk Kk, and from the
-
=
linear
independence
of
4.6
Jk (t) I (t fields Kk
(a, bl)
and the fact that Jk (a)
Let
and V be
infer that the vector
we
f bi 1
a
smooth extension to
b be
a
strictly monotonically increasing
-4
has
vector field
a
such that
V(t)
=
(b
-
vector field
t)W(t). (bi
given by Vi (t) [bi, b]. This gives a
with
Vi[,,,bi)
VI[a,bi]
=
a.
along 7 which is tangent to Z at at b, there exists a well defined
at b. Since V vanishes
fVil
0
=
(ii):
for t E
==
235
are smooth one linearly independent near a. J' with near a. Consequently, the function along -/ w (Kk) k
wk(W.L) (near a)
=
and energy functionals
Hence there
are
forms w'
Vk
G
Length
=
a
sequence and vanishes
vector field W
Let Vi be the piecewise differentiable -
t) W (t)
for t
cz
[a, bi)
and Vi (t)
0
=
sequence of piecewise smooth vector fields and VI[bi,b] 0. Since b is the first conjugate ==
L, point part (i) of the theorem implies that _TZ,f-y(b)j (V;17 Vi)
: 0. Hence
L,-Y TL,-y __4 V 's ' ,J-y(b)j (VI V) (i --- oo) implies that 97,L,,y Z,I-y(b)j 'Z,J-y(b)J (V11 4i ) that non-trivial there To exist with variations serni-definite. see positive -
vanishing
index form let J be
to Definition 4.6.5. Since J
L,-y
1 ,J
fb
(J J) 7
-Y
a
JJ-
=
Jacobi field we
orthogonal
to -y
according
have
V J)
+
(R(J, ) J,
)) dt
a
.Y(J(a), J(a)), (a)) =0 since J is
b
a
Jacobi field
I
f ,V J). V V J, J) J,
-
-
(R(J,,:y) , J)
dt
a.
n 77
(I(J(a), J(a)), (a))
(I(J(a), J(a)), (a))
-
7
I(J(a), J(a)), (a))
Let c E (a, b) be the first focal point of Z along -y non-vanishing Jacobi field according to Definition 4.6.5. limt, V J(t) 74- 0 and the piecewise smooth vector field
(iii):
J be
a
V (t)
satisfies field
By
limt,c
zAV V(c)
along
J (t)
for t
E
0
for t
cz
V J(t)
:4
=
0
and t Then
[a, c], (c, b]
0. Let J > 0 and W be
a
vector
-y which satisfies
W(a) W(b)=O,
(W(t), (t))=O,
the definition of J
have
-217 (,AV(c), W(c)).
we
IL ,7
33, JY (b) I
This in turn
ILI,'Y (V + 6WI V + 6W) &Y (b) I
(VI V)
implies
W(c),AV V(c))>O. 0 and
_[L,-y . I_Y(b)J(V) W)
4. Pseudo-Riemannian manifolds
236
L
L,,y
=
=
I ' O(b) I (Y) V)
26I "71-Y (b) I (VI W)
+
-2,q6 (.6V(c), W(c))
+
62,L,-y
621L,-y
Z,{,Y( b) I (W1
+
W)
(W W).
Zj-y(b)l
7
sufficiently small we obtain sign(IL,-y Zj-y(b)j (V+6W, V+JW))
Hence for J
sign(-qJ). Since we can replace W by IL;-y fails to be semi-definite.
-W this equation
that
implies
Zj-y(b)j
In order to have to
an
(M, g)
Theorem 4.6.2. Let Z be
fold, like
of Theorem 4.6.1 for null
analogue
geodesics
we
need
the energy index form.
use
a
(Riemannian case)
be
a
Riemannian
causal
or
Lorentzian mani-
a
or
M [a, b] (Lorentzian case) geodesic
and -y:
submanifold,
Riemannian
--+
be
a
with
space-
(a)
I
(T-y (a) -") (i)
-
The
submanifold
Z does not have
focal points along
-
if
and
only
if the index
-
E -y, I
form
' ,'J-y(b)j
E,,y, I
S
POS itive
semi-definite,
a function vo with implies V vo , i.e., V corresponds to a reparameterisation of (ii) The point -y(b) is the only focal point of Z along -y if and only if E"L the index form ' &y(b)j ZS S emi-definite and
1 j-y(b)j (V V)
-
=
I
that there exists
0
=
.
-
there is
-
a
non-vanishing
IE`Y, -(Lb
vector
field
V:
[a, b]
M
-*
0 and has values in satisfies Z,f-y(b)j (V V) , I is There a of Z along -y with point -y(c) focal (iii) E,-y,-L not is index the semi-definite. form , ,J-y(b)j if
along
-y which
=
I
c
<
b
if and only
If -y is timelike (in the Lorentzian case) or spacelike (in the Riemannian case) then the proof is completely analogous to the proof of
Proof.
Theorem 4.6.1. We
can
therefore restrict to the
case
that -y is
a
null
and Z is spacelike. Again, there are Jacobi vector fields which satisfy assertions (a)the proof of Theorem 4.6.1 (i). Since every Jacobi field satisfying
geodesic
(i):
(c) in (a)-(c) loss of
must be
generality
4.6.1
we
V (b)
==
a
linear combination of the Ji
that J1 (t)
=
t (t). Exactly
obtain for every vector field V
along
as
we can assume
in the
-y with
proof
V(a)
E
without
of Theorem
Ty(",)z
and
0
E,-Y,-L
{-y (b) I
(V V)
=
jb (,b'Ji, b'Jj)
dt,
a
where t
-4
v'(t)
equation (Ji, whence
are
functions defined 0
_rE,,y,-L E'{-y(b)j(V1V)
implies > 0.
by V(t)
=
En-1 v'(t) Ji (t).
that each Ji must be
i=1
The
null,
spacelike is spacelike
Moreover, the Jacobi field Ji
or
for
4.6
By the
i > 2.
E
have
we
the
We
independence of the Jacobi fields i)n-1 VI ( V ) > 0 unless i)2 -y (b) I linear
at every 0. Since
=
therefore V
E,-y,l
' ,J-y(b)j (V V)
0
=
7
v'J1
=
which is
implies that V2
parallel
237
point
a
V(b)
0
Vn-1
0 and
=
to
(ii) and (iii) (ii), (iii).
of assertions
proof
of Theorem 4.6.1
proof
and energy functionals
'Y,-L
equation The
Length
is
to the
completely analogous
I
geodesics in the case that one Z1, Z2 degenerates to a point. This is achieved by there is a variation of -/ through timelike curves from the
extend Corollary 4.6.4 to null
can now
of the submanifolds
showing that spacelike submanifold that
we
cannot
use
Z to
the
-1(b)
if
Z,J-y(b)J
argument
same
as
semi-definite. Observe
i 11 ot 'S in
Corollary
4.6.4. While
we
f with E(f (s, .)) < 0 for all 8, be clear that these varied curves are everywhere timelikel.
would obtain the existence of it would not
a
variation
(M, g) be a Lorentz manifold, Z be a spacelike subM be a null geodesics which intersects Z orb]
Lemma 4.6.15. Let
manifold, and
thogonally. Z to
from
-y: [a, E, -y, I
ff ,J-y(b)j
first focal
a
(0, b
J c
not
-y(b) arbitrarily
Proof. The strategy there is
is
c)
-
semi-definite then there
a
timelike
curve
close to -y-
of proof is as follows. Theorem point 7(c) (c E (a, b)) of Z along
will construct two vector fields
we
is
of -y with 71(a,c+61 such that for every variation f
4.6.2
that
implies
-y. For
small
some
(t) and A(t) along (t) and f,(O,t) =
f
(0, t) f,
V
=
A (t)
we
have
(ft (t, s), ft (t, s))
to
f (s, a)
join Z
curve can
E Z
from Z to
curve
for
s
> 0
and
a
variation
which,
sufficiently
addition, f (s, c + 6) 7(c + 6) for all s. It is then possible and -y(c + 6) by a timelike curve arbitrarily close to -Y. This in turn be slightly deformed in order to arrive at a timelike
small. We will show that there is such satisfies
< 0
in
=
-y(b).
Observe that A cannot be chosen
completely independently
of
.
f
fact,
-y(a)
at
1( , ). I 2
E
Z
From the
(
we
proof
d' ds
2
( ft ft ,
have must have A'
))
Lemma 4.6.7 f
we see
f
V at fs, V
at
atf,') (
f
f
a 'fs
Vf atfs + R(f,, ft)ft,
f,)
+
,
+
V at f",
f.
)'
LI)
that
(R (f,, ft) f, ft)
Vf
_Y (h I
V at V
1 '=() +
(V a 'f')
=
ft)
.
+
f
f
V'9t
V
a'fs
,
ft)
In
238
4. Pseudo-Riemannian manifolds
holds. If
we can
construct -vector fields
, A
with
?
V, V
+
) >0_ V4,
R( , ) ,
__
V
)
A,
> 0
then any corresponding variation f satisfies (ft, ft) < 0 for small 8 > 0. Let -y(c) (c c (a, b)) be the first focal point of Z along -y and let J be a Jacobi field according to Definition 4.6.5. It follows from Lemma 4.6-11
(ii)
that this Jacobi field is
If there would be
field
J(t)
-
t
'
(t)
d
a
point d
would have
everywhere orthogonal
(a, c)
C
with
J(d)
at d and
a zero
of Definition 4.6.5. Hence there would be
=
a- (d)
to -Y.
then the Jacobi
all the conditions
satisfy
focal point -y(d) of Z along in contradiction to the definition of c. We have therefore
-y before
-y(c) J(t)
shown that
( (t))-L \ R (t)
E
The derivative of J at
c
a
for all t c-
satisfies
V.,J(c)
(a, c). E
T7(,)M \ R (c)
since J
is non-trivial and not
such that
is
c
parallel to . This implies that there is a J > 0 the only point in (a, c + J] where J is tangent to - . Hence spacelike vector field U along -y with value in ( )' and a
there exists
a
function p:
[a, b]
-
-
-
-
R such that
I for all t E [a, c + (U(t), U(t)) J(t) p(t)U(t) for allt E [a,c+J], for alltE (a,c). >0 W(t) for all t E (c, c + 0 W(t) < =
=
We will
be
-4
a
now
by slightly stretching U. 6 (W + O)U. From
construct
function and consider V
V4 + R( , ffzy
=
=
Let
b: [a,
c
+
J]
-->
R
=
V V, (OU) + OR(U, OU
+
20V U + O(V V U + R(U,
(o
+
0)
We get
(V V4 There is
a
for all t E
+
R( , ) ,
)
=
numb er A 1 > 0 such that
[a,
c
+
J].
Let A2
( -
+
0
(A 1) 2
> 0 and
0(t)
(V V, U
<(V V U =
A2(eAl'
R(U, ) ,
+
+
-
U))
R(U, ) ,
U)
e,\ la) Then .
-
It
we
obtain +
0(17 17 U + R(U, 0
) , U)
((/\1)2
VVU
+
A2 (AI)2e\ la for all t E
(a, c+ J)
itive. Observe that
.
We set A2
0(a)
=
+
U))
R(U,
+
A2(A1) 2e Ala
> 0
=
-W(c+ J)/(e'\, ('+6) e-\Ia) which is pos+ 0(c + J) 0, and W(t) + 0(t) > 0
0, W(c + 5)
-
=
Length and
4.6
for all t G
(a, c].
erwise
we
could
field
satisfies
for all t E
We
6
replace
(a)
(a, c
can assume
J).
+
by
a
that c+6 is the first
We will
after
zero
c
239
since oth-
smaller number. To summarise, the vector
J(a), (c+J)
=
energy functionals
=
0, and
V
I y V4
)
R( , ) ,
+
> 0
construct the acceleration vector field
now
A. There is
basis el,
a
.
.
.
( (a))
e,,- 1 of
,
(a) (i) e,, -I (h) spanjej,..., e-dim(E)l 6ik for all i (iii) (ei, ek)
-L
such that
=
T-y(.)-,
=
E
=
n
-
k E
21,
n
-
11.
-I and (e, ej) 0 Let en E Ty (a) M be the null vector with ,e, en 1) for all i E fl,...,n-21. We denote by ei(t) (i E fl,...'nj) the parallel =
=
-
transport of ej along -y (i cnj) such that f dim(Z) + 1, .
.
.
,
There
1( (a), (a))
=
,-k
numbers
are
(k
kek.
Enk=dim(Z)+l
G
Let
n-1
+ 6
C
I:
A (t)
t ek,
-
y(t)eni
+
k=dim(Z)+l
where + j
t
T6 c +
a
C
A(t)
+
We have y (a)
P(C
=:
(I ( a W i
-n
(a))
i
+
and
+
2
V (C + 6)
=0
=
(W
+
*)IC+jP
Hence the vector field A satisfies
J)
=
0. We will
now
show that
+
OIC+8
A(a)
V A,
=
)
=
0.
A(a)j+
V
=
and
ff( a,
V ,
)
< 0
.
A(c
We calcu-
late n-1
V A,
V (C+J-t ek) k
E
c
+
a
-
V (Pen)
)
e-n- 1
k=dim(Z)+l n-1
(C+6-t)'
>]
+ J
ek
k=dim(Z +1
( a W i
i
(a)) c
and
our
claim follows from
+ J
+ -
(a a
7
(a)
-
Aeni
en-1
+
A
Pseudo-Riemannian manifolds
4.
240
+
(1 (Ja, J,), (a))
=
(If7 OSf" ft) fS,
-
now
0 (a) ( (a) +(a))
+
I (a, 0)
Vftf,
0 (a) ( (a)
+
(a))
+
I (a,O)
-O(a) (a)
+
O(a)( (a)
show that there is
a
variation
=
We will
+2 V (a) ( , 0
(a))
+
O(a) (a)
=
f
> 0.
[a, c + J]
x
--+
M
such that 1.
f, (0, t)
2.
f(s,a) E Zfor all sE f (s, c + J) -y(c + 6) for
3.
By
=
J(t),
VfS (0, t) fs
construction of
which
and A such
a
4 0. Hence we will are arbitrarily close to
VA(O)A
from Z to
curves
(-E, c)
Let p:
be
variation must
obtain timelike -y which in turn
j(s, a)
variation such that
For
c
We will
now
=
p(s)
this variation
variation vector field
so
=
the variation vector field
of
j
is
j(s,t),
=
7(c
=
A
f
where
yk (V)
.
.
.
Vk for
=
s
(jdds Z) (O't).
f x1,
let
to
TxM let A be the geodesic given by A(s) d2 =
dS2
(Ai (8))
d2 =
j dS2(expx (SV)) + pik
a2
(exp,,)' v implies
a
2
'j9Y_ IyIF
(exp.,)' (0) A
=
=
k v
-Fjk(x).
+
I
X"
a
6) for all (s, t) Z(O,t) 07(t). -*
=
Restricting the 0 implies that
=
In order to cal-
Y1
any vector =
y' I be
+
j (S) k (S) Fik j
j (S) k (S) k rj'k A3 (S) A (S)
=
I (O't)
( Z) dS2
equation
(O't)
a
For any Then the
v.
exp,,(sv).
If follows that the
(V c), expY(t) (Z))
and
be
+
calculation 0
J)
exist
that the modified variation has
given by
coordinate system of TM such that E
< 0
+
smooth map V:
t) Tz(,,t) expY(t) -!iZ(s, ds
culate the acceleration vector field
v
a
J)
+
c
(a),
=
of Z. Let
and acceleration vector field A.
equation 9, exp(Z(s, t))
given
j(s,
and
-y(a), A(O)
=
connection
small there is
E
modify
-y(c
implies that, there also
p(O)
in Z with
sufficiently (- E, E). V(s,t) E Ty(t)M with expY(t)(Z(s,t)) s
satisfy (ft, ft) from Z to
curves
-y(b).
a curve
0 where V is the Levi-Civita
=
[a, c +
(-c, c).
E
s
timelike
all
s
=
for all
A (t) for all t E
=
4.6
holds and
completely analogously that the
f (s, t)
=
Z(s, t) +s(6(t)
expY(t)
has variation vector field
tion, each of the
S2 + 2
(A(t)- A (t))
6 and acceleration vector field A. By construc1--4 f (s, t) starts in Z and ends in -y(c + 6).
points
The existence of focal points
pseudo-Riemannian
variation
(t))
_
241
t
curves
4.6.3 Existence of focal
the
and energy functionals
Length
depends
three
on
of the submanifold Z. A
typical
near -y, the result is the
factors, the curvature of length of -y, and the shape
manifold
following.
4.6.1. Let Z be
a spacelike hypersurface and -y: [a, b] I geodesic with ( , ) If (a) E (Ty (a) Z) q E I 1, 11 and Ric( (t), (t)) > 0 for all t and the mean curvature vector field H of Z satisfies (H-Y(a), (a)) =: c > 0, then there is a focal point of Z along -Y before -y(a + 1/c), provided c > 11(b a).
Proposition M be
a
=
-
-
-
Proof.
Let
lation of ej at
a
+
11c
f el,.
be
en-11
..'
a
Ty(a)Z and Ej the parallel transj (t) : (1 + ca ct) Ej (t) vanishes
basis of
along -y. The vector field and is tangent to Z at t
=
a.
=
-
Since
n-1
,L,-y
E(
C
+
-
ja+l/c (I
+
ca
-
Ct) 2 R(Ej, )Ej, )
dt
n' a
i=1
( (a), I (Ei (a), Ej (a))) a+l/c
=,q((n -
-
I )C-
( (a), (n
-
1
(1
+
ca
_
ct) 2Ric (,:y (t), (t)) dt
I)H-y(a))
a+1/c
(I
+
ca
_
Ct)2 Ric ( (t), (t)) dt
a
we
have found
and
a
vector field
positive index
is semi-definite
before
-y(a
+
or
11c)
with
in the Lorentzian
negative case.
index in the Riemannian
If follows that I L,-y
indefinite and therefore the existence of
Z&Y(a+1/c)j focal point
a
I
4. Pseudo- Riemannian manifolds
242
The
inequality given
of radius
field is
in
Proposition
and with inner normal
r
4.6.1 is
sharp. Consider
Then its
n.
mean
sphere
in. The geodesic -y with (a) = n,, satisfies -1 and intersects the centre of the at sphere -Y(a+r). Clearly,
given by H ,
(H, (a))
a
curvature vector
=
=
r
the centre is the first focal point of the sphere along -y. There is an analogue of Proposition 4.6.1 for null geodesics and spacelike submanifolds of codimension 2. This analogue will become important in
9
Chap.
on
[a, b]
in
general relativity.
4.6.2. Let Z be
Proposition and -y:
singularities
--4
b be
null
a
a
t and the
mean
with
geodesic Ric (
for all
spacelike submanifold of dimension
(t), (t))
curvature vector
(H-y(a), (a)) then there is
11(b
-
a
focal point of Z along
(a)
a
basis
and
en-,
=
E
(T-Y (a) Z)
> 0
field
=: c
H
of Z satisfies
0,
>
before 7(a
7
If
+
11c), provided
c
>
a).
Proof. The proof choose
(a)
n-2
analogous
is
jej,
en-21
-,
Of
to the
T-y(a)Z.
timelike unit vector en, both
a
en-1 + e, We denote the
Ek. Then
proof of Proposition 4.6.1. We are a spacelike unit vector
There
we
orthogonal to Z such that parallel translation of ek along -y by
have
n-2
Ric( , )
E (R(Ei,
=
) , Ej)
+
) , Ej)
+
En-1)
-
(R(En, ) , En)
i=1 n-2
1: (R(Ei,
=
R(En-j, En)E,,, En-1)
i=1
(R(En, En- I)En- 1, En)
-
n-2
E (R(Ei,
=
This
Ej).
implies
n-3
E, -y,
I
,),((l
Z,I-y(a+l/ c
+
ca
-
ct) Ej, (I
+
ca
-
ct) Ej)
n-I C
+
fa+1/c (1
+
ca
a
(a), I (Ei (a), Ej (a)))
Ct)2 R(Ej, )Ej, ) dt
4.6
(n
-
I)c
Length and
ja+l/c (I
-
+
ca
energy functionals
-
243
Ct) 2Ric ( (t), (t)) dt
n' a
(a), (n
I)H-y(a))
-
a+1/c
(I
+
ca
_
Ct)2 Ric ( (t), (t)) dt.
a
Hence
1',
is not
focal point before
7(a
+
positive semi-definite and therefore there
is
a
11c).
possible to prove the existence of conjugate points of geodesics 7, sufficiently long and if suitable curvature conditions hold along -y (cf. Proposition 4.6.3 below). This result is central to the singularity theorem which is presented in Chap. 9. Since in the case of a single geodesic we do not have initial conditions which guarantee focusing in one direction, the proof of Proposition 4.6.3 will be much more delicate than the proof of Propositions 4.6.1 and 4.6.2. The rest of this section will be devoted to proving the following proposition:8 It is also if they
are
Proposition fold and -y be a
Riemannian
a
or
Lorentzian mani-
in the Riemannian > 0
for all
t and
if
to such that the map R:
is not
be
complete geodesic which is spacelike the Lorentzian case. If Ric( (t), (t))
and causal in there is
(M,g)
4.6.3. Let a
J-
( (to))
identically
-4
( (to))
1
Then -y has
zero.
a
proof of this proposition requires conjugate points. seen
Rv
R (v,
pair of conjugate points.
The
We have
-4
v
,
before that there is
more
an
(n
results from the
-
theory
of
l)-dimensional subspace
along -y which have values in ( )' and vanish at a given the In case that -y is a null geodesic, there is always a linear compoint. binations of these Jacobi fields which is equal to the trivial Jacobi field of Jacobi fields
t
--*
(at + 0) (t).
This uninteresting
the factor space ( (t)j instead of the
-L
(t)) /R (t)
=
subspace disappears
f [v]
:
v
E
(,,:y (t)) J-, v
if
-
one w
#
considers v
-
w
II
orthogonal complement ( (t))-L. This space coincides with the orthogonal complement if -y is timelike or spacelike. If -Y is a null geodesic
then
working with the factor space has two advantages: As we above, [(at + 0) (t)] [0]. More importantly, the metric
have indicated
g induces a metric of degenerate.
We follow also
a
closely
=
[g]
the
much shorter
understand.
on
which is
positive definite instead
presentation in (Beem and Ehrlich 1981). There proof in (Hawking and Ellis 1973) which I failed
is to
244
Pseudo-Riemannian manifolds
4.
Definition 4.6.6. Let -y: vectors V, a
E
R with
v
C-
W
[v]
:
M be
-->
called
are
and t E
geodesic
a
equivalent,
v
-
+
w
[a, b].
Two
if there is a number of equivalence classes
w,
a (t). We denote the space ( (t)) I by [ (t)] ' and set [ ] -L UtE [a,b] NO] _L
E
v
(t))
[a, b]
-L
=
*
*
A map R
along along 7.
[ (t)]'
[A](t):
x
...
-L
[ (t)]
x
-y which is linear in each
(mol-L)
x
of
x
its entries is called
From the definition it is clear that any tensor A of [A] at t via
(mol
x
...
tensor class
a
( (t))-L
induces
a
tensor class
[A]([vi], where
w ( )
cp' =
.
-
-,
is defined
[v,,', k,,'], by
[W']([v])
0). Conversely,
[ o'])
-,
=
V'i,(v)
=
A(vi....
for all
Wil
...
(P r)
I
( (t))-L (In particular,
E
v
vs,
i
any tensor class is induced
by
tensor in this
a
way.
The metric and the covariant derivative in direction gous
objects
Lemma 4.6.16. Let -y be
andAbe
induce analo-
for tensor classes.
any tensor
geodesic
a
field along
and
[A]
be
a
[A]= [A].
-y with
tensor class
along
Then
V A is
[A]
:=
well
defined. If -y is a null geodesic
[g] Qv], [w])
:=
g (v,
The operator
Proof.
defined w) [R]: [ (t)]-L [ (t)]-L, [v]
and positive
-4
For any one-form W
(Vxt),P) Mt)
[g]: [ftL
then the metric
is well
+
0
-4
definite.
[R(v, ) ]
1--4
satisfying W( )
R, [v, w]
x
is well
defined.
have
we
f (tMt))
V (t) (w (V (t)
f W (W)
+
(V
-
(t) (V W
+
f W W)
V (t)(w(V(W) -
(V (t)V(t)
W
V (t)MVW) where have
we
have used that
[V, (V(t)
+
+
-
V,
f (t) (t))]
=
df (t) (t)
w
+
(Vxt)V(t))
f
V (t) (t)) =
V WMWI
0. For any vector field V
=
[V, V(t)
+
df (ffy
+
fV
]
along ==
-y
we
[V V(t)]-
Hence the first assertion holds for
1-forms,
for functions
derivation this proves the first claim.
along
-y. Since
V
is
a
The second claim follows since there is
such that k E
g(ei,ej) n 11.
=
6ij
for
i,j
E
vector
basis el, 21 and
a
11,...'n
fields,
-
.
.
.
,
a
consequence of
R( ,
(trivially)
e,,- 1 of
9(en-liek)
-
The third assertion is
and
0.
(t))
-L
0 for
4.6
and energy functionals
Length
Definition 4.6.7. A Jacobi tensor class is
[]
J-
along
which
for
7
[A]+ [R] [A]
if
and
there is
only
[01
a
tensor class
a
[A]:
holds.
[A]: [ ]J-
Lemma 4.6.17. A tensor class tensor class
=
245
[ ]' along
-->
A
field
tensor
and has the property that t i-+ AV (t) is ' vector field V with values in ( )
[A]
a
along
-y is
Jacobi
a
-y which induces
Jacobi field for every
parallel
.
Proof. Suppose that A is a tensor field along -y such that AV Jacobi field for any parallel vector field V with values in immediately from 17
V [A]
[R]
+
are
is then
[A]
exactly (2n
a
+
Jacobi tensor class.
.
v
T,'-'v'Ej
=
and the
observe that there
Conversely
-
.
.
,
and define the tensor field A: where
a
+
=
2) linearly independent Jacobi values in ( ) -L. Let f El, E, I
which have
is
It follows
[A]) [VI [ (V V A) V] [R(AV, [17 V (AV) R(AV, ) )] =
that
( )'
E
vector fields
Ji along
be
frame of
parallel
a
by Av
Aj
are
=
E'_1 i,j=i AiviJj i
(constant)
real numbers. It is
( )J-
the vector field AV
clear that for every parallel vector field V is a Jacobi field with values in ( )-L. Hence
[A]
G
[A]
is
a
Jacobi tensor class.
Since the differential equation 0 implies that the space of + [R][A] Jacobi tensor fields is 4(n 1)2 -dimensional if -/ is timelike or spacelike and 4(n 2)2 -dimensional if -y is null every Jacobi tensor field can be =
-
-
generated by
some
It is clear from the tensor class with
fields
proof of
constructed above.
I
Lemma 4.6.17 that the columns of
to
respect
as
parallel
a
basis of
( (t))-L
are
a
Jacobi
just
Jacobi
in this basis.
expressed
Corollary
tensor field A
4.6.6. Let
-/:
[a, b]
M
--
be
a
geodesic
and to, t,
E
(to =A ti).
[a, b] (i)
For any
pair of
[Ao]: there is
a
tensor classes
[,,:y(to)]J-
-4
[ (to)]-L ,[Aol: [ (to)]
unique Jacobi
[A] (to) Ao; (ii) Assume that
tensor class
[A]
with
[ (to)]--L [A](to)
=
[Ao]
and
=
[Ao]: be
a
-y does not have
[, /(to)]J-
given pair of
tensor class
[A]
--+
conjugate points and let
[ (to)]J-,
tensor classes.
with
[A](to)
=
[Aol
[A,]:.
[ (ti)]J-
Then there is
and
[A](ti)
=
---+
a
[ (ti)]-Junique Jacobi
[Afl.
4. Pseudo-Riemannian manifolds
246
Proof The sitions
assertions follow
immediately
from Lemma 4.6.11 and
Propo-
2.9.2, 2.9.4.
1
The
following lemma is clear from the definitions and the fact that non-vanishing Jacobi field which is parallel to has at most one zero:
a
Lemma 4.6.18. Let -y be a geodesic. Two points -y(c), -y(d) are conjugate if and only if the Jacobi tensor class [A] which satisfies [A](c) [0], ==
[A]](c)
=
id is
singular
at d.
Definition 4.6.8. Let -y be a geodesic and a tensor class along -y. Then the adjoint of
denoted
by [B]
[a, b] [A] (to)
-y:
=
M
and
be
[A]
assume
is
a
Jacobi tensor class
that there is
( AV, 17 AW)
non-singular
-
=
v,
w
E
=
( )-L.
geodesic
[a, b] with [A] [A] -I is
at t then the tensor class
0
( )--L.
-y with values in
The
17 AV, AW) (17 AV, 17 AW) AV, V V AW) (17 17 AV, AW) 17 AV, 17 AW) +
-
A(to)
a
E
at all t.
=
and
is
along
number to
a
Proof. Let V, W be parallel vector fields along equations
'7
be
to
-
[0]. If [A]
self-adjoint
[ (t)]J[g].Y(t)
-+
*
Lemma 4.6.19. Let --+
[B](t): [ (t)]-L [B] with respect
-
(-R(AV, ) , AW)
(Av, 17
imply that
(AV, -R(AW, ) )
-
17
Aw
Av, Aw
)
=:
0
for all vectors
It follows that
17 AA`v,w
(17 A(A'v),A(A-1w)) (A(A-1v), 17 A(A-lw) v, V AA-lw) .
self-adjoint tensor class [A][A]-' has a direct geometrical interpretation in terms of congruences of geodesics. Let -y: 1a,b] -4 M be a spacelike (in the Riemannian case) or timelike (if M is Lorentzian) M, (81,...,sn-l't) 8n-1, t) geodesic9 and f: R'-' x R f(81
The
-4
,
)
'
The
interpretation
in for
lightlike geodesics
is
slightly
I
*
I
less direct.
1
4.6
be
a
smooth
that each
(n
247
1)-parameter geodesic variation of -y. We may assume t sn- 1, t) satisfies (ft, ft) E I 1, 11 and f (s', the vectors I ft, fi, a f,. 1 1 are linearly independent. ft(s',...' sn-1, t) is a well defined vector field near -y(a). i-+
-
.
.
.
,
.
Then U
energy functionals
-
geodesic
that at t
Length and
.
.
,
-
The covariant derivative of U geometric properties of our congruence of geodesics. The function 0 div(U) measures the divergence of neigh=
dO is the infinitesimal rotation bouring geodesics. Analogously, w and a, the traceless, symmetric part of VU, the infinitesimal, volume sn- 1) 0 preserving distortion of neighbouring geodesics. At (sl, =
=
.
we can recover
The Jacobi field
V ft fs, ft)
=
an
such that
fk(O,..., 0, t)
a
arbitrary function,
--->
( )'
It follows that
( )J-
(V A)A-lf,i
=
is
(V ':Y A)A-1v
(V A)Ej
for all
f fl, E
v
Definition 4.6.9. Let Then the
=O for all
tensor
by
t +
h(sl,
.
.
.
,
sn- I ) I
always parameterise our geodesics parallel along
alongy that
maps
f,,,,, I
.
=
f Ej(t)jj=j,...'n-1 -/ and denote
by
Ej into fi, (0.... 01 t). 1
span
Vftfi
( )--L
-J-.
This motivates the
[A]
be
expansion 0 of [A]
vorticity
t
for all k and t. Let
V (AEj)
0 the
0.
=
Jacobi tensor class and that
a
Since the vector fields
1
which is
the tensor field
[A]
we can
( (t))
E
orthonormal basis of
( )'
,
=
where h is
A:
V fS ft, ft )
=
0 implies (fi.(0,...,O, a), (a)) [a, b]. Since we can replace the parameter
Hence
be
.
fi satisfies
Vft Us, ft) t E
.
this information in terms of Jacobi classes.
a
is
we
=
Vfsift
VfSi U.
=
conclude that
following
Jacobi tensor class
along
VvU
definition. the
geodesic
-y.
defined by
QA] [A] 1), -
=
tr
of [A] by
w
2
and its shear tensor
a
by
([,4][A]-' ([A][A ([A] [A] ([A] [A] +
2
The is
following singular.
+
lemma
implies that O(t)
0(t) id
if
0 n-2
if
;T-I
id
tr([AI [A]-')
E
0.
diverges
where A
4. Pseudo-Riemannian manifolds
248
Lemma 4.6.20. For any
0
Jacobi vector class
a
-_
[A]
we
have
-(det([A]))*, det([A])
where det is any parallel determinant function. (In particular, choose the determinant function induced by the metric [g]). Let
Proof.
r
n
=
-
1 if -y is
spacelike
timelike and
or
r
n
=
-
one
can
2 if -y is
null. Since the space of parallel determinant functions it is clear that the formula in the assertion is
along -Y is 1-dimenindependent of the
sional,
choice of det.
Assume that
[A](to) is non-singular. There is a parallel linear tensor [A] (to) [B] (to) id. Let [C] [A] [B] and f [Ei] I j= be a parallel orthonormal basis of [ ] 1 We choose the determinant function defined by det([D]) where Det is the standard deDet(QDji)i,k), k terminant in Rr and [D] [Ek] At t to we have [C] [Ek] [D]ik [Ei]. [Ek] and Det([El],..., [EI) 1. This implies class
[B]
such that
=
=
.
=
=
=
=
=
(det[C])'It=t,,
(detQC][Ej],...,
=
[C][E,,-,])'It=t,,
r
[i ] [Ei], [Ei+,],..., [Er]) It=tO
det([El],..., [Ei- 1], i=1
n-1
E [C]i det([El],..., [Ei-I], [Ei], [Ei+,],..., [Er])It=t,,
=
i=1
tr([C])lt=t,,. Since
[A]
(det [A])'
=
=
[C][B]-1 (det [C]
we
obtain therefore at t
det [B]
-
=
At to
we
==
(det [Cl)'
1
tr Qi ] [B] ([i ]) det [B] tr Q-A] [B]) det [C] -det [A] -
=
1)
tr
have
-
1
=
=
[C]
Lemma 4.6.21
=
id and therefore
tr
=
to
det [B]
-
1
[B]) det [C]
([A] [A]
(det [A])
*
=
(Raychaudhury equation).
-
1
tr
-'
det ([C] [B]
[C]) det [C]
-
1
det [A].
QA] [A] -') det [A].
Let
(M, g)
be
a
I
Lorentz-
Riemannian
manifold and -y be a causal geodesic if (M, g) is Lorentzian and a spacelike geodesic otherwise. If [A] is a Jacobi tensor class then its expansion satisfies ian
or
-Ric( ,,:y) -Ric
-
tr(W2) tr (w 2)
-
-
tr(,72) tr (U2)
_
_
n1102 n1202
if
if
E
f -1, 11, 0.
4.6
Proof. Let Let
r
=
Jacobi class -
n
[R]
spacelike or timelike and r parallel orthonormal frame of (
a
have
we
QAJ [A] 1)'
Q'41 [A] 1) 2and
=-
-
1
-
tr
=
r
=
vector
n
-
-
0
Y g (R(Ei,
find
) , Ej)
tr
-
w
+
+
a
-
Ei'=, g (R(Ei, ) , Ej) vector field
parallel spacelike field E, such that I El, E, I
E,,. Then
we
[Al
is
a
[A] [A]
tr([A] [A] 1)2
1 it is clear that
we can
2 if -Y is null.
Since
(([A] [A]-')
i=1
is null
-
-
r
If
n
)-L.
[A] (- [A]
+
r
=
=
249
therefore
(tr ([A] [A]-'))' -tr([RI)
[A] [A]
-
-
_
energy functionals
1 if -/ is
-
be
jEjjj=j,...,,
Length and
a
=
id
)2)
-Ric( , ).
E,_1 and
a
orthonormal and
are
If -Y
timelike
En
+
have n
Ric( , )
=:
E g (R(
, Ej) , Ej)
n-2 =
E g (R(
, Ej) , Ej)
i=1
+ g -
g
(R (En- 1
+
E, E,,- 1) (E,,- i
+
(R (En
+
En, E,,) (En
En), E,)
-
I
-
I
+
E,,), En- i)
n-2 =
E g (R(
, Ej) , Ej)
+ g
(R (E, E,,-,) E, En-1)
i=1 -
(R (E,- 1, En) E,,- 1, E,,,)
g
n-2 =
E g (R(,:y, Ej)
, Ej)
,
i=1
where
we
have used the symmetries of R and the fact that n-1
tr(B)=
E g(BEj, Ej)
-
g(BE, E,,)
i=1
for every linear map B. Hence in either case,
r
=
n
-
I
or r
=
n
-
2,
we
get
Ric( , -:y)
-
tr
w
2+
2 a
0
2
id +
+
r2
(W+U)
+WU+UW
r
definition we have tr(w) 0. For any tensor (B] we have tr(a) tr([B]2) tr(([B]* )2) + tr([B]*[B]) [B]*)) tr(([B] + [B]*)([B] tr QB] [B] *). Since the definition of the trace implies
By
=
-
-
250
4. Pseudo-Riemannian manifolds
n-1
tr([B]2)
n-1
E[91 ([B]2[Ei], [Ei])
=
=
E[g] ([Eil, QBI *)2)[Ed)
i=1
i=1
tr(([B]* )2) and tr ([B]
conclude that
we
*
[B])
tr QB] QB]
=
tr(wa)
tr(aw)
=
Lemma 4.6.22. Let -y be Jacobi tensor classes is
parallel along
is
(R(v, ) , w)
=
[v], [w] implies [R]
[B]
[A]
-
*
[h])
=
In
=
[R]*.
Hence
+
([A]*), [B]
-
[A]* [B]
QA]) [B]
-
position
a
obtain
[h]
QA]
QR] [A]) [B]
-
-[A]*[R]*[B]
=
[g] Qv], [R] [w])
-
QA]
[B]
[A]
*
=
we
==
[A]* [_b] *
in
[B]
fact,
(R(w, ) , v)
([A] *) *'[B]
=
=
are now
tr (B
spacelike geodesic and [A], [B] be
or
self-adjoint.
-
We
=
-y.
[g] ([R] [v], [w])
(([A]
-
-y. Then the tensor class
Proof. First observe that [R]
for all vectors
[B]) [B] 1) 0.
=
timelike
a
along
*
to prove
+ +
Proposition
*
[A] [R] [B]
[A]*[R][B]
=
0.
4.6.3
Proof of Proposition 4.6.3. Let -/: R --+ M be a complete geodesic and 1 if -y is spacelike or timelike and r n 2 if -y is null. We n choose to E R such that R(-,, (to)) (to) 4 0. The symmetries of R imply then that the induced operator [R]: [ (to)]' [ (to)]J-, [v] 1-4 r
=
=
-
-
--+
[R(v, (to)) (to)] of
conjugate
which
satisfy
does not vanish. We need to show that -Y has a pair points. Let J be the space of all Jacobi tensor classes [A] w
0, [A] (to)
=
=
and tr
id,
We will first show that each
[A]
E
QA] (to))0 J satisfies det[A](t)
=
0 for
loss of
generality) that [A] E J_ Since t>to. Suppose (without a is self-adjoint, tr(a2) > 0 and the Raychaudhury equation implies 6 < _102. If there is a t, > to with O(ti) < 0 then an integration some
the shear
-
implies
1
0 (t)
vanishes for
r
> -
1
0(ti)
some
t
+ =
t -t'
n-1
t2
>
for all t > ti. Since the
t, the expansion
Consequently, det([A]) vanishes
O(t)
right
must
at t2- If there is not any
hand side
diverge ti
>
at t2. to with
4.6
0(ti)
then the
< 0
imply 0(t) that
a
=
is
inequalities 0(to)
=
-[R]
([A][A] -1)2
-
to in contradiction
to
this would
imply R(., to. The proof
on
completely analogous. For each i > to let [Bj] the unique Jacobi
[Bi] (i)
Assume for the moment that tensor class
[B]
[B] (t)
with
0 and
=
=
[B]. Moreover, [B] (to)
id
=
r
[Bil
0 for all
have
From the
the existence of
a
J+
Jacobi
Vt 1 >
0
vanishing vorticity
W so
the Jacobi tensor class
(still
E
id.
=
proved
implies that
we see
[A]
for
limi __,.[BE](t)anddetQB1(t1))=7
to. Since all Jacobi tensor classes
-.102
<
tensor class which satisfies
[Bi] (to)
have
we
251
0 for all t > to. Because of
=
assumption
our
:5 0 and
Raychaudhury equation
[A] [A]-'
0 and therefore also
energy functionals
tr([A](to))
=
0 for all t > to. From the
=
([A][A]-')* t >
Length and
[B]
has
must
proven) fact that [B] (t) J+ \ J-. It follows that tr([b] (to)) > 0 and therefore that there is a i > to with tr(jB ] (to)) > 0. id this implies that the expansion O[B,] of [BJ at to is Since [Bi] (to) strictly positive. From the inequality 6[B,,J :5 n-1 (OB f')2 we obtain lie in either J_ is
J+.
or
non-singular for
t >
to
we
to be
infer that
[B]'
E
=
1
I
to
>
O[Bf (to) Since
O[Bf W
tt
(t det([Bi] (ti))
this
oo)
oo
--
n-1
satisfies
-
t
for all t < to.
+ n
-
I
the existence of
implies
0. Hence there is
a
a
ti which
non-vanishing, parallel vector
field V such that
V(t)
(, (t))-L
E
for all t
and
BjV(ti)
Since the non-trivial Jacobi vector field defined
i
both t, and
by
J
0.
BiV
vanishes at
geodesic -y has a pair of conjugate points. We still have to show that [B] does exist and that [B] (t) is
singular
[Bil
for all t
which
Let
[A](to)
[A] ==
f
>
to. In order to do
depends only
on a
so we
will first obtain
single, given Jacobi tensor uniquely
be the Jacobi tensor class which is 0 and
t > to since
Let
our
[AI(to)
[A] (to) be
=
a
respect to this frame
=
non-
formula for
class. determined
by
id. This tensor class is
0 and -yl [t,),,,) does not
parallel orthonormal define
we
a
a
non-singular for all have conjugate points.
frame of
tensor class
[ffj- along
-/. With
[C] by
t
([C] [vl)'
=
[A] (t)
Observe that this definition is two
parallel orthonormal
(QA] [A]) 1) 3k (s) ds [V] k. *
independent
frames
are
-
of the chosen frame since any
related
by
a
constant orthonormal
4. Pseudo-itierriani-iianniaiiifolds
252
matrix D and since such matrices
[C]
that
satisfy D*D
=
id. We will show below
[Bj].
=
But first
we
need to check that
(14 1 (t) Io
i
is
Jacobi tensor class.
a
(([A]*[A])-')j (s)ds[V]k
1,41 (t)
=
[C]
k
t
[A]3 (t) (QA] [A]) 1) 3kk (t) [VI k -
[A]3 (t) QA]
=
IV])'
(([Al')
([C] IV])
and therefore
[i ] (t) IV]
[A] [A]
[C] IV]
[A][A]-1
+
[A] [A]
[A] [A]
[A] [A] [A] [A]
-
*
1
-
QA] [A] 1) *) QA] -
[A] [A]
is self
[A] [A] [C] + [R] [A] [A] QA] + [R] [A]) [A] [C]
[R] [C]
=
[C]
is indeed
Now
[Bi] and
we
a
show
[C] (([A] [Bi]
follows
=
=
[Bj].
once we
Since
[A] [bi]) I t.
=
id
[a, b].
In
hand, -
[Qi) 1 (i)
([A] *) We
-
can now
0
[( ](i)
implies
This
employ
=
[Bi](i)
[( J(i)
the
equality
Lemma 4.6.22
[A] [bi] *
-
id
[A] (i) [bil (i) *
particular, we get id equivalent to [Bi](i)
[bj (i).
0
imply
*
at all t E
[Qi)
have shown
([A] *) [Bi]
0. This in turn is
0
Jacobi tensor class.
[C] *
-
adjoint by
[C]
1
and
IV]
[C] IV],
where in the last equation we have used that Lemma 4.6.19. Hence we get +
[C] IV]
([A][A]-'[C] ([A]*)-) IV]
[Al QA] *) 1 IV] [C] IV] + ([A] [A]
-
([A]
+
-
[Bil (i)
On the other
[A](i)[A]-'(i)[C#)
=
since
completes the proof
of
[C]
-
=
QA]*)-1(i) [Bil.
the formula t
([Bil)
(([A] [A]) 1) 3k (s) ds *
[A]j'3 (t)
-
t
in order to show that
[B]
limi_,,,, [Bi]
exists if for
some a
<
to the
M does not have any conjugate points. We geodesic segment 7: [a, oo) will prove this by showing that both limi_(,. [Bi] (to) and limi_,,. [bi] (to)
4.6
(cf. Corollary 4.6.6).
exist
[Bil (to)
=
=
We will
limit for every
frame
along
The existence of the first limit is trivial since =
now
IV]
-y and
E
([bj] IV], IV]) [g] ( Bi] IV], IV]) have
[ (to)]-L.
[A] (to)
[g]
the numbers
show that
=
Oil (to) IV], IV])
[g]
253
i. By Lemma 4.6.19 and [Bj] (to) id it follows that [bj(to) for all i. Polarisation implies then that [Bi](to)
uniquely determined by
[ (to)]J-.
energy functionals
id for all
QBi])*(to) is
Length and
6j
id
a
Since we
where
we
use
IV]
E
well defined
(orthonormal) parallel
an
have
(QA] [A]) -') 3kk (s) ds [Vj k IV] 1
[A]j (to)
*
t
[g]'(([A]
(t) IV], IV])
t
(([A]*[A])-')' (s)ds[V]k[V]1 k
t -
and therefore for all
L
>
[g] (A+ I (to) IV], IV])
[g] (QA]*)-1(t)[v], IV])
Igj A- I (to) IV], IV]
-
t+
6ii
k 1 (([A] [A]) 1)'k (8) IV] IV] ds. *
-
t+
[g] (([A] [A]) 1(S) IV] k[Ek](s), [v] [Ej](s)) *
The last
expression *
[g] (([A] [A])
is
-
non-negative
IV], IV])
since *
[g] (QA] [A])
=
-
1
and
[g]
is
[g] ([A] QA] [A])
([bi] (to) IV], IV])
give
an
<
for every
-
thereby ensuring that the index form
_TE,,y,-L
IV]
G
1
-Y(a)I&YN)l
is
*
IV], [A] ([A] [A]) -IV])
(
[g] [bi] (to) IV], IV]
F-
for all
We will
IV]
E
)
is
show that This will
function
(Al (to) IV], IV])
limit exists. Theorem 4.6.2
positive definite
applying the piecewise
now
[-Y(to)] J-.
monotonically increasing
[g]
IV])
*
[ (to)]'.
([& (to)] [vj, IV])
[g]
upper bound for the
i
Hence
-
positive definite. Hence the function i
monotonically increasing
[g]
*
IV], [A] [A] QA] [A])
*
=
ds.
in all the
implies
that the
cases we
consider.
smooth Jacobi vector field
BaWP -YI[t,to]V
for t G
Bi(t)P -/I [to,t)Vfor
t E
[a, to], (to, t].
254
4. Pseudo-Riemannian manifolds
to I
E"Y'l we
f-Y(-)M-Y(tO)1 0 <
IE
obtain
-y I
J, J)
It V J, V J) (R(J, ) ft V V J R(J, ) , J)
dt
J,
+
"a.
(zA(V (to)J), J(to)) bj(to)v, Bi(to)v) (ba(to)v, Ba(to)v) [g] ([bi] (to) IV], [Bi] (to) IV])+ [91 ( A (to) I IV], [Ba I (to) IV] -[91 ([bj1(to)[V1, IV]) [91 (AI(to)[VI, IV]) +
-
-
a
+
-
-
+
implies the desired bound. Let [bo]: ( (to)' unique selfadjoint tensor class defined by
which
[g] Then
[bj(to)
--->
(Iboi IV], IV]) [bol
=
jim Igl t-00
( (to)'
be the
(Ibil (to) IV], IV])
and, consequently, [B], where [B] This by [B](to) id, [B](to)
Jacobi tensor class defined the existence of
--
=
is the proves
[B].
We have still to show that construction it is clear that
[B]'k (t)
=
[B] (t) is non-singular [B] is given by
[A] i (t)
J00 (([A]
*
j
[A])
k
for t > to. From
our
(s) ds.
t.
Let
V(t)
IV] =
cz
[ (t)]-L \ 101
v.
Then
[g] ([A]
and V be the
parallel
vector field
along
-Y with
[B] IV], IV])
fo" [g] (([A] [A]) IV] (s), IV] (s)) foo [g] QA] QA] [A]) IV] (s), [A] QA] *
ds
t
*
*
[A])
IV] (s))
ds
> 0
implies that the operator QA]-'[B])(t) composition of non-singular operators
is is
non-singular.
and therefore also
Thus B is the
non-singular. I
5. General
relativity
p. 210
Einstein's equation is of the form Dg T, where D is an operator acting on the Lorentzian metric g and T an expression which describes the distribution of matter in the universe. In Sect. 5.1 we motivate
1
=
tensor field which symmetric (0) 2 find an expression for Dg.
that T should be
a
and in Sect. 5.3
we
is
[I
divergence-free,
5.1 Matter
Chaps. 1 and 3 we did not explicitly consider gravity. However, one of the main insights of Einstein was that gravity and the geometry of spacetime are closely linked. His argument is very simple and runs roughly as In
follows.
subjected to a fixed external "force field" depends on its initial location, its initial velocity, its mass, and its charge (i.e. its "sensitivity" to the force field). For instance, a particle in an electric field which is initially at rest will move to one side if it is positively charged, to the opposite side if it is negatively charged and not at all if it is neutral. To be more concrete, consider a reference frame (-r, t) in a Galilei spacetime and suppose that there is a The movement of
non-relativistic has the electric
a
which is
particle
particle (m, -y) which is located charge e. Then the formula Y
an
electric field
and
-
:4
M
in
=
e
-
(t(t,!)
Similarly, let 6 be a gravitational field and charge"i of the particle. Then
holds.
g be the
"gravitational
M.
holds. It is
an
experimental fact that the quotient -L depends on the analogous quotient A- is a universal constant and M
particle can
be set
mere
than 1
whereas the
M
=
1
(EUv6s 1896).
Einstein concluded that this fact is not
coincidence but reveals that a
It is
a
gravitation is an acceleration (rather and therefore geometrical. He therefore replaced something force)
usually called the passive gravitational
M. Kriele: LNPm 59, pp. 255 - 286, 1999 © Springer-Verlag Berlin Heidelberg 1999
mass.
p.
270]
5. General
256
relativity
14
the equation
6 by
=
the
geodesic equation
V
0 and the "force
=
the connection V of
field" 0
by spacetime. This point of view physical interpretation of inertial observers: They are simply observers which are freely falling. to
leads
those
a
experimental fact that the matter distribution 2 spacetime determines gravity. Hence we have to look for an equation of the form It is
an
Dg where
(M, g)
is
=
T,
(5.1.2)
n-dimensional Lorentzian manifold 3, D is
an
kind
some
of operator acting on the metric g, and T contains the information on the matter distribution. The "correct" form of T cannot be derived. First
all, it is beyond doubt that matter cannot be described by a smooth object in spacetime but instead demands a quantum description. This implies that we can hardly expect a description from fundamental, physically suggestive principles. T will therefore be a classical approximation, i.e. something phenomenological. Consequently, our final form for Equation (5.1.2), Equation (5.3.11) will appear to be grounded less firmly than the spacetime structure. However, the reader should recall that in the derivation of the Lorentzian structure of spacetime we already assumed that light can be described in an entirely classical (i.e. non-quantum) of
way.
The
matter models
only
relativistic point
had considered
we
particles (cf.
44)
p.
so
which admit
far where a
specialstraight-forward
generalisation. Definition 5. 1. 1. A
particle
is
pair (m, -y), where
a
particle and -y is a curve in M with g( of t E M, representing the history of the particle. the
Exactly v
at
x
as
=
in the
7(0)
momentum serve as a
=
Lemma 5.1.1. Let
for
mass
for
all
x
T.
defining =
-y(to)
Cz M
.
Since
the
-1
=
and
Jw1,...,w,,J be n linearly 1. Then m (to) and m
=
by
=
-
dependent timelike vectors with (wi, wi) determined by the numbers E,,,, Ew.,,,. Proof.
(t), (t))
M
guidance
mined
> 0 is
special-relativistic analogy, an infinitesimal observer the energy E, -mg( (O), v) and the spatial 1 -"v. The (O) following simple observation will
measures
(O)-L
m
19(wi, Ew,,,
is
and
.
.
Tx*M, m
be calculated from
m can
inare
,
basis of
a
-
is -m
uniquely 2 =
deter-
g (M , M ).
1 2
Here
we use
the term "matter" in
a
rather wide
sense
encompassing
all forms
2 of energy. This is motivated by the special-relativistic equation E mc 1: velocity of light) which asserts that (rest) mass is simply a form of (c =
=
energy 3
(cf.
Sect.
1.4.3).
spacetime we live in appears to be a 4-dimensional Lorentzian manifold. 4. However, in this book we will not specialise to n The
=
5.1 Matter
In other
words,
we
only need
E:
fv
E
in order to
T, M ,,
the
recover
to know the energy function
g (v,
:
257
v)
1
R,
w
complete information about
a
1-4
E,,
single particle.
Since g is a smooth object, we would expect T to be smooth also. This indicates that point particles which are not depending smoothly on the coordinates of M cannot be used to constitute T. The to obtain
a
smooth matter distribution from
to consider averages instead of individual
Definition 5.1.2. A congruence of c: M --+ R is a function and U is
g(U, U) The
=
a
simplest way particles is
collection of
particles.
particles is a pair (,E, U), where future directed vector field with
a
-L
integral
curves
of U
are
identified with the world lines of the particles 6 with the energy density, measured by
and the energy density function comoving observers. To keep the
presentation simple we will restrict to a A0 0, i.e. that there exists locally a special to U (cf. Theorem 2.5.4). which Z is orthogonal hypersurface spacelike If B c Z is a compact region then an observer flowing with the particles measures for the energy of those particles which pass through B the quantity case
and
assume
dO
that
E
=
=
JB
'EME,
where jLZ is the induced volume form single observers must be identified with
Definition 4.2.2 4
(cf. a
timelike
curve
Since
a
rather than
a
congruence of curves this expression should be understood as an approximation for small B. It is clear that we recover the definition of a point
particle if the compact set supp(,E) n Z shrinks to a point and the energy density E increases adequately. A different congruence of observers, represented by a vector field V 1 and dO A 0 with g(V, V) 0, will measure a different energy =
content,
EV
=
JB
EV/-tzv
I
v
where -
-
-
4
Ev
is
a
spacelike hypersurface orthogonal
pzv the volume form induced
BV
=
fx
E
EV
:
3
a
V, particle through
to
V,
on
x
which intersects
ZJ,.and
Readers who have not read Sect. 2.5.4 may wish to do so now. Alternatively, they may (for the time being) refer to the footnote in Definition 4.2.2. In the
following
we
in order to avoid
will make
clumsy
use
but
of calculus for differential forms
straightforward calculations.
(Sect. 2.5)
258
-
5.
Ev is
a
General
relativity
function which
depends
on
the congruence of particles
(6, U)
and the observer .el j V, We will
with
now
motivate
transformation law
a
E
-4
ev
through comparison
special relativity
Fig.
5.1.1.
A
localised
congruence
Consider Minkowski space
(A', TI) V:
where
x
and the inertial observer field -4
V
-1. Assume also that the congruence is a vector with q(v, v) i.e. restricted to any spacelike hypersurface Z, Ev has compact support (cf. Fig. 5. 1. 1) The inertial (or freely falling) observers t F-4 x + tv v
is
=
localised,
with
common
rest space
EV
=
x
JX+V-L
+
vj-
EV
/-tx+V-L
measure
the energy
=1.+V-L
EV
(V IPA-),
where MA,,. is the volume form of q. We assume now that the congruence of particles is freely falling, i.e. the field U satisfies the geodesic equation
17UU
=
0. In the limit that
supp(Ev)
n Z
shrinks to
a
point
we
would
the energy associated with a single freely falling particle. Since 17UU 0 we can choose linear coordinates (x 0 xl .... Ix n-1
recover
=
such that
U=00,
This
(aO 11V1101)
V=
+
gives EV
(aO ilvim) J[1'? +
'EV +V
V-1:T,1 V71
-
Matter
5.1
f"+v X
1-ITe
JjVjjdxO A dX2
-
(dxl
F- V I
fx+v
A
...
A
...
A
259
dx"'-'
dx'-1
Evdxl
A
...
A
dXn-
I
I
Evdxo
dX2
A
...
A dx-
X+V-L
X1+
f.XO Evdxo, +
Evdx
V-1 Fv(x 0, x1) (minimal) values
where
hyperplane,
we
VI-1
I ive =:
fEvd X2
of x
A
...
A dx
n-1
and
denote the maximal
4
restricted to the support of ev in
have xO
JIVIlx'
x
In this
+ V
and therefore X1
Let -/: t
-
rest frame
sured is
by
Xn-')
(t, X1,
falling particle -
with rest is
m )
Evdx
(m, -y)
and let
mass m.
to
+
fX
I
Ev
be the
Then its energy =:
the infinitesimal observer
-m
v
(,Oo, ao)
( fl
corresponding freely
-
=
measured in its
m.
The energy
(ao
-
own
mea-
+
given by
to (Vm )
Ev
An
=
-
-F,
analogous relationship should also hold for our smooth congruence can be used to smoothly model a point particle.
U since this congruence Hence we should have
E,
+F&I. f 1 IFVF X
Since
x
-+
Ev
(x)
was
arbitrary this equation implies
transformation law indicates that EV Postulate 5.1.1 The map T is measured by an
(Tensorial
'Ev
T7-,FV1-11-1,11
depends quadratically
This
V.
character of energy momentum). field and the energy density
symmetric Q-tensor 2 infinitesimal observer v is given by T(v, v). a
on
E.
260
.5.
elierai
4
'.1"Vity
In the is
a
special case of our congruence of (non-interacting) particles there simple, well defined tensor field TU namely ,
TU Observe that cv
TU(V, V)
=
U,
c
=
U1.
0
is in accordance with the transformation
law derived above. The
lemma indicates that it is
following
energy densities c, in order to
Lemma 5.1.2. Let T be
determined Let
Proof. all
u
with
by
the values
T, S be
(u, u)
two
a
v
2
all vectors
u
tensors with
symmetric
-I and let
=
(')-tensor.
symmetric
T(u, u) for
enough
the tensor T
recover
be
to know all
possible
(cf.
5.1.1).
Lemma
Then T is
with
g(u, u)
T(u, u)
v/ V_--(v, v, ) satisfies g (v/ V"----(vv, ), v/ V _--(vv,)) T(v, v) S(v, v). Since the space of all timelike vectors =
for every vector w Hence we obtain
a
J > 0 such that
I
T(w, w)
=
2 1 =
2
(dd-t T(v (d S(v
+
v
tw,
-1.
S(u, u)
for
timelike vector. Then the vector
a
=
=
uniquely ==
-
1 which
yields
is open there is + tw is timelike for all t EE [-J, J].
+
v
tw) jt=0
2
dt2
and the claim follows from the
tw,
+
v
+
tw)
)
S (w,
W)
1t=0
polarisation identity.
Conservation of energy and momentum is another fundamental propmatter which we wish to encode in our theory. We will find an
erty of
infinitesimal formulation which momentum
(in special cases) recovers conservation of (cf. Equation (1.4.12)). In Sects. and 1.2.1 and 1.4.3 we have
simply stated conservation of momentum. These conservation laws can actually be derived within the theory of point particle mechanics. This is the content of the Noether Theorem which is covered in textbooks on mechanics. The main non-mechanical input for the Noether theorem is the Galilei group (in the non-relativistic case) and the Poincar6 group (in the relativistic case). Recall that the Poincar6 group is the set of all
isometries of Minkowski
spacetime. In order
mulation of conservation of momentum
Killing
vector fields which
1-parameter
a
Killing field.
Proof.
be
regarded
to find
an
infinitesimal for-
will therefore have to as
infinitesimal
employ analogues of
groups of isometries.
Lemma 5.1.3. Let T be be
can
we
since T is
a
Then div
symmetric
(T( ,
divergence-free
(') -tensor with div(T) 2
0. we
have
=
0 and
Matter
5.1
div(T( , -) )
=
(TabQ
V,,
=
(VaT ab) 6
+ T
ab
(Va6)
Now the symmetry of T and the anti-symmetry of imply that the second summand also vanishes.
lEtItER
Let
be
a
foliation of M into
=
Ve (cf.
T
ab
261
(VaCb)
Lemma
-
4.5.2) I
spacelike hypersurfaces with future
normals nt. A world tube with respect to lEtItER is an open subset )/V of M with piecewise smooth, timelike boundary such that the intersection
WnEt is connected for all then the
we
t. If W is
a
world tube with respect to
I-TtItER
denote th& subset
boundary
UtE [ti,t2l )/V n Zt by IlVt,,t, and the part of which is not contained in Zt, U Zt, by )/Vti,,,,.
0 -tensorfield with div(T) Corollary 5.1.1. Let T be a symmetric (0) 2 and be a Killing field. Let tj < t2 and VV be a world tube with respect to jZtjtER such that supp(T) nl/Vtim, 0. Then the following conservation =
law holds.
(nt2
(nt,, T
nt, _j pm
2nwt,,t2
t,
nwt,
nt2 -j AM
"2
Proof. We have
(nt, T( , -) ) (nt ipm)(VI,..., Vn_j) 0) ym (nt, V1 (nt, T -(T( , for any
Zt,/, Wtim.
=
Vn-1)
1.tm)(Vj,...' V'_j)
I)-tuple of vector fields tangent to Zt. Hence pulled back to -T( , .)0 Jym. This (and supp(T) n get Ot, T( , .)0) nt Jym 0) imply
(n
we
_j
1
-
=
I (nt,T( ,-)O)ntJpm=j
ntV T( , .)0) t2
nw,,
nt, J ym
t2
nt2 'j AM t,
where
we
nw,,,, t2
have used that the future and past boundaries Zt2 and Zt, have orientations. The assertion follows now from the Theo-
opposite induced rem
of Stokes 2.5.5 since
d( nt, T( ,
(ntipm))
=
-div(T( , -)O)pm
o.5
fZ, Ot, T
Hence the
quantity
parameter
t defined
Readers who have of GauB.
by
0) (nt J pm)
the foliation if
skipped Sect.
div(T)
independent
is =
2.5 may instead
0. We will
apply
the
of the time
now
identify
integral theorem
262
5.
this
General
relativity
quantity with
a
component of special-relativistic
momentum in the
context of Sect. 1.4.3.
We will
that spacetime is isometric to Minkowski spacetime represented by a spacelike hypersurface Zt, and after a time t2 represented by a spacelike hypersurface Zt, We will study a matter model which consists of k freely falling congruences of particles in the region before Zt, and after Zt, In between these hypersurfaces interactions or collisions may take place. Hence in this region we will (at this point) neither make an assumption on the matter model nor on the before
assume
time tj
a
metric.
To be concrete, consider the set A', a point 0 E A', and a nonvanishing constant 1-form -r. This 1-form defines a foliation of A with affine hypersurfaces Zt Ix E A : 7r(x-o) tj. Let 77 be a Minkowski metric such that Zt are spacelike hypersurfaces and let V be the timelike, future directed constant vector field which is orthogonal to all Et =
and satisfies M
q(V, V)
=
-
=
1. Assume that the
spacetime (M, g) satisfies
A' and 9jfxEM:7r(X) '-[t1,t21j 7711XEM: W(X) 'E[tl,t2l 1. Let (Ei, Ui)i=l,...,k be the congruences of particles defined at all points x with .7r(x) =
==
(tl,t2).
Assume that
isfy supp(Ei)
n
Let T be
a
Ix
VUiUj
E M
:
7r(x)
G
(0)
symmetric
0 and that the energy densities Ej sat-
=
Itli t2jj
is compact.
tensor field with
2
div(T)
=
0 and
k
Tx
and
=
that
assume
E ci (x) (Uj)
(M, g)
0
X
admits
(Uj)
for
Killing
a
7r
X
(x)
(t 11 2)
vector field
. Corollary
5. 1.1
implies that
f
(V,
,
f
V J pA,,-
T
(V,
,
T
V _j /-tA,,,
t2
Conversely, ity
it is clear that
is valid for all such
div(T( ))
=
0 must hold if the
integral equal-
particle flows.
Since the vector fields Uj
constant for
are
x
Zt (t
E
[tl t2l) i
we
I:k dEi(Ui)Ui Uj are get div(T) point linearly independent then this equation implies dEi(Ui) Oj.e. the =
=
0. If the vector fields
at each
=
energy
density
following are
we
of each
will
particle
assume
flow is constant
that this is also the
(pointwise) linearly dependent. (mi),, (Tni)2 defined by
Then the
along case
where
(Zi)a
=
f(_,').
is chosen such that
+ WO
if these vector fields
in-going and out-going
masses
(Tni)a
its flow lines. In the
,6i Ui
J tLA-
rest
5.1 Matter
-
-
supp(Ei) ?T((Zi)l)
are
n <
7r(X) E [tI t211 t1, ?T((Zi)2) > t2 Ix
E
M
:
=
7
263
0 and
well defined constants.
Fig.
5.1.2.
tion of the
T ransforma-
mass
density in
special relativity
is constant for t > t2 now that the Killing vector field ,. For each vecti, and denote these constant vector fields by 6 0 field Uj and a E f 1, 21 there are vectors (e-i) a with (V, (ei) a)
Assume t < tor
:::::::::
1
((ei)ai (e-i)a)
=
1, and I
A)
The
integrals
(V, T( ,
(V
z- "-
+
Oj (ei),,).
in the formula above reduce to
V J PAII
Ei
(V, Ui) (Ul a)
V J /-tA,l
k
a)
6i
fz
V _j ttA,,,
t(II
k
(Ui7 a)
Ei
Ui
J PAn
k
(Tni) (Ui a) a
where Hence
we
we
have taken the
contraction into account
conservation of
special-relativistic demand the second matter postulate
recover
motivates to
length
i
(cf. Fig. 5.1.2).
momentum. This
264
5.
Postulate 5.1.2 T has
(Infinitesimal conservation law). 0. vanishing divergence, div(T)
The tensor field
=
Postulate 5.1.2 is
interpreted
as an
infinitesimal formulation of
conser-
vation of energy and momentum. That these quantities are conserved is intuitively clear from the absence of a perpetuum mobile. However, the
infinitesimal formulation
implies a true conservation law only if spaceKilling vector field. In general, this is not the conservation of energy can only hold infinitesimally.
time is endowed with case.
It follows that
a
(0)
Definition 5.1.3. A symmetric -tensor field with div(T) 0 is called 2 an energy momentum tensor. It is sometimes called stress energy momentum tensor
5.2 Some If T and g
or
stress energy tensor.
matter models
specific
are
=
simultaneously diagonalisable,
-1
0
0
1
0
...
6
0
0
P,
T=
g
0
...
(5.2-3)
0 0
then
...
0
1
0
0
0
...
Pn-1
the energy
density with respect to the flow of interpreted as principal pressures. (i 11, 11) To motivate this interpretation we will simplify to a perfect fluid, i.e. a matter distribution for which all principal pressures are equal. E
is
interpreted
matter and pi
5.2.1 The
E
perfect
as .
.
.
,
n
-
are
fluid
Definition 5.2.1. Let E,p: M --+ R be smooth 1. Then vector field with g (U, U) =
T
functions and
U be
a
-
=
(c +p)0
0
is called the energy momentum tensor with p 0 is called dust.
0 +pg
of
a
perfect
fluid. A
perfect fluid
=
EW & U5 considered in Observe that the energy momentum tensor T the motivation of Postulates 5.1.1 and 5.1.2 describes dust. =
Lemma 5.2. 1. Let T be the energy momentum tensor Then div(T) 0 is equivalent to
Of a perfect fluid.
=
dc(U)=-(E+p)div(U), where iru-L denotes the
projection
(,E+p)VUU==-7rU_Lgrad(p), to the
orthogonal complement of
U.
5.2 Some
It is
Proof.
straightforward
specific
matter models
265
to calculate
(divT),, =
9C bVcTab cb
=9
(ac(E+P)Uaub+(E+P)((Vcua)ub+ua(VcUb)) +OcPgab)
=
d(E+p)(U)U,,+(E+p)(VUU),,+(E+p)div(U)U,,+grad(p),,
=
(dE (U)
(E
+
+
(E
+
p) div U) U,,
A (VUU)a
+
+
The assertion follows since
projected
to U
((0
0
VUU
0
+
g) (grad(p),
(Ub
1 U and
(9
Ub
+
is the metric
g)
1
I
-
The vector field U is the
velocity
of the fluid
particles
and
the
e
en-
ergy a comoving observer would measure. If the divergence of this vector field is negative then the particles are getting closer together and conse-
quently the
energy
density should
increase. This is
in the first
expressed
equation. The second equation states that the spatial acceleration of the fluid particles is proportional to grad(p). This indicates that p should be
interpreted
as
the pressure exerted
Perfect fluids
are
the fluid.
phenomenological
models and the equations im-
develop shock waves. It is therefore often fluids exhibit properties which are not shared by real
0 tend to
plied by div(T) argued that perfect matter. However, perfect fluids =
on
are
prevalent
in
cosmological
models of
the universe.
5.2.2 The collisionless gas
An attempt to arrive at a more realistic matter model is to consider a relativistic gas. The idea is that we do not have a congruence of particles but that each individual
particle
can move
energy momentum tensor is then obtained
velocities. Let
the canonical defined can
by
a
where
=--
JP+
P+(x)
over
all
particle
(xo.... Xn-1) be a coordinate system of M and choose coordinates (xo.... Ix n-1 ,Po Pn-1) of T*M which are ....
==
Pa(a)dxa
be described
Tab (X)
in any direction. The
by averaging
by
an
for every 1-form
a
E
T,,,M. A relativistic gas
energy momentum tensor
PaPbf(x, P) (- det((gcd)c,d=0,-..,n-1))_'! dP1 2
A
...
A
dPn,
(-) C
T*M
denotes the set of future causal 1-forms and
f:
P+(x)
--4R+
is a density function. We assume that for lp,, I --- oc the function f (x, -) is sufficiently rapidly decreasing so that the integral is well defined. Observe
that the n-form
266
General
5.
relativity
det;((gcd)c,d=O,...,n -1))_'! dP1 2
does not
depend
the choice of coordinates
on
A
...
(xl,
Xn- 1)
tivistic gas is collisionless if the Liouville equation where I ab
XH
PaaXb
=9
-
2
dp,
A
df (XH)
-
The rela-
==
0
holds,
19x,ga6PaPb19pc .6
Using a system of normal coordinates it is easy to see that df (XH) 0 0. implies div(T) If U is a vector field and f was replaced by the delta distribution =---
=
j(,1-CUa "gas"
_
pa)
would obtain dust. Hence dust may be viewed as a are all aligned and move into a preferred direction
one
whose molecules
determined
by
the vector field U.
Analogously,
relativistic
a
photon
gas is
given by
an
energy
momen-
tum tensor of the form
Tab (X)
fPO
::::
PaPb A (X
P)
+
Po+(x)
where
C
Vpl 0 (X),
Tx*M denotes the submanifold of non-vanishing future v 1-form on PO+ P+ (X) is a non-vanishing, oriented n 0 (x),
null Morms at x,
-
0
and fo:
P+(x) 0 We
R+ is the photon density function with respect
-+
assume
that for
IPal
oo
the function
fo
vp+(x). rapidly decreasing so that the integral is well defined. The following lemma implies that the energy density a relativistic gas is always positive. --->
is
0
to
sufficiently
associated with
Lemma 5.2.2. Let T be the energy momentum tensor gas (respectively, photon gas) with f ':: 0 (respectively,
T(u, u)
> 0
for all timelike
spectively, fo)
vectors
u
unless the
vanishes.
Proof. This
is clear since for each vector
for
is
T(u, u)
of a relativistic fo ! 0). Then density function f (re-
u the integrand in the definition I positive unless f (respectively, fo) vanishes.
Lemma 5.2.3. Let T be the energy momentum tensor Then tr(T) 0.
of
a
photon
gas.
=
Proof.
The assertion follows from
tr(T)
==
gab
,
+
PaPbfO(XiP)1,'p(+ (x) 0,
0
since the 1-forms p,
are
JP0 91 (p, A +
A (X, A VP,+ (X)
=
0
0
null.
Readers who have
knowledge of mechanics will notice that XH is just the Hamilton vector field to the Hamiltonian function H (x, p) IgabPaPb. The =
2
equation df (XH)
(Ehlers 1973)
for
=
0 expresses then conservation of mechanical energy
details).
(cf.
5.2 Some
5.2.3 The
An
field
can
matter models
267
field
electromagnetic
electromagnetic
specific
be described
by
a
2-form F which satisfies
Maxwell's equations,
dF
div(F)
=
0,
(5.2.4)
=
J,
(5.2.5)
where J is
interpreted as an electromagnetic current one form. The first be geometrically explained within gauge theory (A small equation volume which contains the essentials of gauge theory is (Bleecker 198 1)7). The second equation does not have any content without a prior interpretation of J. For our purposes it is sufficient to note that J is linked can
to other forms of matter.
Remark 5.2. 1.
instead of If there is
fields,
Using
div(F) no
=
the
Hodge
star
operator
we can
write *d
interaction between
i.e. if matter is
neutral,
electromagnetism
then
1
(Tel)ab
we
are
leading.
=
J
and the other matter
dF
=
0,
(5.2.6)
div(F)
=
0.
(5.2.7)
=
4-7r
(gcdF
I
ac
-
Fbd
many mathematical texts
on
-
one
(F,
F)gab)
may
(or
lack of
replaced by a
given by
(5.2.8)
-
justify these formulas.
"gauge theory"
For "mathematical convenience"
the Lorentzian metric of spacetime is This leads to equations which are of
_
4
We will sketch in Sect. 5.3.1 below how There
F
have the set of equations
These equations are called the source-free Maxwell equations. The electromagnetic part of the energy momentum tensor is
7
*
J.
a
which
are
very mis-
physical knowledge) Riemannian metric.
very different nature from those
which describe
physics. Only in very special cases (a prerequisite is that all is it possible to convert results of the Riemannian theory to the Lorentzian theory using an analytic extension argument according to which one can "rotate" a Riemannian theory into a corresponding Lorentzian theory, where both theories are embedded in a complex theory. In the literature on quantum field theory this rotation is known as the Wick rotation. The Riemannian analogue of gauge theory is mathematically (but not necessarily physically) of interest because it is linked to the well developed theory of elliptic partial differential equations. Gauge theory, on the other hand, is linked to hyperbolic partial differential equations. To sell the Riemannian analogue as gauge theory has presumably the advantage that functions
are
analytic)
it makes it easier to
get funds for research in
pure mathematics. On the
other field
hand, it does confuse people. A pure Mathematician who worked in a closely related to this Riemannian analogue and who saw work using
the Lorentzian metric instead of a Riemannian metric once even asked whether this Lorentzian approach would also be useful to physics!
me
5.
268
General
relativity
Lemma 5.2.4. Let F be
by Equation Proof. Since 1 Fac VbFca
5.2.8. Then
dF =
47rdiv(T)b
=
0
we
a
we
closed
cd =
gcd FacFbd
Va
9
Fbd VaFac
4
+
assume
F(.,
that Tj is given
'-,div(FO))
41r
V,Fab
=
F(., JO)
0 which
=
implies
gives
-
19ef9cd FecFfdgab
4
gcdF
, a,
ga Fbd
-
-
-9
4
ef
cd
9
Ffdgab Va F.
9ef 9cd Fecgab Va Ff d gcdFacVa Fbd
==
F(., div(FO))
+
=
F(., div(FO))
+ F
=
=
have VaFb, + VbF,, +
-F ac Vo Fb,. This
=
2-form and
div(Tj)
have
a,
VaFbc
-
-F ec VbFec 2
+1 Fec VbFce 2
F(., div(FO)). I
Corollary
5.2.1. Assume that the
source-free Maxwell equations hold.
Then
div(Tel)
=
0.
Remark 5.2.2. Recall that in the derivation of the Lorentzian struc-
spacetime we assumed that light rays can be described by null geodesics. Since light is electromagnetic radiation we should now check that this identification is consistent with the description of electromagnetism in this section. However, this would require a proper discussion of electromagnetism which is beyond the scope of this book. Readers with knowledge of electromagnetism may consult (De Felice and Clarke 1990, section 7.8) for the identification of light with lightlike geodesics. Here we can only say that nullgeodesics can be taken as a description of light
ture of
rays in
an
limit.
(observer-dependent)
5.3 Einstein's
equation
Recall that the equation which links geometry and matter should be of the form
Dg In the
preceding
two sections
we
=
T.
have motivated that the
of this equation should be a symmetric, divergence-free Now we will find an expression for the left-hand side.
right-hand side
(0) 2
tensor field.
5.3 Einstein's
theory
In the Newtonian
of
gravitation, gravity
equation
269
is described
by the
equations
where
0
I
==
0
A0
=
ko,
is the Newtonian
=
potential
grad(o),
(5.3.9) (5.3.10)
for the
gravitational field. Equation
a second order partial differential equation for the Newtonian potential 0 and describes how it is related to the mass density
5.3.10 is
varrho of the universe. Recall that
17
=
we
0 which is
5.3.9
have
replaced Equation equivalent to a
a It follows that the Christoffel
by the geodesic equation
Fbc symbols rba,,,
have
r6le similar to the
a
force field 6. One obtains the Christoffel symbols from g differentiation, just as one obtains the gravitational force field 0 from the Newtonian'potential 0 through differentiation. This indicates that 0 corresponds to the metric g. Since the Newtonian potential is related to the matter distribution via a second order partial differential equation, we expect that g F-4 Dg is likewise a second order operator.
gravitational
via
(Gravitation
Postulate 5.3.1
is determined
given coordinate system, D: function Of ged; aaged, and '9a'ybgcdIn any
Theorem 5.3.1. Let
08 and Dg be
a
(0) 2
Postulate
If,
in
Dg
is
a
a
2
nd
-ord.
pde).
pointwise smooth
be a Lorentzian manifold such that dScal: field which satisfies Postulate 5.3. 1 and
(M, g)
tensor
div(Dg) (cf.
--+
g
by
=
0
5.1-2). is linear in
addition, Dg
aaabg
then there exist constants
A, y
E
R such that 1
Dg
Proof. By Corollary
=
/-t(Ric
-
2
4.3.1 and the
Scalg)
+
Ag.
linearity assumption Dg
must be of
the form
Dg Lemma 4.3.1
implies 0
=
clRic + C2Scalg +
C39-
now
==
div(Dg)
cl
=
8The condition states that dScal is vanish identically.
(2 C2) +
not the
dScal.
null-function,
i.e. dScal does not
270
5. General
relativity
Hence the result follows
dScall., =h
by
assumption that there is
our
an x
E
M with
0.
1
Remark 5.3. 1. The assumption that
is linear in its
Dg
highest derivatives
is rather awkward. Lovelock not in
(1972) has shown that in 4-dimensional (but (!)) Lorentzian manifolds this assumption is
higher dimensional Unfortunately, his proof is
not needed,
far too involved to be reproduced
here.
Remark 5.3.2. Observe that
did not
we
even
need to
assume
symmetry
of
Dg, i.e. Postulate 5.1.1 is superfluous. However, the symmetry assumption was important to prove the conservation property Corollary 5.1.1 which motivates the
In
conclusion,
equation
as
our
requirement div(T)
postulates imply
that
:=
0.
gravity is governed by Einstein's
defined below.
Definition 5.3. 1. Einstein's equation R is given by
(with cosmological constant)
A E
I
Ric
-
2
Seal g +
Ag
where T is the energy momentum tensor tion. In the above
form, Einstein's equation
the Gravitational constant and the
=
87rT,
(5.3.11)
describing
is valid in
velocity
geometrical
of light
1984, appendix F) for explicit translation rules
the matter distribu-
are
units where
(cf. (Wald units).
set to I
to other
Remark 5.3.3. Einstein's equation itself does not indicate any
special
value for A. In the past, astronomical observations seemed to very
of
small,
if not
zero.
gravitation arises
if A
imply that JAI is theory (c: velocity of light) if and only
It should also be noted that the Newtonian
as a
limit for
c -4 oo
0. This
implies that A must be very small if non-zero (cf. (Hawking and Ellis 1973, p. 362), (Sandage 1968)). On the other hand, I have been told that to present day cosmological =
data point to
a non-zero
value for A.
Some of the theorems which will be presented do require A
=
0,
1
Ric
-
2
[p.
255
and
in
-9
1
p. 270
Seal g
=
87rT,
much of the literature Einstein's equation is used
(5.3.12) synonymously
with equation 5.3.12. 9
Our
guide ends with Einstein's equation. For what follows we will also use skipped in order to get to Einstein's equation
the material which has been
5.3 Einstein's
5.3.1 The
formulation of Einstein's
Lagrangian
In this section
equation
271
equation
alternative way which leads to Einstein's equation
an
approach also aids in finding an appropriate energy momentum tensor. Unlike the rest of this book, this chapter rests on an underlying principle which is difficult to verify directly. This section can be omitted on first reading and is not required for any other part of this book. is sketched. This
It appears that all fundamental
dynamical equations in physics admit a Lagrangian formulation. According to this formulation, a physical system R where E is an appropriate is described by a Lagrange function L: E generalisation of a vector bundle over spacetime M which contains the possible physical states of the system. Such a setup is motivated by classical mechanics. One can calculate the movement of a point-particle -y: [a, b] -- A' with mass m which is subject to a conservative" force field through the variation of an associated Lagrange function. Let L: A' x R 3 -- R be given by L(x, v) --+
T
T
(VI V)R3
-
V(x).
Then
rn (t) (cf. Equation (1.2.7))
-y satisfies the
a curve
if and
-gradV1.y(t)
=
only if
d
L(-y(t)
T-r
r, ["
equation
-rh(t), (t)
+
+
Th(t))dt
=
(5.3.13)
0
b]
for all smooth maps h:
[a, b]
-4
R 3 with
h(a)
=
h(b)
=
0. In
fact,
we
have d
Ir
)
L (-y (t) + J,=0
-r
h (t),
(t)
+
-r
h (t)) dt
la,b]
f[a, (M (:Y (t)' f[a,b] (m ( )
h (t)
b]
)
-
R3
dV(h(t)))
dt
d
dt
-
-
m
( (t), h W) R3
dV(h(t)))dt
quickly. Since we will discuss
physical applications which make use of all skipped, a continuation of this guide would lead to a lot of skipping forward and backward. On the other hand, the reader should have by now enough physical motivation in order to read the mathematical sections which we have skipped without getting bogged down. Still, the reader is advised to read on and to skip back only when needed. On these occasions skipped material should probably be read section-wise. Here "conservative" simply means that there is a function V: A R with P -gradV. the mathematics which
10
(t), h(t))R3
we
now
have
---->
=
272
5. General
m
relativity
m (t)
( (t), h(t))R3
+
gradV,-,(t), h(t) ) R, dt.
the first summand vanishes for all choices of h with
Assume, there scalar product
(a, b)
is
a
is
non-degenerate
to
E
m (to) (say
>
0).
Let
h: [a, b]
By continuity there
(m (t)
m (to)
there is
+
an
h(b)
=
=
0.
0. Since the
E
0
any smooth function with
neighbourhobd (t-, t+)
a
h(a) 4
grad Ry(to) with
ho
gradV,7(to), ho W =h
R' be
---*
is
+
with
C
(a, b)
h(to)
=
ho.
of to such
)R3
> 0 for all t E (t-, t+). Finally, let gradVjy(t), h(t) R+ U f01 be a smooth positive function with non-empty sup,0: [a, b] in port (t-, t+). Setting h 7ph the integrand (m (t) + gradVjy(t), h(t) )R3
that
+
--
=
non-negative and strictly positive in an open set. Hence the integral must be positive in contradiction to Equation (5.3.13). This proves m (t) + gradV,.y(t) 0 for all t. is
=
The existence of
Lagrangian formulation is widely seen as fundageneral (classical) physical systems. The physical state of an elementary particle" is described by the section 0: M -- E of an approa
mental for
vector bundle. Its
governing equation should again be determined integral equation whose integrand is built from 0, its derivative, and physical fields which interact with 0. To make this program work one first has to define the derivative of a section in an arbitrary vector bundle. It turns out that one can generalise our treatpriate
the variation of
by
an
ment of derivatives of vector fields
of
general
vector bundles
ble to define
(cf. Sect. 2.6) and define connections (Kobayashi and Nomizu 1963). It is also possi-
notion of curvature for these
generalised connections and one can interpret this general relativity curvature F in terms of physical fields which interact with the given elementary particle. To obtain the complete system of equations one writes down a Lagrange function which depends on 0, its derivative, the curvature F, and perhaps other physical expressions. We denote all these physical inputs collectively by 0: M ---> E and by T -4 0, 0+T a has compact support. The one-parameter family of sections such that equations which have to be satisfied by the physical system are then given by a
analogously
-
to the
case
of
-
=
d
(.Cum)
a--r for all variations
This
the
The mathematical
0, of 0. Lagrange function C.
It is far
0
M
recipe
is referred to
the variation
of
properties of equations
of
as
beyond the scope of this book to explain what this actually is. Our purpose is to vaguely set the following discussion into context. Readers who want to know more are referred to books on gauge theory. only
5.3 Einstein's equation
273
this type are similar to the properties one encounters in the theory of minimal submanifolds (cf. Lemma 4.4.8 and the discussion following this
lemma). We will
now
discuss
an
formulation, the physical
this
example of
system consisting of electrodynarnics and gravity. To keep things simple we will assume that there are no other physical inputs. In particular, there
electromagnetic sources, i.e., there summands,
are no
no
charges.
Our
Lagrange
function will consist of two
L
where
Lgrav
=
Lgrav
+
Lei
-
stands for the contribution from gravity and Lei for the
contribution from
electrodynamics.
Remark 5.3.4. If we had included electromagnetic have to add at least two more terms: -
-
sources we
would also
A term Ckin for the elementary particle which is analogous to the term M ( ) 6 in the mechanical example above, and a
term
Lint which describes the interaction of the elementary particle electromagnetic field.
with the
suspected that the electro-magnetic field generalised connection 2( (Bleecker 1981) The connection Q1 can be identified with a 1-form A which, however, is not invariantly defined. (This corresponds to regarding the Christoffel dA. The first of symbols as tensors.). With this 1-form A we have F then trivial is a Maxwell's equations, dF 0, consequence. According to The reader may already have F is in fact the curvature of a
=
=
the program above we have to vary L with respect to the 1-form A. This 0. Gravity depends will give the second of Maxwell's equations, divF =
on two geometric quantities, the torsion-free connection V and the metric g. We will use the Palatini formalism, i.e., we will independently varyC
independently with respect sense only
variation makes
to V and g. Observe that this
if
we
do not
assume a
independent priori that V is the
Levi-Civita connection. It will turn out that the variation with respect to the connection will fix the Levi-Civita connection and that the variation 12 with respect to the metric will give Einstein's equation. The simplest non-trivial, invariant function L,j which can be defined
is
(modulo
constant factors and modulo the addition of
a
constant
term)
given by -
,ce, (A, v, g) where
(F, F)
=
=
-
167r
F, F)
=
-
-
167r
(dA, dA)
,
gacgbdF,%bFd,
Alternatively, one could assume that the connection is the Levi-Civita cononly vary the metric. We have chosen the Palatini formalism
nection and
because this is
more
akin to the treatment of other gauge theories.
274
General
5.
relativity
Lemma 5.3. 1. Let B
'r," (-A,-'[)
C
and consider the vaxiia-tio- ---
be
terisor
a
4-
,
I
(C.iym)
o
0,
gence
7
where
47r
operator From
Proof.
tbBd
dBbd
I
d
167r
d7-
f
-
g and
(Tiv
is the diver-
we
(dA,, dA,)
get
pm
A M
-L
I (F, dB) -1 IM 9ab9 (Fac)(' bBd I-tm
87r
Am
cd
=
of
t.
tdBb
-
-
-rB, V, g). Then
A M
is the Levi-Civita connection
associated with
with compact support
f ((&F)(B )[tm
=
M
holds,
field
4, 77,
dj,' ;-
-
87r
tdBb)l-tM
gabgcd (Fac)tbBdAM
47r
A M
47r
M
47r
M
(tb (_,abgcd (Fac,)Bd) (F (-, BO) 0)
-
gabgcd tbFacB d) AM
(&F ) (B)) Mm.
The first summand vanishes because of the theorem of GauB and the fact that B has compact support. Since the
respect
fmCal (A, 17, g)lLm
I
does not
depend
on
17 the variation with
to V vanishes.
Lemma 5.3.2. Let h
C
sym(T2(M))
be
support and consider the variation
a
tensor
(A, 17, g,)
field =
with compact
(A, 17, g
+
-rh).
Then d
('COM)
I-F holds,
where Tj is
.
(Tel)abh abltM
0, 2
M
A M
given by Equation (5.2.8)
Proof The equation
0
- 'dT
((g-r) ab (gT )bc) Ir=o
=
habgbc +gab dr d(gr)bc I-r=O
implies d h ac dT
Recall from the
(gr)ac.
proof of Lemma 4.6.20 that the derivative of det(,P,), whereWT 1-parameter family of matrices, is given by (det(W,))* det tr(( W-') W. This implies is
a
5.3 Einstein's
(det(g,))*
=
tr(h) det(g,),
where in this formula tr is the metric trace of
Using d
these two formulas
we
275
equation
covariant
a
(') 2
tensor.
calculate ,
(j
(41
07-) (tLM)
0
A M
d
16,1 -
7r
M
167r
M
dt
-r)
I 7 =0
Iet _(gr) (FacFbd (gr)ab(gr)cd V_-_det(gr))
dxl
A
...
A
d Xn
I-r=o
2FacFbdgcd h ab.",det () g Ad Xn det(g)tr(h))dx (FacFbdgcd (F, F) gab) habMM.
(FF),I.
+
2 1
A
...
1
_
87r
M
4
I
Remark 5.3.5. We have thus obtained the form of the energy momentum tensor
by
variation of the
-(I 67r)
-
1
electro-magnetic simple Lagrange function
(F, F)
with respect to the electro-magnetic potential A. For other matter fields analogous results hold. In this sense it can be said that variational tech-
niques aid For the
in
finding
gravitational
the correct energy momentum tensor. term
we
set
1
Lgrav (A, V, g)
=
167r
(Ricabgab
-
2A),
where Ric is the Ricci tensor with respect to the connection V. This is again the (modulo constant factors and summands) simplest invariant function which
can
be build from the metric g and the connection V.
Recall that for any two torsion-free connections V, the difference is a field K which is symmetric in its covariant indices. In index-
(l)-tensor 2
notation, K fields
V,
is
W. We
tVWa
given by VVWa
simply
Lemma 5.3.3. Let C
write V E
T2"(M)
+ K.
be
a
in its covariant entries and which has
V +,TC and d
d-r
(
0,
=
(A, (V,), g)
(,CgavAM) M
-
0,)
1,=O
we
VcWd for all vector + Ka cd
have
tensor field which is symmetric compact support. Setting (V,)
5. General
276
relativity
161 1M(K
bd
d 6a c
+ K
d
d
cgab
2K acb
-
CcabAM
,7r
where K is
defined by V
'
+ K and
is the Levi-Civita connection
of g. Proof. Since the Ricci we
d
dT Let
tensor is the
only quantity
which involves
(V,)
obtain
(f
Lg,,
(A, (V,), g)
o
A M
x
E
M and consider
(Ric,)aU
=
ac (Fab c
d (Fab
+
+
7Cab) c 7
-
'9b (1-ac c
+
c
c
-Ca b) (Fdc
x
(Ric7-)ab 17.=Ogab4M.
dT
A M
and for
r
=
x.
From
7Cac) c
7Cdc)
+
and the fact that the Christoffel calculation that at
16-7r
IT=O
normal coordinate system centered at
a
d
+
d
(Mm),)
d
(Fac +
-
symbols vanish
d
c
-Ca c) (-Vdb
7
at
0 the derivative of
x
we
Ric,
is
+
see
c
-Cd b)
7
without
given by
d
j,-(Ric-r)abl,r=o,x Since this is
VcCablr=O,x
equation which points of M.
tensor
a
must hold at all
We will
c
=
connection
7
re-express
VbCaclT=O,x-
is
independent of coordinates
c
with respect to the Levi-Civita
c
now
c
-
VeCab-VbCac
and the tensor field K. From the definition of K c
VdCab
tdCab + KdeCab c
=
c
c
e
e
-
KdbCae
e
we
it
get
c
KdaCbe
-
and therefore ab
9
(Vecab
ab
C
-
VbCac
9
( ccab
VbCac ab =
9 +
=
KcceCab
Kb`eCac
+
-
KcebCae
KbecCae
+
-
KceaCbce
KbaCcce)
( CCab tbCac) -
gab (KceCab C
div(tr2,3C +
(Kc gab Ce
-
-
-
KceaCbce
-
KbceCaec
+
Kb'aCcce)
trl,2C) 2
gdb Ked a
b
+ Kf hg
fh6a) Cab, e
trij denotes the (metric) trace over the ith and jth entry. Since the a divergence with respect to the Levi-Civita connection I vanishes integral by the theorem of GauJ3.
where
first summand is its
-
+
5.3 Einstein's
Lemma 5.3.4. Let h E
support and g,
d,r
=
sym(T20(M))
g + Th. For
0,
be
field with compact
tensor
a
(A, V, g,)
we
have
M
I =
-
167r
fm (
Proof. We can split the integral separately, d
=
277
equation
(j
(CgravAM)
0
A M
d 167r
M
A M
d 2A
T-F
cd
gab+ Agab hab Am.
parts which will be considered
into two
I,=O
((RiCab (g,)ab
-
((Ric,)ab ((g,) V
1Riccdg
-
2
d
I
161r
-
d -r
07-)
RiCab
det(g,))
2A)
ab
det(g,)I,=O)dxl
V/---det(g,)) A
dxl
A
...
A
dx n
I-r=O
I-r=O
A d xn.
...
Exactly as in the proof of Lemma 5.3.2 we see that the second summand in the integral equals -Agabhab. For the first summand we calculate d
TT
(gr)ab
-det(g,))
I-r=O
d d-r
((g,)a b) Ir=o
det grT) + 1
-
bd
ac
-g
g
hbd V- det(g)
(g,)ab d7-
det(g,),,=o
-
+ -g 2
ab
cd
g
hed V- det(g)
and therefore
d
RiCab
-
dt
det(g7. ) det(97')) ((g,.)ab,/-
Ir=o
RiCab+
The
following corollary
Corollary be
a
hab V
---det(g).
is the main result of this subsection.
5.3.1. Let A be
a
I-form, V
Lorentzian metric and set F
Einstein's equation magnetical field,
IRiccdgcd 9ab 2
=
be
a
torsion-free connection,
g
dA.
and Maxwell's
equations for
a
source-free
electro-
278
General
5.
relativity
I
Ric2
are
I
Scalg+Ag
(gcd
=
2
equivalent
1
FacFbd
(F,
-
4
F)gab)
dF
7
(41 +,Cgrav)
d-7-
Proof. We
can
Since C is
arbitrary
MAM),
-
Taking 0
=
bd
d ja c
which
add6b c
+ K dd c
gab
2K acb
-
K
ad d
+ nK
ad
+ 2K
d
dcgab
-
d+2K dda
-
+
0
=
2nK ddc
-
a
Ccab
0
-rC, g +,rh) where
and b. This is
2K acb c we
-
4K dcd
a
=
2K bca
=:
equivalent
to
(5.3.14)
0.
get
and b
2(n
separately.
0
2K ad d- 2K dda
the trace with respect to
now
in
symmetric
are
the trace with respect to b arid
=
Taking
c
Cab + K
C
=
consider the variations with respect to A, V, g Lemma 5.3.3 implies at each point x
(K d6a
0
A M
=
bd
=
d
for all variations 0, (A, (V,), g,) (A + -rB, V B, C, h are tensor fields with compact support.
K
divF
0,
to
(
for all tensors
=
we
-
=
(n
-
I)K
ad d-
get
2)K ddc-
These
equations together with the symmetry of Ka bc in b and c: imply 2) that all traces of K vanish. Hence Equation (5.3.14) simplifies K acb -K bca and Kabc is a tensor with the properties
(for to
n
>
=
Kabc We will it is
now
=
-Kbaci
Kabc
=
Kacb-
show that this tensor vanishes. Since it has the property that in two indices and anti-symmetric in two other indices,
symmetric the expressions
sym(K )
and
alt(K5)
0
=
Kabc
+
Kbca
+
Kcab
+
0
=
Kabc
+
Kbca
+
Kcab
-
both vanish. This is equivalent to
Kacb
+
Kbac
+
Kcba
and
These
(Kacb
equations imply Kabc + Kbca + Kcab Kabc + Kbac + Kc.ab symmetries of Kb, 0
=
=
and
we
have V
=
t
Kbac + Kcba)-
+
0 and
=
therefore, using
the
Kcab. Hence K vanishes
5.4 The Einstein
equation
equation dF
The
=
as a
systemof partial differential equations
0 follows
trivially
from the definition F
The second part of Maxwell's equations, divF from Lemma 5.3.1 since B is arbitrary. Since h is
=
0 follows
279
=
dA.
immediately
arbitrary and we know that 17 is in fact the Levi-Civita validity of Einstein's equation follows immediately from
connection the
Lemma 5.3.2 and Lemma 5.3.4.
1
The process to
as
leading to Einstein's equation via Corollary 5.1.1 is referred varying the total Lagrangian 41 + Lgrav with respect to the metric
9.
We have chosen
have
concrete
a
models admit
a
electromagnetism for our matter model in order to example. To my knowledge all fundamental 13 matter Lagrangian formulation such that
d d-r
) f
((Linatter + Lgrav)AM)
for all variations of the metric is
the
=
0
equivalent
to Einstein's
equation for
matter model.
particular Lagrangian
That the
trivial and I have
means
5.4 The Einstein
of
O-r
0
A M
Ir=o
partial
ansatz described in this section works is no
explanation
by
no
for it.
equation as a system equations
differential
Physicists are accustomed to the fact that (classical) physical systems depend on initial conditions and then evolve in a determined manner which is governed by second order hyperbolic differential equations. Since the energy momentum tensor T contains the metric, the Einstein Equation (5.3.11) cannot be simply solved for a given T. Instead, one has to convert the system. of Equations (5.3.11) into a system of partial differential equations for g and some matter quantities. The analogue in relativity would therefore be to fix an (n dimensional Riemannian manifold
(Z, 'g)
which represents
instant of time. This manifold will be isometric to
surface in the solution. Since Einstein's
system
we
specifies
would need to
prescribe
equations symmetric (0) 2
are
functions
or
tensor fields which
our
hypersurface.
represent the initial
an
initial
spacelike hypera
second order
tensor field k which
the normal derivative of the induced metric
the second fundamental form of
at
a
a
_r
g or,
equivalently,
We also need to fix matter distribution
Z, possibly also their normal derivatives. A perfect fluid is a macroscopic concept and the Lagrangian formulation does not work well in this case. See (De Felice and Clarke 1990, chapter
6.5)
for
a
discussion.
280
5.
Geneml rela"'IMI,
The character of this system of partial differential equations will crucially depend on the form of matter assumed. In particular, one can choose unphysical matter models which lead to spacetimes in which it is possible for information to travel faster than light (cf. Corollary 7.4.1). It is also possible to choose unphysical matter models which do not lead to a hyperbolic system of differential equations. Another problem lies in the fact that we have always the freedom to change coordinates. Hence the choice of coordinate system may also have an effect on the kind of system of partial differential equations we
will end up with.
Nevertheless, in most situations of interest, it is possible to obtain a well-posed system of equations. We will show this for the special case that T 0 and A 0. In order to avoid subtleties arising from the theory of partial differential equations we will assume that our initial data 0g, k are analytic and that Z is an analytic manifold. (This restriction allows us to appeal to the relatively elementary theorem of Cauchy-Kowalewskaya.) We will also fix coordinates in which the equations are especially simple. In Chap. 6 we will study the more general case of a perfect fluid. However, we will impose strong symmetry assumptions in order to simthe system of equations plify the problem drastically (cf. Sect. 6.2) will be reduced to a system of ordinary differential equations. Chapter 7 contains an intermediate treatment. We will again consider a perfect fluid but use weaker symmetry assumptions which lead to a system of partial differential with two independent variables. This system of equations is substantially simpler than the general system depending =
=
-
on
4 variables. We will therefore be able to
an
analytic)
existence theorem
(cf.
give
Theorem
a
smooth
(rather
than
7.4.1).
Since the
analogous but considerably simpler discussion in Chap. 6 already exhibits some of the key concepts of the initial value problem for Einstein's equation, the reader may wish to skip the rest of this section on first reading. Let
(M, g)
be
Lorentz manifold Z C M be
smooth, spacelike hypergeodesic - ., with nx. There is a neighbourhood of Z which is foliated by these x(O) geodesics. If this neighbourhood is chosen small enough it is also foliated by spacelike hypersurfaces of the form Zt 1-yx(t) : x E Z1. If one views M as being foliated by spacelike hypersurfaces Zt (ZO a
surface with normal 14
n.
For each
x
E
a
Z consider the
=
=
Z)
with induced metric
fundamental form kt
as
-"'g
then
one
can
the t-derivative of
view the associated second
-I-'_1,g: 2
Lemma 5.4.1. Let Z be
be
a
be
a
14
a spacelike hypersurfaces of (M,g) and JZtj neighbourhood as constructed above. Let x1.... X n-1 coordinate system of Z centred at x E Z.
foliation of
Here
we mean:
a
I
g (n,
n)
=
-
1,
g (n,
v)
=
0 for all
v
G
TZ.
5.4 The Einstein
equation
Then there is
a
xi, j:n-1 i=1 q,j (t,
systemof partial differential equations
as a
neighbourhood
U
of
M such that g
(E
x
-dt2
+
Xn-') dx'dxj.
Moreover, the second fundamental form kt of Zt X
=
281
is
given by kt
c9t z`9
find a neighbourhood U of x such that for every point exactly one point -' E ZnU and one geodesic -y, through: which satisfies , (0) n,. and intersects Z exactly once without leaving U. This gives a chart (U, p) defined by W(y)
Proof. We
can
y E U there is
=
where y -/, (t). It follows from =
given by Ztg
functions. At t
=
0
construction that the induced metric
our
En-1 i,j=
=
X1 I gij (t,
I ....
Xn-1)dx dxi,
where gij
are
on
Et is
suitable
have for each -- E Z
we
n-1 =
gj
-dt2
E
+
Xn-l(.: ))dx dxi
gij
i,j=l
(O)
since
nx _L
=
E
ZO. From
=
=0
tz
0
(:4' axi)
V%% ,axi at,
we
get
(t)
Zt for all
I
V'9X
i,
)
( X' V%axi
+
V
a,
t. This
2
at' Vataxi 0
a,Xi
implies the claim
for the metric compo-
nents.
From Lemma 4.4.6
get
= Vaxiat' axi (Vaiaxi" a",j at gij axi' va"9xj ) at
kt (axi, axj)
0
=
The assertion
we
kt
=
=
-
-LCOtEtg 2
follows
now
*
gij
-
kt (axi ax i ) ,
from the symmetry of
-
kt and
from
V at_,"19) ii
=
V at E19) Oxi. axi) I
=0
'9t -'""gij
-
-,"'gGeataxi, axl)
=0
-
E"g(axi.' xat,9xj)
=
atgij. I
We denote the Levi-Civita connection induced Ricci tensor of
(Zt, Elg) by
El Ric.
on
Zt by EtV and the
282
General
5.
relativity
Lemma 5.4.2. Einstein's equation with vanishing cosmological constant for vacuum is equivalent to the following system of equations.
Otat-"'gij
=
0
=
1
-2EtRiCij Z'Scal
-
Zt,
+
-
4 0
Proof. to Ric Zt
=
(2 atzt'9ijatZt'9k1 19tZt'9ikatZtgil Z"gjl z119 k1, (19tZt'9ij19tZt'9k1 atZt`9ik19t 5,tgil) gii Etgk1
-
_aX,(Etgjkat Zt'gjk)
Ztgjkax., Z tgik
+
Einstein's equation is given by Ric 0 which is equivalent s`1g 2 0. The GauB equation (Proposition 4.4.1) and Ric 0 imply =
-
=
=
Ric (U,
W)
(R (at, U) W, at)
=
+
-
tr (kt) kt (U,
W)
Z"g (kt (U, -) 0, kt (W,
(R(U, at) at, W)
=
(19t -`9ij at Zt'gkl 5'
4
-
at
5t'gik 19t Et,gjl Z1,gj1 zt,gklUiWj
In order to
simplify the term (R(at, U)W,,9t) we may assume that U, W be extended to vector fields U, W which are everywhere tangent to Et and commute with i9t. Using Lemma 4.4.4 we obtain can
=0
1--*1--l
(R(U, at) at,
W
VU Vatat -Vatvuk
(V"tVUat, W) Vqtkt (U, -) 0, W)
W) V",Vatu, W)
I
at
-
2 ==
(.Ca --,.g(U, W))
+
t
X'9tSat-,",g(u, W)
-
2
2
Ot at -"gii
ISa Ety(u, Va W)
2
-C
+ 4
t
t
at -'"t,9 A
+I19t 5t'9ik19tEt9j1g
-Pat -Y,119 (WI -) 0)
k1
) Ui Wj.
4
Hence the spatial components Ricij 0 of Einstein's equation alent to the first system of equations in Lemma 5.4.2. From ==
Rijkl Etgik Etgjl
and the GauB
=
Rijkl (9
=
Scal +
equation
Ric (,9t,
at)
=
we
-
+ at (g
2ffic(Ot, at)
=
Et
Scal +
are
equiv-
t)j1
+ at & a
2(Ric
-
ScIal g) Ot o9t)
get
I(Scal + 2
at)ik(gV
tr(kt )2
_
-
11kt 112)
5.4 The Einstein
Hence Ric
systemof partial
differential
283
equations
=
f El,
Let
as a
0. implies -"I Scal + -14 (tr (-C at Ztg) 2at Ztg 11 2) E,,- 1 1 be an orthonormal frame of Zt and U be a
0
=
equation
field which is tangent to Zt. Then the Codazzi equation
vector
(Proposition
4.4.2) implies Ric (,9t,
U)
1: (R(Ei, Ot) U, Ej)
=
i=1
(V U (kt & at) (Ei, Ej), at)
(7 Ej (kt
0
at) (U, Ej),
at))
n-I
E ( ((V U kt) (Ei, Ej) i9t, at)
=
-
( T Ej
*
kt) (U, Ej) i9t, at
)
j=1 n-1
((-"'VUkt) (Ei, Ej) Ot, Ot) VE, kt) (U, Ej) at, at)' 9
2
Hence
Ricti
equation
=
-"Idiv (kt) (U).
U
tr (kt) +
(
ax, (Zjg'k,9t
0
(i
E
1,
n
Zt
gjk)
+
11)
is
-
Zt, jk
g
a.X.1
Zt'gik) Ui.
equivalent
to the last
system of I
in the statement of the lemma.
The first system of differential equations in Lemma 5.4.2 consists of n (n 1) coupled differential equations for the 2 n (n 1) unknown func-
1
-
-
n tions gij 1, 11). One would expect that these equa, gji (i, tions would uniquely determine g and that therefore Einstein's equation =
-
.
.
.
would be over-determined. Since over-determined systems of differential equations have only very few solutions (if any at all!) and are in general
incompatible with initial value problems, Einstein's equation seems (at first sight) to be very different from other equations in physics. However, the following lemma shows that the over- determinacy of this system is of a very special nature and in fact compatible with a (slightly restricted) initial value problem. Lemma 5.4.3 and Theorem 5.4.2 below hold for initial data which
necessarily analytic. However, we cannot anymore appeal to Cauchy-Kowalewskaya.
not
the
since
the
is then much
Theorem 5.4.1
(Cauchy-Kowalewskaya).
Let F: R 2m- I
Rkand fo: R'-1
---
--->
are
difficult
proof relatively elementary theorem
Rk be analytic
more
maps.
of
Then there
284
5
is
a
f:
U
liahviay
r,
0 and a unique analytic map neighbourhood U c R- of x-R k which satisfies the system of partial differential equations =
ax f=z
719X-.-1f).
rn
f (0, x1.... XM-1
and the initial conditions
Proof (sketch).
fO(X1....
I
I
XM_ 1).
The idea of
proof is to determine the Taylor series of f successive differentiation of the system of partial by jxm 01 differential equations and then to show that this series converges. A formal proof can be found in (Dieudonn6 1971). 1 at
X
E
=
The theorem of
Cauchy-Kowalewskaya rests on the fact that an analytic by its Taylor series and it does not hold when "analytic" is replaced by "smooth". In the non--analytic case
function is determined the word
the structure of the system of partial differential equations matters for both, existence and uniqueness of solutions. This fact indicates that by restricting to the analytic case one may (in general) obtain results which are
misleading
because
Lemma 5.4.3. Let
foliation
as
E
at,9t-'_1,gjj If
at Z
f 1, =
do not
generalise
to the smooth
case.
(M, g)
in Lemma
(i, j
nents gij
they
be a real-analytic spacetime, {ZtltER be a 5.4-1, and assume that the spatial metric compon 11) satisfy -
-
-
.
,
-2z'Ricij
-
(2 atgij'9tgkl
-
19tgik'9tgjlg ji
)9
k1
ZO the "constraint equations"
=
0
hold then
=
'Seal
+
(tr(k) )2
(M, g) satisfies
cosmological constant, Proof. Let
Ric
_
11kJ12,
Einstein's =
0
=
-dtr(k)
vacuum
+
'div(k)
equations with vanishing
0.
Ric be the Ricci tensor associated with g -dt2 + gij dx'dxj. this tensor satisfies Ricij 0 for all spatial compo=
By'assumption, nents. The
2r,'d,cRiCdb)
=
0 implies therefore gab (19a RiCbc identity div (Ric 12 Seal g) 0 which is equivalent to -,9,Scal =
-
-
=
n-1
0
Since
Ricij
=
-,9tRictc
aiRicic
-
0 this is
-
aRictt
linear system of
-
2rdcRiCdbgab
partial differential equaequations are equivalent to Rictt 0 at Z. Hence we have Rictt Ricti 0 everywhere by the uniqueness-part of the theorem of CauchyRicti I Kowalewskaya. =
a
tions for the unknown functions =
=
n
Rictt, Ricti. =
The constraint
=
5.4 The Einstein
equation
as a
systemof partial differential equations
Theorem 5.4.2. Let
(Z, -g)
manifold field which satisfies
and k
Riemannian
0
ZScal
=
(tr(k) )2
+
be G
_
an
(n
-
I)-dimensional real-analytic
sym(T20(Z)) ilk 112,
0
=
285
be
real-analytic
a
-dtr(k)
+
tensor
Zdiv(k).
n-dimensional
real-analytic Lorentzian manifold (M, g) _1'9 and t*ko t*g k, where ko is the second fundamental form of t(Z). Moreover, if (1 1, j) is a second Lorentz manifold with these properties then t(Z) C M and E(Z) C 1 1 have neighbourhoods which are isometric. Then there is
and
an
Fix
Proof.
an
immersion
a
t:
Z
M such that
--
=
=
coordinate system for Z and consider the system of
partial
differential equations
Ot(kt)ij
-[-"1Ricjj](Ztg, ((ht)[kj)k=1,...,n-1 ((ht)[kl])k,1=1,...,n-1)
=
-
Ot-"',gij ,Ot(ht)[k]ij ,9t(ht)[kl]ij where
braic
=
=
=
((kt)ij(kt)kl 2(kt)ik(kt)jlzlgjl) -
Zt k1
g
2(kt)ij, 2,%k(kt)jj, 2a,,ka.,1,(kt)jj,
[ZtRicjj](E1g, ((ht)[k])k=1,...,n-1, ((ht)[kl])k,1=1,...,n-1)
is the
alge-
expression defined by 5'
["7tRicjj](Ztg, (aXk -t9)k=1,...,n-1, (aXkaXI, Ztg)k,1=1,...,n-1)
Z'Ricij.
=
Cauchy-Kowalewskaya implies that for any real-analytic set of initial values 1,-rOgij, ko, ((ho)[k])k=1,...,n-1, ((ho)[kl])k,1=1,...,n-1j7 0 and a unique solution of the system there is a neighbourhood U of t which is defined on U and has these iniof partial differential equations The theorem of
=
tial values. Since
at-tgij
=
the equation
2(kt)ij
at(ht)[k]ij
=
2ax1c(kt)jj
implies 0
=
at (ht) [k] ij
-
a,, k at Z"gij
=
at ((ht) [k] ij
-
aX
k
Z"gij).
of this equation we see that (ht) [k) ij aX -"tgij if that the In same see 19Xk,9X,,-,"gjj 19XkZ"gjj. (ht)[kl]ij waywe (ho)[k]ij if (ho)[kl]ij aXkaXI -Togij. It follows that the solutions Z1,gjj of this
From
integration
an
k
=
=
system of equations also solves
atatzlgij if and
=
only
atE"gij
-2ztRicij
-
(2 19tZ t'9ij19tZt9k1
5'
-
19trtgik at tgjl
t'gjl
Z
19 W.
if the initial conditions
=
2(ko)ij7
(ho)[k]ij
=
aXkZ('gij,
(ho)[kllij
=
aXk19X1,ZOgjj
5.
286
General
relativity
hold. Now the assertion follows
directly
from Lemma 5.4.3.
1
Theorem 9.4.1 is local in character. Also note that the coordinates chosen tend to
develop singularities due geodesics t
and observe that the
A discussion of the smooth
to --+
focusing effects (cf. Proposition 4.6.1) (t, x1, x') are length maximising.
case can
.
.
.
,
be found in
(Hawking
and Ellis
1973, chapter 7). An improved but mathematically more sophisticated theorem is presented in (Hughes, Kato, and Marsden 1977)
6. Robertson-Walker
6.1
cosmology
Homogeneity and isotropy
It is very difficult, and one cannot make with any certainty assertions about the universe as a whole. This is so because we only know a very
small portion of the universe. Hence any cosmological model reflects our own prejudice. Nevertheless, there are certain assumptions which seem to have
high degree of plausibility. After having built
cosmological model Although imperfect, this approach seems to have given us much deeper understanding of the development of the universe than would seem possible at first sight. The first cosmologists placed the earth at the centre of the universe. Copernicus' revolutionary model gave us a much more humble place in the earth was reduced to being just one of its planets. the solar system This model of the universe had such a success that nowadays we not only take it for granted but don't even sincerely doubt that there may be other (more advanced) forms of life in the universe. At Copernicus' times, such a thought would have been considered blasphemous. The a
one can
compare it with the few data
we
a
do have.
-
monk Giordano Bruno
(1548-1600)
truth of such ideas.' Our in
spacetime
is in
places anywhere a cosmology.
was
burned because he asserted the
leads
us to think that our place modesty and that there are no exceptional way exceptional, We will this fundamental idea to build use spacetime. new
no
in
Let x E M be our event in spacetime (M, g) and Ux E TxM be the velocity vector of our world line. If there is not any point (or direction) in spacetime which is special then the universe should be isotropic, i.e., it should not be possible to distinguish any direction in Ux-L by physical measurements. Although a glance at the nocturnal sky indicates that this is at odds with experience, on a sufficiently big scale this assumption coincides very well with observation. The galaxies seem to be randomly distributed, and since at night we mainly see a part of a single galaxy (the Milky Way), our first impression is not very representative.
He is sometimes
styled
and
an
important forerunner of enlightenment. (Bruno 1584) quite unscientific.
must confess that I find his book
M. Kriele: LNPm 59, pp. 287 - 305, 1999 © Springer-Verlag Berlin Heidelberg 1999
I
6. Robertson-Walker
288
cosmology
The mathematical interpretation of the isotropy assumption is that which impossible to construct geometrical objects, using Ux and U-L X
it is
breaking
are
this symmetry. Let E be
(any)
3-dimensional subspace of
(n I)-dimensional space U.- and u, v, w be vectors in E. Since U R(v, U) is a vector, isotropy about Ux and the fact that (R(v, U) U, U) 0 imply that R(v, U)U I-t(x)v for some scalar I_t(x). Consider any 2the
-
=
=:
plane P spanf u, vJ in E. The sectional curvature K(P) should be inof P since otherwise therewould be a plane PO of maximal secdependent tional curvature which in turn defines a distinguished direction POJ- in E. Since by isotropy there should not be any distinguish'ed direction in E we conclude that at x the equation R(u, v)w n(x) ((v, w) u (u, w) v) + =
=
c(u,
v, w) U holds We show -
R(P) lies in
U.J-
=
that
now
((U,V)2
c
=
-
0. The vector
11U11 IIVII)-1/2 R (u, v) U., .
_
and
depends only on P (rather than on the representatives 1 by irp the orthogonal projection U -- P. For each vector 1 let P,, be the plane in E orthogonal to n. Since n E E with g(n, n) 7rp. (R(P,,)) lies in P,, and is therefore orthogonal to n we obtain a vector field 93: n --+ 7rp,, (R(P,,)) on the 2-sphere f n E E : g (n, n) 11. Since the 2-sphere is compact the vector field 93 has constant length since otherwise there would be a distinguished direction of maximal length in violation of isotropy. By Theorem 2.5.11 this is impossible unless the vector field vanishes identically. By isotropy, it should have no component in P since otherwise there would be a distinguished direction in P. It follows that R(P) is orthogonal to P. Further, its length cannot depend on P since otherwise there would be a distinguished plane (and therefore a distinguished vector) in u,
v).
Denote
=
=
E. Let u, v, U1 i U2 E E
W E
we
E.
From,7r,;Panfui,U2J (Rspanf ul, U21)
0 for all vectors
==
obtain
; 02 (R(u, v)U.,, w)
=
(R(u, v
+
w)U-,, w))
=
(R(u, v
+
w) U, v
+
-
(R(u, w)U.,, w))
w)
(R(u, v
+
w) U, v)
;02 1-
This
implies that
-1
(R(u, v) U,,, v))
the tensor field
-
(R(u, w) U, v))
.
(u, v, w) (R (u, v) U, w) is antisym(R(u, v)w, U,,) c(u, v, w), c is a 3form. The first Bianchi identity (Lemma 2.8.2) yields (R(u, v)w, U,') + 0. This motivates the (R(v, w)u, U,,) + R(w, u)u, U,,) 3c(u, v, w) metric. Since
R(u, v)U,,, w)
=
=
following definition.
t-->
=
-
=
6.1
Definition 6. 1. 1. Let
(Uo, U.,) about U.,
if
at
(M, g)
be
a
Lorentzian
spacetime (M, g)
1. The
the curvature tensor
x
Homogeneity
and
isotropy
89
manifold and U., E M, infinitesimally isotropic
is called
satisfies
R(u, v)w
=
n(x) ( v, w)
R(v, U.,)Ux
=
tt(x)v,
u
u, w) v)
-
w E UxL, where y(x), K(X) E R are independent of u, v, w. spacetime (M, g) is called infinitesimally isotropic if there exits a normalised, timelike vector field U such that (M, g) is isotropic about Ux for all x E M. If (M, g) is infinitesimally isotropic, U is called a cosmological observer field.
all u, v,
for
The
Given
infinitesimally spacetime,
an
there may not be
a unique cosmologspacetime all normalised
ical observer field. For instance, in Minkowski
timelike vector fields In the
following
are
cosmological observer
we
will
fields.
that spacetime is
assume
infinitesimally
isotropic. While it can be argued that infinitesimal isotropy about our own velocity vector is backed experimentally fairly well, it is a very questionable extrapolation to assume that spacetime is infinitesimally isotropic. On the other hand, this extrapolation seems to be exactly the
Copernicus. Hence to demand that (M, g) is infinitesimally isotropic appears very plausible to us. (Our acceptance of such a postulate is in striking opposition to the response a medieval scholar lesson learned from
would have
given).
Lemma 6. 1. 1. Let
be
(M, g)
infinitesimally isotropic
about U,,.
Then
R(u, v)U,, for
all u,
v
E
U,,J-
=
0 and
R(U, u)v
=
-I-t(x) (u, v) U.,
and the energy momentum tensor is given
TX
=
0E W
+
Xx)) UX'
(9
U X
+
by
P(x) g,
where
Proof. -
87rc(x)
=
I(n 2
8,7rp(x)
=
(n
Let u, v,
w
E
(R(u, v)w, U.)
tion is
a
obtain
0
2)(n
2)
-
I)r,(x)
(_2 (n
=
-
-
A,
3) n (x)
+ p (x)
The first assertion follows from and
(R(u, v)Ux, Ux)
=
0.
+ A.
(R (u, v) U, w)
The second
(R(Ux,u)v,w) R(v,w)Uu) (R(u, Ux)Ux, v) y(x) (u, v). Taking the
consequence of
(R(Ux, u)v, Ux) we
Uj.
=
-
-
=
=
=
asser-
0 and
trace of R
290
Robertson-Walker
6.
Ric (Ux,
Ux)
Ric (u,
cosmology
(n
=
v)
-
((n
=
-
1) p (x),
2) n (x)
Ric (Ux, /-t (x))
-
v)
0,
=
(u, v)
,
and therefore Ric
((n
=
2)/,t
the
Taking again
n((n-2)n-/_t) tensor is
-
=
(n
+
trace
-
we
2)r,) 0 have
0
0
Seal
+
((n
-
2)K
-(n
=
(-2n+2)jL+(n-1)(n-2)K-
/_t) g.
-
2)(y
-
n)
+
+
The energy momentum
given by
now
I
8-ffT
Ric
=
-
2
(n
=
+
-
Seal g +
2)(/,t
((n
+
K)O
2)K
-
Ag
-
ft
0
&
-
-1 ((-2n + 2),4 + (n 2
=(n-2)(,u+r,)0O0+ (n-2) It follows that the energy
81r(e +p)
=
(n
-
2)(tz
+
r,)
density 8irp
and
E =
3
+ A
9
nn+(n-2)tz+A
g.
-
1)(n
2)r,)
-
-
2
and the pressure p
'(n 2
-
2)(3
-
are
n)r, + (n
-
given by
2)/-t
+ A.
1
Lemma 6.1.2. Let
(M, g) be infinitesimally isotropic and U be the cosmological observer field. Further assume that n > 3 and that E + p : 0. Then Uj- is an integrable distribution. The hypersurfaces perpendicular to U are totally umbilic (cf. Definition 4.4.7) and r, is constant on these hypersurfaces.
Proof. Let X, Y, Z be equation
vector fields which lie in
VUX (analogously
-
(X, vuu
U
=
Uj-
at
x
and
satisfy the
0
Y, Z). This can easily be arranged by considering vechypersurface Z which lie in Uj-. They may then be uniquely extended into a neighbourhood of Z using the above differential equation. From VU (X, U) 0 it X, VUU) U, U) VUX follows that X, Y, Z are everywhere perpendicular to U. Moreover, the derivative VUX is parallel to U since VUX (VUX, U) U (analogously for Y, Z). We will now exploit the second Bianchi identify (cf. tor fields
for
along
a
=
=
-
=
Lemma
2.8.1),
(VUR) (X, Y) Using
-
+
(VXR) (Y, U)
+
(VyR) (U, X)
the formulas in Definition 6.1.1 and the property
==
0.
6.1
Homogeneity
VUX' VUY' VUZ 11 we
and
isotropy
291
U
calculate
(17UR) (X, Y)Z
17U (R(X, Y)Z) R(VUX, Y)Z dr,(U) ((Y, Z) X
=
-
+
K
(VXR) (Y, U)Z
-
(Y' Z)
(X, Z)
-
VU X
VX (R(Y, U)Z) R(Y, VXU)Z
=
R(X, VUY)Z (X, Z) Y)
-
((Y' Z) VUX
0 + P
-
R(X, Y)17UZ
-
/-t
-
((VXU, Z) /-t(- (U, vxz)) Y Y
K
-
dl-t(X) (Y, Z) (K
+
+
(VXY), U)Z
i
R(Y, U)VXZ
-
VX (A (Y' Z) U) -
VUY)
(X, Z) VUY'
tt
R(7ru
-
-
-
-
(Z' VXY) U) (Y' Z) VXU) p Y' VXz) U
-
U
P) ((Y1 Z) VXU
Z' VXU) Y),
-
and
(VyR) (U, X)Z
(VyR) (X, U)Z
=
-
=
-d[t(Y) (X, Z) (r,
-
Inserting these equations 0
=
dti(X) (Y, Z) +
0
=
Since
n
+
> 3
U
-
Z,
Y I 0
=
dy(Y) (X, Z)
K.) ((Y1 Z) VUX
(X, Z) VYU there
[1) ((X, Z) VYU
are
-
into the second Bianchi
dr,(U) ((Y, Z) X -
with X J-
(A
+
U
-
+
-
we
obtain
VUY), + n) ((Y, Z) VXU (6.1.2) (VXU, Z) Y).
(y
-
Vyu' Z) X
identity
U
(X, Z)
X, Z) Y)
Z' VYU) X).
-
pointwise linearly independent vector fields X, Y Equation 6.1.2 implies
Z. For these vector fields
(/,t
+
K)
((VYU, Z) X
-
Vxu' Z) Y).
It follows
immediately that for orthogonal vector u, v the expression. Hence, restricted to Uj-, the bilinear form VO is (u, VvU) 1 a multiple of g restricted to U-L. In particular, VO restricted to U vanishes.
symmetric. We will now show that the tensor field V((n + I-t)U) symmetric in all of T, M. Since U is symmetric on U-L and V ((n + M) U) (d/-. + dy) 0 U + (n + y) V U we only have to show (VX ((n + /-t) U), U) is
is =
=
6. Robertson-Walker
292
VU((n + y)U), X). from
Equation
(1,t
=
+
we
Z and
X 1 Z at
choosing
get
x we
-(p
=
+
n) (X, VUU).
implies
-
-1 2
(n
=
equation with dlt(x)
-
3)dr,(X)
-
have used the formulas
we can
=
n) (VUX, U)
dp(X) and therefore
Y
Setting
6.1.1
dl-t(X) Lemma 5.2.1
cosmology
+
+
dp(X)
provided by
-(M
=
p) X, VUU),
(E
(ft
n) (X, VUU)
+
+
n) X, VUU),
Lemma 6.1.1.
obtain
we
where
Combining
dn(X)
=
this
0. Now
calculate.
(17X((r, + y)U), U) (17U((K + A)U), X)
=
-d(r,
=
(y
It follows that V
((r, + /,t) U)
d((n + M)W) dt (K+p)W.
0.
=
=
to U for each
multiple
of g
By
The
p)(x)
+
is
-dp(X)
=
(p
+
n) X, 17UU),
17UU' X).
K)
+
=
symmetric
in all of T, M and that therefore
the lemma of Poincar6 there is
hypersurface Zi
Ix
=
G
M
a
Tj
t(x)
:
function t with is
orthogonal
f. Its second fundamental form, k(u, v) (u, VvU), is a restricted to Zi which implies that Zi is totally umbilic. I
Theorem 6.1.1. Let
(M,g) be infinitesimally isotropic and U be the cosmological observer field. Assume that e + p0. Then there is an (n 1) -dimensional Riemannian manifold (A , ,) of constant curvature R x M, g-dt2 + a2(t),, where F E f 1, 0, -11 such that (locally) M -
=
e/O
=
Proof.
K
_
I(dn(U)/(K + /_t) 4
2.
It is clear that g (r + A)-2 dt2 + t for some t. Let u, v E Uj- be vectors with 1. Then Equation (6.1.2) gives with v
mannian metric
(v, v) dK(U)
=
=
=
(K + /,t) ( VvU, v) + (VuU, u ).
-
form of the
hypersurfaces perpendicular 1
k
The GauB
=
(u, VVU)
2
X,
u
Y
=
to U is
given by 1
+ /,t
2
OtKg.
equation gives
Rz, (u, v)w
R(u, v)w
(/,z
1
4
+
(
r,
k(u, w)k(v,
+ A
Rie-
(u, U)
I V,
=
Z
Hence the second fundamental
dK(U)g K
t-dependent u
)2)
-
((v, W)
k(v, w)k(u, u
-
(u, W) V)
.
Homogeneity and isotropy
6.1
By the
constant in the
hypersurfaces.
R
a:
R+ is
--->
constant curvature
We will
the
proof
therefore t
=
E
can
I 1, 0, 11
1 n
-
(dr,(U) T4
2
-
-Z
+1z
)
is
words, the hypersurfaces Et have + M)-2 dt2 + a2 (t),, -
l)-dimensional
metric of
-
-
show that the factor
now
dt. To this end
a
e
In other
the factor
therefore write g function and , the (n
constant curvature. We
where
(Proposition 4.3.4)
Lemma of Schur
293
1/(n
+
M)
can
be absorbed into
have to show that M only depends on t (recall from of Lemma 6.1.2 that dn(X) 0 for each X E TZt and that we
=
r,
const,
depends only
so
on
t).
Since
n
at k
=
atK
hypersurfaces
'
t. On the other
on
2 a
-
2
depends only U1
is constant in the
at n. This implies that
is
2
hand,
a
g.
direct calculation gives for
v,w E
k(v, w)
=
(V,'7wU)
=
(r-
+
(/-t
+
V,VwOt)
(K
+
Y)
Ocgat
+
atgca
=
1
=
Comparing the
P)
2
-
depends only we
/-t)gabva rclt wc
+
1 =
2
(N
+
A)'Otgea vawc
r,)aata,. two formulas for k
t. Hence
on
take
(K
19agc't) VaWb
we
obtain that
(a0tK -ata which
=
+
(N(t) +p(t))-ldt
as our new
K
is the differential of
a
function I
time coordinate.
Corollary 6.1.1. If (M,g) is infinitesimally isotropic then it is also spatially homogeneous, i.e. for hypersurfaces Zt orthogonal to U and all x, y E Zt there exists for any pair of orthogonal frames of Y: Zt, TyZt M which maps one of the frames into the other. an isometry M "
-->
Proof This follows is
a
from
Corollary 4.5.1
and Lemma 4.5.5 since
(Eta 2 (t)6 I
Riemannian manifold of constant curvature.
Corollary 6.1.2. Let (M, g) be infinitesimally isotropic and U be the cosmological observer field. Then there is an interval I (t-, t+) and a spaceform (Z, ,) of constant curvature E E I 1, 0, 1 such that the universal cover of (M, g) is isometric to (I x Z, -dt2 + ,) and U at. 4 then there are local coordinates (t, r, 0, p) such that If n -
=
=
g
=
-dt2
+
2 a
(t)
( -1-r2 dr2+ I
r2
(d02 + sin2 (O)d 02)
)
-
(6.1.3)
294
6. Robertson-Walker
Proof The first part
cosmology
of the
obvious since I
corollary is
x
Z is
simply
con-
nected for any spaceform Z. The second part follows immediately from the classification of 3-dimensional Riemannian manifolds with constant curvature
(cf.
Lemma
A spacetime
(M, g)
(6.1.3)
1
which is
finitesimally isotropic spacetime
4.5.5). locally
Robertson- Walker
or
isometric to
Lorentzian manifold is called
cosmology.
a
4-dimensional in-
a
Robertson- Walker
The metric given
by Equation
is called the Robertson- Walker metric.
6.2 The initial value
for
problem infinitesimally isotropic spacetimes
solve Einstein's equations Ric -1 Scal g + Ag 87rT for 2 Robertson-Walker cosmologies. While, in general, Einstein's equations give rise to a system of partial differential equations, in the case at hand In this section
have
we
=
-
shown that the unknown functions
already
depend on only one ordinary instead of a partial system of differential equation. This simplifies the problem greatly. However, even this simple case exhibits typical aspects of Einstein's equation.
we
variable. We will therefore obtain
an
We denote the derivative with respect to t with
Lemma 6.2. 1. Let be tangent to the vature
(M, g)
be
a
hypersurfaces are given by
Robertson- Walker spacetime and u, v, w Et which are orthogonal to U. The cur-
expressions
R(u, v)w
=
( ()2 a2
-'
+
2
) ((v, w)
u
-
(u, w) v),
a"
R(v, U)U
R(v, w) U
=
0,
a"
R(v, U)w
v,
(v, W) U,
a
a
a"
Ric(U, U)
=
-(n
-
1)
Ric(U, v)
,
=
0,
a
(' )) ( W) (n 2) (' )) 6
w)
Ric (v,
(n
+
-
a
Scal
Proof.
=
(n
1)
-
a
(2
2
+
a
V,
2
-,
+
+
-
a2
a
_2
The constant curvature metric 1 9z
=
-
-
dr
-2 61
has the curvature tensor Rz (u,
4.3.3).
2)
2
+r
v) w
2
==
(d02 + sin 2(O)d W2) E
((v, w)
u
-
u, w) v) (Proposition
Hence the formula for R follows from Lemma 4.4.14. The other
formulas
are
direct consequences from Lemma 4.4-15.
1
6.2 The initial value
Corollary
If
6.2.1.
for
problem
the energy momentum tensor is T
(c
=
then Einstein's equation is
p)0
+
equivalent
87rE + A
(n
2
295
infinitesimally isotropic spacetimes
-
1)(n
&
0
+ pg,
2)
( (a/)
given by
to
-
6
(6.2.4)
+
a2
a2
and
87rp
A
-
=
-(n
-
3
n
(
2)
+
E
+
(6.2.5)
2
a2
2
a
((a?
Proof. Recall from Lemma 6.2.1 that the Ricci tensor Ric restricted to spatial subspace U,,-L is a multiple of the metric g restricted to this 1 Scal g + Ag 87rT subspace. It follows that Einstein's equation Ric 2 restricted to this subspace,
the
=
-
a// +
(n
-
2)
LaL)2
e
+
a2
a2
a
n
2 2
is
equivalent
+
(n
-
2)
(a? +
a2
a
+
2
A)
g
8-7Tpg
Equation (6.2.5). From Lemma 6.2.1 we get Ric(v, U) v J_ U which implies that the only other non-trivial Einstein's equation is given by evaluating it on the pair of
to
0 for all vectors
component of vectors
(U,,, U,,). at/
(n
-
1)
n-1 +
a
which is
We obtain
2
equivalent
to
(2
at' +
(n
-
2)
a
Equation (6.2.4).
(a' )2 +
a2
a2
)) -A)gjE
=
87rEgIE 1
equation is not a well posed system of differenhave only two equations for three unknowns, we Instead, equations. of the function a appear in our system derivatives Moreover, only a,,E, p. of equations. The first problem has a direct physical resolution. Just specifying a perfect fluid is not enough to specify a matter-model completely. Rather, perfect fluids give a framework which is fitting for many different matter models. In particular, vacuum is a (very degenerate) perfect fluid, and so is dust. In order to arrive at a determined system of equations we therefore have to specify an additional relation between the energy density c, the pressure p, and the metric described by a. In the following we make a rather simplistic assumption, namely that there is a given equation of state: p f (c) for some smooth function f : R --> R. Having made this assumption, we still have the problem that we have Observe that Einstein's
tial
=
6. Robertson-Walker
296
cosmology
equations for a rather than a system of two differential equation for a and E. We can resolve this problem by replacing one of our equations with an equation of motion (Lemma 5.2.1). two differential
R Corollary 6.2.2. Assume that there is a smooth function f: R with f (e) > F_ for all 6 E R. Let ao G R+ \ f01 and 6o c R and assume -+
-
2 (ao )2 + 2) (81rEo Tn-_(n 1)
that
-
-
A)
(a, c) of Einstein's equations a,
Then there exists
> 0.
e
-
such
a(O)
e(O)
and
ao
=
unique solution
a
=
E0.
The
functions
satisfy
E
(i) (n
-
2)
a'
-8ir
a
(f(c) n-3.) +
2
+
n-1
A,
n-1
a
(ii) (n
E+P
a
and
a'(0) Proof. Assume
2(ao)2_ (8-7rEo
(n
1)(n
-
-
+
2)
A)
-
solution of Einstein's equation.
EquaEquations (6.2.4) and (6.2.5). Equation (ii) follows from Lemma 5.2.1. Finally, the equation for a'(0) follows immediately from the equation for E in Corollary 6.2. 1.
tion
(i)
is
first that a,
linear combination of
a
For the
converse
unique solution a, [--2(ao)2
a'(0)
=-
is also
E are a
E
notice first that for
initial conditions there is
our
which satisfies the system of equations
(i), (ii)
a
and
We have to show that this solution
(87rEo A) -(n2) solution to the system of equations given in Corollary 6.2-1. It +
V7
-
n
a
is clear that this
system of equations is satisfied
at t
=
0. We will show
that the first equation is satisfied for all t. Since the second equation is a linear combination of the first equation and equation (i), it must then also be satisfied for all t.
Defining
87rc + A
-
2
(n
1)(n
-
have to show that 0 vanishes for using equations (i), (ii) gives
87rE'
-
2
(n
1)(n
-
-
-
Taking
2aa'
2)
-
8-7r(n
(ii) a'
-
2
the derivative of
+
6)
-
(N
-
1)(N
-
2) a
(a/ )2
a'/
(a
+6
a2
a"
87r(n
-
I)E
+
(n
-
1)(n
-
2)
+ a
a
-(N-1)(N-2)a' a
0
a4 a'
1)a(E +p)
a
((a')2
a/
(i) =
all t.
2aa"a2
1 =
2
a
we
0'
2)
-
a'/
(a
(a/
)22+6 a
87r(n
-
3)E
+ 2A
and
6.3 Geodesics and redshift
297
2a/ a
Since
0(0)
=
0 and 0 is
a
solution of the differential equation 22a 0, ordinary differential equations implies that
the fundamental theorem for
0
must vanish for all t.
I
This solution of Einstein's equation is is often
for the are
not
that
typical in two aspects. Firstly, it exchange part of the original set of equations 0. Secondly, Einstein's equations motion, div(T)
advantageous
to
equations of a free system of differential equations but =
we were
not free to choose
equation for
a.
In other
a'(0)
even
words, only
constrained. Recall
are
though
had
we
a
second order
restricted set of initial values
a
had the chance to lead to solutions of Einstein's equation. The system equations which was solved was derived from Einstein's
of differential
equation but
not identical to it. We had therefore to show that the
solutions to this system are also solutions to Einstein's equation. We did so by deriving an additional linear differential equation and used our
constraint
this equation
the choice of
(i.e.,
a'(0))
implies that the original
phenomenon has
to show that the solution of
set of
equations is satisfied. This settings where we
direct counterpart in more general have to deal with systems of differential equations. a
6.3 Geodesics and redshift In 1929 Hubble made
galaxies
are
moving
a
cosmological discovery which implies
that distant
(and each other)
proportional
away from
us
at
a
rate
to their distance. This astronomical fact shattered the
long cherished idea physical processes take place. 2 It is instructive to describe Hubble's discovery in slightly more detail: Each star has a spectrum of light which contains characteristic gaps due to absorption of light of certain frequencies in the atmosphere of the star. Since we have physical explanations for these absorptions, we can calibrate these patterns and thereby obtain information about the chemical composition of the star's atmosphere. Hubble discovered that for stars in galaxies which are not too close 3 these gaps are shifted towards smaller frequencies. Moreover, this shift is proportional to the distance of the galaxy. From his observation it was then that
our
universe
was an
eternal
arena
in which the
2Einstein introduced his cosmological
constant a decade earlier because he wanted to have static solutions in accordance with the prejudice of his time. Had he not done so, there would have been another striking prediction by
general relativity. very nearby galaxies
3For
overshadows this effect.
the
(local)
movement of the
galaxy relative
to
us
298
Robertson-Walker
6.
cosmology
concluded that all
galaxies are moving a-way from each other. (Everyone analogous effect: If a fast car is approaching one has the impression that the noise of the engine is higher pitched than when it is moving away: In other words, if the source and the detector of a sound move away from each other, the frequency of the sound appears is familiar with
to be
an
smaller).
In this section
we
will show that Hubble's
discovery can be undercosmology (cf. Corolof the great successes of general relativity
stood within the framework of Robertson-Walker
lary
6.3.2
below).
This is
one
and the
isotropy assumption. Recall from Sect. 1.4.3 that we can describe photons by null geodesics. The energy of a photon -y measured by an observer u is given by E hv (u, ). Here v denotes the frequency of the photon (as measured by u) and h denotes Planck's constant. In Robertson-Walker spacetime we have a natural unit vector field U which is approximately tangent to the world lines of the galaxies. We will therefore define the energy of a photon using this distinguished observer. In this section we will always refer to this energy. Let -y be a photon which moves from x e M to y c M. In general, it is possible that its energy is not constant along the world line of the photon. This is traditionally expressed using the fractional increase z of the world lines of
=
the associated
being
=
11v
detected at y is
(t, Y), (t, Y)
I
Z
-
hIE: z
of
a
photon originating
at
x
given by A (Y)
AW A(X)
*1 Y) If
=
The redshift factor
Definition 6.3.1. and
A
wavelength
=
-
the events
occupied by two galaxies, then the a (t) dz (Y, Y), given by d ((t, i), (t, Y)) where dz (Y, Y) is the distance of Y and Y in (Z, ,). We will show that there is a constant H such that we have approximately z(x, y) Hd((t, Y), (t, Y)) for galaxies which are distant enough for Hubble's disE
x
are
distance of these events at time t is
==
==
covery to hold but still
this end
we
close that it is sensible to linearise
so
must first calculate the null
geodesics
z.
To
in Robertson-Walker
spacetimes. Lemma 6.3.1. Let
(M, g)
finitesimally isotropic
8
is
a
aeodesic if and
::=
x Z, -dt2 + a 2(t) manifold. The curve
((t-, t+)
Lorentzian
F-+
-Y(s)
=
(t(s), 1(s))
only if
d2t
a' dt +
d 82
'Y,
a(t)a'
=
0,
+ 2 a
a,
0
,)
be
an
in-
6.3 Geodesics and redshift
hold,
where
(t(s), I(s))
t is
denotes the induced covariant derivative a
null
geodesic then the
conservation
on
Z.
299
If y(s)
equation
dt
a(t(s))
const
holds.
Proof. The first part follows immediately from the corresponding formugeneral warped products (Corollary 4.4. 1). Assume now that 7
las for is
a
null
geodesic.
equation and
(4, -4)
dt
dt
2
From the first
a
ly
(dt/ds)2
=
we
obtain
d
TS_ (a (t(s))
)2
a'(t)
a(t)ddS22t
+
0.
=
Corollary 6.3.1. Let (M,g) ((t-,t+) x Z,-dt2 + a 2(t)g,) be infinitesimally isotropic Lorentzian manifold. The curve s 1-4 -y(s) (t(s), I(s)) is a null geodesic if =
(i) -F (ii) and
is
1-4
unit
a
there is
a
speed geodesic
constant
c
(Z,
in
such that t (s)
8
=
C
ft.
a
an
(i) di,
1(s)
di
(ft,,
ly
a(i)
Proof Assume that 'y' is a unit speed geodesic and that the integral equations (ii) hold. The curve -y(s) (t(s), I(s)) is a null curve because .
=
of
-(dt/ds)2 =
=
ds
26
d
( J_T
1 +
d
z,
2 (dt/ds)2
dT
,
integral equation
equation d
0
a
d
(dt/ds)2
It follows from the first
Hence the first
+
d-r in
d
v
Y7
(ii)
that
dt
a
))
=
0.
adt/ds
is constant.
in Lemma 6.3.1 follows from
dt
(a (t(s)) a-s )= a'(t) ( TS_ )2
and the fact that -y is
Z,
V/ r
null
curve.
+
a(t)ddS22t
The second equation follows also
direct calculation:
t ,=
I
-
Y
c2a2
t7
d
='
a7r'y
(
I
d
a2 d-F
)
-2 da d
c2a5 d-r d-r
z,
-2 a
da.:, ds
by
300
6.
where To
Robertson-Walker
have used
see
that all null
to notice that
where
d' ds
we
e
is
a
(up
to
1 d' ads
ale-*
tange,,-iat
from
(M, g) c
x
be described this way it is sufficient any null vector can be realised as 9t + e
can
niultiple)
6.3.1. Let moves
I -
geodesics
a,
unit -vector
Proposition photon which
cosmology
to 'S4,
be
infinitesimally isotropic
M to y
and 7 be
a
M. Then
c
1.
Z
a(t(x)) Proof -
A
From Lemma 6.3.1
.
jd;t ds
(U, =
we
implies that A/a
ka in the definition of
a(t) ds!
get that
h/(a(t) jdt,) ds
=
is constant. Hence E
k is constant.
=:
Inserting
proves the claim.
z
Corollary 6.3.2. Let xo (to, Yo) E M. Then the frequencies emitted by nearby galaxies (situated at y (t,:V) E M) appear to be red-shifted at xo by z Hto d((to, Yo), (to,: o-)), =
=
where
Hto
=
a'(to)/a(to)
Proof. We
assume
Hence
obtain
we
is the Hubble "constant" at xo.
that dz (
Fo, go) < 1 which in turn implies It a(to) + (t to)a'(to) and therefore
a(t) a(to)
1
z
-
-
+
I
;:z
t <
to
we
-(t
-
time to the Hubble constant
1
to)Ht(,.
have t-to
6.4 The age of the universe and the At
to I < 1.
-
to) Ht(,
speed of light is I and implies the assertion
Since the
which
(t
a(to) (t to)al(to)
;:z
a(to) I +
-
-
-d((to, Yo), (to, jjo-)
big bang
Ht,, a'(to)/a(to) is positive. In the introduction of the preceding section we have given a heuristic argument which indicates that universe is expanding because of the redshift of light emitted from nearby galaxies. In this section we we will show that the observation of Hubble implies that there has been a big bang (if infinitesimal isotropy holds). In order to do so we need an assumption which eliminates the unphysical cases that our
=
6.4 The age of the universe and the
-
-
is
a
is not
We will therefore
they
that E, p: I
assume
become unbounded
hold then there would exist
a(I
such that at
spring
301
simply not defined on its maximal domain or that everywhere differentiable where it is defined.
a
unless
big bang
Z)
x
f
C
an
x
R
can
be
continuously extended,
0. If this
assumption would not Z, dt' + ,) of spacetime would miraculously disappear or
or a
(i x
extension
Z matter
-
into existence.
((t-, t+) x Z, -dt2 + a2 (t)&) be an inC2-maximally extended Lorentz manifold of difinitesimally isotropic, Theorem 6.4.1. Let
mension
n
> 3.
(M, g)
=
If
(i) there is a to E (t-, t+) with Hto > 0 (ii) E, p are continuous on (t-, t+), (iii) e +A/ (87r) > 0, (iv) there exist constants c such that E n
-
3
p
n
then t- >
limt,t-
a
-oo.
In
2(t)(6(t)
-
I
addition, +
+
c
we
A/(81r))
-
:5
< c_
=
and p
Al (81r) AI(8-x)
limt-t_ a(t)
have oc.
For t+ there
lim.
a'(t)
satisfy
< -
=
0 and
are
the
limt-t_ a'(t)
following possi-
bilities.
1
8
lim.
t+, t
-1
1
(t),
lim.
t -t+
a'(t)(E(t)
+
A/(8?r))
t-t+
00
1
0
00
00
0
0
finite
0
-00
00
00
0
a
t+
Proof If c can be extended beyond t- or t+ as a bounded function so call p. For given E, p, Corollary 6.2.2 (i) can be viewed as a linear differential equation of second order for a. Consequently, if t is finite, a could be extended
as a
C2- function if it is
(Dieudonn6 1960, limsupt-t, a(t)
not infinite
or implies liminft,t, a(t) 10.4.6). (M' g) is C2- -maximally extended by assumption.
This
(iv) imply
=
that 8,7r
(P
+
n-3. n-1
)
2A/(n
0
-
1)
is
=
Conditions
positive
Remark 00
since
(iii)
and
Hence a" (t) < 0 -
by Corollary 6.2.2 (i). The function a' is therefore monotone and we can replace lim inf (and lim sup) by lim for a and a'. First we investigate what happens near t-. The inequality Hto > 0 implies the inequality a(to) > 0. Since a"(t) :5 0 for all t, the graph of a lies below the graph of the map t -4 a(to) + a'(to)(t to). This linear for all t
-
graph
(a 0)-axis a(to)/a'(to), to).
intersects the
is finite: t- E
there is
a
[to
-
6 > 0 with
=
at
a(t) < to which implies that tto a, (to) Since p A/(87r) > c- (,E + A/(87r)), -
-
6. Robertson-Walker
302
cosmology
J
3
(I
+p
E
c-)(f, + A/(87r))
+
)(E
>
+
A/ (87r))
n
2 + J
A/ (87r)).
+
Corollary 6.2.2 (ii) implies (E + A/(87r))' -(n 1)(E +p)a/a < -(2+J)(e+A/(87r))a'/a and therefore ((E+A/(81r))a2+6)' < 0. The equa0 implies now immediately that limt,t- O(E(t) + tion limt-t, a(t) oo. From the equation for the energy density c in Corollary A/(87r)) Hence
-
==
=
6.2.1
(a/)2 diverges
infer that
we
For t
t+ there
also.
possibilities. If a has no maximum oo. Since ((E + implies that t+ A/(87r))a2+J)' < 0, the function (e + A/(87))a2+6 is decreasing which 0. The equation for the energy implies limt-t+ (E(t) + +A/(8ir))a2(t) density E in Corollary 6.2.1 implies now that < 0. The assertions about -->
limt,t+ a(t)
then
are
oo.
=
several
a"
0
<
=
=
-
limt-t+ a'(t) maximum if
follow from the 6
=
equation. Note that
same
a
must have
a
1.
If a has a maximum at some tj E (t-, t+), 87rE(ti) + A -I(n1)(n 2 1. Since a"(ti) < 0 we have a'(t2) < 0 for some 2)E/a 2(tl) whence E by time reversal t2 E (ti, t+) and we can apply the same argument t-. This proves limt-t+ a(t) as for t 0, limt-t+ a(t) -oo, and I limt-t+ a2 (t) (E (t) + A/ (87r)) oo. =
-
=
-
-
-+
=
=
=
Corollary
If our universe is described by a Robertson- Walker cosmological constant, then it is younger than 11Ht.
6.4.1.
model without
a numerical estimate note that according to measurements of luminosity of stars (so-called "standard candles") located at different 1.7 10-18s-1 (Wald 1984, p. 114) (see distances one arrives at Ho
To obtain
the
=
-
Since I year the universe should be younger than also
(Weinberg 1972)).
=
60 .60 .24 .365
11HtO
=
2.
1010
s we
obtain that
years. Wald
(1984)
also quotes experimental evidence which points to values for HO which are twice as high. This would imply that the universe is less than half as
old
as
indicated above.
Part of Theorem 6.4.1
can
be
generalised
to
locally spatially homoge-
universes which are not necessarily isotropic (Rendall 1994). With 0 cf. Remark 5.3.3. respect to the significance of the assumption A neous
=
p-A/(81r) is sharp. This E+A/(8ir) from 3-dimensional dust spacetimes. We solve the system of
Example 6.4.1. The inequality can
be
seen
k
c=
=
< C_
<
-
of Corollary 6.2.2. Both equations decouple Equation (ii) implies that there is a constant ka2. Equation (i) reduces to a" r-Aa, whence a(t) k-e-11 A'. From our initial data ao, 60 we obtain a'(0)
0 and
R with
n_3 n-1
(i), (ii)
differential equations since p
-
n
=
3.
=
=
k+ell At + V/-(87rf-o + A)(ao)2
=
=
-
E
and therefore
6.4 The age of the universe and the
-
a(t) for A >
ao
cos(vAt)
+
V(8 7r co
+
A) (ao)2
-
s
big bang
303
sin(VA-t)
v/A_
0,
a
(ao A
(t)
-
V--A V (8,7r co
+
A) (ao) 2
e
/=_At
2A
(ao A
+
vf-- A V (8,7r co
+
A) (ao) 2
I-A
e- v
At
+ 2A
for A <
0, and
a(t) for A
that
=
(,Eo
=:
0. If +
ao +
e
V8 (ao)27r co
1, A
=
A) (ao )2
=
<
-
E
t
0, and the initial data
I then
we
ao, 60
are
obtain solutions without
chosen such
singularities.
Remark 6.4. 1. Observe that the metric of Robertson-Walker spacetimes non-singular Lorentzian manifold
differs from the
((t-, t+)
x
Z, -dt2
+ I
-Isr2 dr2+
2 r
(d02 + sin 2(O)d W2)
)
only by an overall-factor a 2. (We say that these pairs of spacetimes are conformally equivalent). This implies that there are past light cones which do not intersect. In particular, there are regions in the universe filled with particles which may never have had a chance to interact. This raises a serious problem. In physical theories, homogeneous states are usually obtained by the statistical description of microscopic states. In particular, homogeneity is always a result of prolonged interaction. If there are regions in the universe which cannot have interacted with each other in the past, then we need a new explanation why they are so similar that we can describe the universe as homogeneous. Physicists are currently trying to find an answer to this question by arguing that during the early phase of the universe the expansion was much faster than would be plausible if one considers ordinary matter. Under this assumption, these regions would have had a chance to interact after all. However, the physicists favouring this inflational universe are Still4 very far from a satisfactory physical explanation for the occurence of this "inflationary" phase in the history of the universe.
'
I write this in 1998.
304
Robertson-Walker
6.
6.5 A
cosmology
simple model for the universe
In this subsection
we
Einstein's equation
will discuss two be
we
live in
especially simple
sol-ved.
cases
in which
of
explicitly They are special imporwith qualitative expectations of the they are bn development of the universe. At present', there seem to be two principal types of matter which fill the universe. To simplify matters we may assume that today most of the matter consists of galaxies5. Galaxies do not can.
tance since
move
much relative to each other and
interact. It is therefore
are
too far
good description
a
apart from each other
to model them
dust. The universe is also filled with radiation which the
ing form of matter
to
by pressure-less
was
the dominat-
stages of the Universe. This 'cosmic
during early background radiation has been discovered by'Penzias and Wilson (1965) '. The most important feature of this microwave background radiation is that it is (almost) completely isotropic and therefore cannot be explained by a confined source which is located somewhere in microwave
the universe. The radiation is
now
very weak
(having
a
temperature of
Kelvin) and its energy is completely dominated by the energy contribution from the galaxies The microwave background radiation can about 2.7
-
photon gas. The energy momentum tensor of a photon gas is traceless by Lemma 5.2.3 which, together with infinitesimal isotropy, implies E (n I)p. We will not consider the composed system which consists of dust and radiation but only the much simpler cases where we have pure dust or pure radiation. At the end of the section we will give a heuristic justification. be described
by
a
=
Lemma 6.5.1. Let
-
(M, g)
finitesimally isotropic sume that the function are equivalent.
Z, -dt2 + a2 (t)g,) be an inmanifold of dimension n > 3 and asnon-constant. Then the following statements =
((t-, t+)
x
Lorentzian a
is
(i) P 0 (ii) ca" m is constant, (iii) (n 1) (n 2) ((a' )2 +6) =
=
-
-
=
16,7rma3-n
+ 2A.
0 is equivalent to (n Proof. The equation p I) a'/a +,E'/e 0 because Corollary 6.2.2 and a' 54 0. An integration shows that this equation is equivalent to can-1 m for some constant m. The equivalence of (ii) =
-
=
of
=
and 5
(iii)
is clear from the formula for
E
in Lemma 6.2.1.
1
However, it is expected that most of the matter is "dark" which is an euphemism for "we cannot directly observe it and do not know anything about it".
6
The discoverers where concerned with the development of a new satellite communication system and found that their new high precision antenna seemed to have
an
unexplainable background
noise.
6.5 A
In the
simple model
physically interesting
in Lemma 6.5.1
(iii)
we
live in
305
4, A 0, the differential equations explicitly and their solutions are given
case n
be solved
can
for the universe
=
=
by
lc(l 2
I c
is
a
-
lc(,o 2
-
constant of
finitesimally isotropic sume that the function are equivalent.
sin(79))
-
integration.
(M, g)
Lemma 6.5.2. Let
=
((t-, t+)
Lorentzian a
Z, -dt2
x
manifold of
a2(t)g,)
+
dimension
is non-constant. Then the
(i) E (n I)p (ii) Ean =,M is constant, (iii) (n 1)(n 2) ((a')2 =
_71-c(O sinh(i9))
cosh(d))
-
(9c/4)1/3t2/3 !C(l cos('0)) 2
0
where
t
a
E
-1
n
be
an
in-
> 3 and
as-
following
statements
-
+
-
-
6)
161rma2-n
=
+ 2A.
0 (n 1) is equivalent to (n 1) + Proof. The equation 6 obtain this anda' 0. we 6.2.2 equation Integrating : Corollary Ean m for some constant m. The equivalence of (ii) and (iii) is clear =
-
-
n
a
e
because of =
from the formula for
In the
case n
explicitly
=
4, A
1
in Lemma 6.2.1.
c
0 the
=
and the solutions
resulting differential equations given by
can
be solved
are
t
E
1-4
a
(t)
-
C. 'F_j + (1 + t/C)2 0 (4C2)1/4VFt CVj_ (I t/C)2 1.
-1
-
-
where
c:
is
a
constant of
We will make the of matter is 6.5.1 and
integration.
assumption that the
negligible.
6.5.2, radiation dominates
dominates at late times imation to
use
interaction between both
Observe that due to the formulas
((a
>>
1).
at
early
Hence it
the radiation model for the
model for the present universe.
times
seems
early
(a
to be
(ii) < a
types
in Lemmas
1)
and dust
good
approx-
universe and the dust
7.
Spherical symmetry
chapter serves two purposes. Firstly, a large isometry group simpliproblem of solving Einstein's equation considerably. Virtually all explicitly known solutions of Einstein's equations for physically plausible matter fields have a high degree of symmetry. Secondly, a spherically symmetric spacetimes are very good descriptions of non-rotating, isolated stars and therefore of astrophysical interest. (If the star rotates
This
fies the
the rotation axis breaks the
symmetry).
In Sect. 7.2
we
will
see
that
unique 1-parameter family of spherically symmetric solutions to Einsteins's equation for vacuum with vanishing cosmological constant. The parameter can be interpreted as the mass of the isolated star. If the mass of the sun is chosen, one obtains an excellent model of the gravitational field in our solar system. Some aspects of this model have been there is
a
verified
experimentally.
These solutions also form the basis for much of
intuition of black holes.
our
In this
chapter
we
will also discuss the initial value
that the energy momentum tensor represents gether with a non-interacting electric field (cf. Sect. case
more
general
less arrive at
case we a
will not obtain
explicit
a
problem for the perfect fluid to-
7.4).
solutions
we
While in this will neverthe-
non-trivial existence theorem for the considered class of
spacetimes. This section will hardly be of primary interest to a geometrically oriented reader. Since the discussion uses elements of the theory of systems of hyperbolic partial differential equations even physically oriented readers may wish to skip the proofs on first reading. The validity of the physical conclusions from this (and also the following) chapter depends very much on the question of whether the corresponding properties of our explicit solutions are stable under perturbations. We know only very little about the stability of Einstein's equations. Since they are highly non-linear it is well possible that these properties have little to do with our actual universe which has only "approximate isometries".
M. Kriele: LNPm 59, pp. 307 - 355, 1999 © Springer-Verlag Berlin Heidelberg 1999
308
7.
Sphericai sprimetry
7.1 Pseudo-Riemannian manifolds with
spherical symmetry
A property in R 3 is spherically symmetric if it is invariant under rotations about the origin. The rotational isometries defined below form a group
which is our
(locally)
isometric to the rotation group main justification of the following definition.
SO(3).
We take this
as
Definition 7.1.1. A
pseudo-Riemannian manifold (M, g) is called spherically symmetric if it has a dense open subset M' such that (MO, g) can locally be written as a warped product (Z x S', gz + r'dS?'), where dQ' is the metric of the 2-dimensional unit sphere, (Z, gz) an (n 2) dimensional pseudo-Riemannian manifold, and r: Z R a positive function. The sets f xJ X S2 (x E Z) are called spheres of symmetry and those isometries which map all spheres of symmetry into themselves are called -
-
---+
rotational isometries. The set C
Lemma 7.1-1. Let
(Z, gz) res
(M,g)
be
M
=
\
M' is the. centre of symmetry.
spherically symmetric spacetime and manifold orthogonal to the sphe-
a
be the 2-dimensional Lorentzian
of symmetry. For each frame f U, QJ of (Z, gz)
(U'U)=-1' there
are
g
and U
=
U'Q)=O'
(Q'Q)=I'
adapted coordinates with respect =
-
e2v(t,q)dt2
e-',9t, Q
=
e2,\(t,q)dq 2
+
to which
r2 (t, q) (d02
+
with
sin20 dW2)
+
e-'\'9q-
In these coordinates the energy momentum tensor T is 1
87rT (U,
U)
(1
=
r2 +
(U
+
2(-Q
*
r)
*
Q*
2
(Q
_
r
r) 2)
,
(U* A)(U* r))
+
given by
-
A,
+
A,
r
87rT(U, Q)
=
-
2(U
Qr
-
(Q
-
-
OW
-
r)),
r
1
87rT(Q, Q)
=
-
+
72 (1
+
2(-U
(U
*
r)2
U
*
r
e
-
+
(Q
(Q
9
9
r)2)
v)(Q
e
r))
r
87rT(V, X)
81rT(I'22, r
(911 r
)
=
=
0
for
81rT(aI(0) r
=
all X
Q,, Q +
sin
0
V
+
U
*
E TZ
and V E
TZ-L,
a, 1
r
sin (0)
(Q U
*
*
r
V)2 +
_
Q
*
U
Q
U,, A
r
-
_
(Q
(U e
9
A)2
v)(Q
*
r)
7.1 Pseudo-Riemannian manifolds with
(U
-
87rT('90, r
01* r sin(O)
A)(U
o
+
spherical symmetry
309
A,
0.
Proof. The existence of the adapted coordinates (t, q, 0, o) is clear from 1 Ric Seal g + Corollary 2.4.2. Hence we only need to calculate 8,7rT 2 Ag. By Lemma 4.4.15 we have for X, Y tangent to Z and V, W orthogonal =
-
to E
1
Ric(X, Y)
=
Ric(X, V)
=
Ric(V, W)
=
2
-Scalz (X, Y) 0
I(V, W) r2
Since
ScalE
=
VVr(X, Y)
=
+
X
VVr(U, U)
=
U
9
U
*
r
VVr(U, Q)
=
U
o
Q
o
r
VVr(Q, Q)
=
Q
o
Q
9 r
which in turn
A) (U
9
r).
-
-
impliesAr
Hence
we
r
9
-
I
(grad(r), grad(r))
+
r2
r
4
_2Y
e
Ar
(
-
2
Seal
-VVr(X, Y),
-
2
2
'Ar
_2 (grad(r), grad(r))
-
r
VXY
-
*
v)(Q
o
r),
(Q
*
v)(U
o
r)
(U 9,\)(U
o
r)
-U*
obtain
r we
*
(Q
=
=
Q
Uor+Q oQ
o
U
9 r
or+
-
(U *,\)(Q
(Q*v)(Q or)
2
-
-
72
2
4(-U*Uor+Q*Qor+(Qov)(Qor)2
r))
0
72 (_(U
-
0
02
+
(Q
or
)2))
-
A
I
(I + (U r)2 0
(Q
_
0
r)2)
2
_r(Q*Qor-(U*A)(Uor)) 87rT(U, Q)
=
-
2(U
Q
o
o r
(Q
-
-A,
v)(U r)),
o
o
r
87rT(Q, Q)
=
IScalz -2(Q*Qor-(UoA)(Uor))
2
r
2
+
-
_2
4(-U
o
U
or
+Q
o
Q
*r+
I -
2
(Scal-v
(Q ov)(Q or)-
r
2
(U
(U
(ScalE
r
(U 0'\)(U 2
r),
1 +
r
2 +
9
get
IScal_r -2(U*Uor-(Qov)(Qor))
87rT(U, U)
(V, W),
o
A)(U
9
+ (U 1(1 2
r))
9
r)2
-
-
-2(-(Uor )2 +(Qor )2)) (Q
o
r)2)
+ A
9
310
7.
Spherical symmetry 2
(U Let X
9
r
(Q
-
v)(Q
9
r))
o
+ A.
0 and (X, V) 0 (TZ) 1. Since Ric(V, X) 0. Since there is an isometry which maps 87rT(X, V) a" aP we have 87rT(00,'90) 81rT( sin (0) sin ). The Ricci (0)
TZ and V c
c
it is clear that '90
U
9
a,
into
=
=
=
=
T
r
sin(O)
r
r
1
r
r
tensor restricted to (T.,Z)J- is a multiple of the metric which implies '91" 0. Finally, we the last component of T which needs to ) r r sin (0) be calculated is given by
T(-'91-,
=
1
-(I(-U*Uor+Q*Qor+(Qov)(Qor)
87rT( L11,21-1)= 2 r
r
r
I -
(U
9
A)(U
9
r))
1 2
(Q
9
v)(Q
2-(-(U
72
(-
+
o
r)2
9
po
+
r(-Uo Uor+ Q
-
72
o
(U
2
r)2)
4
2
(Scalz
-
+
+
r)
r)2
(U
-
(Q
+
A)(U
9
r)2))
0
9
Q
or
r))
o
+A
U*U*?-+Q*Qor
r
+ _
(Q
o
v)(Q
(U
0
U
Scalz
2(U
=
9
r)
A +
0
where in the last equation
9
we
-
(U
(U 0
A)(U
o
A)2
_
Q
r))
9
0
Q
0
V
_
po
V)2)
+
A,
have used
U *,\ +
(U
0
A)2
_
Q
0
Q
0
1',
_
(Q
0
,,)2)
(cf. Proposition 4.3-5). We will if
now
re-arrange these
in
form which is
more practical complete generality but the following genericity assumption on the matter model.
equations
a
wants to solve them. We will not do this in
one
rather
assume
Recall that for any normalised timelike vector v the number T(v, v) represents the energy density measured by u. This number should be
positive. For null Given
a
pect T(N, N) to
vectors N
obtain then
T(N, N)
> 0
by continuity. plausible to ex0 for all null vectors N. In this case it is always possible and T simultaneously (Greub 1981, Chapter IX 3). we
non-extreme matter distribution it is therefore
diagonalise
>
g
Lemma 7.1.2. Let
(M, g)
be
spherically symmetric Lorentzian mantensor field which is also spherically ifold symmetric (0) 2 T Assume and that symmetric. g can be simultaneously diagonalised. and T be
Then there exist coordinates g
==
-e
a
a
2v(t,q)dt2
+
(t, q, 0, W)
such that
e2,\(t,q)dq2+ r 2(t, q) (d02
+
sin20 d(P2)'
(7.1.1)
7.1 Pseudo-Riemannian manifolds with
T
=
E0
0
0
+
Q
PradQ
+ Psphr
spherical symmetry
+ sin 20
2(t, q) (d02
d(P2),
311
(7.1.2)
:= e-',Ot and Q e-AOq are invariantly defined if E '7 -Prad87rT is equivaMoreover, Einstein's equation, Ric Scal/2 g + Ag, lent to the system of differential equations
where U
=
-
(U
1 +
Q*Q*r= (Uer)(U* A)+
9
r)2
-
(Q
*
r)2
2r
'A
47rr(,E Q*Uor
(7.1.3)
+
87r
(Q*r)(U* A),
=
(7.1.4) (U r)2
1 +
e
(Qor)(Qov)-
U*U*r=
_
(Q
0
r)2
2r
A
41rr(Prad
-
-
-
8,7r
),
(7.1.5)
_(U*,\)2+Q*Q.V+(Q.1.,)2+
UOUOI\=
47r (E
-
and the equation
-
Prad +
2p,ph)
1+(Uor)2
=
0,
is
equivalent
2(E
+
Psph)
U
U
+
Prad)U
A
0
(Q
*
r)2
(7.1.6)
7
of motion, div(T)
-
r2
-
e
to
r
(7.1.7) r
Q
0
(E
Prad
+
Prad) Q
* v
-
2 (Prad
-
Psph)
Q*r
(7.1.8)
r
Lemma 7.1.1
Proof. -
-
0 T(X, V) and T (V, V)
for all X E T
=
Since
implies that
=
T (W,
by assumption
W)
Z,
T satisfies V E
there is
a
Q,
V, W
E
(-- Z)
frame which
mentum tensor T and the metric g
fields
(TM)'
for all unit vectors
diagonalises the simultaneously there must
U tangent to Z and functions r, E, Prad i Psph: Z g
T
=
_U5
=
EW
(j5
W
+
+
Q
(&
Pra&
-->
energy
mo-
exist vector
R such that
Qb +r2dQ2, (D
Q
+ Psphr 2dp2.
The existence of
adapted coordinates follows now from Corollary 2.4.2. equations are equivalent to Equations (7.1.3)-(7.1.6) follows immediately from the definitions Of E7PradiPsph and Lemma 7.1.1. Observe that for any vector fields X,Y tangent to Z the decomposition VXY -"VXY and that for any vector fields V, W tangent to the of spheres symmetry the decomposition VVW (V7 W) grad(r) + That Einstein's
=
S2
VVW
div(T)
=
holds.
(dE(U)
Using Proposition +
4.3.5
we
ediv(U))W +,E(VUU)
obtain
+
(dPrad(Q)
+
Praddiv(Q))Q'
312
7.
Spherical symmetry
+
+ Psph
+ Psph
=
I
Prad(VQQ)5 + (Psphdiv(
(de(U)
( (
i9o)) r dO r
I
V
r
ao r ao
r
sin(o)
I
V
+
I
+ Psph div r
sin (0)
a ,)r sin (O)d(p
aw
aw
E((U A)
2(U
+
9
+ Prad (U
*
A)) U
5
r
2
(dPrad (Q)
((Q V)
+ Prad
(Q Or)) +,E(Q. v))Q5
+
0
r
cos(O) Psph
dO
1dr + 0 + o
-
-
r
=
(dE(U)
+
E
((U A)
2(U
+
0
Idr
-
sin(O)
r
*
dO) U
r))
+ Prad (U
*
A)
+
r
+
-
(dPrad (Q) 2p.,ph r) Q5,
+ Prad
r
2Psph
U5
r
((Q V) 0
2(Q
+
e
r)) +e(Qe v)
r
r
where
we
have used
S2
div(,9o)
Proposition
4.3.5 to calculate
dQ2(S2,V ao 190,ao
+
dQ2(S2,7
9W sin(O)
ao
sin(O)
cos(O) sm(0) S2
div(
aw sin(O)
The functions U
dQ2(S2,Vao sin(O)
ao)
=
0.
A and
Q * v are well defined invariants since the Q given by [U, Q] (Q * 1/) U (U . A) Q. For spherically symmetric spacetimes we can define an invariant tion which can be interpreted as a mass. mutator of U and
is
Definition 7.1.2. Let Then its
mass
function
=
(M, g) m
is
The term "mass function"
sUmptions that there that it is
exists
be
com-
-
func-
spherically symmetric spacetime. defined by m :=:= 112 (1 (grad(r), grad(r))) a
-
can a
be motivated under the additional
centre of
symmetry, that Q
* r
as-
>_ 0, and
possible to diagonalise, g and T simultaneously. Recall from special relativity that mass and energy are equivalent concepts and that the energy of a point particle measured by an observer depends on this observer.
7.1 Pseudo-Riemannian manifolds with
In order to determine the in
of
mass
a
would first fix
313
spherical symmetry
object consisting of spacelike hypersurface repeach particle which intersects material
spacetime particles resenting an instant of time. The mass of this hypersurface would be measured by the infinitesimal observer represented by the normal of the hypersurface (cf. Sect. 1.4-3). The sum over all these numbers is then the mass of the material object with respect to the chosen hypersurface. Since we assume that we can simultaneously diagonalise g and T there must be a timelike eigenvector of the linear map Tab 9bc. Observe that we
a
it is orthogonal to the spacelike const the hypersurfaces t words, hypersurfaces of this should that This indicates use we defined. family are invariantly it is
unique if
-Prad. In this
E
case
const. In other
t
=
hypersurfaces in order to define mass. Multiplying Equation (7.1.3) with Qer and inserting Equation (7.1.4) * r obtain we (Q * r)2)) =2Q*m Q e (r(l + (U * r)2 ft(c + A)r2Q 81r =
-
and therefore r(t
m(to, qo)
=
47r
qo)
in
A C+ 87r
)
r2dr
where t
to is fixed.
=
(7.1.9) This
integral
can
also be written
A
as a
volume
integral,
) sin(O) ),r2 sin(O)(Q r)e- 'dq (6+ ) (Q r) (U 1-tB)
JB (
m(to, qo)
C+
r2
_Ir
dr A dO A
dW
A
9
A
dO A
d o
A
J
0
8?r
Ix
where B is the ball
E
M
:
t(x)
=
to, &)
< qo
1. Equations Q
*
r
=
I
the energy
0 would imply we integral 2 in view of E can also be interpreted as a me density c which 0 then T represents a smooth mass density. If, in addition, Prad Psph of freely falling particles and we would obtain the 3-parameter family smooth analogue to the motivation using individual particles above. In general, however, Q * r = 1. This reflects that one also has,to take into account the energy contribution of the gravitational field.
and A
had just
that
=
over
an
-
,
Lemma 7.1-3. Let
U
m
be the
* m
=
mass
-47rr
function of (M, g). Then
2(U r) 9
Q*m=47rr2(Q*r)
(Prad A)
A
-
E+
8,7r
87r
314
7.
Proof.
r)
-
Spherical symmetry
We have Us
(U
o
A) (Q
QOM
Q. +
Qer
r). By
o
r
=
Q*Uor+[UQjor= Q*Uor+(Q*v)(U*
Lemma 7.1.2
(r2 (I + (U r)2 o
(yQ
U
o
Qor +
m
o r
-
(Q
Q
r)2))
o
r
o
r
(Q
-
calculate
we can
o
r) Q
9
Q
a
e
A
r((U r)(Q r)U o
o
r)
(Q
-
r)(U
o
o
r)U
o
r)
A
r
A
-
(Q
47r(Q
o
r)Tn
r)r
o
+
47rr(Q
r)
o
81r)
+
e
(6+ A)
2
8,7r
and Uor U
0
M
+
m
=
r
((U r)U o
o
U
o
r
(Q
-
o
r)U
Q
9
r
Uor
(Uor)(Qor)Qov-(Uor)m,
m+r
=
r
-
-
47rr(U (Q
e
9
r)
r)(U
o
A
(Prad
87r
r)Q
+
o v
(Q
(Q
9
r)Q
o
m,
+
r
-
m(U
9
r)
-
47rr(U
o
r)
r
=
While
-47rr'(U r) o
(A:
orthogonal coordinates
nates introduced below
are
U
o
r
r)2(U O'\))
Uor =
o
(Prad
A -
87r
A ad
-
87r
are
often very useful double null coordiadapted to the geometry of
much better
spherically symmetric spacetimes.
(double null coordinates). Let (M,g) be a 4-dimenspherically symmetric Lorentz manifold. Then there exist local coordinates (u, v, 0, o) and functions G: (u, v) --+ F(u, v) E R, r: (u, v)
Lemma 7.1.4
sional
r(u, v)
E R such that
g
The v
i--*
function
b(v)
=
G(u, v)dudv
G is
unique
+
2 r
up to
and
In these
Fu'
=
(U, V)
(dO'
+
sin2(O)d W2)
interchanging of coordinates, the Christoffel symbols
9,, (In r),
the
transformations of
form
u
-4
ft(u)
the coordinates.
I'&
are
given by 1
=
FufV
==
9u (In G),
FouO
u
==
sin
2
(0)
rWp
2rc9vr G
7.2 The Schwarzschild solution
F,v,
I
Since in
Proof.
point exactly
I
Fv00
Ov (In r),
=
r, '.
=
0, (In G),
=
=
sin
I"
Wo F11W
=
ow
r.V,
sin(O)
I
2
(0)
315
2r8,,r
FVW V
G
sin(O) cos(O)
two-dimensional Lorentz manifold there exist for each
a
linearly independent, lightlike directions, the existence immediately from Definition 7.1.1 and Corollary 2.4.2.
two
assertion follows
ft, f) be coordinates and 6(fi, 0), (ft, f)) be coordinates with G(fi,,b)dftdb + 2 (,a, f)) (d02 + sin2(O)d W2). Since the warped product invariantly defined we have G(u, v)dudv G(ft, ))dftdb. At each point Let
g
=
is
of
=
a
2-dimensional Lorentzian manifold there
tions whence
we can assume
functions fu,
fv with a,
(without and
fuau
=
of GauBian vector fields vanishes 0
and therefore
VU au fV 19,11
=
avfu
=
we
af,
0. This
=
=
are
exactly
generality')
f,,9,.
two null direc-
that there exist
Since the commutator
obtain
fu A fV) 191,
=
I
a,,f,
loss of
-
implies
fl, (C')V fu) au
f fu(u)du
fi
and b
f f,(v)du. It is
formula
to calculate the Christoffel
straightforward
Fbac
I =
2
gad (abgdc
+
Remark 7.1.1. The function
acgbd
-
M
r:
adgbc)
-4
R
symbols using the I
-
gives the
area
of the orbits S., via
the equation Area(S.,) 41rr 2and is therefore invariantly defined. That G is almost an invariant is one of the two main reasons why double null =
(u, v)
coordinates
are a
very
practical choice. The other reason is that in (M, g) is explicitly described.
these coordinates the causal structure of
7.2 The Schwarzschild solution In this section
we
ically symmetric
will solve Einstein's equation for the case of a spherspacetime. These solutions describe the grav-
vacuum
itational field caused
empty
space. As the
by
sun
a
single non-rotating star which is situated in slowly and space is almost empty,
rotates rather
these solutions describe gravitation in the solar system very well. Theorem 7.2.1 uum
(Birkhoff).
spacetime. Then either
given by
m
=
11(2VA-)
or
M' C M such that each
with local coordinates '
Otherwise
we
x
Let 1
r
=
E M'
(t, r, W, 0)
exchange
A
there is
ft and f;.
a
(M, g)
be
a
spherically symmetric vacmass function is
is constant and the constant mo and
admits which
a
a
dense, open subset neighbourhood
local coordinate
satisfy
7.
316
Spherical symmetry 2
2mo
r
9=
A
3
r
Proof. Lemma
dr2
dt2+
_
7.1.3
2mo
r2A
r
3
+
r2
* M
9
( 3)
=
U
in
a
and
Q
*
m
=
Q
6
constant ?no such that
m
mo +
=
open set then from the definition of
an
2(0)d(P2)
Ar
( 3)
0
6
Hence there is
+ sin
implies Ar
U
(d02
m we
A3
If
.
6
get
(Qor)2 mo +
2
(Uor)2
=
Ar3
which
6
const. Since T 0 we can choose coordinates implies that r which simultaneously diagonalise g and T. Equation (7.1.3) implies then
in turn
r2=
=
1/A
Let
and
obtain
we
m
that
us now assume
orthogonal
are
=
coordinates
r/2
=
=
1/(2v/A).
gO (dr, dr) -(Ugr)2 + (Q * r)2 : 0. (4, T) such that r 4. Since T 0 this
There
=
=
=
choice
of coordinates
trivially diagonalises g and T simultaneously and we can generality that in Lemma 7.1.2 we have t t, 0. Hence Equation 4. We immediately obtain U * r e-'atq q (7.1.4) yields U e A 0 and Equation (7.1.3) implies without loss of
assume
=
=
=
=
=
m
Q*Q*r= Since
Q
=
e-Aar
this equation is
e-
and
can
be
_2_
integrated e-
to
2A
mo
Ar
2
r2
3
equivalent
2Y-
Ar
Mo
arA
_2
+ 3
2mo
r2A
2m =
-
3
r
where
to
give
A(t)
=
2A
Ar
A(t)
-
r
A(t) is an integration constant. Equation (7.1.5) implies e 2A arV A-- and therefore e -2AarV -e 2X,9rA which in turn yields v -
=
-
-
=
=
3
r
B(t) (1
-
r
_
A. After
(Q,,,r)2)
the assertion is
a =
re-parameterisation of L 2
(1
-
e
-2A)
L(l 2
-
t
A +
we
can
choose B
=
2m/r) implies A(t)
0. =
Tn
1 and
proved.
Observe that any spherically symmetric vaccum spacetime is automatistatic in the region gO (dr, dr) > 0. In the region gO (dr, dr) < 0 it is not static but has a fourth spacelike Killing vector field.
cally
spacetime has (for A 0) first been obtained by Schwarzschild static, spherically symmetric vacuum equation. (1916) Birkhoff then showed that staticity was not needed as an assumption. This
who solved the
=
7.2 The Schwarzschild solution
Definition 7.2.1. A
ishing cosmological The coordinates
spherically symmetric
constant
(t, r, W, 0)
In the rest of this section
The
regions
r
these coordinates
r
at
2m. Below
we
will
A
assume
0.
=
be matched
> 2m cannot
geometrically determine
null coordinates of the solution. This will show that
and that there exists
singularity spherically symmetric
Fig.
vacuum
7.2.1. Schwarzschild
Proposition
F:
R+ \
naively using
101
a
-+
r
=
singularity
useful double
2m is
a
spurious
unique, inextensible solution of the
equation.
spacetime
in Schwarzschild coordinates
r -->
more
structure),
With
7.2.1.
f : R+ \ f Of
van-
a
7.2.1 which represents the causal time it has been believed that there is a physical
some
=
will
is called
(cf. Fig.
and for r
we
and
< 2m
spacetime with
vacuum
Schwarzschild spacetime 2 called Schwarzschild coordinates.
(M, g)
are
317
(-oo, 1),
r
F-->
f (r)
=
-
2m
(
2M
Ir
32m 3 e-'l
R+ \ fol
r
F-4
F(r)
=
)
e
r/(2m)
(2m)I
r
let B ,chw
(X, Y)
E R
Then the Lorentzian
(R
2
X
0. satisfies Ric related through =
2 :
XY < I I and
S2, 9B,;cl,,
+
(f-I(Xy))2 (d02
The coordinates
ln
( X) Y
In the literature this
F
of
(XY)
dXdY.
manifold
(t, r) of
+ sin
2(0)d(P2
Theorem 7.2.1 and
(X, Y)
are
t
XY
2m'
=
f (r).
name is usually reserved for a subset of the maximally extended Schwarzschild spacetime, the shaded region in Fig. 7.2.2.
318
7.
Proof.
We
Spherical symmetry
restrict to the base manifold
can
field N which is,
!, yic',r,,s
B,,I,w. For
any null vector
of synr).nnetry the
equations
2rn/r) (N') 2+ (I 2m/r) 1 (Nr) 2 0 holds, whence we have Nt 2m/r) 'Nr. Double null coordinates can now be obtained through integration of these two vector fields. Since fo (1 2m/r)--ldr
(I (1
-
-
=
-
-
-
r
an
r
-
+ 2mIn
=t+
r
(2.
1)
-
we
define
coordinates
our
by
r
(r+2mln (2m 1))
t
-
,
-
(r
This gives ddk (dt + (I 2m/r)-ldr)(dt d t2_(1_2m -2 dr 2 and therefore 9B,,,,,,, (1 2(r + 2rn In( lm - 1)) we obtain =
-
=
+ 2mln
r _
2m
1))
(I 2m/r)-ldr) d dk From Y
-
-
(
-
.
-
2m
e(,k- ')/(4m) which
-=
er/(2m)
r
r
(2m
2m
e/ (2m)
f (r)
2m
r
implies
9B,_j_
We set X
=
=
(
2m 1
-
2me -r/(2m)e(
ddk
r
e-'k/(4m)
,
(4m) d,dk.
r
Y
=
-ekl (4m )
and
finally
obtain
32M3 e-'/(2m) 9B,,cllw
Furthermore, Ox and Oy
Y).
are
f (r)
both future oriented
=
XY.
(this
has been the
the minus sign in the coordinate transformation for It remains to show that the inverse of f exists for all r > 0. But this
reason
follows
for
dXdY,
-_
choosing
immediately
from
f(r)
r =
_
-ZM-2 e
-"--
2-
< 0.
1
The coordinates
provided by Proposition 7.2.1 are called Kruskalcorresponding spacetime is often called Kruskal-Szekeres-spacetime. This spacetime is locally isometric to the metric given in Theorem 7.2.1 but the global structure is different from the global structure obtained by using Schwarschild coordinates (cf. Fig. 7.2.2). Nevertheless, in this book we will refer to the inextensible spacetime given in Proposition 7.2.1 as Schwarzschild spacetime. Szekeres- coordinates and the
Remark 7.2. 1. The motivation for the Schwarzschild spacetime is to dea non-rotating star. During the lifetime of the star
scribe the exterior of
the radius may change (typically, it may shrink and 0). If we denote the r-component of the star at t will need for each t
Fig.
only the part r region can be
7.2.2. The white
purposes.
perhaps even reach by rstar(t) then we > rtar (t) of the shaded region in completely discarded for physical
7.2 The Schwarzschild solution
319
Fig. 7.2.2. Schwarzschild spacetime. Radial null geodesics are the const. The region covered by const and Y straight lines X =
=
Schwarzschild coordinates is shaded
We will
glimpse
investigate this solution
now
at what is known
as a
singular
will show that this spacetime is need to calculate its
geodesics.
Lemma 7.2.1. Let
(M,g)
....
be
in
black hole: In
detail and get a first Proposition 7.2.3 below we
more
but inextensible. But first
pseudo-Riemannian manifold and diagonal,
a
we
(xi,
be coordinates such that gab is
Xn)
n
gabdxadXb
=
1: gadxadxa. a=1
Then the
geodesics
^1(s) of (M, g)
s
are
given by
n
d
i-s (ga
0
7(s), a(s))
=
2
1: 19agb
2
0
'Y (S)
( b (S)) (no
summation
over
a).
b=1
Proof. Then
We we
suspend
have gab
the summation convention if the
=
repeated index
and obtain ga ja b d
V
aa
7,
aa
aa
ds
(ga a)
_
b c C)b,
d
,s (ga a)
c_ (a.-ga jab+ 19agbe 2
-
19bga ja) c
is
a.
320
7.
Spherical synimetry d
Proposition
d
1- b
TsS (ga a)
caagbc
2
7.2-2. Let
s
(s)
-->
be
-=
a
1
jS (ga a) -
geodesic
( b)
-
2
2
49a 9b
in Schwarzschild space-
time
(R2 with
S2, 9B
X
f-2 (XY) (d02
_I_ +
+
sin2 (O)d
77 E f 1, 0, 11 and assume that y-(O) Then there exists a rotational isometry 0 such that 7 =
fx
-
dt
2m
2
E, E2
=
_
r o
(x) is
2rn
=
given by
7r
0
L,
2
(dra-s )2 (I 2m) (_,, +
:
dW ds
r
02))
+
L2 /r 2),
r
where
Proof.
E,L
constants.
are
We will
the metric is
the coordinates
use
diagonal
(t, 0, p)-components
we
can
provided by Theorem 7.2.1. Since apply Lemma 7.2.1 and obtain for the
of
d
dt
((I -r 2m) js- ) (0) ds) a-s ( ( is- )
=
0,
=
0,
=
r
_
is'
d
r2
sin
dW
2
d
2
dO
r
2
ds
There is
rotational isometry 0: x 0. Then 0 o -y last equation. The first two equations
sin(O) cos(O)
dW ds
such that 0
o 0 o %0) ir/2 unique solution of the can immediately be integrated. To derive the fourth equation in the assertion of the proposition it is more convenient to use the conservation property ( , - ) 71 than to use the r-component of the geodesic equation in Lemma 7.2.1. In fact, it follows directly from
and
an
dO(O.,y(O))
=
=
ir/2
=
is the
=
2m) (dt )2 ds
r
after
+
2m)-1 (dr )2 r
ds
2
+
r2
(d o ds
inserting the equations for dtlds and dWIds.
Lemma 7.2.2. Let Then there is
a
null
(M, g)
geodesic
be in
spacetime which is locally extensible. (M, g) which is incomplete and extensible. a
321
7.2 The Schwarzschild solution
(M, g)
Let
Proof.
be
(M, g),
local extension of
a
future
x
time
M
\ M, oriented) E
and y E M.
broken null
(not necessarily in geodesic from x to y. This broken geodesic y must intersect OM C M that of loss assume Without can we a point z generality 'Y (t) 7 (0). M this geodesic is incomplete for t < 0, Itl sufficiently small. Since z Then there is
a
or
=
I
and extensible.
Proposition 7.2.3. Schwarzschild spacetime is inextensible and geodesically incomplete. A future directed null geodesic is incomplete if and only 0 and the if it enters the region f 1 2m/r < 01. It then approaches r 48,tn2 Rabed scalar Kretschmann Rabcd /r6 diverges along this given by =
-
curve.
-2', 71 Ropop Rabcd
implies Rrero
Lemma 4.4.14
Proof.
==
=
Rrtrt
2m-r
r), and 2(0) Rotot 'T (2m _-i_-_!1__(OJ Rptwt components which are not related to these com-
2rmsin
0 for all other
i- 1703 Rwrw 1
-
,
ponents by the general symmetries of the Riemann tensor. It follows Rabcd Rabcd 48M2/,r6 and therefore that any curve -y(s) with (7Bo,,_h,7'YS2) in B chw X S2 r o - (s) --+ 0 is inextensible. A curve -y is extensible if and Only if 'YB.,,:I,w is extensible in B ,ch, and 'YS2 is extensible in S2 By Lemma 7.2.2 we only have to study null geodesics in order
that
=
=
.
to prove that
(M, g)
is inextensible. If 'YB.,,h, is extensible then dr/ds 7.2.2 and r 74 0. By Corollary 4.4.1 'YS2 is a
is bounded
by Proposition
pregeodesic
with bounded acceleration in
fore also extensible. Hence
we can
in this 2-dimensional
null
geodesics (X, Y) these geodesics
are
the reflection isometries
only
(Bschw gB_11w) and study spacetime. In Kruskal coordinates
given by X
(X, Y)
1-4
compact manifold and there-
a
restrict to
=
7
const
or
and
(Y, X)
Y
const. Because of
=
(X, Y)
--+
(-X, -Y)
Y > 0. The
region
r/2m
(i)
Y > 0 is the >
r/2m,
(ii)
1,
=
=
r/2m
(iii)
1,
each of them being invariant under future directed null
const, Y
we
const, geodesics of the form X disjoint union of three different subsets,
need to consider future directed
<
1,
geodesics
X
> 0.
We have to estimate the affine parameter of our null geodesics. If -Y is a null geodesic given by X const, Y > 0 then there is a function =
Y
h(Y)
--4
with
h(Y)Vayi9y)
==
=
h(Y)ay
h(Y)(h'(Y)
and
'7h(Y),9y(h(Y),9y)
h(Y)FyYy)ay. 1, where c: (gxy) +
From
=
h(Y)(h'(Y),9y +
Fyyy
=
9y ln(gxy),
-
we
obt ain therefore h (Y) In
-y(s)
=
region
h(Y)
=
c
is
a
const ant.
(ii) gxy is constant which implies that -y satisfies const and is therefore future complete.
=
Now consider
regions (i), (iii). Since
dr/ds
=
VE2
-
L2/r2 (I
-
2m/r)
-+
JEJ
> 0
1-Y ds
o
322
for
7.
r --
In
for all
Spherical symmetry
the parameter
oo
(i)
case
s.
we
have
(I
diverges if and only if r diverges. 2m/r) < 0 and the square root is well
s
-
r
dY/dr
defined
The equation
h(y) dsldr
=
32m
3e'/ (2m) (E2
-
L 2/r 2
(1
-
2m/r))
-1/2
diverges 3. Hence s diverges for Y --> oo and complete. In case (iii) it is clear from XY f (r) m-) er/(2m) that 2m (1 our future directed null geodesics X const, Y > 0 are approaching 0 and are therefore inextensible and incomplete. r I implies that geodesic
diverges
r
the
if Y
-y must be future
-
r
=
=
The
region
r
black hole is
< 2m, X > 0 is the simplest model loosely characterised by the fact that
of
a
black hole. A
light ray which enters it cannot leave it any more but instead reaches the edge of the universe before the affine parameter of the corresponding null geodesic has reached the value oo. 4 Since a black hole does not emit a single light ray one is tempted to say that it is black, whence the name coined by J. A. Wheeler. However, this name is slightly misleading, since the black hole is not in the past of any observer who is situated outside this region. Rather than appearing black it is simply invisible. An observer who enters the region does not have a very low life expectancy. The longest timelike curve within the black hole region is given by X Y, X E [0, 1]. In Schwarzschild coordinates this corresponds to the path t 0, r E (0, 2m). Hence the observer's life is bounded by a
=
==
0
As
7.2.1
=
f2
0
9B _ lw
((9r, ar) dr
m
Experimental
In this section
=f2m
V"2-m/r1-1 dr -
-
=
7rm.
tests for the Schwarzschild solution
will
investigate the region 2m/r < I which may be a non-rotating, spherically symmetric star of mass m. The discussion applies in particular to the gravitational field produced by the sun which was Schwarzschild's motivation for solving Einstein's equation in this special case. considered
as
we
the exterior of
3This property could also have been seen geometrically: The lines X const < 0, Y > 0 intersect all the hyperbolas r const > 2m. 4 A widely accepted general definition of black holes does not exist. The definition we have just given has the disadvantage that any RobertsonWalker solution which satisfies the assumptions of Theorem 6.4.1 and e I is a giant black hole. In this special case one would have to replace the condition that the null geodesics in the black hole don't reach the affine parameter oo by the condition that they don't end in the cosmological future singularity given by t t+. =
=
=
7.2 The Schwarzschild solution
Since the exterior
onal to the
Ro9t E) (,9t)
contains the timelike
Killing
field
09t orthog-
spheres of symmetry it admits a natural infinitesimal split of spacetime into space and time. Moreover, the distribution
integrable whence we obtain geometrically defined hypersurfaces constant time. These hypersurfaces are given by
(,Ot)' of
1
region
323
is
Zt,,
I (t, r, 0, p)
=
t
:
=
to, r
1.
> 2m
pullback of the metric to Et does identify all Et through projection along the t coordinate. A timelike curve in spacetime corresponds to a curve in the Riemannian manifold (Zo, (I 2Tn/r) -'dr' + r 2(d02 + sin 2(O)d W2)) at
Since
is
depend
not
a
Killing t and
on
vector field the
we can
-
which represents space. In
our
case,
we
may
imagine the non-rotating located in the centre
> 2m. It is
radius
system but the Schwarzschild solution is of
r
star to be the
sun
with
0 of the coordinate
=
only
course
valid for
>
r
2m.' It follows that the region r < 2m can be excluded from introduced above. our discussion and we can utilise the spacetime split It is natural to identify this spacetime split with the infinitesimal splits defined by our own world lines. "Space" has then its intuitive meaning. While in general timelike geodesics represent freely falling particles, in asteroids and our case they should be interpreted as planets (or perhaps
rsu,,
>
satellites)
-
Proposition 7.2.2 we can assume that light ray is contained in the plane 0
Because of
single planet
or
Lemma 7.2.3. Let
-y(s)
=
(t(s), r(s), O(s), W(s))
be
the movement of =
a
a
7r/2.
geodesic. Then
we
have
(
1
r2
dW) 2+ (-77L2
I
dr
+
2m)
2
p(s)
=
7.2. 1. Let -y (s)
dr/ds
and
d olds
in
1
be
a
geodesic.
Then
11r(s) satisfies d
-77m
dW2 Proof. Substituting p(s)
(-,qlL 2+02)(1
-
2mg)
=
=
+
+ g
L2
11r(s)
E 2IL 2.
3me2.
in Lemma 7.2.3 gives (do/d(p)2 + Differentiating this equation implies
1
the assertion.
5
for
(t (s), r (s), 0 (s), W (s))
=
L2
r
Proof. This follows by dividing the equations Proposition 7.2.2.
Corollary
E2 =
Hence it does not matter that the Schwarzschild metric is not defined at
the centre
r
=:
0 where the
sun
is located.
7.
324
Spherical symmetry
Bending of light rays. Since (null) geodesics are influenced by curvature, according to general relativity, light rays should appear bent near regions where gravity is large. In particular a light ray passing the sun at a short distance should appear to be slightly bent. The experimental verification of this effect was one of the first tests of the theory. To describe this effect we need to determine the angle a under which central a object appears to an observer in Schwarzschild spacetime. This angle can then be compared with the corresponding angle determined by the background metric dr2 + r 2dS?2 of space (cf. Fig. 7.2.3)
ly
Fig.
7.2.3. The size of
a
central star in Schwarzschild
spacetime
Lemma 7.2.4. Let -y be a lightlike geodesic and -y(O) Then a (ILIIE) /(-Or, (O)) satisfies ro sin(a)
VF1
=
=
=
-
(to, ro, Oo, wo). 2m/r.
"
Proof. Since (s) dtlds,9t + ly the null condition (I 2m/r) Jdt1dsJ' + 0 implies 1-412 2m/r) -1/2 E. Since (I V1 2m/rdt/ds ly the is I a a, angle given by sin(a) IL/rl/((l ld ods,9,1111(s)j aw ==
=
-
-
=
=
2Tn/r)
-1/2
-
-
-
I
E).
Corollary 7.2.2. Let 7 be a lightlike geodesic with past endpoint -Y(O) (to, ro, Oo, Wo). If -y) passes the boundary of a centred star of radius r,, 3m, then the angle a,, defined in Fig. 7.2.3 satisfies
sin(a.) Proof. Assume
(dr/ds) ,
=
r.
I
ro
I
-
== -
=
>
2m/ro 2m/r*
that -y passes the boundary of the star at s s*. Then r has a minimum there. Proposition 7.2.2 implies =
0 since
r*EIILI
=
V1
-
2m/r*
7.2 The Schwarzschild solution
and the assertion follows
by inserting this equation
325
into Lemma 7.2.4.
1
angle a. of the star is larger than one would expect physics where sin(a,,) would just be given by r,,/ro. This effect has been verified by a British team lead by Arthur Eddington (1882-1944) which measured the bending of light rays close to the sun during the total eclipse in 1919. They used the limiting behaviour given Observe that the
in non-relativistic
below. 7.2.4. Let IY be
Proposition
region 2m/r > minfr o -y(s) : s E RI along respect to the flat metric gflat and intersect at
an
A
angle
inextensible null
an
not enter the
Then there is
1. -y. =
+
which does =
Furthermore, dr2 + r2dQ2 which
4m/ro
=
geodesic
minimal radius ro there are two lines with a
are
asymptotes of -y
o( 77) ro
r o -y(so) exists since -y is inextensible Proof. The minimal value ro and r o -y > 2m by assumption. We may choose our spherical coordinates 0, o so that the light ray lies in the plane 0 7r/2 and the equation oo the coor0 holds. Proposition 7.2.2 implies that for s (p oy(so) =
=
-4
=
dinate W converges to limits V. The lines (with respect to gflat) which pas's through the origin under these angles W are therefore parallel to asymptotes of -y. The differential equation provided by Corollary 7.2.1 can
be solved
and has the solution
exactly 9
(P (Q)
1
d
=
_ 2 We
are
angle
+ 2
M0 3
+
(ro)-2
interested in situations where the ratio
A
=
2
lime-o o(p)
vanishes when
with respect to the parameter x Differentiating the function m/ro
=
m/ro
F-
W
C9(mo/ro)
mo(ro)-3
m/ro
is small. Since the
0
will linearise
=
we
3 (_ 2
+ 2
MJ3
0.
3 -
ro
ro
11,ro
W(g)
and then take the limit
gives
0
aw
m,
2
-
+
(ro)-2
2
-
mo(ro)-3)3/2
dp
and therefore
19 W
,9(mo/ro) Hence
we
have zA
0
),o=0,m0/r0=0 =
4mo/ro
+
=
ro
JI/ro ((ro)
o(mo/ro).
3
-
ro-
-2 -
3
2+)3/2
-4
2
I
326
7.
Spherical symmetry
The
perihelion precession of Mercury. Mercury moves around the sun describing an orbit whicli i.,., *1 to an ellipse but not closed. 6 the in 19th Already century one has ineasured the angle between consecutive. local minima of the distance between
Mercury and the sun and explain this angle within the Newtonian theory of gravity. 7 While such a "precession" occurs if one takes into account the gravitational fields caused by the other planets, this does not give a quantitative explanation of the measured value. The first outstanding success of general relativity was Einstein's demonstration that his theory could explain this discrepancy. 8 In order to calculate the "missing angle" we have to compare the Newtonian solution of the two-body problem (the sun, Mercury) with timelike geodesics (Mercury) in the Schwarzschild solution which dehas tried to
scribes the
sun.
In Newtonian
gravitation,
spherically symmetric
star
particle in according
a
moves
the
gravitational field
to the
ordinary
of
a
differential
equation
d21(8)
M
_ (S) 111(s)II,
-
ds2 Lemma 7.2.5. Let
ential equation
o
is
1(s) -
(1/9N, W)
are
and
Le
polar
R
:
Then the
equivalent
ds where
> 0
(7.2.10).
Equation (7.2.10)
dW
m
-->
curve
R 3 be y
(7.2.10)
-
a
solution
contained in
is
of the differa plane and
to
d 2ON
2
+ ON
,
coordinates
:--:
m/L
of this plane
2
(7.2.11)
.
and L is
a
constant.
"
Proof Equation (7.2. 10) implies that (s) x ds (" x being the vector product in R 3) is constant with respect to s. Hence I is contained in the plane spanned by -!L (O) and (O). If (r, W) are polar coordinates ds of this plane (x r cos W, x2 r sin W), Equation (7.2. 10) is equivalent cross
=
to 2
d 2,r(s)
r(s)
-
dP
(
ds
)
M
r(s)
r2 (8)
The second equation implies that 1 /r, the first equation is therefore
dV(s)
2 r
==
ds
equivalent
d2 0(s) -
+2
dS2
dr(s) =
ds
L is constant.
0.
ds
Setting
2 to d ON /d (p2 + ON
=
ON
m/L
2
1
6This
true for all
is
but in the
case actually planets, Mercury especially pronounced. 7In order to do so, Astronomers have assumed the existence of a further planet. However, this planet has never been seen. 8 He did this using the equation Ric 87rT before he arrived at his final theScal/2 g 87rT. This was possible since for these calculations ory with Ric only the vacuum equation is needed. =
-
=
of
the effect is
7.2 The Schwarzschild solution
Equation (7.2.11)
is
an
inhomogeneous
constant coefficients. It is easy to
see
327
linear differential equation with (in the generic case L = 0)
that
' there exist constants C1, C2 such that ON (W) + cl sin(W) + C2 COS (W) L2 Our polar coordinates are only fixed up to a rotation in the plane. Hence =
-
without loss of
we can assume
generality that
there is
constant
a
e
> 0
such that M
ON (
This solution is
periodic.
O)
f2 (1
=
We could
+
now
COS(W)).
e
attempt solve the corresponding and to calculate the difference of
in the Schwarzschild
solution, equation angle W (modulo 27r) between two consecutive minima of the coordinate radius as a Taylor polynomial inmo/ro. However, it is difficult to use this strategy in practice because it would involve integrals which are quite complicated. We will therefore employ a different method and obtain approximate solutions from approximate differential equations. Observe that Equation (7.2.11) is the Newtonian analogue to Corollary 7.2.1 and that both 2 2 equations differ only by the quadratic term mg m/r which is very small. The idea is now to view the Newtonian solution as an approximation to the relativistic equation. Inserting the Newtonian solution into the quadratic term gives a third equation =
d2Oapprox dW2 which is also
linear
a
2
Tn
M
+ Papprox
+ 3Tn
=
1-2
( -L-2 (1
+
e
COS(W))
inhomogeneous differential equation
with constant
coefficients. It appears to be a better approximation than the first differential equation since the term 0 has been replaced by the term Un(pN )2
approximation for 3mg 2. While this argument justification appears to be too complicated to be
which should be
a
better
only heuristic
a
real
is
worthwhile in
our
context.
This third equation gives
3Tn3
'M
Papprox
=
1-2 (1
+
To calculate the
angle
g(W).
of the function
d0approx dW
e COS
(W))
+
at the
perihelion
Un 3
sin (W) +
-
y-2
e2
e2
2
6
+
we
cos(2W)
+ eW
sin(W)
have to calculate the minima
The equation
Tne =
L4
(I
!
_L4
(! sin(2W)
+
3
sin(W)
+ W
cos(W))
0 as was to be expected. gives that Qappmx has a perihelion at Wo A comparison with ON indicates that the next perihelion should be at =
27r + 6 where J is small. Hence
with respect to
(27r
+
6) cos(27r
+
6)
we can
-
neglect
and obtain
f 3
sin(2J)
+
sin(6)
7.
328
SpIlericai symmetry
3M3e
me
0
L2 With
J and
tan(6)
s,n(6)
(27r
+
L2
neglecting
6) cos(6).
+
J with respect to 27r, this equation
implies 67rM2 J
L2 which gives
a
correction to the Newtonian value in very
good agreement
with observation.
7.3
in
Quasi-linear hyperbolic systems two independent variables In this section
we
of
equations
prove a theorem about hyperbolic systems of partial in two independent variables which will be applied
differential equation in Sect.
7.4.
The material is very technical and of a different mathematical topic than the rest of this book. The reader may wish to skip this section
For the
on
first reading.
following
call j'(f ): R1 projection R2
theorem
R1
x
Rk,
R, (t, q)
we
_,
X
q is
i-->
The system
Otf
+ A
o
some
notation. If
.
C1(R2
Definition 7.3.1. Let h E
Rk,Lin(Rk,Rk)).
f : R1 -4 Rk we (x, f (x)) the O-jet of f The canonical denoted by pr2-
need
Rk,Rk)
x
and let A
(E
C1(R2
X
of differential equations
jo(f),Oqf
=
h
o
jo(f)
quasi-linear system of hyperbolic equations in two variables if for O-jet (t, q, F) E R2 x Rk the linear map A(t, q, F) has k linearly independent left eigenvectors. The directions R(at + Ap9q) where Ai are the left eigenvalues of A are called characteristic directions. The (unparameterised) integral curves of the characteristic directions9 are called the characteristics of the system of differential equations (and the given is
a
every
solution). The aim of this section is to prove the following fundamental existence and uniqueness theorem for quasi-linear systems of hyperbolic equations in two variables.
C' (R2
Theorem 7.3. 1. Let h E
Rk, Lin(Rk, Rk))
atf Here
we
mean
x
Rk, Rk)
and let A
E
C' (R2
X
such that
integral
characteristic directions
+ A
o
jo(f)aqf
curves
=
h
o
jo(f)
of vector fields which
are
tangent
to the
7.3
in two
Quasi-lineay hyperbolic systerns
independent variables
329
quasi-linear system of hyperbolic equations in two variables. For any function fo E COO ([a, bj, Rk) there is an open neighbourhood U of 101 x (a, b) C R2 and a unique smooth solution f : U --> Rk of the system fo (q) for all q E (a, b). of differential equations such that f (0, q)
is
a
=
proof of Theorem
The main part of the
7.3.1 is contained in the
following
lemma. Lemma 7.3. 1. Let
h,
/\ E C' (R2
R2
x Rk, Rk) of the
at least two
and
a
<
numbers
b E R. Assume that
A(t, q)
are different. point, (t, q) function fo E C' ([a, b], Rk) there is an open neighbourhood U of 101 x (a, b) C R2 and a smooth map f : U -4 Rk such that
at any
E
For any
(i) f (0, q) (ii) 9tfl +
fO(q) for /V
-
all q
jo(f))aqf'
Moreover, the solution
is
(a, b),
G
=
h'
jo(f). (i
o
kj).
unique.
Proof. We will first transform the system of differential equations into a system of integral equations and then employ an iteration technique in order to solve the system of integral equations. Assume that f is a solution to our system of partial differential equathe k J'we denote by t tions. For (s, p) c R2 and i E f 1, -yis' ( P) (t) integral
curve
of the vector field
From the definition of
d dt
fi
0
i, -Y(s'p)(t))
-Y(',,P)
=
df(at
=h i0
we
obtain the system of
fi (S' P)
f' (0 'Y' ,
+
A'
o
A'
at
o
jo (f) aq
with
-y (S'
0)
p)
s,
and
+
jo(f)aq)
=
atf'
+
A'
o
jo(f)aqf'
jo(f)
integral equations ') (0))
+fo
h'
-
jo (f)
o
-y',,P) (T) d-r,
(7.3.12)
t
,Y('S,P)(t)=(S,P)+(t,f A'-jO(f)o-y',,,P)(T)d-r).
(7.3.13)
S
E 11'...'kj) which Conversely, if there are continuous maps t) (i also satisfy the system satisfy this system of integral equations then of differential equations (ii) This follows since differentiation of Equation (7.3.12) implies the differential equation (ii). In order to solve the system of integral equations (7.3.12), (7.3.12) we will employ an iteration procedure. Let
_ILy
-
FO'(s,p)=fO'(p), and
r(,,,P),O(t)=(s+t,p),
330
7.
Spherical symmetry S
F,',,+I(s,p)
=
fo'
F(`-,P),.+1(t)
=
(8,P)
h
+
i o
jo( F,) r ',,P),,.,(-r)dT, o
t -
I
(t,
'
A'
jo(F )o r(,,, P),,,, (r) d -r).
o
..
We will show that these sequences of functions have well defined limits. limiting functions will then solve our system of differential equa-
These
tions. We will prove the existence of
unique limits by showing that the
sequences M
Fm' (8 A
=
1
Fo(S A
+
I
E (Fj(s, p) Fj'_ -
1
(s, p)),
j=1
pr2
(T s,p),m(t))
=
s,p),O (0)
pr2(1
i
(pr2 (' ,,p)j (0)
+
-
pr2 (ri s,p)j
-
1
(0)
j=1
be
majorised by an absolutely converging series which in turn implies they converge absolutely. To achieve this it is important to obtain first bounds on Fj1 and
can
that
P), j
-For any d >
C,
C2
a
> 0
let
jjjf0(q)jj + jjDfo(q)jj Jq E [a, b]j, =sup f jjh(jo(f)(t, q))jj + IIA(jo(f)(t, q))jj + IlDh(jo(f)(t, q))jj + JJDA(jo(f)(t, q))jj sup jf (t, q)l < 2Cj, =
sup
:
i
(t,q) C
C1
+
[-6z,d]
C-
x
[a,b]j,
C2)
and
(t, q)
E
(-a, a)
(a, b) I a
x
t a
-
C
+t C
enough.'o
pr2
for
tion of U&,, 10
Our bounds
(ddt r(',,P),m (t))
r(i ,,P),,,(t)
E
imply
-t -
C
A'
problem imply o
in the
jo (F, 1) -
U, Let (s,p)
V(',,P),m (t)
The choice of & is less to
b
t < q < b +
We will solve the initial value small
< q <
C
E
F(",P),M- 1 M I
U&,,,,. Inequality
U&,,,.
Hence
significant. The only
guarantee the existence of the
region U&,,,,
constant
<
if
C
a
is chosen
(7.3.14)
7.3.14 and the defini-
during the
iteration process
purpose of its introduction is
C2.
7.3
Quasi-linear hyperbolic systems
do not leave the region Let
we
(S
jF
SUP
=
M
Fm(s,p)
C')P
bounds
our
[-a-
t E
:
and
independent variables
s,a-
C1, C2
S],i
E
are
331
valued at all
fl,...,kl
1
I
a
6'"
SUP
=
M
f ap
Assume that a
pr 2
(F(,,p),,n (0)
supflFji(s, p) I "
-
5p-
U&,,,
in two
p)
F,' ,+
=
(foi
-
OP
-
:
[-a
t G
(s, p)
U&,O,, j
E
J" 5p- (h
o
jo(Fm)
o
0
Of0i
+
a
S], i
-
kj
E
10,..., mll
E
2CI. From
<
F( ,,,p),m) (-r)dT 8
9
pr2(F(s,p),m(O)) ap
aq
s,
-V(i"p),m (0))
'9
+
-
D3h D2Fm
a
D2h
+
N) i 5P pr2 (-V(,,p),m
d-r
2(r s,p),m(T)) 7pr P
and t
D 2 /V
1 +
7pr2(r s,p),?,n+1(0) P
ap
(F s,p),m (0)
pr2
a +
D3A'D2Fm
-pr
P we
get
J +, Let if
a
jF
< <
M
are
CJmr
-
1
1+2C
tO,...,mjj
<
IF',+,(s,p)
m.
=
-
M
2. Since
JOF
<
< I + -
imply
a(CJr
M
jF+j
<
M
Jor Fj(s, p)
C and
CJQ").
+
M
3C,
Jr+j
2
=
M
I these bounds
(s, p)
E
U&,
0,,
j
E
F,n(s,p)l -
fs 1h ojo(F,)
CIr(""P),M(O))
f
jr+1
2C,
M
+
CJmFJmr),
We estimate for sup
IfOI(F(",P), (0)) +
+
Then these inequalities
.
3C and jr rn
valid for all
<
a(CJm"
+
sup
-
fo o
F S,P),rn-I(OM
F(,,,P),,n(7-)
-
h
ojo(Fm-,) r(,,P),m-j(T)jd-r
F(""P),M-I(O))l
f maxf 1 (Dh) (j'(Fm) F(.,,P),m (t)) 11 o
Jj(Dh)(jO(Fm-j) r(",,,P),m-j(t))jjj, -
o
7.
332
Spherical symmetry
i
[O,s] I lljo(F,) 0F(S,p),,rn(T) -jo(Fm- I) oF(,,,P),,n-,(-r)lldT
t Ez<
CIr(II,P)'III(O))
r(Is,P),III-1(O))I
-
(Ipr2 (F s,p),m (7))
+ C
(7
pr2
-
0
k
+
E lFmj(s7p)
o
-
F,'n-,(S,p) 1'(,,p),m-ll)dT o
j=j
C1r
F,
S,P),M(O))
(S,P),M-1(0))l
(IPr2 (F 3,,p),. (T))
+ C
pr2
-
(F S,p),Tn- (T)) I
n
k
+
E (IFmi (sip)
r
0
i
,S,P),M
-
Fjn
(s, p)
o
r(,,, P), I M
j=1
+ <
I Fmj- 1 (s, p)
CIF('. "P),M(O))
F( ,,P),,n Fj,,F(,. "P),M-1(0))I
o
-
-
1
(s, p)
o
F(,,P),,n- 1 1)) dT
S
+Cf
3
3
((1+3kC)Ipr 2 (F(S,P),
M
JFmj(sjp) r(.,,P),m
Fmj-,(s,p) -V(',,P),ml)d-r,
7,))
-
pr
2(r .s,p),m-1( M T
0
k
+
o
-
o
j=1
where in the last
inequality
I Fmi- 1 (s, p) o-V(
<
we
have used
Fmi- 1 (s, p) sup JjD2Fmj-j(SiP)jj IF(s,p),M 3CIF'(S,P),M - S,P),--11.
,,P),m
:5
Analogously
we
o
-
r(s,p),m-l
i
-
obtain the estimate
Iri(s,p),-+'
(SIP)IMI
-
S
:5Cf
((1+3kC)Ipr2(F(3s,p),m(T))-pr2(r(3,5,p),m-1(7-))I
0
k
E I Fj,, (s, p) F(,,P),m Fmj-
+
o
-
1
(s, p)
d-T.
o
j=1
Let
F= sup
EM
fiF;,,(s,p)
=:
sup t E
Jpr2 (T s,p),m W)
[-a
-
s,
a
I(s,p)
Fm-,(s,p)l
-
s], i
E
i
i
r
EM
-
-
E
U&,o,, i
E
Pr2(r s,p),m-1(t))I J(s,p) f 1,
.
.
.
,
kj
I
-
k1j,
f E
Quasi-finear hyperbolic systems
7.3
From F
Em+j
estimates
our
<
F
holds. In m
=
fact,
I if
EF+1
<
rn-
,r+,
we
=
M
<
aC(1 +3kC),Er +aCk 6F M
M)
enough and
(8,P)
:
U&,,,,j
E
E
j0,...'mjj
<
2C1
< aC, so the inequalities hold aC and El" have EF< 1 1 Er < 1. If they hold for 6F we get M , M -
1/(2k)2
(1
+
a(l
+
3kC))C(2kCv"a-)mv'-a + akC(2kCvG-)'
(1
+
a(l
+
3kC)
aC(1 (a(I
+
3kC)
+
JFm (s,p)
+
,
kv/a-) -1(2kCv/a-)m+l
+
2k
<
a
F, ,, -,(s,p)l
-
(2kCVa-)m+l kvla-) -1 2k
3kC)(2kCV,'a-)mv/a- + aCk(2kC.\/a)m
Hence it is sufficient to choose
Since
-
-
a
-
(2kCVa-)'V"a-
<
M
supjjFj'(s,p)j for
333
inequalities imply
-
is chosen small
<
M
<(2kC-,Fa)m andEr
M
a
M
show that these
(5
if
er +1
(I+a(1+3kC))CEI'+ak CCF, now
independent variables
obtain
we
M
We will
in two
Jpr2(F(s,p),,rn(t))
minf 1/(2k)2, (2k- 1)2/(1 +3kC+k)21. 6F and
<
-
pr2(F(s,p),m-1(t))1
<Em
the sequences M
Fm' (8 P)
FO'(s, p)
1
+
E (Fj
(s, p)
-
FjL
1
(s, p)),
j=1 M
pr2
(r(,,p),,n (0)
pr2
(r(,,p),O (0)
i
E (pr2 (r(,,p),j W)
+
j=j -
converge if the series
jorised by
a
pr2
,E-7 TOO 3 3=
(r( s,p)j
and
convergent geometrical
-
1 (0)
J E' j= I 3 series if
exist. These series a
<
are ma-
1/(2kC)2. Moreover,
is easy to check that M
supf jFj(8,p)j
:
(8,p)
E
U&,,,,j
E
f 0,..., mjj:5 C1
+
EJ j=1
if
(C1)2 a
<
4k 2C2(1 +
C1)2
<
2CI
it
334
Spherical symmetry
7.
holds for every is chosen smaJI
This proves uniform convergence of
m.
enough. This solution is cleariy differentiable with respect
F,,,,
F" (8 P)'M
if
a
to t and continuous
with respect to q. In order to show that it is C' with respect to q and t we can apply the same to argument to the prolonged system of differential
equations which
(1
E
11,
.
.
.
r
,
-
is obtained
11)
by differentiation
differentiate equations
E f 1,-, 11) by replace (,Yt) f gives k(r 1) equations of the form
and
g"
(ii)
of
and
t
(ii)
as
follows. For 1
1 times with respect to t
(at)lf
by
atg "-. t
This
-
atgi,l
+
t
which
add to
we
Aiaqgti'l
Gt1'i(f'g1'1'...'gk,1 t t
=
system of differential equations. Differentiating
our
1 times with respect to q
equations (ii) 1 at (aq)lf' + A Oq(aq) f' the kr equations
atgi,l q to
our
+
obtain equations of the form (aq)lf). We therefore also add
we
Gl,'(f, 0j, q
=
Aiaqgqi'l
Gl,'(f q
1, ,
gq
1,
....
gk,l) q
system of differential equations. Choosing the additional initial
conditions
Aaqg"'-'
(g"')o(q)
t
t
(g" 1) o (q) (g',')o(q)
+
I G"' t (f, g" t
=
-X'(0, q, fo (q) 19q fO
=
(0jfO(q)
t
q
+
h (0,
q,
gk,l)) O,ql t
fo (q)),
system of differential equations which is of the same form the first components f' of the solution of this system coincide with the solution of our original system. Moreover,
we as
g',' q
obtain
a
(i), (ii). By construction, =
(Ojf', g"'
Cr (U&, ,
t
(,9t)lf'
prove
uniqueness of the solution. Assume that
=
are
continuous for all 1 which
implies
R).
We will
now
I
(f,
i
both
(',,,Y))
Then
we
satisfy the system
of
integral equations (7.3.12), (7.3.13).
get
S ,
(h'(-y'o" Y) (7-), f (-Y('s, Y)) (-r)) 0 8
<
C
f
(1,y' ",',y)(T)
0
and, analogously,
-
+
-
h'( ',, Y) (T),
Y)
(-F)))) d7-
7.3
independent variables
in two
Quasi-linear hyperbolic systems
335
8
<
f
C
+
o
Hence
writing d(s, y)
=
1-/".,,Y)
-
'('s,Y) I + I f (-y ,'Y)) j ( ,'Y)) I
we
-
obtain
S
maxfd(-r, y)j
:5
rEO,s
rEO,5 <
0
maxj2Cs maxf d(-r, y)jjj
can
2Cs
=
maxfd(-r, y) If. -rEO,s
-rEO,s
,rEO,s
This equation
d(-r, y)d-rj
maxj2C
only hold if d(s, y)
0 for
=
s
I
I/ (2C).
<
lk (t, q, F) I be a basis of left Proof of Theorem 7.3. 1. Let f 11 (t, q, F), their denote and A of eigenvalues by A' (t, q, F), (t, q, F) eigenvectors Ak (t, q, F). Multiplying the differential equation from the left by l' we A or, equivalently, obtain Vatf + A'l'aqf .
.
.
,
.
.
.
,
=
d
at (1'f )
+
A'aq (1'f
d
(dq )
(A)
+
(dt )
f
(lih) (A)
+
(Dili
D31il9tf)f
+
(Dlli)f
li
+
+
+
li
Aif
+
Ai(D21i)f
Ai(D21i +
+
D31iaqf)f
(D31ih)f.
This system of differential equations is of the diagonal form treated in the Lemma 7.3.1 above. Hence the assertion follows directly from this I lemma.
[a,,3]
C
assumptions of Theorem 7.3.1 let f of the quasi-linear system of R' depends only on the initial data
7.3.1. In addition to the
Corollary
[a, b]
and to > 0. The solution
hyperbolic equations at (to, qo) E restricted to [a,)3] if and only if all the subset
Proof
101
x
characteristics intersect
f 01
x
R in
[a, 0].
This follows
immediately
from the
integral representation (7.3.12),
(7.3.13). The
following corollary will only
Corollary
be used in Sect. 9.5.1.
7.3.2. In addition to the
assumptions of Theorem
7.3.1
as-
that A is constant and that the map h is linear in f. The characteristic directions do not depend on the solution. More-
sume
b]) be the set of all (t, q) Ei R x R through (t, q) intersect the set f01 x [a, b]. solution f which is defined on all of 0+ ([a, b]) over, let E) Qa,
istics
such that all characterThen there is
a
unique
336
7.
Proof. It
Spherical symmetry
is clear that the characteristics do not
depend on the solution depend on it. We denote the characteristic through (s, p) to the ith eigenvalue A' by -y(',,P) ,The non-trivial part of the corollary is to prove that the solution which (by Theorem 7.3.1 may only be defined locally) extends to all of 0+ Qa, b]). This can be achieved by applying a slightly refined version of Theorem 7.3.1 repeatedly. To this end we will since A does not
need better estimates in the proof of Theorem 7.3. L Assume that the initial dat& satisfy I fo'(q) I < C1 for all q E [a, b]. Since h is linear with respect to f there are functions h, such that
h'(jo(f)(t,q))
E1=1k hj'(t,q)f1(t,q).
=
It follows that there is
a
constant
K G R such that k
K >
(Ihl'(t, q)l
supf
+
JA'j
+
IlDh'l(t, q)JI)
Observe that K is well defined since 0 ([a,
b])
:
(t, q)O([a, b])J.
is compact. Unlike in the
proof of Theorem 7.3.1 the inequality defining K is independent of the solution. We set U,, < t < al. I (t, q) E 0 Qa, b]) =
Recall that
F,',,(8iP)j
r(,,P),,,
in the
proof
-y(',,,P)
for all
The estimate for
m.
of Theorem 7.3.1
simplifies
I F,',,+ 1 (s, p)
-
to
=0
F,',,+I(s,p)
-
F,,,(s,p)l
< k
s
(h ',F,',,
+
<
-
h
'Fl,
o
-y
P) (,r)
dT
F
akKEm7
where F Em=supfjF 1,(S)P)-Fm1_j(S1P)j: (s,p)c2([a,b]),1cf1,.-.,k}j-
This estimate E
F < I
akKC,
Using Fm' (8 A 1
implies the
we
recursive
inequality EF +1
=
Fo'(S) A
<
M
:5 C1 (akK)" and therefore get EF m +
p) Ej' _ (F(s, j
-
1
akKEF. ROM M
J Em j= , 3
Fj' (s, p))
we
< akKCI -
1-akK'
obtain the
bound M
JFm'(s,p)j
<
C,
+
EJ 3
C, <
-
-
I
-
akK
j=1
It follows that for
a
<
11(2kK)
there is
a
solution defined in all of
U, Since the number 11(2kK) is independent of the solution we obtain our global solution by successively solving 2kK maxf s : 3p with (s, p) E 2 ([a, b]) I initial value problems. The proof of uniqueness and differen1 tiability is exactly as in the proof of Theorem 7.3.1.
7.3 The initial value
for
problem
337
spherically symmetric spacetimes
Remark 7.3.1. Observe that Corollary 7.3.2 is still correct if we replace curve 101 x [a, b] by an arbitrary curve C which is intersected
the initial
by each all
(t, q)
curve
characteristic at most E R x
once
and
replace V Qa, b]) by set 0 (C) of through (t, q) intersect the
R such that all characteristics
C.
7.4 The initial value
problem spherically symmetric perfect fluid spacetimes with non-interacting elect romagnet ic fields for
In this section
we
(7.1.3) -(7.1.6))
in
time which
problem (cf. Equations spherically symmetric space-
discuss the initial value some
represents
a
generality for perfect fluid.
a
Since the section is quite technical and requires the results Sect. 7.3 the reader may wish to skip it on first reading.
of
good description for the exterior of an isolated, spherically symmetric, non-rotating star. Here we wish to solve Einstein's equation for the interior of such a star. The complete model of is then usually oban isolated, spherically symmetric, non-rotating star at the boundsolutions exterior the and interior the tained by matching the for 7.5 Sect. in be done will This special case of static ary of the star.
The Schwarzschild solution is
a
stars.
The system of Equations (7.1.3)-(7.1.6) is highly non-linear and rather complicated. Observe that the assumption of a perfect fluid (Prad allows us to integrate Equations (7.1.7) and (7.1.8) directly sub'-,:
Psph)
stantially simplifies the problem. This simplification is unaffected when field. we include a non-interacting electromagnetic first will In the following we study electromagnetic fields in spherithen discuss the initial value problem and cally symmetric spacetimes for a spherically symmetric spacetime which admits a perfect fluid and a non-interacting electromagnetic field. Readers who have not read Sect. 5.2.3 on Maxwell's equation may 0, wish to skip the material up to Lemma 7.4.2 and assume Tj =
e
=
b
=
0 in the
following
discussion.
Recall from Sect. 5.2.3 that the
electromagnetic field
where F is
a
are
equations for
an
given by
2-form. The
tum tensor reads then
free Maxwell
source
dF
=
0,
div(F)
=
0,
electromagnetic part 1 Fbd 41r (gcdF
(Te0ab
`:`
ac
of the energy 4
(F, F) gab)
-
momen-
Given
a
338
7.
Spherical symmetry
spherically symmetric spacetime (M, g) source-free Maxwell equations course, the matter represented
one
and any 2-form F
could define T, natter
by Tnatter
satisfying the
==
T
-
TI. Of
would in
general be quite exotic. Moreover, it is possible that neither Tmatter nor Tel are spherically symmetric. This discussion indicates that we should impose additional conditions in order to describe physical electromagnetic fields. Given that (M, g) is spherically symmetric, the most natural additional assumption
on
the energy momentum tensor Tel would be to demand that it is invari-
ant under rotational isometries and that F is well defined in
large
open sets
containing complete spheres
x S2,gZ +,r2df22) be a spherically symmetric, manifold, yZ be the volume form of (Z, gZ), and form of (S2, dS?2). If F E f22(Z X S2) satisfies
Lemma 7.4.1. Let
(Z
Lorentz
4-dimensional
ILS2 the volume
dF and
=_.L (gcd FacFbd
(Te0ab
sufficiently
of symmetry.
"
-
47r
div(F)
0,
=
(F, F) gab)
4
0,
=
is
spherically symmetric,
then
there exist constants e, b with F
e(7rz)*pz + b(7rS2)*/IS2.
=
2
,r
The
corresponding I
Tel
=
87r
(
energy momentum ten8or is
62
+
b2
Ub
0
H
+
Q
1ao
of T,.l
H
U, Q, E2, E3 1, where I
and E 3
-
r
Spherical symmetry
+e2 + b2r2df22
Qb
Proof. We consider the orthonormal frame E2
given by
r
-aw. sin(O)
implies
T,l (E2, E2)
=
Tel (E3 E3) i
and
Tal(U, E2)
=
Tel(U, E3)
=
Tal(Q, E2)
=
TI(Q, E3)
All other components of Tl are unconstrained Since 47r(Tel)ab gcd FacFbd -1 (F, F) gab 7 =
-
4
0
=
41rT,,l (E2, E2)
=
-F(E2, U)2 -
=
-
+
(-F(E3, U)2
-F(E2, U)2
+
(cf.
=
Tal(E27 E3)
Lemma
7.1.1).
get
we
4,7rTel (E3, E3)
F(E21 Q)2 +
+
F(E3) Q)2
F(E2, Q)2
+
F(E27 E3)2 + F (E3,
E2) 2)
F(E31 U)2
-
F(E3, Q)2
0.
7.4 The initial value
for
problem
spherically symmetric spacetimes
339
and 0
F (E2
47rTq (E2, E3)
=
Multiplying
the first
equation
obtain
we
0
U) F (E3, U)
equation with F(E3,
-F(E27 Q)2F(E3, Q)2
=
(F(E3, U)2
+
(F(E3, U)2
-
-
+
+ F (E2,
U)2
and
Q) F (E3 Q) i
inserting the second
F(E2, Q)2F(E3, U)2
F(E3, Q)2)F(E3, U)2
F(E31Q)2)(F(E3i U)2
+
F(E2' Q)2).
2 0. InF (E2, Q) Q) 4 0 we have F (E3, U) F (E2, U) 2 0 gives then Tel (E3, E3) serting this into Tel (E2, E2) 0. In par0 which in turn implies F(E2, U) F(E3, Q) F(E3, Q)2 0 which contradicts the F(E3, Q) ticular, we have shown F(E3, U) F(E3, Q)2 7 0. assumption F(E3, U)2 0 we get F(E3 U) If F (E3, U)2 qF(E3) Q) where F(E31 Q)2 0 implies that the the first In case Tel E (E2, E3) equation 1, 11. f ,q F (E3, Q) 0. In the second case F (E2, U) 77F (E2 Q) or F (E3, U) 47rTel (E2, E2) we obtain the same conclusion from the equation 0 47rTel (E3, E3). 0 imply now The equations Tel (Q, E2) Tel (Q, E3)
If F (E3,
U)2
F (E3,
-
=
=
=
-
-
=
=
_
=
=
=
-
=
=
-
i
i
=
-
=
=
=
i
=
=
=
0
=
-
F (Q,
U) (77F (E2 Q)) 7
=,qF(U, Q)F(E21 Q) 0
=
=
This is
a
-
-
+
(- F (E3 Q)) F (E21 E3) i
7
F(E2, E3)F(E3, Q)
U) (,qF (E3, Q)) + (- F (E2, Q)) F (E3, E2) 77F(U, Q)F(E3, Q) + F(E2, E3)F(E2, Q) -
F (Q 7
F(E2, Q), F(E3, Q). Since the F(U, Q)2 + F(E21 E3 )2 we have 0-11 We F(E21 E3) F(U, Q)
linear system of equations for
determinant of the associated matrix is either
F(E2, Q)
F(E3, Q)
=
possible
have therefore two such that either F F
=
AU5
A
Qb
In the first
+
=
=
or
let
x
F(,,_,( ),,y)(-U + 77Q, -)
GZ
=
A X
=
A
dO
=
A, b: M + f3r sin(O) (0 +,qQ ) A d p
There
Ar(W +,qQ )
i3-r 2 sin(O)dO
case
0
cases.
are
functions
--
R or
dW. S2 Then .
2r(x)(A(7rz(x), y)dO + f3(ir-r(x), y) sin(O)d p)
sphere of symmetry Sx := 17r_T(X)l X S2. Since non-vanishing vector fields, this 1-form must 0 and therevanish at some point yo of S.,. This implies 0. Since T,,l is spherically symmetric we obtain fore (Tel)(,,(x),yO) defines
a
S2 does
1-form
on
the
not admit any
=
=
The tion.
equations Tel (U, E2)
=
Tel (U, E3)
=
0 do not
give
any
more
informa-
340
Spherical symmetry
7.
(Tel)(,, (,),y) have Tel
0 for all
=
In the second +
we
obtain dF
+
=-
div(FO)
Q
A
d(O
Q )
A
x we
finally
field is given by F= AU5A 0 and d(sin(O)dO A dW) 0
=
apAdW A 0
+
at (r2 f3) sin Odt
dO A
A
dW
=
Qb
A
'9q (,r2 j3-) sin Odq
+
dO A
A
dW.
0 is satisfied if and only if A depends only on t and r2f3 depends only on 0 and W. Since div(U) U * A + 2U In r, Q v + 2Q * In r, and [U, Q] (Q v)U (U e A)Q we get
It follows that dF
div(Q)
S., By arbitrariness of
E
electromagnetic
Since
d o-
A
aoAdO A 0
=
q whereas
the
case
j-3r2 sin(O)dO
Q5
(7rE (x), y)
0.
=
--
e
=
e
-dA(u)Q
=
+
Q
-
U)
(&
dA(Q)u
r2 sin 2(o)
A(div(U)Q
-
' sin(O)
2
r
(ao),9v
00 0,9W
+
3
* d
-
B
div(-A(U o Q
=
1P
=
(
d
-
-
aw
-
div(Q)U
+
0
ao)
[U, Q])
3
r
2
' sin(O)
(aw) ao
B * r
2
(div(ao)a,
sin(O)
A(Q*InA+ Q +A( U*InA
=
*
-
+
(
(.90
*
in(Ar2)
div(F)
=
d o.
(Ub r4
0 is
ln(A r2)
equivalent
aWj3-
0,
=
+
"V
Ub
(D
1
2e2
4
r4
_
=
0,
(D
Qb)
=
r
b2 +
r2
(- 2b2) (-Ub (e2 b2) (_Ub b2) (d02 74 ( r4
I
0
+
r4
2
I
e2
Ub
Ub
+
2
r2
+
2
&
(d02
2
+
sin2
+
Qb
+
sin2
Qb
(9
0
Q )
Od W2).
C90
r2 sin (0)
0.
(7.4.15)
0
A
sin(O)
are
Qb
(
to
r
+
+
A)Q
_
e2
47rT
2U*lnr + Ue
-
constants e, b E R with F 2 2e2 2b and We calculate (F, F) = +
It follows that there b sin(O)dO A
*
1,,)U
e
b
sin(O)
Q
Q
-
)av sin(O) 2 Sin(O)
+
0,
=
U*A
B r2
Hence the equation
u
2Q olnr
v+
-
div(,9V)ao)
-
=
-,T
Q
+
get
Od(P2)
Qb
+
2 r
(d02
+
Sin2 -Od W2)
341
spherically symmetric spacetimes
for
problem
7.4 The initial value
1
Observe that
assuming dF
A where A is
=
1-form would have lead to b
spherically independent non-vanishing vector
some
0 since there is
=
no
field tangent to S2. Lemma 7.1.2- Let T
sin20 d 02)
+
-(Qor)
Q
r
-
(U
*
Q
r)
9
Prad
62
-
9
Prad
2
(7.4.16)
7
(U
e
r) (Q
9
r)
Prad
-
Q
0
where
As
m
47rr 2(Q
r)(6
+
Psph
+ Prad
e
r
(7.4.17) (7.4.18)
Psph) r
r
+ b
e
+
+
(7.4.19)
87rr4
87r
2(1+(Uor )2 -(Qor )2).
m
consequence,
a
0
2(E
-
E
r
A *
r2
V
+
27rr 3
817
U
Q
M
Psph
+ Prad
E
+
to
-
+Prad)
U
Psphr2 (t, q) (d02
+
equivalent
Prad r
+ Prad
E
Q
0
are
2
A
(Prad
47rr
-
Q
Pra&
+
+ Prad
6
e
0
0
T,,I. Then Einstein's equations
U*Uor=
U
60
=-
of Einstein's equation
equations of motion,
the
U
r
U9IE=-(IE+Prad)U*/\-2(E+Psph)
Q*Prad=-(C+Prad)Qov-2(Prad-Psph)Q*r r
hold.
Proof.
The energy momentum tensor is
e2 T
+
2 + b
87rr4
)0
(Psph +
0
+ b
e
+
0
r
8,7rr4
given by e2 +
+
(Prad
2(d02
-
+ sin
b2W
87rr4 2
0
d(P2)
Q
.
Writing e2 + b 2 C
We
can
+
87rr4
apply
e2 + b2
e2+ b2 I
Prad
=
Prad
81rr4
Ps-ph
Lemma 7.1.2 with E,PradiPsph equations of motions (7.1.7),
=
Psph +
87rr4
replaced by E,Prad7Psph-
(7.1.8) are a consequence of they system of differential equations (7.1.3)-(7.1.6). Equation (7.1.6) is the only equation in the system (7.1.3)-(7.1.6) which involves the function Psph- Since this equation as well as Equation (7.1.7) can be Observe that the
342
7.
Spherical symmetry
solved for
Ps~ph we can replace Equation (7.1.6) by Equation (7.1.7) in our 1,,, i- ovr a, cc)ijsequence of Equations system. Using Equa ion (7.1,S) i
(7.1.3)-(7.1.5), (7.1.7)), the definition of m, (Q v)U (U A)Q imply [U, Q] 4,
=
U*Uor=
and the commutator relation
*
-
Qeprad +2(prad -P-h)(Qor)/r SP
-(Qor)
Tn
_2
F+Prad A -
41rr(Prad
-
817
Q
-(Q
0
Prad
2
(Q
0
F+Prad
r
)2 p-rad
-
Tn
PsPh
r2
F+Prad
r
A -
U
9
Q
9
r
47rr(Prad U
=
Q
=
-(Us r)
s
-(U
e
*
-
+
r
81T
(Q
v)(U
*
r)
a
(U
-
9
Prad
2
(U r) (Q
E+prad
proof
-
Psph
+Prad
of Lemma 7.1.3 that
=
we can
L
+ Prad 7
=
E
=
Q
0
Q
9
e2
Prad
Prad
r
replace Equation (7.1.3) by Q m 47rr'(Q * r) follow now from Equations (7.4.16)-(7.4.19) 9
0
r)
___
It follows from the calculation in the
Q
s
Qeprad +2(prad -P-h)(Q*r)/r sp E+prad
r)9
F + Prad
A)(Q
9
Prad
Q
-
+
( b2) 8,rr4
e2 +b 2 =
Prad + 4
e2 Prad
-
Psph
=
Prad
-
Psph
(Q
81rr5
-
+
r)
V
2
87rr4
e2 +b 2
47rrp~ rad
Finally
(7.1.7)
observe that
and
(7.1.8)
as a
hold
=
47rrPrad
27rr3
consequence of these
unchanged,
even
equations the Equations
if e2 + V
=A
0.
The system of differential
Equations (7.4.16)-(7.4.19) is singular at r b 0 this singularity is only a cocase e ordinate singularity, provided, the initial data can be smoothly extended to r < 0 as symmetric functions. A proof, however, would require the solution of a mixed initial-value-boundary-value problem (cf. (Courant and Hilbert 1962; MiRler zurn Hagen, Yodzis, and Seifert 1974)). If e2+b2 7 0 then there is physical singularity at r 0. This follows immediately from 0. It
can
be shown that in the
=
the fact that the invariant function g
2
=
(T, T)
blows up
as r
-4
0. In the
7.4 The initial value
following
theorem
we
problem
for
spherically symmetric spacetimes
will solve the initial value
problem
in
a
343
region
which does not contain the centre of symmetry r 0. The initial value problem is not yet "well posed". In order to transform the system of differential equations into a quasi-linear hyperbolic =
U 9 r and I' system of equations we write y Q equations two our system of equations we obtain =
U
e r
0
Prad
F
P
(Prad
r
A
Adding
these two
87r
-
Tn
Psph
+ Prad
e2 +b
r
2
2
,
(7.4.21)
+
27rr3
-
Q
(E
E
)
-
0
Prad
2yi
+ Prad
r
-Y
0 6
Prad
-2-
+ Prad
47rr
E
U
0
y
-
0
r.
(7.4.20) C
U
*
y,
Q
U
=
+
Prad)
Q
Prad IE
y -
-
Psph
(7.4.22)
+ Prad
2 (E +
Psph)
Y
(7.4.23)
r
and
Q
r
Q*Tn
(7.4.24)
=
=
47rr 2j,
(E
A +
+
8?r
,e2 + b
2
(7.4.25)
81rr4
The first 4 equations constitute a quasi-linear hyperbolic system of equations for r, y, r, E. It will turn out below that last 2 equations will hold everywhere if they hold initially. This leaves us with two undetermined
functions, Prad, and Psph- In order to arrive at a well posed system of equations we could either augment the system with two more differential equations which relate the pressures Prad Psph to our remaining quantities or we could impose functional relationships. We will opt for the latter possibility and assume Prad PsPh since in this case we can solve Equations (7.1.7) (7.1.8) explicitly which greatly simplifies the problem. Furthermore, we will assume a functional relationship Prad P(C) which describes the physical properties of our fluid. This equation is referred to.as an equation of state. 7
=
7
=
Theorem 7.4.1. Let M
=
R
q, W, A E R, and let p: R smooth function.
t,
0,
x
R+
-*
\ 101
R be
a
S2,
coordinised
by functions given monotonically increasing x
For any e, b E R and any smooth functions : R+ --- R, e: R+ -+ R+ e(q) + p o (q) > 0 for all q E R+ there is a neighbourhood U the hypersurface Ito I x R+ X S2 and a unique Lorentz metric g on U
such that
of
such that
(i)
g satisfies the spherically symmetric Einstein equation with cosmological constant A for a perfect fluid with equation of state p(0, a source-free electromagnetic field with parameters e, b;
344
7.
Spherical symmetry
Oi) Elt=t" (iii) rlt=t, to prove that for all ri
Proof. It is sufficient neighbourhood Urir2 defined In
on
a
h(q)A,
Uri,r2
such that
(rj,r2)
x
>
S2 and
X
0, r2 > r, there is a unique solution
a
g
*
first step
v
ftol
of
will show that
we
-
with
(, )
reparameterisation A has a unique solution
up to
-
the initial value
H(q)v (r,,E) coincides
--+
problem
at t
to. To this end
=
we
introduce
U * r and augment Q * r and y dependent variables F the system of equations by the definition of y. This gives the system for of Equations (7.4.20)-(7.4.25). The additional initial values F and from "constraint the calculated are initial, equations" F, y necessary two
new
=
r
(I
2
2
+
P In the first step
_
=
we
P2)
=
41r
=
Ir
le 2 +b
3
r
&2 df
A
+
2
6
0
2
-
Q- .
will show that there is
a
unique solution
In the second step
to the initial
will show that
problem (7-4.20)-(7.4.23). Equations (7.4.24),(7.4.25). Equations ordinary differential equations (7.1.7)), (7.1.8) are consequences of Equations (7.4.20)-(7.4.23) and can be solved independently. In order to simplify the formulas, we define the "baryon number density" n by
value
we
this solution also satisfies
de
E
P(,E)
+
=
*Tn and the dj
E
Jo
F+P(z)
n
asymptotic behaviour c(n)/n ln(n) which in turn implies
-4
(,E
1
0).
It follows that
U U E
+
In(n)
*
P(O
and n
Q
9
In E
n
PW
+
Q n
E
+
P(E)
Since, Equations
+
dp dE
P(E) are
U 9
In(n)
E
equivalent
+
P(E)
to
0 E =
=
E
Q -p(E)
1)
1+
(7.1.7), (7.1.8) U
n2
n
QOE E
(E+P(E)) !dEn Q
Q.E+,Q.,s dc
n
+ P
-
U
e
A
-
U
o
In(r 2),
problem
for
+
Q
7.4 The initial. v6l--- e
Q
*
ln
(6 P)
we
P
-Q
-
C
n
345
spherically symmetric spacetimes
V,
+P
get C
h(q)r'n h,
where
(7.4.23)
H is
are
+P
H(t)n'
constants of integration.
equivalent
r
and e-v
(7.4.20)-
The system of equations
to
0
0
0
0
0
0
0
FA
Y'
0
0
0
yA
F,
0
B
0
0
C/ Hny e2 +b 2 M -rT + -T"7ra-
Hn
-+P
-
rb 2
4,rr(p
A 7wr
0
-2Hn2 r
Ota, a'
where et
Oqa,
:=
and
dp
(E+P )2 nr
A
=
hH
B=hH
(nr)2 r
dE
The matrix
has left
eigenvalues
=
12
(1, 0, 0, 0),
0
0
0
0
o
o
rA
0
0
0
yA
O
B
0
0
and left
ai
al/2
11
0
=
=
eigenvectors 1i gi ven by
O
a3/4
(07 Yi -ri 0)
==
1
IV-ABr,' 13/4
=
(0) 1,0,
lAr,
V
B
)
The left eigenvectors: I I 1 12 1 13 , and 14 are linearly independent unless r 0. Hence our system of differential equations 0 or 0, E + p(,E) de admits a unique, local solution in U,,,,, (cf. Theorem is hyperbolic and =
d2,
=
7.3.1). By
the
=
uniqueness of the solution
we can
all of R+ and
choose
a
collection of
the solutions with
patch together. This gives a unique solution in a neighbourhood of the entire hypersurface Itol x R+ X S2. We will now show that this solution also satisfies Equations (7.4.24) and (7.4.25) everywhere. We have chosen our initial data F, y so that they hold at t to. From
such intervals
respect
(rl, r2)
which
to these intervals
-
cover
346
7.
Uo
Spherical symmetry
(.V-Qor)
=
UoF-Qoy- [U,Q] + -U C + P
+P
(U
+
A)(Q
o
or
"
2--L
-
(U
+
o
A)r
QoP
-Y
+Qou
A(Q
F)
+ U
we
obtain
F- Q or
a
V)Y
0
U
o
o
A)-V
0 E
(E+P A(Q
9
+ U
r
-
o
A + 2
Y)
r)
linear differential equation for the function (r Q 0 r). Since initially this equation therefore holds everywhere. The other -
0
=
constraint
that
9
+r
(U
-
U
(E+P
Q
--I
r
equation (7.4.25)
is
slightly
more
complicated. QP U +
Equations (7.1.7), (7.1.8) imply [U, Q]
6+P
Observe first
(
U"
e+p
+
It) Q. r
The proof of Lemma 7.1.3 shows that the equation U * m 41rr 2F(C + ,2 +b2 is of a consequence Equations (7.4.20)-(7.4.23). We have --g ?r=r2 ) =
'r
U
9
Q*
47r
J(
A C+ 87r
0
)
f2
U
47r
9
E
+ 87r
Q
op
+
(U
41
M) A
=
) r2r+
8,7rry.P
+ 47rr
+ 47rr
Q
E
(E+
2
2
+
T7r
0P
b2F
Q0UoM
2r 2
2y) (Q
0
M)
r
YI,
r2 +
) (U r) 2Y (P 0
87rr4
87r
47rQ*
-
A
e2 -
-
+ -
b2))
jrr4
87r
2 e
2 + b
-
r3
87r
YF
4
+41rr
2
(E+
A + 87r
-e2 +
b2) (-YQOP)
87rr4
E
87rr4
87r
5
+
+ b
e
r
A
(41rr 2Y (P e
M)
e2 + b 2
A
87rryr
2+
+
) +
e
E+P
2
=
+ b r
0 C
+
( U *,E)r
6+P
U
(
-
+P
C
e'
-
2
A =
I
df
+P
YI
Q +
F
(Q
M)
7.4 The initial value
for
problem
3
4.7rr2
(,E
2
Q
P)
+
347
spherically symmetric spacetimes
Y-2(c + p)YF r
2 22
A
87rrFy
p
+b
e
-
8,rr4
87r
3
+47rr2 (Q
A-e,2+b2
(P
Y)
,
ftH
87r 5
4
+4,7rr
11
e2+ b2
2"
y(Q A +2yr
r3
4
QOP E
+P
3
e2 b2 A+
(47rr (P 2Y
Q-Y +
87rr4
87r
F
(Q
Tn)
2
A
8,7rrry
+
c:
_
87r
(E +P)
+P
-
87r
3
+(Q
Y)
-
47rr
2E + P F 3
+47rr 2(Q
A-e2+ b2+Q
(P
y)
87rr4
87r
*Tn
F
4
+47rr 2Y(Q
-
+A+
A
e+
b
87rr4
87r
+P
E
5
4
+1 E
QOY r
We get
a
(
(P
-47rr
2
-
A 81r
A-e2+b2+ e2+b 87rr4
87r
+ 87rr4
yr(-l
-
1
+2)
+
8 7r
e2 +b2
0
(6+ A+ -e2+ b2) Q.,M).
linear differential
and 4,7rr 2r(C + satisfied.
+P
2
87rr4
equation for
Q
9
m
47rr 2r(C +
holds
A 87r
everywhere
e2+b2
+ 8.7rr4
since it is
Q
,M
initially
348
Spherical symmetry
7.
Observe that the monotonicity condition
of differential
hyperbolic system
a
p is necessary to obtain
on
equations. The function
physically interpreted as the velocity of sound. physically well justified assumption.
Vie
can
be
This indicates that -P > 0 de
is
The characteristic directions
7.4.1.
Corollary
of the Einstein equation
are
U, z=U+ It
Proof AB.F ev
d2
follows that for =
> 1
FLdpcPQ,
information
h2H2(nr)4/(6+p)2 de2
V dir: Q). The second dp
U
= U-
can
=
FLdPEPL
travel
faster than light.
e-2ve2A 2 implies dE
assertion follows from
0tv1'A__BFaq
Corollary
7.3.1. 1
Remark 7.4.1. The system of differential equations is especially simple 0. Then it reduces to the following system of case of dust: p(E)
in the
=
ordinary
differential equations. U
0 r
Y, 2 + Y 2- F
U
62 +
2r3
t. The
depend
decouples and
be calculated from U
7.5 Static
on
o E
6
energy
Q9*Y -
density
6
2c Y. r
perfect
Most stars do not
fluid stars
change
therefore reasonable to
very much
assume
tions should be
an even
better
all of their fuel. In this section
over
long
time spans. It appears
that their interior
static, spherically symmetric solution
ity has
2
equation for the
where F does not can
rA
+ 2r
to Einstein's
description
once
be described
by
equation. Static solu-
these stars have burned
will show that the
we
can
assumption of static-
an absolute upper bound for the symmetric star. Further, this bound is so
consequence: there is
striking static, spherically small that it is exceeded by a multitude of known stars, which indicates that many of these stars will collapse into singularities once their fuel is mass
a
of
a
exhausted. ,
In this section
we
will model
a
non-rotating
star
by
a
spherically
sym-
metric, perfect fluid spacetime. Under the assumption of staticity, Einstein's equation for a perfect fluid reduces to the following ordinary static
differential equation.
7.5 Static
be
(M, g)
Theorem 7.5.1. Let
perfect fluid
stars
349
spherically symmetric 4-dimensional
a
spacetime which is C' and piecewise smooth, and assume that there exists a timelike Killing vector field U such that the energy momentum tensor
(,E +p) 0 0 U5 +p g, where E, p are given, smooth function. given by T 0 then p has a well defined centre of symmetry r solution the If the Tolman-Oppenheimer-Volkoff equation satisfies is
=
=
dp
m
(p +
dr where
m(r)
=
4,7r
87r
-
f''or
Conversely, let
A
3 (r) + 47rr (P r(r 2m(r))
+
c:
R+
2 &.
911
R+,
-+
R+
p:
-->
R+ given continuous func-
tions such that
(i) ( and p can be extended to R- as smooth, even functions, (ii) E and p vanish for r > ro, (iii) E(r) + p(r) > 0 for r < ro, (iv) E and p are smooth for r < ro, (v) E and p satisfy the Tolman- Oppenheimer- Volkoff equation, :
m(r)
41r
=
A
jr (,,(,)
+
7r
0
)
where
2
a unique 4-dimensional, spherically symmetric LorentzC' -manifold (M, g) which is piecewise smooth and satisfies
Then there exists ian
(a) (b)
Ric
There is a
-
U is an
a
-!Scalg 2
8-x(E
=
timelike
mass
mo +
o
r)U'
vector
field.
+p
r
mo > 0 such that
spherically symmetric
and
o
Killing
0
U5
(p o r) g,
+
for r > ro this spacetime is isometric to spacetime with cosmological constant A
vacuum
: 6-3'
Proof. Equations (7.1.8) and (7.1.5) imply
(Q
-
r)
Q-P +P
E
Since
by
(
definition of m,
Q
*
r
Oppenheimer-Vo,lkoff equation For the
converse we
Volkoff equation
==
A
m
_2 +
F F_ 2M, 1
I
-
follows from
81r
the
d2
10m
Q
9 r
validity
1Q r
r
Tolman-Oppenheimer-
exist functions
0
(
is
r(I
v(r), A(r)
trivially
(Q 2
of the Tolman-
!QR:2 Qer'
=
r
isfy Equations (7.1.3)-(7.1.6). Observe first that Equation (7.1.4) (7.1.3) is equivalent to
ar
-
have to check that the
implies that there
r
47rr(p
0
r)2
which sat-
satisfied. Equation
7.
350
which
A(t)
Spherical symmetry
implies m(r)
is
L(I
=
constant of
a
(Q sr)')
-
-P
A(t)
e-
2
n. Fro-in nz f in-Itc-gratio 9) g. .1
1
") + A (t),
p-,, et A
-,,m
(t)
where
=
0 and
therefore e
2m
2A(r)
r
)_
(7.5.26)
The 4-dimensional solution should be smooth at tation
by
is
7r
an
isometry
under the transformation
being
even
functions of
r
r
-->
But this is
-r.
0. Since
equivalent
We also need that A satisfies
r.
=
a ro-
it is clear that the metric must be invariant
A(O)
to A and
V
0 because
=
for any other value we would get a conical singularity. In fact, consider a centred sphere with area A(r) and (geodesic) radius R(r). In the limit 0
-->
r
obtain
we
47rr2
A(r) W2 (?-)
lim R-0
3
lim
=
47r
2
3
2(Q 9,r)-2 jr=0
4-Tr -
-e
=
(Q
r-O
&)2
2A (0)
3
which reduces to the Euclidean relation in the tangent space at the centre of symmetry if and only if A(O) 0. 12 ==
To
see
of g to
r
=
2 x
=
that
r cos
our
are
0, x3 2
=
e
=:
r
e2A
'dt2 +
Observe first that there =
xi
extendibility
=
r cos
0
cos
sin 0. Then the metric is of the form 3 -
1
are
3
7-
3
Ei= I (X i)2
(XI, X2, X3)
sufficient for the smooth
0 consider the coordinate transformation
0 sin
g
conditions
i x
xjdx'dxj
E(dx
+
i,j=l
)2
i=1
smooth functions
Fj,
in
a
neighbourhood
of
0 with
3
F,(XI, X2, X3)
==
V(
3
E(Xi)2),
(Xl' x2,x3)
E(Xi)2)
if and
only if v and A are even functions. Assume now that A is even. 2A I is a series in the variable y3 Taylor series of e _i=l (x T. The equation A(O) 0 implies therefore that the quotient
Then the
_
=
E3 (Xi)2 i= I
e2A
I
(Xi) well defined at
(x 1 ,x2,x 3)
12
would get the
If A(O) > 0 can
by
we
=
0 and smooth.
analogue
of the
be visualised in the 2-dimensional
a
circle.
2
case
tip of a 3-dimensional cone. This with the sphere being replaced
7.5 Static
A(O)
We obtain r
that the
integrand
=
0 from
=
Equation (7.5.26)
0. The function A is
order at
Tt
of
m
is
Equations (7.1.5) simplifies
to
c
+
-
2A
e
1
=
_
We
therefore determine
can
that
v
is
(P- ')
47rr
-
Q
vanishes to third
(7.5.26)
and the fact
=0
8,7r
is
m
m
351
stars
even.
r2
2m/r,
-
since
because of
even
M
2A,9' V
e
which, using
81r
perfect fluid
+
equivalent
A
47rr3
(p
r(r
2m)
up to
-
to
-
87r
constant of
a
integration. Observe
since the function
even
A
m(r) + 47rr 3 P r(r 2m(r))
87r
-
integrated is uneven. The Tolman-Oppenheimer-Volkov equation implies now the equation of motion (7.1.8). Since this equation is independent from Equations (7.1.3)-(7.1.5) but Equation (7.1.8) is a consequence of Equations (7.1.3)-(7.1.6) we can derive Equation (7.1.6) from the system of Equations (7.1.3)-(7.1.5), (7.1.8). Let to be
=
we
extend A
-
6 2mo
using e2A(r) (1 It is C1 if and only if
beyond
then A is continuous.
M(r)
ro,r
r
If
Ar03
lim
,
MO
ro
=
_
_
Since
-+
Equations (7.1.3)-(7.1.6)
7.1.3 and infer that A is
which is for
v
r
ro,
equivalent
such that
we
to
<
r0Q
9
m(r)
E(ro)
=
=
ArO
2")
have e2v (ro) I
2mo _
r
The function
v
is
-
_.Ar2 for 3
r
9r(e 2v)
r-ro,r
we
lim
can
47rr02(6_(ro)
3
ar(e 2v)
=
+
-A-) 87r
> ro
2(e 2v(ro))
2Aro
2mo =
2
Jim r-ro,r
2
integration
and extend the function
It follows that
T 0)
A"02 =
2
3
We calculate
r-ro,r
to Lemma
> ro
2
C1 if and only if lim
appeal
r
-
0. We choose the constant of ro
such that e 2v(r)
=
satisfied
are
C' if and only if
for
Ar2
'
lim r
Ar2)-1 3
r
0rV
1/
v
is continuous.
7
352
Spherk,. ',
m(ro)
2(1
-
ro
+
ro
(
ro
-
81r
-
22 (m(ro)
22
47rro3 (p(ro) ro(ro 2m(ro)) +
4-7rro3(p(ro)
Ar03 mo +
+ 6
A 87T
4,7rro 3p(ro)
Ar03 -
2
v is C' if and only if p(ro) 0. Uniqueness follows from Theorem 7.4.1.
Hence
=
Observe that we have used all our freedom in order to construct a spherically symmetric C' spacetime with exterior vacuum solution. In general, it is impossible to achieve higher differentiability. Even if staticity is not assumed, a smooth matching of the interior solution to the vacuum spacetime would imply properties of the boundary which are so strong that they could be used in order to calculate the collapse of the star explicitly (Kriele 1995). Independently of the pressure, there is an upper mass limit for spherically symmetric stars. Theorem 7.5.2. Let with
fluid spacetime dE
Tr
< -
(M, g) be a spherically symmetric static, perfect positive, radially decreasing energy density (e ! 0,
0).
(i)
Let r, > 0. Then the
the A
inequality m(r,) > 0
'97r
then the
more
-
e, > 0.
equation of state p: c i--4p(E) and let an m, E R which only depends on the equation low for ene7gy densities, p: [0, F-,] --+ R such that m(r) :5 m, E
R+.
equivalent to Q 9 Q * v + (Q 9 v)' (together with Equation (7.1.8)) implies
Proof. Equation (7-1.6)
p)
which
e'Q
Q
an
Then there is
of state for all r
47r(c
-
Assume that there is
(ii)
+
associated with the radius r, satisfies 5r,/9. If, in addition, we assume that p stringent inequality m(r,) :5 4r,/9 holds. mass
<
9
is
e'
v
(-I (-3m r
e
r4
where for the last 0
we
equality
we
2m +
r3 47r +
+ r
have used
have r
m(r)
=
47r
10 (E( )
A +
87r
)
41r(E
2&
) A))
+
p)
=
-2m/r3 +
I
r2
FQ
87r
Equation (7.1.5). Since de/dr
<
perfect fluid
7.5 Static
A
c(r)
> 47r
Hence
Q
(.Ie'Q v)
8.,
and
< 0
e
e
for
+
*
=
E(r)
47r
+
87r
) r3/3.
integration yields
an
r
1ev(r) (Q
A
2&
353
stars
V) Ir >e'(rr-) (Q
*
V) I
r
for
r
Q
From
< r,
o
v
'9Q ar
=
obtain
o r we
rev(rc:)
Oev >
wr
-
>ev('-) (Q
e'(0)
0
0
We re-express Q 9 r in terms of m using the resulting equation. This gives
ev(r')
(Q
-
TA r)
0
V),
m(I 2
_
(Q r) 2)
and
0
integrate
r,;
Or,
f V-1--L rM(1)
(7.5.27)
dr.
2m(r)
0
r
In order to estimate this
for all
(0, r,). Comparing
E
r
integral
m(r)
show first that
we
!
m(rc')
the derivative of both functions
we
Tr
obtain
that the function
d
f(r)
(m(r)
jr-
r2(4,7re(r) satisfies df (r)
=
dr
with
(0, r,) m(O)
the existence of
47rr2 dc(r)
r
G
(ri, r,)
implies
e'(0)
<
=
f (r) <-0
an
r2
rl E
obtain
m(rc)
our
Inserting this
we
ev(r')
3
T(r,)r2 rc3
<
with
m(r,)
m(r)
m(r,)
(Q
0
V)r(:,
0
Or,
f (ri)
(_r_)
<
Since
for all
r
M
(E
r.,
an
for all we
fo r,rr) VF,-- m-r-
rl
E
r
( , I
2m(r,)
-
f (r)
2)3 < 0
(r2j rc)
in
r2 cannot exist
E
(0, r,)
obtain
dr
2m(r)
r,3
(rc)
(r2)
< 0.
m
Hence such
-
r,
an
Because of
(ri, rc).
E
Inequality (7.5.27)
_
(Q
r
with
(0, rc)
E
Thus if there is
for all
)3.
rc
<
-
r
(0, r2)
m(r)
estimate into
I
if (r).
m(rc)
estimate
ev(rc:)
<
dr
an
47rE(r)r2
3m(r,)/rc3)
0 then
f (r,
contradiction to which
+
r,3 -
0 the existence of
=
implies for all
if (r)
)
-
LM-(r -
r
354
7.
Spherical symmetry
and therefore
0
M
K)
(using Equation (7.1.5))
Fl-_Er-c M(,
v
e
(0)
-
-
(r.)
r
M:,Fr.
m(rc) rc
r,
(l_F2)r C
2
m(r,)
C
2
2
(I
+
1,_2)
-
A)
47rrc
P 8-7r
+ 47rr C3
C
(P
rc),
-
87r
(7.5.28) where order
Arl
have set Fc
we
polynomial
r,. The
in
The
zeros
3( 2 1 It follows that the
m(rc)
0,
c
m(r,)
Since
-
9
M(rc)
<
is
67r(rc )2
positive
2r, (1
-
9
2r, ( 2
-
we
(pP
1
_
a
third
7r. 7
Inequality (7.5.28)
(P A)
side is
are
1 +
87r
67r(rc)2
vanishes for
(P_ A))
.
87r
obtain the mass-bound
67r (r,)
x2
right hand
polynomial
67r(r )22
hand side of
right
2rc (1
+
of this
2
A
(P
8-7r
)
+
I +
6,7r(rc)2
P
-
A))
81r
+X
9
where
2
V 1 + 67r (,rc) (pA
x
Since the function
87r
its minimum at
x
2
_
X2 + x has
1/2
it follows that m(rc) : 5rc/9. If we impose in A- > 0 then we addition the energy condition p get mc < 4r,/9. This 87r proves the first assertion. x
=
-
For the second assertion observe that since
m(rc)
>
43ir(rc)3 (6(rc)
+
87r
is
radially decreasing. This inequality and the upper bound for m(r,) restrict all possible values (r, m(rc)) to a compact subset C C R2 which depends only on E(rc),p(rc), and A. We can now solve Einstein's equations for r > rc using the known equation of state. Clearly, m(r) depends continuously on the data r, rn(rc). This implies that mc c
sup f m (r)
:
(r, m (rc))
E
C I is finite.
Theorem 7.5.2
gives
larities in
universe. In Sect. 7.2
our
an
important indication for the we
have
seen
existence of
that all
singunon-trivial,
7.5 Static
maximally extended, with non-constant
r
non-flat
perfect fluid
spherically symmetric
355
stars
vacuum
of Einstein's equation fail to be static in
a
solutions subset of
spacetime and contain a region where curvature diverges. We saw that it is possible to enter this region but impossible to leave it. Moreover, once having entered the region any observer will fall into the singularity
(where
curvature is
is the Schwarzschild
the
vacuum
infinite) mass.
solution also
within the finite time span 7rm where m question arises whether this property of
The
occurs
for real stars which have
energy momentum tensor. Since in Newtonian
central
singularity
in the
vacuum case
which
gravity
can
tempting
artifact of the
vacuum
also have
a
be avoided if the mat-
ter of the star is not assumed to be concentrated in to argue that the
non-vanishing
we
a
single point,
it is
property of the Schwarzschild solution is
an
equation.
Theorem 7.5.2 indicates that this is not the
case.
We have
proved that
upper limit for the concentration of matter in static, spherically symmetric perfect fluid stars, m/r < 5/9, if the energy density of the star decreases outward and is positive. These physical assumptions
there is
are we
an
very weak and seem to be satisfied for all known objects. Moreover, have seen that for any star which is governed by an equation of state
there is
an
absolute
mass
limit. What is more, this
mass
limit
can
be
estimated using only the equation of state for low energies. This means that we get bounds even if we do not know the physical configuration of
extremely dense stellar cores. It has been shown13 that there
are
stars which exceed the
mass
limits
in this section. This indicates that these stars will
collapse into given black holes once they have exhausted their nuclear fuel. In Chap. 9 we will give a very general argument to the same extent which does not rely on spherical symmetry. It should be noted however, that all these arguments in favour of the existence of black holes have loop holes. In this section, we heavily rely on spherical symmetry and the assumption of a perfect fluid. Moreover, it is conceivable that there are non-singular solutions which fail to be static. There are other loop holes in Chap. 9 which
we
will address then.
The argument uses input from book, cf. (Hartle 1978)
physics
which is
beyond
the scope of this
8.
Causality
In this
chapter we link the concept of causality to the conformal strucby the metric and present some elementary causal properties and their interpretation. In Minkowski spacetime, causality is trivial since lightlike geodesics are straight lines. Lemma 3.1.4 shows that the local causal structure of arbitrary Lorentzian manifolds is the same as the causal structure of Minkowski spacetime. All non-trivial aspects of causality are therefore global in character. In this chapter we will also discuss in detail the possibility of "causality violation" due to the global geometry of spacetime. Chapter 8 requires Sect. 3.1 and develops rather specialised mathematical techniques. It contains a number of technical results which are needed in Chap. 9 where the existence of singularities in generic spacetimes is proved. We will restrict to those results which are necessary to prove and interpret these singularity theorems. For a more comprehensive mathematical treatment of causality see (Beem and Ehrlich 1981; Hawking and Ellis 1973; Penrose 1972). For more examples which exhibit the subtleties of causality and singularity theorems see ture induced
(Senovilla 1998).
According we assume
Cx
cone
(at
=
least
from
a
experience no signal is faster than light (photons). As photons move along null geodesics, the integrated light 0, vx future orientedl should expx (Jvx G Tx M : g (vx, vx)
to
our
that
=
locally)
given
determine which events
event
x.
It is therefore
can
plausible
in to
principle be identify the
influenced conformal
structure Q of spacetime with the causal structure of the universe. Postulate 8.0.1 E M
can
objects
can
x
A proper theory of this it is
(Causality
and conformal structure
causally influence y E M if and only if y reach y from x if and only if y E 1+(x).
justification of Postulate 8.0.1 would physical particles and fields. This is
E
coincide).
J+(X).
Material
corresponding beyond the scope of book. While Postulate 8.0.1 will not be important for our theorems, crucial for their physical interpretation. require
a
far
Remark 8. 0. 1. For
arbitrary matter models, Einstein's equation does not cone structure and causality. For inspherically symmetric fluid with equation of state c p(,E)
respect the link between the light stance,
a
M. Kriele: LNPm 59, pp. 357 - 382, 1999 © Springer-Verlag Berlin Heidelberg 1999
i-
8.
358
Causality
satisfying dp/dc > 1 has spacelike characteristics (cf. Corollary 7.4.1). Consequently, the characteristics of the initial value problem associated with Einstein's equation is spacelike whence perturbations of the initial data propagate faster than light. For this reason one usually regards these matter models as unphysical. In fact, to date all classical (i.e.,
non-quantum)
matter models which describe real matter have causal
characteristics. The local causal structure of any Lorentzian manifold is trivial, i.e. the same as in Minkowski spacetime. This follows immediately from Lemma 3.1.4 which is fundamental to this
lary
chapter. The following technical
corol-
will also be useful.
Corollary 8.0.1. Let (M,g) be a Lorentzian manifold neighbourhood of x E C. Let K C C compact and -y be a
and C causal
a convex curve
in
K. Then -y is extensible.
Proof. Let curve
-y
[a, b)
-/:
--*
C be
a
future directed causal
be future extended if
can
liMt-b -y(t)
curve
in K. The
exists. In order to
see
that
this limit exists, let
{'Y(ti)bEN7 fY(Sj)IjEN be convergent sequences with b and x, y be their limit points. For any i limi-oo tj limj,,,, sj there is a j > i with y(tj) E J+(7(sj),C) and for any j there is an i > j with -y(si) (2 J+ (-y(tj), C). Hence we obtain x G J+ (y, C) and y G J+ (x, C). Hence by Lemma 3.1.4 (i) there are two future directed causal vectors v, w with x expx (w). Traversing the expy (v) and y =
=
=
geodesics
t
expy(tv)
i-->
past directed is
a
vector
we see
that at
x
there is also
a
causal
with expx (u) y. Since the exponential map expx of an open set C C TM to C we must have w u.
u
=
diffeomorphsim only possible
But this is
8.1
=
backwards
=
if both vectors vanish.
Causality conditions mathematically construct a spacetime with closed timelike first glance one is tempted to rule out such spacetimes since it seems possible to perform experiments in them which lead to logical contradictions. In this section we will investigate this issue in some detail. We will also define a slightly stronger "causality condition" which will Play an important r6le in subsequent sections.
It is easy to curves.
In
a
At
general
Lorentzian
manifold,
it is
possible
for closed timelike
curves
to exist.
Definition 8. 1. 1. Let
M We say that causality (resp., chronology) only if there exists a closed, non-trivial causal (resp., timelike) curve from x to x. The chronology violating set is given by
is violated at
x
if
and
X G
-
8.1
JX and the
causality violating
Ix
E
M
:
3
a
E
M
set
by
X
:
(=-
Causality conditions
359
1+(X)l
non-trivial causal
curve
-y
from,
x
xJ.
to
A Lorentzian
manifold (M, g) is causal (resp. chronological) if the causality violating (resp., chronology violating set) is empty. If (M, g) is chronological (resp. causal), we sometimes say that the chronology condition (resp. causality condition) holds. set
The term
'causality
closed causal
curves
matical arguments
not
violation' is somewhat is not
contradictory
misleading:
the
itself and there
possibility
are no
of
mathe-
against causality violation.
The idea that there may be closed timelike curves in our universe is new: The concept of cyclic time was a widespread idea in ancient
Greek
philosophy (Kanitscheider 1984, p. 45). These Greek philosophers accepted our fundamental experience of local linearity of time but they compactified the time line to a time circle. Its circumference was identified with the time of
model of
one
revolution of the universe
(according
to their
planetary just according 'arbitrary' laying down'). length of this period is sufficient to explain why nobody of us motion
or
her/his
own
to
The sheer ever
has reentered
We
can
easily
obtain
a
past.
spacetime whose causal
structure is
to the causal structure of this ancient Greek model. Just take
a
analogous horizontal
strip of 2-dimensional Minkowski space and identify the upper and the lower boundary (cf. Figure 8.1 .12) Another very instructive example is .
closed timelike
curve
identify
Fig.
8.1.1. A
dimensional space
strip of twoMinkowski
where
future
boundaries identified.
past
and are
the Lorentzian manifold '
2
Plato, The are
for
instance, chose 10,000 years (Kanitscheider 1984, in this and other figure indicate how both sides
arrows
oriented
p.
55)
to be identified
8.
360
Causality
(R
x
S1, 2dwdt
given in (Misner 1967) (cf. Fig. 8.1.2). given by f (t, W) : t < 01.
first is
+ td
closed timelike
Fig.
8.1.2. Misner's
spacetime
(S'
W2)
The
chronology violating
set
curve
x
R, 2dtdW
+
td(P2)
compactification" arising in these examples is trivial in the examples there is a locally isometric Lorentzian manifold which satisfies the chronology condition. In Lorentzian geometry however, there also exist non-trivial examples where causality violation arises geometrically and not merely topologically. An example which will also be of importance in Chap. 9 is the G,5del solution (G6del 1949).
The "time sense
that in both
Example 8. 1. 1. The G6del solution describes a solutions of Einstein's + Ag Ig equation with dust and postive cosmological constant, Ric 2 8,7reu 0 u where u is a timelike unit vector field and F_ A/(4?r). The metric is given by -
=:=
2 g
=
dt2
-
+
dr2
+
A
(sinh2 (r)
-
sinh4(r))&P2 +
2,v/2 sinh 2 (r)
d odt)
+ dz
2
have r > 0 and identify W with W + 21r. The vector field 9, integral curves and it is spacelike for r < arsinh(l). For r arsinh(l) the integral curves of a. are lightlike (but not null geodesics) and for r > arsinh(l) they are timelike. Since sinh 2(r) is an even function 0 we have only the usual coordinate singularity of r it follows that at r associated with polar coordinates. Hence spacetime has the topology R4 and is in particular simply connected. It follows that chronological where
we
has closed
=
=
violation is
an
inherent property of the solution.
8.1
Causality conditions
361
A
physically interesting solution of Einstein's vacuum equation with vanishing cosmological constant is the Kerr solution. For details cf. (O'Neill
1995), (Wald 1984), (Hawking Despite
and Ellis
the existence of these
1973).
examples
most
physicists regard causalcausality
ity violation as 'unphysical'. The reason for this rejection of violation is the following thought experiment:
Suppose,
you
are
travelling
in
spacetime and reach
a
point
in
past before your departure. Now you decide not to your travel after all and instead to stay home. Contradiction. own
At
glance, the possibility of "free will" seems to be at the centre of However, following (Wheeler and Feynman 1949) Clarke (1977) has re-formulated the thought experiment in terms of a simple machine and has argued that the thought experiment is fallacious: Assume that a
first
the issue.
gun directed at
there is
a
with
shutter
a
which,
if
a
target in spacetime. This target is connected
closed, blocks off the path between the
gun
and the target: If the gun is triggered, the bullet will hit the target which in turn will cause the shutter to fall. A second shot will now be blocked Now
by the
assume
shutter and therefore cannot hit the target (c.f. Fig. 8.1.3). configuration is located in a region with causality
that the
violation such that the shutter falls
along
a
closed timelike
curve so
that
,gers shutter
Fig.
A gedanken experidisprove causality viola-
8.1.3.
ment to
tion
it blocks the bullet to arrive at
a
before
the gun had been
triggered. Again
contradiction: If the shutter is open the bullet
we seem
can
hit the
362
8.
Causality
target. But the target closes the shutter which
in turn blocks the
path
of the bullet.
This
may be resolved
as follows. The angle a, under which by the shutter depends continuously on the shutter's position x at the time the bullet passes the shutter. For simplicity we assume that the shutter will descend with constant velocity v. This velocity is continuously related to the angle a. If the length of the closed causal curve is T we obtain the relation x Tv(a(x)). This equation
paradox
the bullet is deflected
=
has at least In
physical
solution xo which leads to a contradiction-free situation. terms this can be explained as follows: The original contraone
diction is due to the fact that the shutter is
thought to be either up or However, the position of the shutter depends continuously on the parameters of the system. What happens is that while the shutter descends it grazes the bullet and thereby deflects it so that the mechanism works only imperfectly. As a consequence, the shutter is released rather late and not yet in place when the bullet hits it again due to causality violation. Hence it grazes the bullet and we are in a paradox-free time loop. This scenario appears to be highly non-generic but Clarke argues that exactly this is the effect of causality violation: It picks out those non-generic data which are in accordance with the causal anomaly. The gist of the argument rests on the assumption that physical processes are continuous, an assumption which does not hold for quantum mechanical systems. These systems may be in discrete pure states such as spin up or spin down. However, if one tries to set up a quantum thought experiment one is faced with the fact that all predictions are probabilistic which invalidates the whole thought experiment from the outset. There are also arguments against Clarke's resolution of the paradox. Instead of releasing the shutter directly when the target is hit we may have a device which automatically releases the shutter a certain time after the impact. This can be achieved with an electronic switch rather down.
than
a
mechanical connection between target and shutter. It seems now probable that this device always releases the shutter such that
much less
it grazes the bullet when coming down. For Clarke's argument to work the bullet must comes out of the gun so slowly that it just touches the
target but does not really hit it. Otherwise it cannot be explained that the second device is not successful in time which would lead to
a
releasing
the shutter at the pre-set
contradiction.
Whether Clarke's argument is correct
or
not
-
we are
only
able to
conduct local experiments. But causality violation is a global effect, and so the lack of experience cannot give evidence of its absence. Any objec-
against causality violation every-day experience.
tion
of
rests
on
an
(unjustified) extrapolatiom
8.1
Causality conditions
363
There is another point which should be addressed. Causality vioseems to constrain free will. While this is not really a physical
lation
problem, such an effect would have some bearing on philosophical and questions. But an almost trivial observation resolves any possible argument concerning free will at once: If we want to incorporate the notion of free will into a physical description we have to view it at least as a quantum effect (or caused by another yet undiscovered 'mechanism'), but certainly not as something fitting into the framework of classical physics. We only can expect that general relativity is a classical limit of such a theory, It is therefore quite possible that 'free will' is something like a second order effect and that the classical "limit-spacetime" of our world contains closed timelike curves even though we still enjoy free will. With this discussion in mind we should always be very watchful if in order to obtain physical results the seemingly innocent assumption of chronology has to be made. moral
Lemma 8.1.1.
The chronology (resp., causality) violating set consists of connected components of the form I+ (xi) n I- (xi) (resp., J- (xi) n J- (Xi)) 0 1, ). =
-
-
-
Proof. We only show the lemma for chronology violation. The proof for causality violation is completely analogous. Let C be a connected component of the chronology violating set and x E C. Since C is connected there is for each pair of points Jx, yJ C C a (not necessarily causal) curve -/ c C which connects
neighbourhood
of
and y. Since for all z G C the set _T+ (z) is a curve -y is compact, there are finitely many
x
and the
z
zi e -y such that zi+1 E 1+(zi) and the neighborhoods 1+(zi) cover -Y. It follows that there is a timelike curve from x to y. By the same argument
there is
timelike
a
form y to
curve
assertion follows since 1+ (x) n I-
The
following proposition chronological.
Proposition 8.1.1. If of M is non-empty.
Proof.
We
k). a
can cover
(x)
Hence C c I+ (x) n I-
shows that
a
I
with x, E
Ui=1I1+(x,(i)) X, E
a
compact spacetime
M is compact then the
M with
finitely
C
I+
there is
we
1+ (xi)
set
I
an
(i
E
would have
(U JT+ (XI(O)
in contradiction to the definition of xi. If xi
argument
cannot be
chronology violating
many sets of the form
=
i=1
same
and the
1+(xi) f2,..., kJ and 1+ (x,(1)), xi V Ui-=', i+ (x,(i)). This implies
since otherwise
I+ (XO,(1))
(x)
is connected.
If x, is not contained in
permutation
X,
x.
U 1+(X,(i)) i=1
1+ (xi)
to x1 instead of x1. Since there
are
we can apply t he only finitely many
8.
364
one
xi,
Causality
of the xi must be in its own future for otherwise none of the 1+(xi).
we
would have
I
that X.(k) is in
Proposition 8.1.1 is often taken as a reason for dismissing compact spaceas unphysical. While the chronology condition and the causality condition are very intuitive, from a technical point of view, a slightly stronger condition is advantageous:
times
Definition 8.1.2.
neighbourhood
any
any causal
curve
The strong causality condition holds at x E M if for of x there is a neighbourhood U C V of x such that
V
intersects U at most
once.
40Z
(remo,
remove
7-
7
iden
rp.mnvp
ify
8.1.4.
Fig.
A
spacetime which strongly
is causal but fails to be
causal
words, if the strong causality condition does not hold at x, there curves starting at x which come arbitrarily close to x after leaving a giving convex neighbourhood. Hence the chronology condition
In other
timelike
are
is almost violated. In the next section
will
we
see
causality condition. Finally, we wish to introduce global hyperbolicity 3,
the importance of this the strongest
causality
condition which is often assumed. Its relevance stems from the fact that in
a
globally hyperbolic spacetime
given -points properties in the
two
is
compact in
is
the set of causal
natural
topology.
curves
We will
connecting use
related
next section.
Definition 8.1.3. A subset
A is
a
strongly causal
of A
C M is
said to be
globally hyperbolic if
and for any two points x, y e A the set J+ (x) n J-
(y)
compact.
This
name
partial
has been coined
by Leray (1953)
differential equations.
in connection with
systems of
8.2 Cluster and limit
365
curves
a ove
remove
remove
P
identify
remove
Fig. 8.1.5. A spacetime which is strongly causal. An infinitesimally small perturbation of the metric results in a spacetime with chronology violation
8.2 Cluster and limit In this section
based
(Beem
on
practical
It is
to continuous
study
we
sequences
section will be
of this
The results
curves
and Ehrlich
to
generalise
1981). the concept of
every
point
:
y
point on
x
timelike
or
causal
curve
x
7 n C
can
be
-y is called causal
curve
(resp.,
time-
neighbourhood C such that any connected by a causal (resp., timelike) C'
-y has
on
a
convex
which is contained in C.
curve
this definition coincides for Cl-curves with
Clearly,
Lemma 8.2.1. Let
of x,
causal
our
previous Defini-
(iii).
tion 3.1.3
of C
a
curves.
Definition 8.2.1. A continuous
like) if
of causal curves and there limits. fundamental to what follows. It is
x
E
M. There is
constant k >
a
curves
-y in C
can
a convex
0, and coordinates be
coordinate
(xO,
parameterised by
.
.
,
neighbourhood
xn- 1) such that all
xO and the coordinate
t
inequality n
E (,ya(t) 7a(S))2 -
<
kit
sl
a=0
holds
for
all t,
We choose coordinates
Proof.
hood C of
dx' (i
s.
E
x
f 1,
(xO,...'xn-1)
in
a
convex
neighbour-
with compact closure such that dxO is timelike and all n 11) are spacelike. Then any causal curve in C can , -
.
.
.
parameterised by A Let M and p(s) -y(s) (cf. Fig. 8.2.1).
be
be
a
causal
Cl-curve with y(t)
=
-y(t)
Since the closure of C is compact there all causal vectors are also causal with such that constant ko > 0 =
exists
a
respect
to the flat metric
I
-
kodt2 + Eni=1 (dx' )2.
In
particular,
p satisfies
366
8.
Causality
nst
Fig.
8.2.1. The
proof of Lemma
8.2.1
ko
==
En-I(Ai)2.
ko (itO)2
f eo,..., en-11
and write
If
we
11va ea 112
=
denote the standard basis of Rn
VEa-=01 (va)2,
we
by
obtain t
a
(t)ea
-
,a (S) ea 11 2 =11,a M ea <
a
WeaJJ2
-
ft 114a (T)eajj2dr
Corollary 8.2.1. It follows that causal fore differentiable almost everywhere.
=
<
11
1 ita (-r)
V'_1-+ko(t
curves are
d-rea 11 2
-
s).
Lipschitz and
be the space of continuous causal For x, y c M let Co causal (XI y) from x to y and Ct jme (XI y) be the space of timelike curves from
C'. We will
which
are
causal
curves.
now
specify
a
natural
topology
there-
curves x
to y
for the space of
Definition 8.2.2. Let - : [a, b] --+ M, -yi: [a, b] -4 M (i E N) be curves. The sequence f7ijiEN converges to -y in the Co-topology if for every
neighbourhood all i > io. The
V
of -
curve
in M there exists
an
-y is called the limit
io
E N
curve
such that -yj C V
of the
sequence
for
f -Yi jiErq
-
terminology is slightly at odds with the traditional definition of general relativity but closer to generic mathematical terminology. Often, not limits of curves (with respect to a natural topology) but curves which are better thought of as a set of pointwise accumulation points are called "limit curves". We will reserve the term "cluster Our
"limit curve" in
curve" for such accumulation
curves:
Definition 8.2.3. Let -y: [a, b] -y is said to be a cluster curve
-M, of the
-yi:
[a, b]
sequence
M (i E N) be curves. 17ifiEN if there exists a
-4
8.2 Cluster and limit
8.2.2. A limit
Fig. of
a
367
curves
sequence of
curve
curves
-y
-yn
subsequence f -yij jj,2N such that for all x E -y each neighbourhood of x intersects all but finitely many of the curves -yjJ. Following Beem and Ehrlich (1981) we will say that the sequence f7ijbEN distinguishes the cluster
curve
-y.
It will turn out that for curves are
essentially
the
strongly causal spacetimes same
(cf.
from
curve
for
a
An
8.2.3.
Fig.
x
cluster and limit
below. There is
8.2.2)
Theorem
example where
to y is not
sequence of
a
cluster
a
points from
x
no
limit curve
to y
logical relation between limit curves and cluster curves. Consider in cylinder (S' x R, d odt) (cf. Fig. 8.2.3) the sequence of identical const. The curve curves 7,, which connect x with y and satisfy W t(x) and then connects x with y satisfying -y which first traverses t const is a limit curve of -y, but it is not a cluster curve. In Fig. 8.2.4 a
strict
the flat
=
=
=
we a
have
limit
may
a
cluster
even
be
curve
-/ of
a
sequence of causal
general, a cluster curve spacelike (cf. Fig. 8.2.5
curve.
In
of
a
Proposition 8.2.1. If (M, g) is strongly causal and -y is of a sequence f-yjjjEN of causal curves, then -y is causal. Proof.
Since
(M, g)
is
strongly causal,
bourhoods Ci such that
no
causal
we
may
curve can
which is not
curves
sequence of causal
cover
-y
by
a
curves
cluster
convex
enter any of these
curve
neighneigh-
8.
368
causabity
identify
Fig.
A spacetime,
8.2.4.
which is causal but fails to be strongly causal identi
Fig.
8.2.5. Assume that
b1c is
rational and
a/b is ir
rational. Then the
.
c/a
......
Ar
b
R2
from
depicted a-
projecslope
tion of the line with
identify
to the torus
in the
dense
curve
curve
is
a
figure
is
a
7. Hence every
cluster
curve
of
ly
bourhoods twice. Consider
one
such
neighbourhood
and denote it
by
C.
Let x, y be points in -y n C and denote by f 7ij JjEN a subsequence which distinguishes 7. Then there are sequences xj, yj E 7j., n C with xj --> x and yj -+ y. Since yi., E J+ (xi, , C) Lemma 3.1.4 implies the existence of a
causal vector vj with
expx,, (vj)
=
yi.,. These vectors have
an accumu-
lation point v with exp,(v) y. The vector v must be causal since the set of causal vectors is closed. But this implies y G J+ (x, C). If x and y are arbitrary points on -y, we can find finitely many neighbourhoods Ci such that the segment from x to y is covered by Uj Ci. We can now apply the preceding argument finitely often to conclude that x and y are
causally
related.
I
The basic Lemma 8.2.4 below is which
we
an
application
of the theorem of Ascoli
will present first.
Let A C
Rk be
a
compact
CO(A,Rl)
=
ff:
set. The space of continuous functions
A
--
R
k :
f
is
continuousl
8.2 Cluster and limit
then be
can
regarded
normed vector space in
a
as
a
369
curves
natural way. Just
SUP., C A I I f W I I Moreover, this norm is complete, i. e., evCauchy sequence Ifil c C'(A, R1) converges to a function f E ery CO (A, R1) . (f can be constructed pointwise using the completeness of
I I f I lo
set
=
.
R1.) C(A, R1)
Lemma 8.2.2. Let B C > 0
e
every
radius
E
there
and 13 C
are
finitely
Uj(') B'(xj). i=1
Proof If the lemm 'a covering B such that .
6
be
closed set and
a
many balls B
''(xi),...,Bj,()(xj(,))
for
with
Then B is compact.
is not true then there no
that
assume
are
finite subset
open sets
(t
U,
E
I)
B. Let
covers
JBI'(xi),...' B3,(1)(xj(j)) cover B. By our assumption by finitely many U,. (Otherwise we would obtain a finite cover of B by finitely many sets which are in turn finitely covered by sets V,). We denote this ball by B0. Assume that we
be
a
one
finite set of balls of radius 1 which
of these balls cannot be covered
have constructed balls
jBjjj=o,...'k-1
such that
(i) Any two consecutive balls intersect, (ii) Each ball Bi has radius 2', (iii) None of these balls can be covered by finitely There
are
fore also
balls
(xi), B1, ?
.
.
Bk-1. Since Bk-1
must exist at least
one
.
,
many
k(2-k) B-k (Xk(2-k))which 2
cannot be covered
B2"! 2- (XII)which
U,.
cover
B and there
U, there Bk-1 and cannot be by Bk we have induc-
by finitely
many
intersects
k
covered by finitely many U,. Denoting B2"lk(x,) 2tively defined a sequence f BiliENUO of balls which satisfy (i)-(iii). Denote the centres of these balls by yi. For any natural numbers m < n we obtain n
n
JYn
-
YmIl
Y_
<
IlYi
-
Yi-111
fYiIiENUO
is
a
Cauchy
which contains y and is contained in U,,). But this an
to
U,
< 2
-
2-m
i=m+l
i=m+l
Hence
Y_ (2-' + 2- +'))
:5
Denoting
sequence.
a
number
r
E
its limit
by
y there is
N such that the ball
implies B2-1-1 (Yr+l)
C
U,
B2- (Y)
in contradiction
I
(iii). Let A C R k be
a compact subset and fi: A -+ R1 (i E N) be an equi-continuous sequence of continuous functions such that for all a E A the set UEN fi(a) is compact. Then there is a continuous function f : A -4M and a subsequence ffii TiEN Of ffibEN which converqes uniformly to f.
Theorem 8.2.1
(Ascoli).
370
8.
Causality
Proof.
We show first that the subset
space
(CO (A, R'), I I 11,,). By Lemma
any
c
> 0
8.2.2
there is for each E
compact,
we can
c
A
neighbourhood U,,
a
U,, the inequality
kj).
we
a
a
and all y E
compact in the normed
is
only have to show that for finite number of balls with diameter less than E which Let c > 0 and a E A. Since ffiliEN is equi-continuous,
there is
U :Iffjj.
cover
(1
Ui"=o 1 f fi I
-
cover
11fj(a)
A with
-
finitely
of
fj(y)llo,,
such that for all
a
c/4
<
many such
fj
holds. Since A is
neighbourhoods U"',
Since the union k
K
oo
U Uffi(al)l
=
1=1 i=1
is
compact it
be covered
by finitely many open balls of radius E/4. by x, E K (s Cz f 1, rj). We will now construct neighbourhoods of diameter less than c in CO (A, R') which can
Denote their centres
.
finitely
many
all of
cover
U :Iffjj.
These
=
f (a) close
to x,, for all
finite set of all maps
a:
a
f 1,
.
.
,
neighbourhood will be defined by the ref in each such neighbourhood have values near some aj. More precisely, consider the k f I.... rj and let
quirement that the functions x
.
.
.
,
W
hcCO(A,R1)nUjfjj:
V,=
11h(al)
f
E
U Zjffjj.
Since K is covered
radius IE4 there is for each 1 6, (1)
=
si
we see
that
j
Since the sets
Ua ...... U,,, such that
This
kj
x,(,)
11
A there is for
cover
11h(a)
-
V,
sets
h(al)II
<
e/4
V1 E
f
kj
many balls xl,...,X,. of
I I j(aj)
x, such that
an
c/4
<
by finitely
V&. Hence the
E
I E
f
-
x,
-
h, h
and
II
cover
<
6/4. Defining Uj' I f fi 1.
all of
V, and
E
11h(a)
-
a
EE
A
an
h(al)II <,E/4.
implies
11 h(a)
-
h(a) 11
<
11 h(a)
11h(al)
+ < 4
-
e/4
is covered
by finitely
-
=
Hence each set V, is contained in
Uj' 1 1 fi I
h(al) 11
-
+
x,(,)
11
11 h(a) +
h(al) I llx,(,) h(al) I -
-
c.
a
ball of radius
many balls of radius
This
implies
that
and therefore
com-
c. E
pact. The assertion follows space every sequence has
since in a compact subset of convergent subsequence.
now a
a
normed
I
8.2 Cluster and limit
Lemma 8.2.3. Let C be
a
convex
coordinate
371
curves
neighbourhood
with
com-
pact closure and let f-yjjjErq be a sequence of causal curves in Z! which are inextensible in U. If x E M is an accumulation point of this sequence, then there is
causal cluster
a
curve
through
-y
x
which is inextensible in
C.
Proof. By considering a subsequence erality that for each i there is a tj
without loss of gen-yi(ti) --+ x. Any cluster
we can assume
such that
y of this sequence intersects x. We choose the same coordinates as in Lemma 8.2.1 and view the curves -yj as continuous maps from finite
curve
intervals
[ai, bi]
trivially enlarge -yj (t)
=
maps is
to R'. In order to
the domains
-yj (ai) for t E
[ai, bi]
to
[infiEN(ai), SUPiEN(bi)] by setting
=-
for all t. The theorem of Ascoli 8.2.1
UjEjq -yj (t) of these
we
and ^/i (t) -yj (bi) for t c: [bi, b]. This family of Lemma Since C has compact closure so 8.2.1. by
[a, aij
equi-continuous
has the set
of Ascoli 8.2.1
apply the Theorem
implies
continuous
that
converges uniformly that -y is causal let t, s E [a, b], s < t. Since -yi (t) E J+ (-Yi (s), C) for all i, there is a causal geodesic pi from -yi(s) to -yi(t) for all i. By the continuous dependence of the solutions of ordinary differential equations a
subsequence
-y. To
to
curves
a
curve
see
geodesic p starting at -y(s). -y(t) by the convexity of C. This implies -y (t) E J+ (-y (s), C) Since s and t were arbitrary, -Y must be a causal curve. Since -y(a) is an accumulation point of -yj (a) E j \ C this point must also lie in Z \ C. Analogously for -y(b). This implies that -Y is
with respect to initial values, there is a limit It is clearly causal and has future end point .
I
inextensible in C.
biliEjq be a sequence of future inextensible causal If x E M is an accumulation point of this sequence, then there causal, future inextensible cluster curve -y through x.
Lemma 8.2.4. Let curves.
is
a
Proof.
Let C,, be
there exists
a
a
convex
subsequence
neighbourhood as distinguishes
which
in Lemma 8.2.3. Then a
cluster
curve
which is
inextensible in C,,. We may now take the intersection x, of the future boundary of C,, with this cluster curve. Applying the same argument
neighbourhood Cx,
to
a
an
extension of the cluster
convex
induction.
of xi and the
curve
to
Cx
U
Cx,
subsequence, Now
we can
we
obtain
proceed by I
8.2.2. Assume that (M,g) is globally hyperbolic and let fXiLENi fYiJiErq be sequences of points converging to x, y E M such that yj e J+ (xi), y E J+ (x). Let -yj be a causal curve from xi to yi. Then the sequence bibErj has a cluster curve which connects x with y.
Proposition
E 1+ (y). The set J+ (. ) n J- ( ) is' then a Proof. Let Jc' E I- (x) and of and we can assume without loss of generality that neighbourhood x, y
372
8.
Causality
all 7i lie in J+ (.: ) n J- ( ). Since J+ (.- ) n J- ( ) is compact, we can extend all -yj such that they are inextensible in J+ (i ) n J- ( ). By compactness
and strong causality of J+ (.- ) n J- ( ) we can cover this set by finitely many convex neighbourhoods as provided by Lemma 8.2.3 such that no -yj intersects any neighbourhood twice. Now the claim is an immediate consequence of Lemma 8.2.3.
Theorem 8.2.2. Let M let -yi:
[a, b]
(M, g)
M be
-+
[a, b]
f7ifiEN if
and
Proof. Assume first that
(') (ii)
_/ C no
and
x
Jim 7i(b)
=
y.
0 M, 7 E Ccausal (X, Y) is a cluster curve of the sequence only if there is a subsequence OijljEN of f-yjjjErq which in the CO topology of curves.
of -y is compact, there sets
=
X, y E
--
converges to -y
convex
be
strongly causal. For given points sequence of causal geodesics with
a
-yj (a)
.lim Then -y:
1
jCkjk=1,...'j
U1k=1 Ck
causal
C
-y is
are
a
cluster
for each
curve
Of
OiIiEN-
neighbourhood
Since the
U of -X
image
finitely
many
such that
Ui
curve can
enter any
Ck twice.
We have to show that there is
a subsequence 1-yi., JjEN such that for sufficiently large j all -yi., are contained in U. We can choose a finite sequence of points jY(tk)jk=1,...'k with 'Y(tk) E Ck n Ck+l. By the definition of a cluster sequence (and the fact that there are only finitely many such there is a such that each intersects all points) subsequence f7ij JjEN -yi., of the sets Ck n Ck+l. Since all Ck are convex and no causal curve can
Ck, -Yi ., is contained in U1k =1 Ck C UConversely, assume (without loss of generality) that j_YijiErq COnverges to -y in the Co topology of curves. Let h be a fixed Riemannian metric h on M and let for r E N U1 be a neighbourhood of -y which is re-enter any
the finite union of
enter'any Ck
can
respect
to h.
U1k =1 Ck,e.
in
sets
convex
ICk,.' }k=1,...'j,.
twice and all
.1
We' an
Now let
Ck,.i
cauW l
extend the
fUi J,Erj be a y. Since IYi IiEN
such that
no
causal
curve
have diameter less than curves
7i to inextensible
sequence of
with curves
neighbourhoods
of 7
1
such
thatnz, u., ,
of curves,
each ter
r
E
curve
we
=
obtain
a
converges to -y in the
Co topology
subsequence of curves Oi,JrEN with -yj, E Ui for implies that this subsequence has a causal lus-
N. Lemma 8.2.4
A. Since A lies in the intersection of all Ui and this intersection
equals the
set traversed
by
-y,
we
obtain A C -y.
Thi other
inclusion -Y c
follows since A is inextensible and -y does not have self-intersections. I In view of -/
Corollary simply by setting
8.2.1
we can
define the
length L(-y)
of
a
causal
curve
8.2 Cluster and limit
L (-y)
(t))
373
curves
dt.
(8.2.1)
0 Since in any neighbourhood U of any causal curve -y C Ccausal(-70 from null broken the functional to x a cangeodesic length y,
there is
not be lower semi-continuous with
If
(M,g)
chronological,
is not
semi-continuous either. To
see
respect
then the
to the
length
this consider
a
Co topology of
curves.
functional is not upper timelike curve -Y which
has self intersections. Then in any neighbourhood of 7 there are timelike C' curves which contain a closed segment which can be repeatedly
particular, we see that L length. On the other hand, following proposition.
traversed. In 7 has finite
oc
=
for all such
(M, g)
if
is
causal,
curves
while
have the
we
Proposition 8.2.3. Let x, y E M and assume that M is causal. The length functional L is upper semi-continuous in Cocausal (xl y) with respect to the CO topology of causal curves. For the
proof
of
Proposition
Lemma 8.2.5. Let
(M, g)
8.2.3
we
will need the
be causal and
E
neighbourhood U of -y curve parameter t of -y to a function t: U and grad(t),,y - .
-1. Then there is
a
following result.
Ctime (X, Y)
in M and -->
an
with g
extension
of the
R such that dt is timelike
=
Proof.
Let U be
a
finite number of
neighbourhood
convex
sets with
of -y in M which is the union of a be a timecompact closure. Let
like extension of -y which has no endpoint in U (cf. Corollary 8.0.1). Since (M,g) is causal -y cannot have self intersections. Hence we can
simply connected. Let jEo(t),...'En_j(t)j be with Eo along . The map f : V C R'
choose U to be
thonormal frame
(t'x1'...'xn-1)
=
-,
exp,:Y(t)
(En-1 x'Ei(t)
is
i=1
a
local
an ---->
or-
U,
diffeomorphism
local
diffeomorphism near the origin. Hence we can choose U and V so small that f is a diffeomorphism. This and the fact that U is simply connected (otherwise t may be a cyclic coordinate and only defined modulo an additive constant) implies that we can extend the function t to all of U. Since dt is timelike at , by choosing U small enough, dt is timelike everywhere in U. Finally obnear
-/ since exp is
serve
that
dt(Ei)
=
a
dt(D',jT expy(t)
orthogonal grad(t) I dt( o) g(grad(t), ) is
=
to
=
(Tn-
1
xiEi(t)
Ej and therefore we
obtain
parallel
dt(,9,,i) to
E0
nects
x
--*
M be
a
curve
in
0. Hence
0 From I
grad(t),.y
0 Proof of Proposition 8.2.3. Since Ccausal GT, Y) C Ctime (X, Y) show that the length functional is upper semi-continuous
Let -y: [a, b] hood given
=
=
Ct lme (x, y)
7
we
in
and U be the
need
only
Ctime (X, Y) neighbour-
Lemma 8.2.5. Let p be any timelike curve which conand y and is contained in U. Since dt(A(t)) 1 we obtain
by
=
Causality
374
8.
AW
Fgrad(t),graj(W
1 -
This
(grad(t) + v (t))
where
v
(t)
is
orthogonal to grad(t).
implies -
(g(grad(t), grad(t)) ) g(grad(t), grad(t)) )
A)
gi, (t) (A,
g,(t) (grad(t), grad(t))
1
g1j,(t) (V(t),
V(t))
1 g (grad(t),
grad (t))
gl,(t) (grad(t),
grad(t))
g(grad(t), grad(t))' On the other Since c
-4
x
hand, so
L(-y).
=
Since p E
, )
-
E
<
=
we
CtLie(, Y) on
was
-gy(t) (grad(t),
grad(t)).
may choose for
g(grad(t), grad(t))
-j1.,g,(t)(,9t,,9t)
:5
upper semi-continuous
-97(t) (
is continuous
small that -I
-g,,(t)(A,A)
implies
have I
g.,(grad(t), grad(t))
the set U
> 0
we
and therefore
given
a
< -1 +
E.
This
L(p)
arbitrary the length function
<
L is
this set.
Hawking and Ellis (1973) give a slightly different definilength of a causal curve. They first define the length of a timelike curve E Ctlime (X, y) by Equation 8.2. 1. For a causal curve they set L(p) inff (U) : U is a neighbourhood of yj, Remark 8.2. 1.
tion for the
=
where
t (U)
=
It follows that this
sup f L (-y)
length
cocausal(,x, y) with respect
-y C U is timelike and C
functional
to the
L
is also upper semi-continuous in
C' topology of causal
L
coincide. However, if (M, does not coincide with the original (and more
causal spacetimes, L and
ical, L
:
since for any timelike
curve
-y with self-intersection
curves.
Hence in
chronologg) intuitive) definition one has L(7) oo. is not
=
8.3 Achronal submanifolds
and Given enced
Cauchy developments a
set A the set
by A.
1+ (A) consists of those events which can be influboundary aI+ (A) is of special importance.
It is clear that its
we show that 91+(A) must be a Lipschitz manifold. We investigate the "domain of dependence" of A, i.e., the set of those events which are completely determined by the physical data of A. For the following, definition, standard examples are spacelike sub-
In this section
will also
manifolds
and,91+(B)
where B is any subset.
8.3 Achronal submanifolds and
Cauchy developments
375
Definition 8.3.1. A subset A C M is called achronal
if there
timelike
a
1+(F)
curve
which intersects it twice. A set F is called
C F. Past sets
no
if
defined analogously.
are
Lemma 8.3. 1. Let F be
is
future set
future
a
set. Then aF is
an
achronal
Lipschitz
hypersurface. Proof Let
X
E
follows that I-
9F. Since I+ (x) C F is open, 9F must be achronal. It also (x) C M\F. Consider a convex coordinate neighbourhood
U of
x as given by Lemma 8.2.1. The integral curves of axo [t(.,i "-1) through (0, x1.... Xn-1) are timelike. Since the integral curve of 9xo through x intersects both I- (x, C) and 1+ (x, C) there is a neighbourhood V C U of x such that all integral curves of axo which intersect ......
I
V intersect I- (x,
and 1+ (x,
C)
I-t(xl,...,x--,) intersects U in is
achronal,
there is
a
a
C).
It follows that each of these
unique point
y(x1,...'Xn-1).
curves
Since aF
constant k > 0 such that
n-1
ix0 Y(Xi
Xn- 1)
0
for all
0
y
...
I
Xn-1)
n-
......
(X1'...' Xn-1), (&I ....... n-i).
(X11 is
X0
-
-4
1)
k
(Xi
-
_, i) 2
Hence the function
Y(X1.... IXn-1)
Lipschitz.
Definition 8.3.2.
,9nu11F
=
fx
The acausal
E
c9F
:
The
null-boundary of
3
neighbourhood
a
boundary of F
This definition is
is a"F
=
U
a
future
set F is the set
of x with 1+ (F\U)
aF
=
1+ (F)
\ anu"F.
justified by the following lemma.
a future set and x E 0"11F. Then there is an achronal null geodesic generator through x with future endpoint x. This generator is Past inextensible or has past endpoint in aa'F.
Lemma 8.3.2. Let F be
376
8.
Proof.
Let U be
There is
causal curve
Causality
a
of
neighbourhood
a
curve
such that 1+ (F)
x
=
1+ (F
\ U).
points lXibErq 1+(F) with xi --+ x. Let -yi be a from some yi E 1+ (F) \ U to xi and let -y be a causal cluster
sequence of
with future
C
endpoint
x.
This
cannot intersect
curve
1+ (F) since
we x E 1+ (F). It follows that -y lies in the achronal set aF and is therefore an achronal null geodesic. If -y has a past endpoint z E a""' F, we can repeat the argument above to obtain a cluster curve y
would have
then
which has future
endpoint
z
and which is also
achronal null
an
geodesic.
Since the concatenation of I-L and -y must be achronal, they are both part of the same null geodesic. It follows that this null geodesic is either past
inextensible
or
has past
endpoint
in 9a'F.
I
Definition 8.3.3. Let A be
an achronal set. Then the edge of A, edge(A) A such that for any neighbourhood U of x each x E 1:L (x, U) can be joined by a timelike curve which is
all points
is the set
of pair of points x :
in
contained in U and does not intersect A.
Corollary 8.3. 1. Let A be a set. Then 9J+ (A) is an achronal Lipschitz hypersurface. The set OJ+ (A) \ A is generated by null geodesics without conjugate points. These generators are past inextensible or have past endpoint in edge(A). Proof. The first property is clear since J+ (A) is a future set. Let x aJ+ (A) \q and -y be the maximally past extended null geodesic generator with future endpoint x. Let y be the past endpoint of -y. Since 7 is a f there is a neighbourhood )/V of -y \ Jyj which generator of 9J+ (A) does not intersect X Assume that y
edge(A).
E
Then there is
a
neighbourhood
convex
U of y such that for every pair of points z:L in P: (y, U) every causal curve A which connects z- and z+ intersects A. Let z- E I- (x, U),
zo
E 1+ (z-, U) n A There is a to fromzO to z+. W q \ edge(A). y E
1-
(4+, U)
point z+
E
J+(ZO)
proves y
c
W, and A,
I+ (y, U)
E
n
By
1+(z)
a
timelike
W and
The concatenation A of
A,
a
and
curve
timelike
A2
c
a
point
1+(A)
Z
E
A2
n
A.
curve
Consequently,
in contradiction to
from z-
A2
C
intersects A since
the construction of IN it is clear that
intersect A. Hence there is x
n -y n
x
E
/\2 we
cannot
obtain
9J+(A).
This
q \ edge(A).
Assume
now
that y is past
endpoint
of the null
geodesic generator
-Y
and is not contained in X Then y has a neighbourhood U which does not intersect X This implies that for each z E 1+ (A) n U there is a timelike
curve
from A to
have shown I+ (A)
This is
=
impossible by
z
which
1+ (1+ (A))
initially
=
Lemma 8.3.2.
does not lie in U. Hence
1+ (1+ (A)
\ U), i.e.,
y E
we
anull j+ (A). 1
8.3 Achronal submanifolds and
If A boundary of A.
Lemma 8.3.3.
the
A
Proof.
Let
U of
such that
x
x
it divides U one
of them
E
is
\ M.
a
spacelike hypersurface, edge(A)
Since A is
spacelike there exists 0. Since A
(I+ (x, U) u l- (x, U)) n A
(if chosen
Cauchy developments
small
enough)
=
curve
from I-
is
a
subset
of
neighbourhood a hypersurface
into two disconnected components,
containing 1+ (x,.U) and the other
that every causal
a
is
377
(x, U)
to
I-
(x, U).
But this
implies
1+ (U) must intersect A.
I
Another important set is the future horismos of A c M. It consists of those points which can be reached from A via causal but not via timelike curves.
The future horismos
Definition 8.3.4. a
neighbourhood
U
of
Lemma 8.3.4. Let A c M be x
E
of
A is the set E+ (A, a
subset A C M relative to
a
U)
=
J+ (A, U)
\
1+ (A, U).
spacelike submanifold and x E M. Then a null geodesic from A to x U and does not have focal points before
E+ (A, U) holds if and only if there is
which is
completely contained
in
X.
Proof. This follows immediately
from
Corollary 8.3.1, Theorem 4.6.1,
Lemma 4.6.. 15, and the fact that E+ (A, If A is
a
set
determined set of these
we are
the
by points
U)
is
a
subset of aJ+ (A,
U).
I
also interested in those points which are completely physical data at A. If Postulate 8.0.1 holds then this
by the following definition.
is described
Definition 8.3.5. Let A be
a
D+ (A) is the set of all points
x
set. Then the future E M
Cauchy development
such that all past inextensible causal
through x intersect A. The past Cauchy development is defined D+ (A) U D -(A) analogously and denoted by D -(A). The union D (A) is called the Cauchy development of A. curves
=
Lemma 8.3.5. For any achronal set A
we
have I-
(D+.(A))
n 1+
(A)
c
D+ (A). If
Proof.
x
E I-
(D+ (A))
n
1+
then there is
Let -y be a timelike curve from x to y. If x inextensible curve p with future endpoint
a
y E
0 D+ (A) x
D+ (A)
n
then there is
1+ (x). a
past
which does not intersect A.
Since the concatenation of M and -y is a past inextensible curve with future y the curve -y intersects A at some point z. From z E 1+ (x) and x E 1+ (A) we obtain z E 1+ (A) which implies that there is a timelike
endpoint curve
gives The and
with future a
endpoint
contradiction to the
y E
1+ (z) which intersects A twice. This
achronality
of A.
I
following theorem shows that global hyperbolicity (Definition 8.1.3) Cauchy developments are strongly linked.
378
8.
Causality
Theorem 8.3.1. Let A be
hyperbolic First
we
need
to
Lemma 8.3.6. x
Proof. xo a
x.
=
establish the
Any past
int(D+ (A))
E
be
int(D(A))
is
globally
following
which passes
through
two lemmas.
inextensible causal
curve
intersects I- (A).
Let -y be a past inextensible causal curve with future end point The set I+ (-y) n A is empty unless 7 intersects _T- (A). We choose
Riemannian metric h
hood
achronal set. Then
an
empty.
or
fi
M
G
M and denote for
on
Z_ diSth (Z, ')
:
<
Ej by B,(x).
Let
z
E
fxil
M the
neighbour-
C -y, xi+l E
J-(xi)
sequence without
past accumulation point and let yo 'E 1+(xo) n int(D+ (A)). There is a point yj G I- (yo) n 1+ (xi, Bi (xi)). We inductively define a sequence jyjjiC_N with yi E I- (yi-1) n 1+ (xi, B I (xi)) a
and connect the yi these yo E
curve
by
segments is
int(D+(A)),
timelike an
the timelike
y. There is
spacetime depicted
achronal null
geodesic.
=
which
curve
y C
1+(-y).
y must intersect A at
implies xj
E
Since
point I- (y). This gives a some
0.
1
Fig. 8.1.4 violates strong causality along an following lemma shows that for chronological strong causality is violated is always generated
in
The
set where
spacetimes the
curve
(y)
an yj E p n Icontradiction to 1+ (-y) n A
The
segments. The concatenation of
curve
inextensible timelike
by null geodesics. Lemma 8.3.7. Assume that
(M, g) is chronological. If it fails to be strongly causal at x E M, then there is an achronal, inextensible null geodesic y through x along which strong causality is violated. The end piece of p to the past (resp. future) of x is a cluster curve of curves which have intersected (resp. will intersect) arbitrarily small neighbourhoods of x before (resp. afterwards) Proof. Let Uj be
a
basis of
neighbourhoods
convex
of
x
and
fttiliEN
be
sequence of inextensible timelike curves such that pi intersects Uj at least twice. Let xi, yi E Uj n pi, yj E J+ (xi) be points such that the a
segment of pi between xi and yi leaves Uj. We denote by yi] (resp. l,ti[xi -->]) the past (resp. future) inextensible endpiece of Mi with future
(resp. past) endpoint yi (resp. xi). Let /-t+ [x ---+] be a future inextensible causal cluster curve of fyi [xi I IiEN with past endpoint x and f Mi., [xi., _-+1 IjEN be a a distinguishing subsequence. If there is a point z E M+ [x >1 n 1+ (x) then there are ---
neighbourhoods V, V, of V, Let
jo
E
c
1+ (. ) for all 1-7
N be
definition of
x
a
a
and E
curve
such that
Vx and Vx
number such
cluster
z
C I-
that yi.,
there is
a
ii
>
()
for all
E
V,
V., for all j > jo. By the jo with Mij, [xi.,, -+] n V, = _ 0.
E-z
8.3 Achronal submanifolds and
Cauchy developments
379
Let
n V,. We obtain a closed timelike curve by first E Mi.,, [xi.,, from i to traversing Mi.,, [xi.,, yi.,, and then connecting yi.,, E Vx with V,, through a timelike curve. Since this contradicts the chronology of (M, g) we have shown that M+ [x -+] n 1+ (x) 0 and therefore that =
the cluster
curve
/,t+ [x --*]
is achronal.
Let p- 1---> X] be a future inextensible, causal cluster curve of Yij I IjEN with future endpoint x. As for /,t+ [x -->] we see that
x]
is
achronal. The concatenation y of y-
[---> x]
/-t+ [x --4]
and
if future and past
inextensible. If it is not achronal there exist points z- E /,t- [-- x] and + E /-t+ [x -->] with Z+ E I+ (z-) and therefore also neighbourhoods V,and
of
Vz+
z-
and z+ such that all
calVz_ and all
E
chronologically related. Let fpi[--4 yi]jiEN which distinguishes
be
yiJjjEN
are
x].
a
+ (E calVz+ subsequence of
Hence there is
a
jo
E
N
pij--4 yij intersects Vz_ for all j > jo. Consider now the subsequence JI.Li., [-- yiJ ] jj >j,, and recall that the curves Mi., [--4 yi., ] and pi.,[xi., -- ] coincide. We can assume (without loss of generality) that the sequence fpi.,[xi., -->]Ij>j(, distinguishes p+[x Hence there is a ji > jo and a curve Mi,, [xi.,, -->] which intersects Vz+. Since P+ [x --4] is a cluster curve with past endpoint x limxi., this neighbourhood is intersected before pi.,, [xi.,, --4] pij yij, ] intersects Vz- Again such that
=
we
get
a
contradiction to the
chronology
of
(M, g)
-
This proves the first
assertion of the lemma.
The second assertion follows
directly from
our
construction.
I
Proof of Theorem 8.3. 1. We show first that int (D (A)) is strongly causal. By the definition of Cauchy development there cannot be any closed causal curve in int(D(A)). Assume that the strong causality condition is violated at x E int(D(A)) and let U be an arbitrarily small convex neighbourhood of x in int(D(A)). Then there exists a sequence of future directed timelike curves 7i which have x as an accumulation point and intersect U twice. There is an inextensible cluster curve -Y through x. This curve must intersect A since X E int(D(A)). By Lemma 8.3.6, -Y intersects both I- (A) and 1+ (A). Since these sets are open they are both intersected by yi for i large enough. From Lemma 8.3.7 we see that yi first intersects 1+ (A) and then I- (A). Hence we can prolong these curves such that they intersect A more than once. This gives a contradiction to the achronality of A. We show
now
that for any x, y E int(D(A)) the set J+ (x) n J- (y) are no closed causal curves in int(D(A)) the
is compact. Since there
equation we
can
x
=
y
implies J+ (x)
n J-
(y)
=
jxj
which is compact. Hence
that y E J+(x) \ fyj. We may also assume that y E since otherwise X E int (D (A)) and we could apply the time
assume
int (D+ (A))
reversed argument. Let
-
IZi LEN
to show that it contains
a
be
a
sequence in
J+ (x) n J- (y). We need
convergent subsequence. For each
zi
there is
380
a
8.
causal
fZiliEN
Causality
curve
-yj from
x
lies in
D+(A)
U
through
A,
we
If
zi to y.
a
subsequence jzjj JjEN
consider the cluster
curve
of
with future
end point y. Otherwise a subsequence fZi.,JjEN Of fZiLEN lies in D- (A) and we can consider the cluster curve with past endpoint x. Again by
time-reversal, the cluster
we can
to the first
retrict
compact and has
compact neighbourhood U. Since this segment is
a
cluster curve, a
infinitely many subsequence. convergent
Lemma 8.3.8. Let A be
D+(A)=jxeM:
a
Proof. "C":. Let
x
with future
G
a
of the zi., lie in U and must therefore have
set. Then
past inextensible timelike
every
with
curve
Since y E D+(A), segment from A to y is
possibility.
must intersect A. Hence the
curve
D+ (A)
endpoint
\
x.
future endpoint
A and -y be Let U be a
a
curve x
Al
intersects
past inextensible timelike
convex
neighbourhood
of
x
which does not intersect A and y c -y n U be a point different from x. Then 1+ (y, U) is a neighbourhood of x and must therefore intersect
D+ (A). Let p be
a
timelike
curve
from y to
some z
c
D+ (A)
n U which
is contained in U. Since it does not intersect A and the concatenation of p and -y is past
"D:" let
inextensible, A be
y must intersect A.
point such that every past inextensible timelike curve with future endpoint x intersects A. Now let 7 be a timelike curve with future endpoint x and fXiliEN be a sequence in -( which converges to x. Let Ai be a causal curve which is past inextensible and has fux
a
endpoint xi. Since any causal curve which is not an achronal null geodesic can be perturbed slightly such that the resulting curve is timelike, the concatenation of Ai and the future end piece -y[xi -->] of -Y must intersect A. Hence for i sufficiently large, Ai must intersect A. Since Ai was arbitrary it follows that Xi E D+ (A) and, consequently, x E D+ (A). ture
Thejuture boundary of D+(A),
Definition 8.3.6.
H+ (A) is called the future
Cauchy defined analogously.
==
D+ (A)
\
Cauchy horizon marks the predicted from knowing data at A. It properties. x
E
(D+ (A))
horizon. The past
The future
Lemma 8.3.9. Let
I-
H+ (A)
Cauchy
horizon H-
limit of the set which has the
Then
x
c
(A)
can
is
be
following fundamental
1+ (A).
8.3 Achronal submanifolds and
There is
Proof.
U of
neighbourhood
a
x
381
Cauchy developments
which does not intersect A.
Since for any y E I- (x, U) the set 1+ (y, U) is a neighbourhood of x it must intersect D+ (A). Let z c 1+ (y, U) n D+ (A). It follows that y E D+ (A) c 1+ (A) since otherwise we could construct a past inextensible
through
curve
8.3.2. The
Fig.
which which does not intersect A.
z
Cauchy
horizon for
a
set which fails to be achronal
Proposition 8.3.1. Let A be a closed achronal set. Its future horizon H+ (A) is generated by achronal null geodesics which have either no past endpoint or past endpoint in edge(A). Proof.
The set P
=
D+ (A)
U I-
(A)
is
ast set since I-
a
(D+ (A))
c
D+ (A) \ I- (D+ (A)). If y E I- (A) (A). Let y c H+ (A) then there exists a timelike curve which intersects A twice in contradic-
D+ (A)
U I-
=
achronality
tion to the
of P and therefore Let
x
E
an
achronal
H+ (A) \edge(A) and let
converges to in,
a
of A. Hence H+ (A) is
subset of
x.
By
subset of the
boundary
Lipschitz manifold.
Ixi JjErq be a sequence in 1+ (x)
the definition of H+ (A)
D+ (A). Hence for each
a
xi there is
a
none
which
of the xi is contained
past inextensible causal
curve
7i
which does not intersect A. Let -y be a causal cluster curve with future endpoint x. We will show that x has a convex neighbourhood C such that - n C C H+ (A). Since Lemma 8.3.9 two
possible
cases,
x
E
implies that A
\ edge(A)
X
E
or x
H+ (A) E
c
A U 1+ (A) there
1+ (A). If
x
E
A
are
\ edge(A)
we
choose C such that every timelike curve from 1- (x, C) to 1+ (x, C) which is contained in C intersects A. If X E.T+ (A) we choose C c 1+ (A). It follows that -y n C c 1+ (A) U A. This is clear if x E 1+ (A). Assume x E A and that there is an y E 7 n C \ A. Since A is closed
therefore that
a convex neighbourhood U C neighbourhood intersect -yj (for
there is This I-
(x)
since
X
E
A and
y'E
J-
(A).
C of y which does not intersect A. i large enough). It also intersects
Hence there is
a
timelike
curve
A in
382
Causality
8.
U from I-
(x)
Since -yj does not intersect A we can concatenate A endpiece of -yj to obtain a timelike curve from I- (x) to
to 7i
-
with
a
xi E
1+ (x) which is contained in C and does
the future
not intersect A. This
gives
a
\ edge(A) proved 7 n C c P- (A) U A. Assume that there is an y E (1.,nC) \D+ (A). If the segment (- nC) [y
contradiction to
X
A
E
and
we
have
>
---
x] of -y n C between y and x would intersect A then we would obtain a contradiction to the achronality of A from y E (1+ (A) U A) \ D+ (A) c I+ (A) and
(-y
n
C) [y
x] _t_LLy).
--->
Let U C C be
a
neighbourhood
of
y which does not intersect D+ (A) and consider a point z E (I- (y, U) n 1+ (A)) \ D+ (A). Then the concatenation A of a timelike curve from z to y in U and the
segement (-y n C) [x -- y] does not intersect A. Since H+ (A) and A is closed a slight deformation of A results in a causal curve from z to some point in -: E D+ (A) which does not intersect A.
x
(=-
We could
this
prolong
now
curve
to the
past of
z
E M
\
D+ (A)
thereby
obtaining past inextensible curve which does not intersect A. This gives a contradiction to , E D+ (A), whence we have proved (-y n C) C D+ (A). a
Assume that there is
D+ (A). Let A be
a
y E
an
timelike
(7 n C) n I- (D+ (A))
curve
from y to
This
z.
and let
e
z
1+ (y)
n
cannot intersect
curve
A to the future of y because of y E 1+ (A) U A and the achronality of A. If y V A then there is a neighbourhood U of y with U C I- (z) which does not intersect A. Since -y is a cluster curve Of jYijiEN there is an i and a point from E -yj n U. It follows that there is a causal curve G -yj to z
which does not intersect A. If y cz A then
x
A
E
\ edge(A)
by the construction of C. The point y has a convex neighbourhood of U c C n I- (z) which is intersected by infinitely many -yi. Let i E N with -/i n U :A 0, Consider a timelike
E -yj n
be
U, and
a
timelike
from
curve
9
to
z.
which is contained in C. curve p from I- (x) n C to The concatenation of 1-t and the part of -yj to the future of intersects A because of
and xi
V edge(A)
x
C-
1+ (x). The equation 7i
A
n
=
0 implies
cannot intersect A by the achronality of A. that M intersects A. Hence We have shown that in either case, y V A and y G A, the concatenation of is past inextensible the past endpiece of -yj with future endpoint and
and does not intersect A. This gives a contradiction to 0 Consequently, we have shown (-y n U) n I- (D+ (A)) We have 7 n C C D+ (A)
\
I-
(D+ (A))
curve
-yj C
inductively x
endpoint
H+(A) we
obtain
which has either
Since
H+(A)
n
&. The concatenation of y n C and
with future
no
causal
a
endpoint curve
past endpoint
is achronal this
curve
x.
-yc,,, C or
E
D+ (A).
H+ (A). The past endpoint
=
of 7 n C lies in H+(A) since H+(A) is closed. If can repeat the construction thereby obtaining a curve
with future
z
Repeating this
edge(A) we d C H+ (A) n d gives a construction
H+ (A) with future endpoint
has past endpoint in an achronal null
must be
edge(A). geodesic. I
9.
Singularity
theorems
chapter we prove and investigate "singularity theorems". These are usually interpreted as an indication that black holes exist and that there has been a big bang or at least that there are regions in spacetime where general relativity breaks down. They are one of the main motivations for attempting to quantise general relativity. While there is a lot of evidence in favour of this interpretation we will see that there are also open problems which have to be addressed in order to justify this interpretation.
In this
theorems
-
Chaps. 7 and 6 we have seen that spacetimes describing a single, non-rotating star and the simplest cosmological models of our universe contain regions where the curvature diverges. One may think that these singularities are only an artifact of our high symmetry assumptions, but in this section, we will give an indication that a physically realistic spacetime must contain such singularities. More precisely, we will show that there exist causal, inextensible geodesics which are incomplete. Recall that a freely falling particle is represented by a timelike geodesic. If the geodesic cannot be extended to a complete one (i.e. if its future endless continuation or its past endless continuation is of finite length), then either the particle suddenly ceases to exist or the particle suddenly springs into existence'. In either case this can only happen if spacetime admits a "singularity" at the end (or beginning) of the history of the particle. This singularity may be a curvature singularity, there may be a topological obstruction, or spacetime may simply cease to be sufficiently smooth. However, the Schwarzschild and Robertson Walker solutions indicate that these singularities are accompanied with diverging curvature. (But cf. Sect. 9.5.1 below where we present a spacetime which indicates that such singularities are very mild). We will prove a singularity theorem which only establishes the existence of incomplete causal geodesics rather than incomplete timelike geodesics. While the innocent looking extension to null geodesics is necessary for the proof, the name singularity theorem is in this case somewhat misleading, because there exist In
This should not be confused with pair creation or pair annihilation of particles and anti particles, because during these processes nothing really ceases or
starts to exist.
changes
These quantum mechanical
of state.
M. Kriele: LNPm 59, pp. 383 - 424, 1999 © Springer-Verlag Berlin Heidelberg 1999
phenomenons
are
merely
384
9,
Singularity theorerns
perfectly regular spacetimes
with
incomplete
null
geodesics contained
compact subsets. On the other hand, it has been argued that such
ples this
are
very
and that in
special
phenomenon does
not
occur
in
exam-
stable, physically realistic spacetimes
(cf. (Hawking
and Ellis
1973)).
Energy conditions
9.1 In
general, a maximally extended Lorentzian manifold need not contain incomplete causal geodesics. In order to prove a singularity theorem, we will have to make some physical assumptions. There are two sorts of fundamental physical experience which come to mind. Firstly, energy density as measured by the energy momentum tensor is positive. Secondly, gravitation is attractive. Recall that the energy density measured by an observer -y (with g( , ) -1) is given by E T( , ). We feel that this energy density should be positive. Recall also that in the motivation of the energy momentum tensor (cf. Sect. 5.1) we have obtained the energy density E T(U, U) as an average of a positive mass distribution. For our purpose this should be enough of a motivation of the following definition =
=
=
Definition 9.1.1. M
We say that the weak energy condition holds at
T(u, u) For
x
E
if
a
> 0
for all causal
vectors
u
E
TXM.
physical verification of the weak energy condition one would have physical matter models. This is beyond the scope
to consider all realistic
of this book but
far the available evidence points to the fact that the
so
weak energy condition does hold. Gravity is attractive if and only if any two nearby freely falling observers will be forced to approach each other under the influence of the
underlying spacetime geometry. This can be formulated infinitesimally in a rigorous manner. A freely falling observer is modelled by a timelike M. Let f : (-J, J) x [a, b] --> M a geodesic variageodesic -y: [a, b] tion of -y and J f,(0, -) be the variation vector field. Observe that J is a Jacobi vector field. From Taylor's theorem we get with respect to any coordinate
ordinate
expression
can
i
f (0, t) + 8 ji (t) + 0 (S2). This cobe interpreted in Newtonian terms as follows.
system f '(s, t)
=
The observers 7 and f (s, -) have (up to first order) the same rest space and are separated by the space vector sP. Hence up to first order it
speak of the (Newtonian) force F with which -Y acts on -msj' where m is approximately given by P the mass of the observer f (s, .). (The minus sign is inserted because the force vector points from f (s, t) to -y (t) and J points into the opposite direction). Clearly there cannot be a direct translation of Newtonian makes
f (s, .).
sense
to
This force is
=
9.1
(ft (s, t))'
'(t)
for
s
<
limit. In fact, the location of
an
case,
acterised
by
acted
is
this is
on
;::z
385
(as
in this
But for small relative velocities
general relativity.
to
concepts
Energy conditions
1)
there is
a
well defined infinitesimal
infinitesimally nearby observer is charorthogonal to -y and the force it is V J. Since J is a Jacobi vector field,
the Jacobi vector field J
given by
exactly
F
=
-mV
the force in the
following
Definition 9.1.2. Let -y be a timelike neighbouring freely falling observer J of
definition.
geodesic mass m
-1. A
with is
a
pair (J, m), where
J is a Jacobi field along -y with values in J- and m, is a positive number. The tidal force which acts between the observer -y of mass m and its
neighbouring freely falling
observer J is F
=
given by
mR(J, ) -
The component of F pointing towards the observer -y I
F,
-
_0(_j1J)
J)
can
is
given by
M -
VP_1J)
(R(J,, I) , J).
Hence the assertion that / attracts the infinitesimally neighbouring observer corresponding to the Jacobi field J is equivalent to the assertion
plane spanned by following definition.
that the sectional curvature of the
positive.2
This motivates the
J and
is
non-
We say that at x E M gravity is attractive in all and only the sectional curvature of all timelike planes in
Definition 9.1.3. directions
TxM
is
if non-positive.
The requirement that gravity is attractive in all directions is very strong. Also note that apart from our experience in weak gravitational fields we have not much evidence that
On the other
hand,
gravity is really attractive in all directions. gravity must be attractive on average.
it is clear that
reason is that gravity is much weaker than all the other fundamenphysical interactions. Electromagnetism and gravitation are the only long range interactions. On large scales, electromagnetism is not of prithe mary importance because it is attractive or repulsive depending on be of would it for even gravity, configuration. If this would also be true less importance for astrophysical applications. We know however that this is not the case. There are of course several ways of defining averages of gravitation. We will restrict attention to our single observer 'Y together with the associated space of neighbouring, freely falling observers. The average of the tidal forces in every direction is given by integrating the
The tal
2
we neglect the contribution of -y and gravitational field. They are thought to
Observer that server
to the
the
neighbouring obnegligible mass.
be of
386
9.
Singularity theorems
tidal force component in every direction 4 we obtain n
sphere S".
the unit
over
For
==
3m
41r
f
ps2
S2 CT". M
/2
3m
47r
2-7r
-ir/2
O(cos Wei
cos
(R(cos O(cos
0
+ sin
(Pe2)
+ sin
pej + sin
Oe3
)
+ sin
Oe3
OdWdO
cos
ir/2
M
4/37r
fr/2 fO 21r(
2 Cos
W
Cos
20
(R(ei,
o sin W
cos
cos
el)
2o
R(ei, ) , e-2 + 2 W cos 0 sin 0 (R (ei, 63) + sin 0 sinO (R(e2, + 2 sin p e3) + 2
(pe2)
cos
2
O
COS2
0
(R (e2
e-2)
cos
+ sin 2o
(R (e3 i4 )
4/37r f
+ 2-7r
e3) )
cos
Od pdO
ir/2
M
-
,
cos
(7r Cos 3o (R(ei, ) , ei)
+
7r
COS3 0
(R(e27 )`7Y) e2)
/2
0 sin 2o
(R (e-3
e3) )
,
cos
OdO
3
R
(ei,
ej
=
-Ric
This motivates the definition
Definition 9.1.4. The timelike convergence condition holds
Ric(u, u)
', : 0 for all causal vectors
Using Einstein's equation this condition
can
be
if
u.
re-expressed
in terms of
the energy momentum tensor: 1
T(u, u) While this
-
,
n-2
(tr(T)
A -
47r
) g(u, u)
> 0
for all causal vectors
u.
is not implied by the weak energy condition (nor the timelike imply it), convergence condition is often also called the condition. The strong energy following lemma gives a partial motivation for this terminology.
inequality
does
Lemma 9.1.1. Assume that g and T can be simultaneously so that the energy density E and the principal pressures pi
fined. Then
the weak energy condition is E
>
0,
E
+ pi > 0
for
equivalent
all i
C-
to
n
-
11,
diagonalised of T are de-
Energy conditions
9.1
and the timelike convergence condition is
equivalent
387
to
n-1
(n
3)r:
-
+
Let
Proof.
nalises T.
-A
Pi
>
41r
be
jeo,...' en-11 Any
0,
E
be written
u can
n
1
Ei'--i (c') 2<
where c, c' numbers with tion is equivalent to
+
c(eo
+
diago-
En-1 e,),
n-1 6
=
+
i=1
E(Ci)2p, i=1
n-1
n-1
-1: (Ci) 2C +E (Ci)2 (6 + P,)
1
-
11.
-
1. Hence the weak energy condi-
T- Ciei, eo + I: Cie,) i=1
=
as u
n-1
n-1
T(eo
0 <
En
orthonormal basis of TM which
an
causal vector
! 0 for all i
+ pi
.
The condition for the weak energy condition is sufficient since the factors n-1
(Ci)2
(Ci)2
and
positive. It is necessary since we can choose the numbers ci for any given j the numbers 6' J The strong energy condition is equivalent to
=
are
0
or
=
*
n-1
n-1
0 <
(n
-
2)Ric(eo
Y Ciei, eo + 1: ciei)
+
i=1
i=1
n-1
n-1
(n
-
2)T( e-0
E cei, eo + E Ciei) n-1
A
(tr (T)
-
+
g(eO
47r
+
(n
2)
6
+
ei, eo +
E cei) n-1
n-1
n-1 -
E
n-1 c
i
E(C )
2
A
i=1
) (-
(Ci)
1 +
2
i=1
) (- +I:pj '5
A -
47r
j=1
n-1
(n
-
2)
(Ci) 2 (6 n-1
+
(I_
(Ci)2
+
)(
P,) n-1
(n
-
3)c
+
Epj
A -
-
41r
j=1
Hence the assertion for the timelike convergence condition follows same
argument.
by the I
388
It
9.
Singularity
physically
seems
bution of
theorems
plausible
more
to demand that the matter distri-
universe satisfies the
our
I
A
inequality
>
6
0 rather than the
inequality sense the timelike convergence En- P, condition is "physically" a stronger condition on the matter distribution than the weak energy condition. Since the cosmological constant is close to zero and the energy density is much larger than the princic
+
pal pressureS3 the timelike assumption to make. A further condition states that
along
> 0.
In this
convergence condition is still
a
very
plausible
impose is the genericity condition. It geodesic -y there exists a point at which
want to
we
each causal
c d [,,Rbjcd[e f] =h
0
holds. This condition is
only of technical nature, because the set of metthe satisfy strong energy condition is dense in the set of metrics which satisfy both the strong energy and the genericity condition. Here rics which
impose the (very fine) C2 -Whitney topology (for details and proofs
we
(Lerner 1973)).
see
The mathematical
significance of the strong energy condition in conjunction with the genericity condition is clear from the following corollary to Proposition 4.6.3.
Corollary
(a, b)
-y:
for all
--
9.1.1. Assume that
M be
t and the
an
(M,g)
is
inextensible causal
Lorentzian
manifold and let geodesic. If Ric( (t), (t)) > 0 a
genericity condition holds along
-y then either
(i) -y is incomplete, or (ii) -y contains a pair of conjugate points. Proof of
a
We have to show that the
(-:y(to))-L
R: is not
in
identically
,
v 1-4
Rv
:=
R(v,
zero.
c(t) d M [a(t)Rbjcd[e f](O =h c(t) d (t)Rbede -7' 0. Since a symmetric
0 then
bilinear form is
Suppose i
-
=
by
its associated
g(R( , (t)) , (t)) =h
choose
a
( (to))-L
particular
determined
n
--+
If -y is timelike and
with
6ik
genericity condition implies the existence
to such that the map
0.
quadratic form there is a vector Consequently, R( , (t)) (t) 7
(ei, e,)
c
have
( (t))-L
0.
that -y is null and that c(t) d (0 [a(t)Rbjcd[e f](O =A 0. We of T,(t)M such that e, (to) and (ei i ek)
basis
a
we
=
=
0, (e, e,)
=
4,
-
1 for
i, k
c
f 1,
1, n 2 1. In this and the associated dual basis -6n-1 This implies -
=
.
we
.
.
,
n
have
2 1, r,
-
:y'
=
R,
s
E
and
a
Recall that in
physical
units where the
to 1 the numerical value of the energy comparison to the principal pressures.
velocity density
of
light
c
increases
is not normalised
by
a
factor
C2
in
9.2 Closed
4 ' (t)
d (t) [,,(t)Rb]cd[e f](t)
trapped surfaces
4J[,,-1Rb]nn[e6fn_1 I
=
a
jn'Rbnne 6n-I a
=
_
f
+ 6nb
If this
and
expression does must be either
1
'R jna This
-
(n-l)nne
clearly
not vanish then either
vanishes if
terms cancel
n
pairwise if
cancels the third
one
Rann(n-1) 6n-I
a or
=
0. But this
n
-
b must be
'R 6na
equal to n n f
that b
=-
(n-1)nn(n-1)
-
-
I
1.
1. jne
The first two terms and the last two 1. If
e
n
=
-
1 the second summand
and the first summand the fourth summand. Hence
c(t) d M [a(t)Rb]cd[e f] W = Ranne :
_
jn'Rbnnf 6n- I a e
assume
e
a, el.
E
a
_
e
e or
Ranne +
6n'Ranne 6n-1 b f
Rannf 6n-I
f. For definiteness, The formula then simplifies to so
389
implies
0
implies that
in turn that
a, e E f R(-, ) : ( (t))'
n
21
-
--+
and
( (t))J-
does not vanish.
9.2 Closed As in
trapped surfaces
a further preparation, consider an isolated, dense object, say a star, spacetime. If it produces enough gravitation, it will not only attract
the light rays it sends out. (Recall that they geodesics and therefore perceptible to the curvaspacetime). Exactly this situation happens in the Schwarzschild
material are
objects by
modelled
ture of
but
even
null
spacetime. To be
more
concrete, let T be
an
(n
-
2)-dimensional
space-
like submanifold of M. We may think of T as the surface of the star at a fixed time. We can now send out light orthogonal to this surface, either in direction to the centre of the star
Everyday experience suggests
that the
centre should converge while the
light
or
light
into the
opposite direction.
congruence directed to the
congruence directed into the op-
posite direction should diverge. However, this does not take into account the extrinsic curvature of the spacelike hypersurface which represents the instant of time. There are many examples where both Songruences const < 2m, t const in the converge, for instance the surfaces r Schwarzschild solution. These surfaces are in the black hole region of the Schwarzschild solution and the general interpretation is that the gravitation of the black hole is so strong that it forces even initially outgoing light rays to converge. Since the normal bundle of T in M is a Lorentzian plane at each point, there exist two future directed null vector fields N+, N_ along T which are orthogonal to T and satisfy (N-, N+) -1. They are unique up to transformations of the form N i--+ a1N, where oz E ==
=
=
C" (T, R+
\ 10 1).
390
Singularity theorems
9.
Let
x
E
T and
jeAjA=2,...,,,_j
be
an
orthonormal basis of
the requirement that both verge
can
light concruences immediately be expressed by the inequalities
n-1
E
Then
start to
con-
n-1
g(VeA N+, eA)
<
E g(VIA N-, 6A)
0,
A=2
< 0-
(9.2.1)
A=2
This requirement can be formulated in a manifestly invariant way using the mean curvature vector field H of T (cf. Definition 4.4.2).
by
Definition 9.2.1. A closed
(future) trapped (n-2)-surface (respectively (future) strictly trapped (n 2)-surface, closed (future) marginally is a closed (n trapped (n 2)-surface) 2)-dimensional spacelike closed
-
-
submanifold pointing and The
mean
Any
vector
-
T such that the causal
mean
curvature vector
E
where VA
T,,M
(V, eA)
H
of T
is
past
(respectively timelike, null).
curvature vector field is defined v
field
can
be
decomposed
and v
by
as V
H =
(v, N:F). Using
I:n-1 A=2 1(eA eA)
n-2
VA eA this
,
+
V _N
-
+
v+N+,
decomposition
,
we
obtain n-1
H n
2
1:
(_ VeA N-) N+ (Ve AeAi N+) N-) e-Ai
-
A=2
n
2
(trT(X-)N+
+
trr(X+)N-),
(9.2.2)
VN
where X are the null second fundamental forms. Like N, these null second fundamental forms are uniquely defined up to transformations of the form X
-->
a1 X, where
a
E
C' (T, R+
\ f0j).
The null
gAB (X)AB' It is clear that T is a strictly closed expansions are 0 trapped surface if and only if both null expansions are everywhere negative on T and therefore equivalent to Inequalities (9.2.1). =:
9.3 The We
singularity theorem of Hawking
are now
ready
to state the main result of this
Theorem 9.3. 1. A spacetime
(M, g)
and Penrose
chapter,
is not causal
geodesically complete
if
(i) the strong energy condition and the genericity (ii) The chronology conditions holds, (iii) There exists at least one of the following: (a) a strictly closed trapped surface, (b) a compact achronal set without edge,
condition
hold,
9.3 The
(c)
point
singularity
theorem of
Hawking
and Penrose
391
such that
along every past (or every future) inextengeodesic from x the expansion of the null geodesics starting at x becomes negative. a
x
sible null
Strictly
closed
trapped surfaces are expected to surround very dense paradigmatic example is the Schwarzschild solution. Further evidence is provided by some theorems which prove the existence of strictly closed trapped surfaces if the concentration of matter is high (Schoen and Yau 1983; Bizo'n, Malec, and O'Murchadha 1988). Condition (b) is satisfied for spatially closed universes such as the Robertson Walker spacetimes with positive curvature. Condition (c) seems to be satisfied for our point in the universe (assuming a spacetime which differs only slightly from a Robertson Walker cosmology). This indicates that there was a big bang or that there will be a big crunch. (For more details cf. (Hawking and Ellis 1973, p. 358)). Theorem 9.3.1 will follow as a corollary of the following proposition. stars. The
Proposition gether.
(i)
9.3.1.
The
following
every inextensible causal
three conditions cannot hold all to-
geodesic
contains
a
pair of conjugate
points,
(ii) (M,g) is strongly causal (iii) there is an achronal set
A such that E+ (A)
or
E-
(A)
is
com-
pact.
Proof that Theorem 9.3.1 follows from Proposition 9.3.1. Assume, that (M, g) is causally geodesically complete and satisfies the chronology condition. By Corollary 9. 1. 1, the strong energy condition and the genericity condition imply that any inextensible causal geodesic has a pair of conjugate points. In particular, there do not exist maximal, inextensible causal geodesics. It follows that (M, g) must be strongly causal, since otherwise it would contain an inextensible achronal. null geodesic by Lemma 8.3.7. If (M, g) contains a strictly closed trapped surface T, then E+ (T) C aJ+(E) is generated by null geodesics. These null geodesics are orthogonal to T and the definition of a strictly closed trapped surface implies that each of them has a focal point (cf. Proposition 4.6.2). Since T is compact and E+(T) is generated by null geodesics without focal points it follows that E+
that in If
(c)
(M, g)
E+ (A)
lary
case
=
8.3.1
(T)
is also be
the set
contains
E+(x) a
compact. An analogous argument shows
is compact.
compact achronal.
A. This follows since E+ (A)
imply
that
through
every
generator of E+ (A) which intersects also compact.
=
point
set A without
J+ (A) x
edge(A).
\
edge, then
1+ (A) and Corol-
E+ (A) \ A there is a Hence the set E+ (A) is
G
392
Singularity theorems
9.
Properties (i)
-
(iii)
of
Proposition
9.3.1 would therefore have to hold
under the conditions of Theorem 9.3.1 and the additional assumption I that all causal geodesics are complete. The idea for the
proof of Proposition
9.3.1 is
but the
simple
proof itself is
quite involved. We will therefore first give an outline and then establish a sequence of lemmas which will imply the proposition.
Suppose, (i), (ii), (iii) in Proposition 9.3.1 hold and assume without generality that E+(A) is compact. We will show that the horizon H+(E+(A)) is non-compact or empty. Every non-vanishing vector field U must have a future inextensible integral curve 7 in D+(E+(A)). Otherwise we could map the compact set E+(A) along the integral loss of
curve
H+(E+ (A))
of U onto
(and non-empty),
which in turn would have to be compact apply a similar construction to the past of
too. We
E+ (A) n J- (7) and obtain
an
contained in D (E+ (A)). This extensible maximal causal
inextensible causal curve can
geodesic
in contradiction to
In order to carry out this program 1. H+ (E+ (A)) C H+ (,9J+
3. There is
a
A c
:= E+ (A) D-(E-(-'F)).
5. There is
we
n J-
(-I).
is non-compact 7 C
curve
Then there is
inextensible causal
an
wholly an
in-
(i).
will prove the
future inextensible timelike
4. Set Y
M which is
following facts.
(A)),
Cauchy horizon H+(E+ (A))
2. The
curve
then be used to construct
a
or empty. D+ (E+ (A)).
past inextensible
curve
geodesic without conjugate points
in
D(E- ( F)). The last property 4 is in contradiction with Lemma 9.3.1. Let A be
a
(i)
of
Proposition
9.3.1.
closed achronal set. Then the inclusion
H+(E+(A))
C
H+(,9J+(A))
holds.
Proof.
Let
x
H+ (E+ A)) \ H+ (,OJ+ (A)). From E+ (A)
(E
obtain D+ (E+ (A)) C D+ (,9J+ (A)) and therefore Hence there is
a
y E
1+ (x)
n
open set
1-(z)
therefore
D+(E+ (A)).
is
a
a
E
I-
C YJ+ (A) we (D+ ((9J+ (Affl.
D+ ((,9J+ (A))).
We will first show that I+ (x) n
Assume that there is
x
point
z
E
1-(y)
aJ+ (A)
neighbourhood
of
x
E
does not intersect 9J+ (A).
1+ (x) n I- (y). H+(E+ (A)) and
n
Since every past inextensible timelike
Then the intersects
curve
with
(A)) intersects E+ (A) c aJ+ (A) we would find a point E I- (z)naJ+ (A) C I- (c)J+ (A)) nc)J+ (A) in contradiction to the achronality of aJ+ (A). Since I- (y) is a neighbourhood of x E H+ (E+ (A)) and 1+ (x) n1- (y) does not intersect 9J+(A) there is a past inextensible timelike curve future
endpoint
in D+ (E+
9.3 The
-y which has future
singularity theorem
endpoint
of
Hawking
and Penrose
y and does not intersect
393
E-4- (A). From
that -y does intersect aJ+ (A) at some point z. D+((,9J+ (A))) Let M be the generator of aJ+ (A) with future endpoint z. By Corollary 8.3.1 this generator is either past inextensible or intersects edge(A). We y E
we see
will show that both
cases
lead to
Assume first that there is
a
a
contradiction.
point
G
edge(A)
which is intersected
M. This point also lies in A since A is closed. Hence /-t is contained in J+ (A) which in turn implies z E J+ (A) n aJ+ (A) E+ (A). This is a
by
=
contradiction to the construction of -/. Assume that /t is past inextensible and does not intersect A. Since
endpoint in D+ (,9J+ (A)) it intersects the set implies that /.t intersects I- (c9J+ (A)) (cf. Lemma c 9J+ (A) gives a contradiction to the achronality
-y is timelike and has future
int (D+ (aJ+
8.3.6).
(A))).
This
The inclusion 1L
of 9J+ (A).
I
Lemma 9.3.2. Let A be
causal. Then
a
H+(E+(A))
closed achronal set such that J+ (A) is is non-compact
or
strongly
empty.
Proof Suppose that H+ (E+ (A)) is non-empty but compact. Since J+ (A) strongly causal, H+(E+(A)) can be covered by a finite number of convex neighbourhoods Ui with compact closure such that no Ui is intersected twice by any causal curve. Let zi e H+(E+(A)) and Ui(,) be one of the convex neighbourhoods Ui with zI E Ui(I). Because of is
Lemma 9.3.1 there is
a
Lemma 8.3.8, there is
a
point
x,
E
J+ (A)
n
(Ui(,) \ D+(,9J(A))). By
timelike past inextensible curve a, through x, which does not intersect D+ (aJ(A)). Hence a, neither intersects'9J+ (A)
D+(E+(A)). Since a, does not intersect OJ+(A) it is contained in int(J+(A)) I+(A). The curve a, leaves Ui(,) because of its compactThere is ness. a point yl E a, \ Ui(,) c I+ (A). Let 31 be a past directed timelike curve from y, to A. Since A c E+ (A) and E+ (A) is an achronal (topological) hypersurface this curve must intersect D+(E+(A)) and thereforealso H+(E+(A)). Let Z2 cz 13,nH+(E+(A)) andletUi(2) beone of the convex neighbourhoods Ui with Z2 C Ui(2). The neighbourhoods Ui(,) and Ui(2) are different since by construction we have Z2 E J-(Zl) nor
=
and since tion
we
no
Ui
obtain
Jui(k)JkEN
can
be entered
by
any causal
curve
twice.
By
induc-
infinite sequence of pairwise disjunct neighbourhoods in contradiction the finite number of sets Ui. I an
Lemma 9.3.3. Let A be
a
closed achronal set such that J+ (A) is
strongly
that E+ (A) is compact. Then there exists a future inextensible timelike curve -y which is wholly contained in D+(,4'+ (A)). causal and
assume
Proof. Without loss
of
generality we can assume that (M, g) is time a timelike, time oriented vector field V on
oriented. Hence there exists
394
theorems
Singularity
9.
M. Since E+ (A) is
achronal
hypersurface all future directed timelike are initially in int(D+(E+ (A))). If curves with past endpoint H+ of V intersected (E+ (A)) after having intersected every integral curve an
E+ (A)
in
E+ (A),
we
would obtain
a
continuous map E+ (A) -4 H+ (E+ (A)), x 1-4 t(x) > 0 the unique number
where F is the flow of V and
Ft(_-)(x) Ft(,,)(x)
with
H+(E+(A)).
E
This map would be surjective
because,
8.3.8, every past inextensible timelike curve which intersects by the event horizon of a closed set must intersect this set as well. Since Lemma
E+ (A) is compact, H+ (E+ (A)) would also be compact in contradiction to Lemma 9.3.2. Hence there is at least curve
Lemma 9.3.4. Let
(M, g)
be
a
future inextensible
one
integral
int(D+(E+(A))).
-y of V which is contained in
causal
I
geodesically complete
and
strongly
causal spacetime in which every inextensible causal geodesic has a pair of conjugate points. Let A be a closed achronal set with compact fu-
E+ (A) and let ^/ be
ture horismos
future inextensible
a
timelike
curve
in
D+ (E+ (A)). Then there exists tained in D- (E-
Proof. We will first Let
I+ (.
)
x
E E-
would be
past inextensible timelike curve A which is E+ (A) n J- (-y).
a
(,F)),
where F
show the inclusion E-
(-'F) \
F. If there
x
E
curve
I-
neighbourhood achronality
a
through
x
z
a z
E
of E+ (A). Hence I
I+ (x)
to -y. This
n I-
curve
D+ (E+ (A)) and I- (x) n E+ (A) is in E+ (A) n I- (-y) C JF we obtain x
7 C
=
assumption I-
(-y)
=
E E- (.F). Hence
x
OJ-
(-y)
(-().
T U aJ-
C
Jr' E I-
(x)
n
E+ (A) then
(E+ (A)) (x) n E+ (A) 0.
of x and therefore intersect I-
then there is
(-y)
from
(.F)
point
was a
in contradiction to the
If
con-
=
we
and the assertion E-
The set T is the intersection of
E
have
a
(-y).
-
=
Denote
by
must intersect
p
a
timelike
E+ (A) since
0. Since this intersection point
(T)
Ix
(.F)
E
J-
C F
closed and
in contradiction to the
(.F) \
I-
(-y)
C J-
U,9J- (-y) follows. a
compact
set and there-
fore compact. Since 7 is future inextensible, all generators of OJ- (-Y) must be future inextensible as well. Suppose, there was a sequence Oi of
generators of
E-(T)
there would exist
a
with
diverging affine lengths.
cluster
curve
3 Of
PiIiEN
Since _'F is compact,
which would be past in-
geodesic prolongation would be an inextensible 9Jof (7). By assumption this generator cannot be achronal generator which gives a contradiction to the to the achronality of OJ- (-y). Hence E-(-F) is compact and we can apply the time reverse of Lemma 9.3.3. extensible. But then its
1
Lemma 9.3.5. Let C be
a
future inextensible timelike
compact subset of M. If D+ (C) contains a y and D- (C) n J- (-/) contains a past
curve
inextensible timelike curve A, then D(C) geodesic without conjugate points
contains
an
inextensible causal
9.3 The
Proof. Let f YiliErq be point such that
singularity
a
theorem of
sequence of
points
Hawking
and Penrose
395
in 7 without accumulation
I+(yi). Choose a sequence f-TibEN xi E I+(xi+,) for all i. For every i
yi+1 E
that yi E 1+(xi) and a causal curve [ii which
in A such we
obtain
joins xi via C to Yi. This causal curve is contained in a globally hyperbolic set (Theorem 8.3.1) and can therefore be replaced by a maximal geodesic segment pi (Proposition 8.2.2). Without loss of generality we have pi(O) E C. Then the oriented half lines
f (R+ \ f01) Ai(O) -
causal directions
geodesic
p with
any cluster
have
a
i E
A(O)
(i)
accumulation point f in the space of space is compact. Any inextensible
an
E
f is
cluster
a
curve
of maximal
Proof of Proposition
9.3.1
have
NJ
C, because this
geodesics conjugate points.
curve
pair of
:
over
Applications
9.3. 1.
We
of the
is
of the sequence fpibErq. Since maximal, the curve p does not
I
only need
to choose C
=
E-
1
singularity theorem
Consider the Schwarzschild solution. It satisfies all assumptions of Theorem 9.3.1 with the exception of the genericity condition. However, it seems plausible that any generic perturbation of the Schwarzschild solution using a reasonable matter model should result in a spacetime which satisfies all the assumptions. Here it s important that the existence of
a strictly closed trapped surface is an open condition, i.e., if a spacetime which contains a strictly closed trapped surface is slightly perturbed then this surface is also a strictly closed trapped surface in the perturbed spacetime. Hence Corollary 9.1.1 indicates that the Schwarzschild singularity is stable under (physically reasonable) perturbations. In particular, it is not an artifact of the high symmetry of the Schwarzschild spacetime. This application of Corollary 9.1.1 is one of the main reasons why the existence of black holes is widely accepted. (ii) Consider a Robertson Walker solution without cosmological constant and spacelike hypersurfaces of constant, positive sectional curvature. These spacelike hypersurfaces t const are compact achronal sets without edge. The Ricci tensor, which is given by =
Ric =T-
is
tr(T)g 2
-1
.
positive definite for
2 e
(3(c+p)
> p > 0.
+
(E-p))O 00
Hence in this
case
+
(E-p)g
the timelike
)
I
con-
vergence and the genericity conditions are both satisfied. It follows that all assumption of Theorem 9.3.1 hold even if the spacetime is
slightly perturbed. This indicates that (at least for closed universes) big bang is not an artifact of the symmetry properties of the Robertson Walker cosmologies.
the
Singularity theorems
9.
396
(iii)
Consider
again
genericity and the timelike will
now
function t
g
(M, g)
show that
=
F-4
+
of
arbitrary
> p > 0.
Then the
convergence conditions are satisfied. We a strictly closed trapped surface.
coordinates
are
a2(t)
( -1-r2
Ti,, )
and
clearly compact
(
1
N
dr2
r2
+
I
Consider the codim-2 surface surface is
c
(t, r, 0, o)
and
positive
a
such that
a(t)
-dt2
that
assume
contains
6.1.2 there
By Corollary
(M,g)
Robertson Walker solution
a
constant sectional curvature and
=
2
fx
=
(d02
+
t(x)
:
sin2 (O)d
=
i, r(x)
02) J.
=
This
The vector fields
spacelike.
1_6 r2 ar
at a
along Tj,, are normalised null vector fields orthogonal corresponding 1-forms are given by
to
Ti,p
and
the
-1
(N)
a
dr
dt 2 VJ_--E r
2
Hence
0
trT,,,,, (7N)
(VaON
a2r2
VaONI(ao)
-
a2r2
where for the third equation for i E f t, rJ we have r,9 0
we
=
obtain
Hence for these
sin2 (0)
(aw))
Va N
roto
=
-atala
and
ro'O (N) i,
have used
1
29
ii
FO'O
spherical symmetry.
-Ig"ai (a2r2) (2aogjo aigoo) 2 -1/r. This implies
=
2
(1'Ot,9(N)t + Foro(N)r) a
<
1, Ota
inequalities
are
> 0
Since
-
2
a2r2
+
2
2 2-2 a r
we
(ao)
=
a2r2
(_,9ta 2a
a
:F r
V/'J_--Er2
)
.
both expansions, 0 are negative. Since near the big bang we can apply the
satisfied
of Theorem 9.3.1 to infer the existence of
a singular4 the is big bang. singularity just example our spacetime slightly all assumptions of Theorem 9.3.1 are still satisfied. Hence we can conclude that the perturbed spacetime also contains an incomplete, inextensible causal geodesic. It is therefore natural to expect that the big bang is stable under perturbations of the metric.
time
reverse
ity. In fact, in If we perturb
4
our
this
Note, however, that Theorem 9.3.1 does not make any assertion about the location of the singularity. In particular, it does not assert whether it is to the future or to the past of the strictly closed trapped surface.
9.4
(iv) Using
condition
(c)
Singularities
our
causality violations
397
of Theorem 9.3.1 it is also
ments in favour of the existence of
that
and
universe is well described
possible to give argusingularity without assuming by a Robertson Walker space-
a
time. This argument microwave
requires assumptions on the spectrum of the background radiation and is therefore beyond the scope
of this book
9.3.2 General
(cf. (Hawking
and Ellis 1973, pp. 354
problems with Theorem
These
physical applications
which
are
359)).
9.3.1
of Theorem 9.3.1 suffer from two defects
often considered to be
mation of
-
negligible. Firstly, instead
of the for-
singularities curves. Secondly, even if singularities occur, Theorem 9.3.1 does not predict their strength. For a long time, it has been thought that these problems are only technical and that the theorem could be sharpened accordingly. Unfortunately, this is not the case. In Sect. 9.4 we will present an example (due to Newman) that the chronology condition is necessary for Theorem 9.3.1. In Sect. 9.5 we will show that Theorem 9.3.1 may only predict the existence of singularities which are too weak to be taken seriously by most physicists. At the time of writing it is not clear whether it is possible to improve on Theorem 9.3.1 if additional physically realistic assumptions are made. It should be remarked that other singularity spacetime could form closed timelike
theorems suffer similar defects.
9.4 The
Singularities
and
chronology assumption
therefore
can
causality violations in Theorem 9.3.1 is
not be verified
by physical
considered to be self-evident
we
to be debated. It is therefore
have
a
global assumption
and
measurements. While it is often
seen
in
Chap.
8 that this view is
important question whether Theorem chronology condition is dropped. In this section we will give an example due to R. P. A. C. Newman (1989) which proves that the chronology assumption is essential. We will then quote a generalisation of Theorem 9.3.1 which sheds some light on what is going on. 9.3.1 continues to hold
even
an
if the
9.4.1 The G,5del solution The
proofs of Propositions 9.4.1 and 9.4.2 consist of straightforward long calculations. We will not spell them out in all details. However, we will provide enough information such that a careful reader equipped with pen & paper (better: with access to a symbolic computing program such as REDUCE, MAPLE or MATHEMATICA) should be able to fill in the missing details. but
9.
398
theorems
Singularity
In 1949, the famous mathematician Kurt G6del published a new solution to Einstein's equation for a dust matter model with a cosmological con-
completely homogeneous and has the property that timelike curve through each point. Newman's counter
stant. His solution is
there is
a
closed
is
example
modification of the G6del solution.
a
(t, x, y, z) be standard coordinates of R' and w E spacetime (R 4, -dt2 +dX2 + 12 e2' `d y2 +dZ2 -2e1'2w-`dtdy)
Definition 9.4. 1. Let
R+
0 1. The
is called the G6del solution.
The G6del solution is
9.4.1.
Proposition satisfies
spacetime and
Lorentzian
a
2
Ric
Scalg
-
w
g
2
It
corresponds therefore
w
2 -
to
81r
=:
(C)t
at
-
41r
dust solution with
a
negative cosmological
constant.
Proof.
For the first claim observe that
090
9
For the second claim metric
we
(cQ
dX2
+
1 +
+
e2V"2wxd Y2
e"2xdy) implies
+
dZ2.
2
will have to calculate the Ricci tensor. Since the
on
we
-
Vt t
Fttx
-(dt
only one variable this is a simple task and left to only note that the values of the Christoffel symbols
depends
reader. Here
0
(,Ot)
2w ,
Ftyx
rx, Y
=
v"2 w e
F,,Yt
we""-2
'"
Ft YX
72
-
' '2 " x ,
Ftxy
=
Fyxt
the are
vr2wx
=\1-2we
2v 2wx
I'x
-we
yy
2
symbols vanish.
where it is understood that all other Christoffel
The
Ricci tensor reads then Ric
and
we
=
2w
2
(dt2
e2 vr2x dY2
2ev2)xdtdy)
+
=
2U,2 (at
obtain
Ric
The
+
following
-
lemma
Scalg 2
.
W2
implies that
(2 (at)
(,Ot)
b
+
g)
.
the G6del solution is
homogeneous
in
space and time.
Lemma 9.4.1. For any two points p, q q. 0: M --+ M which satisfies 0(p) =
G
M
there is
an
isometry
9.4
Proof. The spacetime
Singularities
M admits the
and
causality
following
four
+ a, x, y,
z)
violations
399
I-parameter
groups
of isometries 5.
01 (a): (t, x,
y,
z)
-
02(a):
x, y,
z)
F-4
03 (a):
-X, Y)
Z)
X, Y + a,
Z)
X, Y)
Z)
X, Y,
+
a)
04 (Ce)
:
(t
7
(t
It follows that for any p, q E M there P
:::::
04 (CQ)
03 (013)
0
0
02 (Ce2)
The
following lemma implies singularities.
o
+ a,
x
are
Z
Z)
ye-
numbers al,
04
such that
01 (al) (q).
I
that the G6del
Lemma 9.4.2. Each inextensible causal
spacetime does
geodesic
(M, g)
in
not contain
is
complete.
Proof. We will first partially solve the system of equations for geodesics. practical to work with slightly different coordinates. Consider the global coordinate transformation 0: M --+ R x R+ \101 x R 2, T, v,"2-wx , i) where t (t, x, y, z) i--> ln(V2_wJc), y /2_ , and To this end it is
=
=
=
-
In these coordinates the metric reads
z
I
-(d+
g
We have the
The
6 These
Killing
d )2
+ W
2' C7- 2
(dj 2
d 2)
+
+
di2.
I-parameter families of isometries
V) I (a):
(i,
, i-)
-4
(
02 (a)
(ij
I
)
-4
( ) (I
7
0
-->
V7
7_)
-4
:
03 (01):
(7-,
04 (a):
(f,
:
,
1
=
(9E,
6
=::
vector fields
+ a, x,
+
, -)
a)_: _, (I
+
7,
+ a,
-)
(f, _: j
_; +
ce).
vector fields
corresponding Killing
sic, (6j,
I
W _ --
j
+
a) , ;E)
are
0 ,
63
4
give four constants Writing -y(-r)
of motion
along
ci is constant.
the
geode-
i(-r))
we
obtain
C,
=
-t
Y WX
This is
a
x 2
yx
2wJU
-
-
-
TW 2,;,
-
-'
linear system of equations for
W2X2
(fx,y-,z-),
C3
=
Z-
and its solution is
given by The isometry group is five-dimensional, but the additional group of isometries is not important for the argument.
1-parameter
400
t= cl
If
Singularity theorems
9.
-
2w -:
C3)
we rename
2w
X
(Cl, C2, C3
2
(C2
C3)i
d,
V
our
without loss of
Values for
projection of
V)
-
geodesics
we
d Z=
-7
: 1)2
+
-y to the
have the
(9.4.3)
--
'
C
generality that
and radius
cle with centre
causal
obtain
C
i into this
/)2 Hence the
C4
X
C
Inserting
: -C3)'
V2-w-i-
2 j;
v/2-C v/2_wJ (.'r
can assume
we
T
v2C!
t
We
+
:
C
,
20)2;,
Y
equation
-
d
-
2
we
V
2 ' -
2
traverses
d2
1
qC2.
+
2
(x, y)-plane
1, 0,
71 E
obtain
+,qC2.
an
arc
of
a
cir-
Observe that for
inequality < -
which
R+
that the circle is
implies
\ 101
the
R. In
2
wholly contained in a compact subset of coordinates: , in remain bounded along
particular, is bounded imply that Since these coordinates are global, it follows x
9.4.1
Equations
,y.
must range from
-oo
The G6del solution is see
this,
we
g
=
coordinate system.
I
oo.
axially symmetric
2(
There is
9.4.2.
E R
2w-
our
that the affine parameter
with respect to any
point. To
will introduce different coordinates.
Proposition
(8, r, 0,,i)
to
in
x
-
R+
d82
\ f01 +
dr2
x
a
S'
dense open set N C M and coordinates R such that 91N is given by
x
+ sinh
2r(I
-
sinh
2
r)d W2
+ d'
2
-
2V2-sinh 2
rd ods).
Proof. Observe that the metric is a direct product, (M, g) (R 3 -dt2 + dX2 + Ie 2Nf2"d y2 2eV2"dtdy. Hence it R, h + dz 2) with h =
=
-
2
sufficient to show that there is
(s, r, o)
hll
=
a
dense open set
9
C
X
is
R 3 and coordinates
such that
2L,)2
(-dS2
+
dr2
+ sinh
2r(i
-
sinh 2r)d
02
-
2N 2 sinh 2
rd ods)
.
9.4
Singularities and causality violations
The assertion follows then with i
12
=
V2w
via the
(s, r, W)
=
cosh(2r)
=
sin(W) sinh(2r)
=
e-
12wx wye
I
(2 ((p
We define the coordinates
equations
ev2wx
tan
z.
+ wt
V2-s)
-
2r
+
cos(W) sinh(2r)
(21W)
tan
(9.4.4)
(9.4.5) (9.4.6)
-
To show that h has the desired form in these coordinates is
ward" but very cumbersome. 6 We first differentiate Equations
vl2weV2xdx
0
+
-
2(sinh(2r)
-
1
0
2
-
sin(W) cosh(2r)dr
cos(W) sinh(2r)dW
(1 (tan(W/2 112wt (1'+ (tan(W/2 wt12 +
-
+
I +
2
(
-2r -
e
(tan(W/2
(I
+
+
)2) s/v/2)) 2) V2
112V2-s)
-
2
+
to obtain
sin(W) sinh(r)dW
+
2
(9.4.4)-(9.4.6)
"straightfor-
cos( p) cosh(2r))dr
+
V/2_W2ev/2-w xydx + evf2-w xwdy
0
401
_ds
(tan(W/2) )2
wt12
_
S
/ N/2-)
wdt
+ 2e
-2r
tan(W/2)dr
+ 1
) 2dW.
V2wx system of equations we can eliminate y and e using Equations The and then be considered can as a linear system (9.4.5). system (9.4.4) for dt, dx, dy which only depends on s, r, W, ds, dr, dW. Solving this linear system gives (after some simplifications)
In this
2
V2-ds dt
-4 W
w
sm(W/2) cos(W/2)e rdr 2 (Or COS2 (W12) +sin (W12)
(- sin 2((p12)
+
4r
2r e
-
e
+ W
v2 dx
1/2
(Or COS2 (W12) 4r
(2e COS2 ( o12) (e 4r COS2 (W12)
w
In
+
COS2 (W12)) dW
(sin(W/2)
)2)
sin2(W12)) dr + sin (W12))
-
2
2
fact, in his original paper, G6del chose to derive this form independently geometric assumptions which led him to the metric.
of the
402
9.
Singularity
theorems
V2_sin(112 o) cos(W/2) (I e4r ) dw (Or C092 (W12) + sin 2 (w12)) -
+1/2
w
dy
sin( o/2) cos(W/2)e 4'dr
4
=
(Or COS2( p/2) + sin 2 (V12) )2 (sin2(W12) + es' COS2 (W12) e4r ) dW +1/2 w (e4r COS2 (W12) + sin%o/2) )2 w
-
We
simply calculate
can now
h
tion
=
e-
2r
-
2
(s, r, p) using (9.4.4) (which is equivalent
ev"2wx
-dt2 + dX2 + Ie 2/2x dy2
=
in the coordinates
(e
4r
our
expressions for
2eV2"dtdy
dt, dx, dy and Equa-
to
COS2 (W12)
(sin( p/2)
+
)2)).
This gives with
A(r, p) B(r,W)
sin2( p12)
=
-sin
=
+
2(W12)
e8 COS2 (W12) r
2
+
e
-
e4
e4 rCos2(W/2)
r -
and using trigonometric identities
ds2 g
-2
C2
-4r
+
v/2-e -2 rA(r, W)
(
+
(8e (e4
W2
(e4
r
+ 2B (r,
COS2 (W/ 2) +
sin2(W12))
2(W12) COS2 (W12) (e 4r)2 COS2 (W/2) + sin2( p12) )2
r
(sin2(W12) e4 COS2 (W12) )2 dr 2 W2 (e4 COS2 (W12) + sin2 (W12)) A (r, W) (B (r, 0))2 + B(r, o)e r
)
-
r
-2
+
W2 1
9
(e4
-4r e
r
COS2 (W12) +
(A(r, W))2 W
2 w-
dW ds
sin
W2
+2
W) V2_
2(
-
2
d '32 + dr
(e4 2
r
I
'r
(sin( o/2) )2
COS2(W/2) sin2(W12) (I
_
COS2 (W/ 2) +
+ sinh
)2
2 r
(1
-
-
sin2( p12) )2
sinh
2
r) d W2
e4r)2
d p2
+d,z2
2V2 sinh 2rdWds).
We will
now
const,
=
s
dates for
show that there
are spacelike submanifolds given by r vanishing expansions. They are our canditrapped surfaces in a suitably perturbed metric.
const which have
strictly closed
9.4
Let
Proposition 9.4.3. 2 surface given by s <
In(I
v'-2).
+
E r
=
R,
case
its
I
2
In this
-
H
Tg,p
Then
.
=
R+ \ 101, and
E
sinh2(r)
N-),
(N+
2V2-sinh(r) cosh(r)
be the codim-
Tg,,
spacelike if and only if vector field reads
is
curvature
rnean
403
causality violations
and
Singularities
where
/l N
a
(r) -01+
-0
f-
v2
are
2
sinh
-
=
,F2
cosh(r)
pair of normalised null
vector
cosh(r)V1 sini7(r)
av
-
fields orthogonal
to
Tg,, .
h + dz 2 is a direct product it is sufficient to prove Proof. Since g the analogous assertion for h. For any x E Tg, the tangent space of Tg,p is spanned by Ov. It is clear that h(i9,, 9v) > 0 if and only if r < 1 sinh- (1) In(I + v'2-). We first calculate N. From =
=
0
we
get
=
h (N,
c9w)
sinh
=
(r) (1
V2 The T--s-inh--2-(r-) NI.
N`
-
sinh
2
(r)) NI'
V2 sinh.2 (r) N'
-
equation
g(N, N)
0
_(NI)2 -
(N )2
+
1 + sinh
sinh
-
Cosh(r)
r
implies N
N
_
V(1-sinh2(r)
(r)
2
(r) ((I
sinh
-
2
(r)))N 0)2
'
(Ntj)2
(r) and therefore
_
I N+,
2
cosh(r)
9,
=
2
_
1
We normalise
+ sinh
2-\/2 sinh 2(r) N'P NI
(Nr)2
N
2
-
sinh
2
a,
(r)
v/'2-2 (r) aw
+
-
sinh
I
by demanding (N+, N-)
N-
=
)
NI.
-1. This is
equivalent
to
cosh 2(r) 1
-
sinh
2V2 sinh 2 (r) I 2
cosh2 (r)
-
sinh
2
(r)
2 -+ sinh
(r) V2-
-
t
-
I
2
(NI
sinh2 (r) 2
2(r) (I (Nt )2
-
sinh
2(r))
I
-
sinh
2
(r)
404
9.
Singularity theorems
whence 2
N
N'
W2
(-V2
+
V2 sinh 2(r)Nf
-
sinh
2
(r)Nj
+ sinh
VF sini (r)
2
2
(N) dr
ds +
(r) (I
-
sinh
d o)
2(r))N '
2
_,v2 sinh (r)
-
ds
_
U)2
cosh
V2
+
sinh 2(r)
2
V2- l
can now
interested in
need
V1
sinh
-
inverse of h is
sinh
(r)) 2
d o)
(r)
=
2
aP,
((N )s-FW8,p (N )rF rOV)
-V '
+
symbols Fs,,,
Fr.
note first that the
(r)
sinh
simple
2 NF2
1
(as)2+(ar)2+
2
(r)
cosh
2
(r)
(aw)2_
cosh
2
(r)
aW as
exercise to compute
1hr(,OwhrW +,OwhWr -,9rhwp)
Fr W (P
2
Ihrr,9rhw o
2
2
9r (sinh 2(r) (I
sinh(r) cosh(r) (1
Ih's(,9whw,
S
r,' W
+
2 =
Hence
trT
2
given by
sinh2z(r)
cosh now a
-
covariant derivative in direction
hPPV,,N (ap)
=
=
It is
sinh
-
dr
721
(r)
dr
v -2-
(r)
2
ds 2
In order to calculate the Christoffel
-
2
=
only calculate the
1
sinh
calculate the covariant derivatives VN. Since we are only and TTg,p is spanned by 19,P we trT,,,,,(X)
tr,T,,,,(VN )
hO
-
(r)+ sinh 2(r) (I v -2cosh(r) cosh(r) V1
sinh
-
V1
cosh(r)
cosh(r)
W2
We
(r)
-
2 sinh
2
-
sinh
2(r)))
(r)),
awh,w -,9,hww)
+
1hs '(Owhww +,9whvw
2
-
Owhww)
0.
we
get
sinh(r) cosh(r) (I
(VN) sinh I
2(r) cosh 2(r) 2sinh 2(r)
-
V2- sinh (r) cosh (r)
-
2 sinh
2(r))
9.4
and the assertion follows
now
Singularities
and
from
directly
causality violations
Equation (9.2.2).
Proposition 9.4.3 implies that the surfaces T,,,,,((,+,/3)/,/2) and have vanishing mean curvature vector field H. 9.4.2 Newman's
Consider the
(M7 (Here
(R we
view
S1, h
x
1
are
spacelike
example
following partial compactification
3
405
S1
as
dZ2)
+
where
the subset
of the G6del solution.
is the natural coordinate of
z
[0, 21r]
R, where
C
the
points
S1.
0 and
identified). Clearly, (1 1, ) is locally isometric to (M, g). Observe corresponding sets Tg,p are compact (and diffeomorphic to tori) In particular, the surfaces Tjn((1+v"3)/v/2) are closed (but not strictly surfaces. It is plausible that a suitable deformation of trapped closed) will result in (M, j) compact surfaces with past pointing, timelike mean 21r
are
that the
curvature vector field. This would
give
example which
an
is
causally
geodesically complete and satisfies all assumptions of Theorem 9.3.1 with the exception of chronology. In order to preserve the causal structure of the G6del spacetime we will deform j by multiplying it with a
-
conformal factor
f22,
k
j?:
-4
R. We need to calculate the
curvature vector field when
mean
is
j
replaced by
=
change
of the
p2j.
Lemma 9.4.3. Let
(M, g) a pseudo-Riemannian manifold and Q: M f?2 g and V (resp., t) the Levi-Civita connection of g (resp., Then for every vector field U and every 1-form A on M we have
R. Let
=
VUA
=
VUA
-
0-
1(dQ(U)A + A(U)dQ
Denote the difference tensor of
VUV.
Let
t
Proof.
centred at
(xl,... x
E
I
Xn ) be
M. Then
(C(,9x,.,,9.j))k
a
we
I 2 x.
Since this is
a
S3
From
ij
09xi ij
(
+
Okl Va Aj
tensor
C . everywhere.
Vj'k
1 2
by C, C(U, V)
=
=
_
k1
A(gradS?)g(U,
normal coordinate system with respect to g have at 0 at x and therefore
('Oxiax.,,)k (Vaxiaxj)k 2
at
and V
-
+
X1,
equation
kl (V, Ij
Q2 gij and
Vijk
ax-i ii
=
-
+
'j
Vj ,, ==
(ta
-i
ajXj)k
ax,, ij)
VaX.Iii
we can
==:
-
VaXAj
infer
_
Vl ,j)
S?-2gii
2Q(ViQ)gjk
we
get
^k
rij
406
9.
Singularity theorems
and therefore ryk
k1
I
9
ij
The assertion follows
(glj Vi Q
directly
now
+ gil Vj S?
from
tA
=
-
gij Vi f2)
VA
-
A(C(.,
Lemma 9.4.4. Let
Ts,, be the projection of Tg, to M and U, V be open and satisfy neighbourhoods of ! g,p which have compact closure of Ti C V. Then there is a function Let f : 1 1 ---> R+ such that f (X) (=- ( '2 DI 2 Oi) suppf C V, (iii) flu only depends on t, (iv) Of (x) < 0 for all x E -
1
Proof. This
is
a
and
simple application
of Lemma 2.1.7.
Proposition 9.4.4. Let In((1 + V3-)/- ,F2), -ro E (0, 1), f be the Lemma I + -rf (x) for all x, function provided by 9.4.4 and set S?, (x) Then the family , :== (flr)2j is a deformation of j which 7- E [0,,To). depends smoothly on -r and satisfies o j. Furthermore, there is a each such that cc -ri , (-r (0,-ro) (0,T1)) contains a strictly closed trapped surface. =
=
=
Proof The first assertion is trivial and we only have to show the existence a strictly closed trapped surface for -F > 0 sufficiently small. Consider the surface We must show that trt', (VN) are both negative. Here of
9 to
0
=
f?,-'N We
denotes
can
Vv
pair of normalised null
vector fields
orthogonal
" V& (N) (a;,)
+
(gwwta Q-1
a
restrict attention to U and obtain
(N) (aw)
g" t' az (N)
+
W
(gwP (Vaw(N) (ap)
-
(a,))
S?-'(-df2(N))gpv
=0
+
gzz
( V"z(N) 1
-
(a,)
2sinh
2
-071(-dQ(N))ga-,p.))
(r)
_+ v,r2- sinh (r) cosh (r) +
Q-'dQ(N)
)
f2-1
1
sinh2(r) dQ(N) cosh2(r)
-
9.4
I
-
2sinh
Singularities
2(r)
causality violations
V1
2
,F2 sinh(r) cosh(r) Inserting
and
cosh2(r)
-
sinh
407
2
(r)
-,/-2cosh(r)
this gives
r
01
2
Q-2.
==
cosh2 (r)
(19, Q)
Vrl7sin]? (r) < 0.
V2_ cosh(r) I
It is clear that for
conditions
(M, ,)
are
small
-r
enough the genericity
still satisfied. Hence for
-r
and timelike convergence enough the spacetime
small
complete but satisfies all assumption of Theorem 9.3.1 with chronology condition. It is possible estimate which kinds of causality violation can invalidate Theorem 9.3. 1. In the proof of Theorem 9.3.1 the future horismos E+ (T) of the strictly closed trapped surface T played a significant r6le. In a spacetime with chronology violations (and in particular in our example) this set is in general empty. However, there is a generalisation of the non-global features of a horismos. is
the exception of the
Definition 9.4.2. Let
(M, g)
be
compact set which is achronal in
a
time oriented
some
spacetime and D be
a
neighbourhood of U of D.
(i)
Let -y: [0, b) be a future directed future inextensible null starting in D. A point x -y(t) is called a focal point, if
geodesic
=
(a) for
all
-y(t+) (t+
arbitrarily
(b)
t)
>
there is
a
timelike
curve
from
D to
-Y(t+
close to -y,
There is not any t- < t such that D and -y(t-) by timelike curves arbitrarily close to -/.
can
be connected
by -yx: [0, b(x)) the maximal geodesic prolongation of the generator of E+ (D, U) with starting point x -y (0) E D which does not have a focal point. The generalised future horismos of D is the closure e+(D) of the set E M : x E Dj. The future focal set of D is defined by generalised (iii)
(ii)
Denote
=
+
f (D)
y E
e+ (D)
:
y is
future endpoint of
e-
(D)
and
f (D) -
are
some
generator
-y.,
of e+ (D)
defined analogously.
It is clear that for every spacetime and every compact set D we have E+(D) C e+(D) and D C e+(D). The future horismos is always a
Lipschitz hypersurface erty is also true for
with induced
e+(D).
degenerate
metric. A similar prop-
408
9.
Singula-rity 'l-beorems
Lemma 9.4-5. Let p C -
-
-
e+(D)
be
a
causal
curve.
Then p is either
geodesic generator -y., of e+ (D) or f + (D) or a concatenation a subset of a null geodesic generator + curve contained in f (D) a
subset
of
a
causal
curve
a
null
contained in
and
of
causal
a
.
If y is a non-trivial curve in e+ (D) which is not a part of a generator of e+(D) and does not lie in f+(D), then it is intersected
Proof
.
generator -y., of e+ (D). Hence there exists a broken, curve \ f + (D) with past endpoint in D. Let y (-= A be this after break and 7, the generator of e+ (D) with future point s
transversely by a
some
A in e+ (D)
causal
y. Using our broken causal curve we see that there is a timelike from D to y arbitrarily close to -y,,. This gives a contradiction to
endpoint curve
the definition of
e+(D)-
I
Hence any closed causal curve arbitrarily close to the generalised future horismos must lie in f + (D). This is exactly what happens in the example of Newman. The null
f + (Tg,p) which r
=
const,
z
=
geodesic generators of e+(Tg,,p) end in by closed null curves of the form s
const. Observe that these
The existence of these
curves am
not null
caustic
a
is ruled
=
const,
geodesics.
basically the reason why Theorem 9.3.1 causality violation. Notice that we can slightly such that f + (Eg,f ) is not ruled by closed causal our example generalise curve but only by almost closed causal curves. We simply replace the identification (s, r, W + 27r, z) (s, r, W, z) by an identification (s, r, W + curves
is
fails in the presence of
=
27r, z) a) such that the quotient a/27r is irrational. It is (s, r, W, clear that the curves -y locally defined by 8 const const, r const, z are not closed but satisfy instead: z
=
+
=
For each
-y(t)
-y(t+)
and each
=
(small enough) neighbourhood
> t such that the
there is
=
a t+ segment of leaves V and then re-enters this set.
between
V of
-1(t)
-y(t)
and
All other properties of
our example are unchanged since the new spacelocally isometric to the old one. In order to state a theorem which justifies the claim that the only impediment to a version of Theorem 9.3.1 in the presence of causality violation is the possible existence of almost closed causal curves in f + (T) we need the following technical definition.
time is
Definition 9.4.3. Let -y be a curve and choose any Riemannian metTic h on M. Let ft: (a, b) be a reparameterisation of -y which satisfies
h(A, A) JA(t) : exists
=
t E
a
1.
(a,
We call -y almost closed if there exists a vector u E that such for every neighbourhood it of u in TM there b) I
deformation A of 1,t
satisfies X(t)
E
WTM(it)
-,:*
in
(t)
7rTM(5-0 E
which
yields
a
closed
3A.
Observe that this definition is independent of the choice of h.
curve
and
ISItrength
9.5
of
Theorem 9.4. 1. A spacetime
singularities and
(M, g)
cosmic
409
censorship
is not causal
geodesically complete
the timelike convergence condition and the
genericity condition
if
(i)
hold,
-
(ii) there exists at least one of the following: (a) a (locally spacelike) strictly closed trapped surface T, (b) a compact achronal set T without edge, (c) a point x such that along every past (or every future)
inexten-
sible null
geodesic from x the expansion of the null geodesics starting at x becomes negative, (iii) neither f + (T) (respectively, f + (f xJ) nor any f (D), where D is a compact topological submanifold (possibly with boundary) with D n T = 0 (respectively, x E D) contains any almost closed causal curve that is a cluster curve of a sequence of closed timelike curves. -
This is
a
proper
(iii) just
tion
generalisation
of Theorem -93.1. The technical condi-
states the situation which
we have already anticipated by example. The proof of Theorem 9.4.1 is far too technical to be reproduced here. It basically consists of a cutting and pasting procedure (Kriele 1990).
Newman's
analyzing
The closed
trapped
surface in Newman's counter
example has the topol-
ogy of a torus. In a physically realistic collapse scenario of a star one would rather expect that there exists a closed trapped surface of topol-
S' surrounding the collapsing conjecture: ogy
star. This motivates the
Conjecture 9.4. 1. A 4-dimensional spacetime (M, g) cally complete if
(i)
the timelike convergence condition and the
following
is not causal
geodesi-
genericity condition
hold,
(ii)
there exists
a
strictly
closed
trapped surface
of
topology S'.
In Newman's is
example the generalised future focal set of T Jn((I+V 3_)/2) generated by closed null curves. This is impossible if T has topology
S2 In
open
spite of this small piece of evidence and the importance of Conjecour interpretation of singularity theorems it is completely whether this conjecture is true or not.
9.5
Strength of singularities
ture 9.4.1 for
In this section
dicted
we
will
by Theorem
and cosmic
censorship
investigate the character of the singularities pregive an example (cf. Sect. 9.5. 1)
9. 3. 1. We will also
410
Singularity theorems
9.
which shows that the theorem
of Hawking and Penrose may only imof "singularities" which are so weak that the energy exists in a distributional sense. Our example is not very physical for a start, it is 3-di'mensional rather than 4-dimensional. On the other hand, it is a good test case for the mechanism behind the singularity theorems.
ply the density
existence
-
The existence of
incomplete causal geodesic does not imply that there singularity. This is the reason why "singularity theorems" are often referred to as "incompleteness theorems". The standard counter example in general relativity is the Taub-NUT spacetime (cf. (Hawking and Ellis 1973, chapter 5.8). The following two-dimensional example is especially simple.
is
a
2 (Clifford-Pohl torus Let (M, g) (R 2\f0J, _UTTV_T dudv). Then the curve -y(t) is an and the incomplete geodesic 0) map 0: (u, v) (2u, 2v) is an isometry. Defining
Example
9. 5. 1
=
.
-4
x
we
obtain
a
-
y :<=
Elk G Z with
Ok(X)
compact Lorentzian torus
incomplete lightlike geodesic
=
Y,
7r: X
(ir(M), 7rg).
_4
The
[X]. curve
in this compact and therefore
7r(-Y)
is
an
non-singular
spacetime. While
Example 9.5.1 shows that there exist spacetimes which are nonsingular and geodesically incomplete, so far there is no example of such a spacetime which satisfies the assumptions of Theorem 9.3.1. In fact, it is believed that the incomplete geodesics predicted by the singularity theorems are of a different nature. The justification for this expectation is the fact that a non-compact or empty "Cauchy horizon" plays an important r6le in the proof of the theorem of Hawking and Penrose. Since the Cauchy horizon is necessarily closed, one would expect that a Cauchy horizon due to compactly imprisoned curves (such as in Example 9.5.1 above) is compact and non-empty. This motivates the following conjecture.
Conjecture there exists
9.5. 1. If the a
assumptions of Theorem 9.3.1 are satisfied, then incomplete geodesic which leaves every
causal inextensible
compact subset in both future and past direction.
Using dicted
an
by
approximation argument the theorem of
being C2- instead
of
gence of the curvature
one can
Hawking &
show that the
Penrose is not
C2 There also exist .
singularity prejust due to g merely
upper bounds
on
the diver-
along any incomplete geodesic -/ which are large enough to allow for the possibility of strong curvature singularities,i.e, curvature scalars which not only diverge but whose integral over an appropriate spacetime region diverges as well. Unfortunately, we have no curvature estimates for anything in between these two extremes.
Strength
9.5
of
singularities
and cosmic
censorship
411
singularity predicted by Theorem 9.3.1 were only only predict the existence of "shockwaves" and we would lack evidence for the existence of serious singularities such as black holes, The following conjecture (formulated by Hawking & Ellis (Hawking and Ellis 1973) with respect to a slightly different singularity If, weak,
in
general,
the
this theorem would
is therefore central to
theorem)
9. 5.2. Assume that
Conjecture
our
interpretation.
(M, g)
is
chronological,
satisfies the time-
like convergence and the genericity condition and contains a closed trapped surface. Then there exists an incomplete, future inextensible geodesic -y and
for
a
Vol(U) < oo and fu f (R)y oo f (R) which is (positively) homogeneous in tensor, f (AR) 1,\ I f (R) for all A > 0.
neighbourhood U
some
of -y such that
=
curvature invariant
the Riemann
=
homogeneity condition is important because otherwise we could appropriate power of a weakly diverging curvature invariant in 1 order to obtain a diverging integral: While fo, -1--dx is finite, the integral _X The
take
fo
an
716dx
not.
is
Hawking & of such will
a
give
Ellis state that while
conjecture' they a
may not be true in the
as a
are
convinced of the
validity
black
present form.
singularity in our universe, we would like to interpret hole, i.e., we would hope that it is invisible just as the
If there is it
they
unable to prove it. In Sect. 9.5.1 below we 3-dimensional example which indicates that Conjecture 9.5.2 are
a
-
singularity in the Schwarzschild spacetime. Otherwise we would not have a chance to globally solve Einstein's equation as a Cauchy problem since the singularity (whose data are unknown) would influence the geometry spacetime to its future. There are also important theorems for our interpretation of black holes which need a assumption similar to cosmic censorship. The prime example is the "area theorem" due to Hawking which states that the area of black holes can only increase 8) (Wald 1984, of
theorem
12.2.6).
Since it is easy to find which contain visible (or
examples
of inextensible Lorentzian manifolds
additional assumptions spacetime must be made in any conjecture which "censors" naked singularities. The following conjecture is due to Penrose.
"naked") singularities,
on
Conjecture
9.5.3
((strong)
If (M, g) is qualitatively physically reasonable then no future causal geodesic -/ lies in the past of any
cosmic
censorship).
stable and its matter model T is
incomplete,
future inextensible
XEM.
7They 8
state their
This is
only
conjecture with respect to a different singularity theorem. general relativity without taking quantum
true in classical
effects into account
412
Singularity theorems
9.
It is not sufficient just to demand
"physically reasonable" solution,
a
matter model
because of the Reissner-Nordstr6m
27n 9
+
-
e2)
-
dt2
r2
r
+
27n
(I
-
+
e2) _1d
-
+r
The energy momentum tensor T e
T
Ub
U5
=
_
87rr 4
where
81r
Qb
2 r
r2
r
(Ric
2
(d02 + sin2Od W2)
(Scal/2)g)
-
Qb +r2 (d02
is
given by
+ sin 2OdW
2))
have set
we
2m
(I
U
e2)
2
19t
;72-
2m
e2
r
r2
Q
I
'21
)
ar
This
spherically symmetric spacetime satisfies Einstein's equations for electromagnetic field (cf. Lemma 7.4-1) which is certainly "physically reasonable". On the other hand, unlike in the case of the Schwarzschild solution (e const < I are space0) where the hypersurfaces r const < I in the Reissner-Nordstr6m solike, the hypersurfaces r an
=
=
=
lution
are
such that
timelike. It is easy to
1+(x)
see
that there exist
points
x, y G M
1-(y) contains timelike future inextensible curves which approach r 0. (For a more thorough discussion of the ReissnerNordstr6m spacetime including its global properties cf. (Hawking and Ellis 1973, chapter 5.5)). This Reissner-Nordstr6m spacetime therefore vin
=
olates
Conjecture
son
and Penrose
the
case
for
an
9.5.3 if it is
(1973)
qualitatively stable.
and McNamara
intuitive notion of
stability.
Calculations
by Simp-
indicate that this is not
(1978)
It is
generally believed
that
generic, physically acceptable perturbation of the Reissner-Nordstr6m spacetime results in a spacetime which is qualitatively more similar to the Schwarzschild spacetime, even though the Reissner-Nordstr6m spacetime itself can be thought of as a perturbation of the Schwarzschild spacetime. a
9.5.1 A
Let
simple, 3-dimensional example
(M, g)
be
a
3-dimensional spacetime and assume that the energy cUb (9 0 where U be the spacetime given by T
momentum tensor is
=
velocity of the dust particles and
e
their energy
density.
We
are
seeking
solutions of Einstein's equation Ric
-
1Scalg
=
81rEU
(9
U5,
2 c: M -4 R is a function. In general, this is still t ,oo difficult even 3. Assuming that there is a foliation of spacelike though we assume n hypersurfaces orthogonal to U simplifies the problem dramatically.
where
=
9.5
Lemma 9.5. 1.
only if
the
Strength
of
singularities
The vector
field Pfaffian system f 0 1
and cosmic
(i.e., dO
U is irrotational is
censorship
=
413
0) if
and
integrable.
Proof. By Lemma 2.5.8 the integrability of f U51 is equivalent to the 0. Recall from Lemma 5.2.1 that U satisfies the equation dO A 0 0. Let X be any vector field. Then we have geodesic equation VUU dW (X, U) VXW (U) VUW (X) g(VXU, U) g(VUU, X) =
=
=
0
-
0 since
=
-
g(U, U)
=
-1. It follows that
=
-
dO
completely determined
is
by evaluating it on vectors orthogonal to U. Let orthogonal to U. Then we have dO A 0(v, w, U) equivalence follows.
w
v,
=
be two vectors
-dO(v, w)
and the
I
(M, g) be an irrotational, 3-dimensional dust spacehypersurface which is orthogonal to U. If at p c Z the second fundamental form of Z is not a multiple of the metric, then p has -dt2 + V2 dX2 + a neighbourhood with coordinates (t, x, y) such that g W2 d y2' where V, W are functions of t, x, y and T 6(t, x, y)dt 0 dt. These coordinates are unique up to transformations of the type x -4 X(x), y -4 Y(y), t i--> t + const. and interchanging of x and y. Lemma 9.5.2. Let time and let Z be
a
=
=
Proof. Since f 0 1 satisfies dW We
can
write g
=
the bilinear form
-dt2 +
0 there is
=
a
function t with dt
=
0.
(2)g,j (t, Xl' X2)dx'dxj, where for each t E2j=l i,
(2)g(t,
is
a
Riemannian 2-metric and Z is
given by
to. Sinceat(2)g is not umbilic at p, there exists a frame lei, e2j Of Z in a neighbourhood of p such that (2)g andat(2)g are both diagonal
t
=
1
jw W 2 I be the dual frame. It follows (x, y) such that at t to both (2)g and 19t (2)g are diagonal with respect to a, a.. In fact, we only need to show that there exist multiples ale,, a2e-2 of el, e2 such that [alel, 012e2] 0. This is equivalent to dai(ei+l mod 2) + 0 e2l) (no summation over i) which is a system of ordinary differential equations and can be solved by Theorem 2.4. 1. With respect to the coordinates (t, x, y) the 0 (i, j E f x, yj) imply equations Tij with respect to this frame. Let
,
that there exist coordinates
=
==
=
=
atat(2)g,j- tr(atat (2)j) (2)g,j
_
1tr (19t(2)g)at (2)gij I
+
4
Since at t
that the has
a
=
to the bilinear forms
right hand side
is also
up to
(2)gk1at (2)gik at (2)g jj
( (tr (at g) )2 -31 (2)
(2)gij
and
diagonal
unique solution and there exists
simultaneously diagonal, (2)g ',9t (2)g have existence. For
+
2
a
19t(2)gii at t
=
are
at Mg
12) (2)gj.
diagonal,
it follows
to. Since the system
solution when (2)g, 19t(2)g
Must be
diagonal
for all t. Hence
are we
uniqueness observe that the frame lei, e2j is unique multiples and permutation, and that the coordinate t is already
414
9.
Singularity theorems
chosen
so
that it is
coordinates
X,
9,y I I Oy
and
or
unique
Y with the
up to same
additive constant. Thus any other properties must satisfy either o9-- 11 Ox an
9., 11,9y and i9y 11 9x. This
I
proves the lemma.
Corollary 9.5.1. Let (Z, Mg) be a 2-dimensional Riemannian manitensor field which is not proportional to fold and k be a symmetric (0) 2 (2)g. Then the initial value problem for irrotational, 3-dimensional dust spacetimes with initial data (Z' (2)g, k) reduces of ordinary differential equations.
to
a
constrained system
we will summarise properties of generic, irrotational, 3-dimensional dust spacetimes using standard differential geometric terminology. Consider a 2-dimensional Riemannian manifold (Z, gz) and denote the set of unoriented lines in TZ by PZ. Then there exists
In Theorem 9.5.1 below
a
natural map -1: PE
-4
PZ which maps an unoriented line 1 E PZ to a section 1 of PE nowhere geodesic if
to it. We call
the line
orthogonal local, non-vanishing vector field L with L(p) cz l(p) Vp we have 1 gz(L 7LL) : - 0 Vp. This condition does not depend on the chosen representative L. It is a local but not necessarily a global genericity condition on 1. Locally, this condition is slightly stronger than demanding that 1 does not have any local integral curve which is a geodesic. Let (z, g_') be a 2-dimensional, spacelike submanifold of a Lorentzian 3-manifold for any
(M, g) with future directed normal n and second fundamental form k(X, Y) -g(17XY, n). We denote the bilinear form associated with the the of corresponding matrix by k2' i.e., (V), (g_r)1mkjjkj,,,. square =
=
eigenvalues kj, k2 of k with respect to gz the principal curZ and the (unoriented) lines spanned by the eigenvectors the
We call the
of principal directions of Z. vatures
Theorem 9.5. 1. Let
(R2, (2)g)
be
a
Riemannian
2-manifold
and 1: R2
which maps each point p (2 R2 into an unoriented line l(p) C and assume that 1 is nowhere geodesic. Let C C R2 be a smooth
TR2
TpR2 curve
which divides R2 into two disconnected regions such that 1 and 1--L intersect TpC transversely at each p E C. Finally, let KI, K2: C --* R be smooth
junctions.
the set of points p E R2 such that the integral l' through p intersect C. There exists an irrotational, 3-dimensional dust spacetime (M, g) and an isometric embedding L: (Z (C' 1, 11), (2)g) ---> Z C M such that the second fundamental form k of Z in M satisfies (a) the principal directions of Z are given by t.1, tlj-, K1, (b) along C the submanifold Z has principal curvatures k, K2 k2 R2. Then (M, g) is inextenC can be chosen such that 0 (C, 1, 1 -L)
(i)
Let
curves
O(C, 1, 1 of 1
and
=
:--
-
=
sible
if (R2,
(2)g)
is
So.
For any p
(ii)
k-
(p)
c Z
and cosmic
singularities
censorship
415
let
(p), k2 (p))
min (ki
=
of
Strength
9.5
and
k+ (p)
max(ki (p), k2 (P))
=
The world line
of the dust particle through p ends in a curvature 1 1 finite proper times k (p) k+ (p) if k+ (p) > 0 > k- (p), -
singularity at
finite
if k+(p)
in
-1
proper time
There
< 0.
-
,
k
(p)
ifk-(p) other
are no
>
0,
at
finite
proper time
singularities are weak in the sense that for all volume, vol(U) fu -, /_det(g,,b)dtdxdy spacetime average of the energy density, All
(iii)
U with bounded
=
I
fu
vol(U)
e
k+ (p)
singularities. open sets oo, the
<
-(gab)dtdxdy, (t, x, y, z) V/Jet
is also bounded.
(iv) The spacetime is non-singular if and only if K, (v) For generic initial data strong cosmic censorship
=
is
0. K2 violated, =
pro-
regards the solution as "qualitatively stable" and "physireasonable".9 cally (vi) Generically, the data P)g, 1, K1, K2 parameterise the set of local, irrotational, 3-dimensional dust spacetimes. vided
one
Properties (iii) and (ii) hold Lim
a more
at each
point
context
(Kriele
and
(t, x, y)
-dt2+V2(t, x,
t
choose coordinates
==
0 1 and
(2)gyy.
assume
y)d X2 + W2 (t,
a2V
In view of Lemma 9.5.2
av ay
=
0
=
or
(x, y). f (x, y)i9., and (L(x, y))--L
function of
=
W
=
given by g VO (X, Y), W(O, X, Y) 0 is equivalent to
at
w'9v, ay
In the coordinates =
0
we can
0. Thus
write V (t, x,
we
y)
==
set
we can
=
=
W-awav
h(x, y)ay,
functions. The condition that I is
f2 Wo av : ay
such that
that the dust-metric is
'90y obtain either
(x, y)
ax, ay span 1 and lj- (see the is diagonal in these coordinates.
x, y)d y2, V(O, X, y)
WO (x, y). Since the constraint Tty 0
(2)g
metric
=
:
can
R 2 the Gaussian vectors
p E
=
,
=
We
(i):
Theorem 9.5. 1.
proof of Lemma 9.5.2). The (2 )gxx and L et V2 W02 0
we
general
1995).
Proof of
Z
true in
have '9VO ay
:
Vo (x, y)
+
C')Y
where
(x, y)
w
we
is
some
can
non-vanishing
write
L(x, y)
where
f, h are non-vanishing nowhere geodesic reduces then to 0. Since 0
=
Tyy
E2
( _)2 LV
_
V
.9y
tq (x, y). Now the equations Ttx
a2V 7tT2=
0
gives The relation of these solutions and cosmic
in
(Kriele 1997).
censorship
are
further discussed
416
9.
theorems
Singularity
Fig. 9.5. 1. The singularity structure of spacetime. The case ki, k2 < 0. The y-component of spacetime is suppressed. The singularity A is given by I + 1 0 and the singularity B is given by Vo+tq t (49Vo/ft) 0. Observe (o9q/ft) -
=
=
singularity A the light cone degenerates in the y-direction and that at B degenerates in the x-direction. Hence there exist future directed timelike curves emanating from the singularity and cosmic censorship is violated that at the
'9W 0
=
_5_X
a2V0
aw avf0
OLq
V
ax
ay
9Y
Re-expressing w (x, y) by Wo partial differential equation
w
=
+
axay
(
W6-1
avo
awo -
'9X
q +
_,9zay
-
a 2q
a2q
f
,
( ) ay
aV0
ay
1
92 Vo
axay -
)
q(x, y).
follows that each
fact, exist
has
if a
a
The lines
integral
ay
(VC1'9V0 W -
coordinate line
x
=
point (Xm, Ym) of C
x0
1, 1j-
are
of 1 and
curve
1j-
linear, hyperbolic
9q
ay
for the function
YX,9Y
obtain the
we
(9.5.7)
Vf0.
w
1
'9 Wo
ax
)
q
transverse to C.
0
(9.5.8)
Moreover,
intersects C at most
once.
it
In
would intersect C twice then there would
in between these intersection
locally maximal distance
=
points which
with respect to the flat metric would be tangent to C in contradiction to our to
(xo, Ym)
dx2 + d y2 At this point a. transversality assumption. We will now show that for given initial values this differential equation has a unique solution in A) (C, 1, 1 ') by reducing it to an appropriate system of hyperbolic differential equations." Let .
'0
Alternatively, we could directly appeal symbol of the hyperbolic equation (9.5.8)
to standard theorems.
is constant it has
a
Since the
unique, global
,rl
=
(9.5.8)
2
+ y and
x
-r
reduces to
x
=
1
we see
that
R is
-*
y. With
-
a2q (&r2 ) 2
2
h: R5
singularities and
cosmic
to these variables
respect
417
censorship
Equation
equation of the form
an
a2q
where
of
Strength
9.5
=
h
(,ri,
'94 r2,
q-,
71 71
,
Equation (9.5.8)
,
I)
71
'94
'94
and Setting F ar2 a7l equivalent to the hyperbolic system of
suitable function.
a
7
is
=
equations
r,
a,ri
+
71
197-2
The characteristic directions
h(-ri, -r2,
097-2
197-1
of the
given by the left eigenvalues
are
ma-
trix
A=
11
=
(I
1
01
0) 12
(0 1, 1), 13
=
0
1
0
1
0
-
1).
These vectors
are
linearly
in-
with Remark 7.3.1
together apply Corollary given initial values. The initial values be, calculated from the initial values for q. If we parameterise,
dependent
and
for 4
can
C
a curve s
7.3.2
we can
in order to obtain
by
0
0
(0, 1,
=
1
7
0
0
-4
a
solution for
A(s)
we
aq
o
get
9q
A =
ds
'94
(d-ri(A)f + &F2( ))
and another linear combination for the normal derivative of q which depends on the coordinate expression for (2)g. Hence we can calculate the initial values for
if
(4, f 9) ,
Since this normal derivative have
T,y
as our
=
initial data q1C
can
=
know q1C and its normal derivative.
Einstein's equation which is
g=-dt2 +(Vo+tq )2 d X2
KjVojC
and
(21) ay
=-
W2 0
1+t
leads to
aq
(aVo
5 y_
ay
immediately that the principal
curvatures
)2
are
ki (x, y)
q(x, y) /Vo (x, y)
19VO
I
and k2 (X) YJ
C
-
1C
solution of
d Y2.
(9.5.9)
given by -
=
a
we
ay
.9y
Ic
automatically this given by
+
(21) Since K2 (aVO )
be calculated from q1C and
0 hold
0 and T
It follows
we
1
9q
-
Y
5Y
for the initial value problem with respect to the initial solution in 0 (C, 1, 1 curve C C Z (Garabedian 1986, section 4.2).
418
9.
If
chooses C
one
Singularity
theorems
J(x,y)
=
:
x
yj
=
0 (C,
1, 1 -L)
(ii):
(R2, (2)g)
R' and
=
The energy
is
density
6(t, X, Y)
O(C,1,1-L)
then
solution is inextensible in t-direction
(M, g)
R'. Since the
=:=
is inextensible
provided
is inextensible.
given by E (x,
y)
(9.5.10)
=
(VO
+
aq + t (9y ay
avo
tq)
where
E(x, y)
2
a VO
aq q +
=
Oy
aWOW
(-5y-)
ay
avr0 a2 WO
Oy Since g is and k that
E
=
OX2
1
VO
W,0
qVOd X2
+
W2 0
(av ay
ay
may become infinite at
'OVO awo'9vo +
+
----
we
have
=
0,
aV1
ay
-2
VO -2W0 -
2
(2)g
-1
V0 dX2 +
==
It follows from
tq
W0I
I
j x- 5x
'9Y
dy2.
VO
--5y- -W
-1
given by Equation (9.5.9) 0
aVO a2V0
-3
10
WO2dy2
Equation (9.5. 10)
aq + t ay
=
0 and that
generically it will be infinite at these points. The first part of (ii) follows immediately. To see that there is no other singularity observe that for 3-dimensional spacetimes the Riemann tensor is completely determined by the energy momentum tensor and that therefore all Riemann tensor components
(iii):
are
bounded where
F-
is bounded.
This follows since C9 VO
eV det(gab)
Wo
EWO
(Vo
Oy E1W0
+
tq)
Oq
(-' Y- ay) + t
a VO
19Y is finite.
(iv):
Since the initial data for
K, VO and rem
for
( 29-)j ay
PDES
==
C
K2
Equation (9.5.8)
are
ava the claim follows from the ay
of the type
given by q1C
uniqueness theo-
(9.5.8).
-1 and g.,., or gyy converges to zero it is easy to see (v): Since gtt that at these singularities strong cosmic censorship is violated unless the =
const. (cf. Fig. 9.5.1). singularity is given by t have assumed that the second fundamental form we (vi): Implicitly and the metric can be diagonalised with respect to a smooth frame. This may not be possible at umbilic points but points at which this problem occurs are isolated and therefore not important for local genericity. Since OV 0 is also a local genericity condition, locally almost every irrota5-Y : =
tional dust spacetime can be obtained in this way. Given C, data are invariants. Hence (vi) follows.
our
initial I
Strength
9.5
Proof. In Theorem
q,
9.5.1
W01.,=_,0
given initial data. Then explicit solution
Wolx=x,,(y)e
=
JX
given. Now
are
ay
as
X
Wo
(22)
419
censorship
have solved Einstein's equation
we
problem where Vo, Wo, q1c,
obtain the
cosmic
irrotational dust spacetimes
3-dimensional, by quadratures.
be obtained
value
singularities and
9.5.2. Almost all
Corollary can
of
we
initial
as an
consider
Vo,
le
Equation (9.5.8) and
solve
we can
Voaql,9y qaVolOy d j9V0/j9y j92V01(9z9yaq1,9y -,92q1i9z9yaV01,9y V0
o
-
*
(9.5.11) 1
we have only considered the case where additional restrictions at points with "I 0. It ay
Remark 9.5. 1. In Theorem 9.5.1 aV0
ay
0. There
7
are
=
Equation (9.5.8) that
follows from
at these
points either
and mixed derivatives of V vanish. For
y-derivatives
-3.2 9Y
0
=
all
or
completeness
we
cases in more detail. If '9V0 does not vanish identically then
will discuss both
we can use Equation 9.5.11 equation. Since Wo is smooth and does not vanish, the integrand in Equation 9.5.11 must also be smooth. It follows that avo 0 at (x, y) I the y-derivative of q in a neighbourhood of f (x, y) : ay
ay to Solve Einstein's
=
must
avolay
satisfy qy
(x, y).
0
52VOlaXay If this is satisfied then
The
+
ay
IL2 a2W and Tyy W Do x,
y)
W2 '92
X.
=
-
at
V
Wo (x, y)
=
_X
Thus
+ ts (x,
,9S
have Vo (x)
=h
normalise Vo (x)
Vx. Thus
0
=
1.
solution.
=
(x, y)
-dt2 + =
(1
+
x.
-4
q
ax on
we
0
x)
=
Vo (x)
+
implies
Z is
=
(9.5.12)
given by
+ W0 can
0.
2(X, y)d Y2
use
our
coordinate freedom to
(x
+
=
we
obtain
tq(x) )2 d X2
fx q( ) '9w'(a
(x, y)
=
0 then
Equation (9.5.12) can immediately integrated 0 and hence for any automatically Txy
by the functions
formations
write V (t,
Now Ttx
=
be
for any given q, Wo. We have functions q (x), Wo (x, y), w (y) g
we can
y).
-
V2 (x) d X2
eterised
is any function of
'9WO Vo
Since the initial metric induced
were s
), where 0
we
tq (x) and W (t,
we
a2q axay
we obtain a local, non-singular simply have specified singular initial data. '9V aV0 0 can be easily solved. If non-generic case ay .9y
Otherwise
TXX
(0,9V
=
'Y)
+
a
solution
(W0 (X, Y)
+
tS(X, Y))2 d Y2,
d +w(y). The space of solutions is paramq (x), Wo (x, y), w (y) modulo coordinate transxo, Y(y)). The energy density is given by
420
9.
Singularity
theorenis
a2 W,, qs
_
,E(t, X, Y) Hence We
obtain the
we
remark)
in
9.5.3. Let
(M, g)
timelike unit
eW with 0
T
=
0
as
length W and dW A W
g
0, then there
=
9.5.1
(or
the
preceding
3-dimensional spacetime. If there exfield U and a function E such that
a
vector
-dt2
=
+
where a, b
are
The energy
i9yki
=
(k2
,9x,9yk1
=
(k2
density
is
exist coordinates
(t, x, y)
e-aa,,
now
e
2a(x,y)(1
e2b(x,y) (1 and
+
-
tki (x,
tk2 (X)
y))2 dX2
y))2dY2
k1, k2 satisfy
kl),9ya, ki)(axi%a
-
+
-
9ya,9xb) -,9xk1c')ya.
given by
k1k2
=
+
free functions
+
6
where el
be
in Theorem 9.5.1.
dt and
=
We will
ts)
provided by Theorem geometrical form.
a more
Corollary a
tq) (WO
+
type of singularities
same
(9.5.13)
'
(1
+
express the metric
can
ists
aX2 =
e2
ebe, (1
=
show that
9
e-b
+
eae2
tki) (I
+
tk2)
9
el
+
o
e2
9
e-'
e- ba
Y
examples do not satisfy the genericity conthey would provide counterexamples (cf. corollary 9.5.5 below). our
dition. It turns out that otherwise to
Conjecture
9.5.2
Lemma 9.5.3. Let
ifold.
be
(M, g)
3-dimensional Pseudo-Riemannian mancompletely determined by the Ricci
a
Then the Riemann tensor is
tensor and
given by
Rijkl Proof. Proposition
=
2(gi[kR,lj
4.3.2
-
choose coordinates
(no summation) gab
a
where na
(X1' X2, X3)
Scalgi[kgl]j-
for every pair of tensors G.', S., E metric g such that g., G,, and S., Ric.,.
In
we can
-
implies that
sym(T20(T,M))
there is
fact,
gj[kR,]i)
=
=
E
(x 1,x2,x3) such that (G.,)ab 1-1, 11. Then we simply set
(Gx)ab
-
1-3 E
3
1
77aJab
(Sx)ab?7c6cdXcX d.
C' d=1
From the
first,
the third and the fourth
obtain that at
symmetry in Proposition 4.3.1
the Riemann tensor of
given point pseudo-Riemannian manifold is already specified by
we
a
x
a
3-dimensional
the 6 components
9.5
Strom-ig-th
of
snigularities
and cosmic
censorship
421
R1212, R1213, R1223, R1313, R1323, R2323-
sym(T20(T,,M))
Since
6-dimensional vector space and every tensor S, sym(T2(T.,M)) can be realised as the Ricci tensor of a metric, the map tr: R., - tr(R,,) Ric,, is linear isomorphism. The tensor is
a
(=-
=
r(RiC)ijkl given
in
tensor
=
2(gi[kR,lj
Proposition
gi[kR,]i) tr(RiC)gi[kgl]j satisfies the equations tr(r(Ric)) Ric. Hence it is the Riernarm -
4.3.1 and
corresponding
Corollary
-
=
to Ric.
I
If (M,g) is a 3-dimensional, irrotational dust genericity condition does not hold.
9.5-4.
time then the
space-
Proof The Ricci tensor is given by Ric r:(elb Oej + e2b (9 e?) It follows directly from Lemma 9.5.3 that the components Rtjkt (j, k E I t, x, yj) =
-
of the Riernarm tensor vanish. Hence for any fixed numbers (xo, yo) the genericity condition is violated along the timelike geodesic t -4 (t, xO, yo).
Corollary
There is
9.5.5.
a
3-dimensional spacetime
(M, g)
which
(i) is chronological, (ii) is geodesically inextensible, (iii) satisfies the timelike convergence condition (iv) contains a closed trapped surface, (v) and contains an incomplete future inextensible geodesic a neighbourhood U of -y such that Vol(U) < oo and fu fy < any
polynomial
curvature invariant
f which
-Y and oo
for
is linear in the Riemann
tensor.
Proof. Consider to obtain
a
mod 1, y)j and the metric
g
is
a
=
a
spacetime (R
closed
trapped
b(x, y)
-dt2
+
e
2 ,
g)
a(x, y)
2a(I
+
given by Corollary 9.5.3. In order let M R x S1 x R I (t, x a(y). Then for each function ki (y),
as
surface ==
tkl)2dX2
we
+
(I
=
+
t(k,
+
=
(kl)'/a') )2dY2
solution with
kl((ki + aykl/,9ya) + ea e2 e2 e-a (I + tki)(1 + t(k, + (kl)'/al)) 0
Let T
:=
ft,x,ylt
=
O,y
fundamental form of T C
=:
It
01
andS
01.
=
9
Sjjw'Ow1 bethesecond
Then up to a positive factor, the :-F S11. It follows that T is a closed
expansions 0:: are given by 0:: k, trapped surface if ki(O) < -IS111. This can always be arranged since ki can be freely specified. The spacetime (M, g) has a singularity at
422
y
Singularity theorems
9.
=
0, t
V01(u)
-I/ ki (0). However, this singularity = , fu Jeltim < 00.
=
<
is
9.5.5
Corollary
that
implies
Conjecture
weak that it satisfies
1
9.5.2 without the
ditions does not hold. In order to estimate the
example
so
00
we
have to examine its
special
physical
genericity
con-
relevance of
our
features.
along the geodesic t F--+ proof of Theorem 9.3.1 is to ensure the existence of a singularity along it. Since there actually develops a 0, the failure of (M, g) to satisfy the singularity at I + tki (0) genericity condition does not appear to be grave. Moreover, it seems very likely that a perturbation of (M,g) through dust spacetimes which are not strictly irrational will not suffer from this defect. On the other hand, we don't know much about the global properties of these perturbed spacetimes. In particular, at this point of time" we cannot exclude the possibility that they form stronger singularities then the special spacetimes we have examined. (ii) Our example is 3-dimensional rather than 4-dimensional. Here it is important to note that Theorem 9.3.1 does hold for 3-dimensional spacetimes as well as for 4-dimensional ones. Moreover, there do exist 4-dimensional, spherically symmetric dust spacetimes which have similar singularities (Miiller zum Hagen, Yodzis, and Seifert 1974). However, these 4-dimensional examples also contain much stronger singularities in the centre of symmetry. One may speculate whether these strong (central) singularities are a typical feature for 4-dimensional spacetimes. In the absence of independent evidence disqualifying 3-dimensional models it seems fair to state that our example indicates otherwise. (iii) The closed trapped surface we have constructed has not much to do with the existence of singularities. In fact, the general solution shows that the singularities depend solely on the principal pressures k1, k2. This indicates that the example is more appropriate to ilThe
(i)
genericity condition
(t, xo, yo).
is not satisfied
Its usage in the
=
lustrate condition
of Theorem 9.3.1 rather than condition
(iii)(b)
(iii) (a). However, the hyperbolic face Z.
Still,
as
an
nature of
Equation (9.5.8) greatly
solutions with compact hypersurimmediate consequence of Corollary 9.5.3 we
restricts the existence of
have the existence of
a
global
3-dimensional dust spacetime for any given Agz, (Z, gz) by choosing k
2-dimensional Riemannian manifold where A is
a
constant. The energy
density
=
E
is
positive
if
is suf-
JA
ficiently large. Our construction of closed
riodicity data I
am
are
trapped surface requires (at least)
a
pe-
with respect to x. Observe that perturbation of our initial does very restricted since the differential Equation (9.5.8)
writing this
in 1998
9.5
not need to
Strength
of
singularities and
cosmic
censorship
423
respect this artificial periodicity. It is possible, however,
to construct closed
trapped surfaces with are stable with respect to arbitrary perturbations of the initial data. Choose a 2-dimensional
Riemannian manifold is small
(Z, gz)
and k
and
closed
a
curve
T C Z. If A < < 1
Agz then T is a closed trapped surface. This construction does not rely on periodicity and is therefore stable with respect to perturbations of initial conditions. The trapped surface has topology S' which is qualitatively different enough
=
from S' and
more akin to the torus S' x S1. This is inevitable works with 3-dimensional spacetimes. Unlike in the case of Conjecture 9.4.1 there there does not seem to exist evidence that
if
one
the
(iv)
topology
One
of the closed
trapped
surface matters in
our
context.
argue that dust, arising as an idealisation from the energy momentum tensor for collisionless gas, is not a very realistic matter can
model.
Moreover,
even
in the
corresponding
Newtonian
theory,
con-
gruences of dust tend to form weak singularities. (Rein, Rendall, and Schaeffer 1995) has shown for spherically symmetric 4-dimensional
spacetimes representing a collisionless gas that one does not obtain singularities before a central singularity has formed. This is in striking contrast to the analogous situation in the case of dust (Milller zum Hagen, Yodzis, and Seifert 1974). One may therefore be tempted to disregard our example as typical for a notoriously ill behaved matter model. However, since in our class of examples we only obtain weak singularities, which are forced on us by the singularity Theorem 9.3. 112, we are still led to conclude that singularities weak
(mathematically) and
completely
due to the
singularity theorems
different from what
one
may be very weak
may expect at first
sight.
Hence whether
or not our example is physically realistic, it indicates that existing singularity theorems are not sufficient to conclude the existence of black holes or the big bang. One additional, physically motivated assumption could be that the principal pressures diverge comparably to the energy density. It would therefore be of interest to study a similar example with T (I + 0)ew 0 w + 13eg (0 E R). For these 3-dimensional spacetimes there exist coordi=
nates
(t, x, y)
as
above:
g
T
We have
some
control
=
-dt2
=
cdt2
over
+
+
V2 dX2
Oc(dX2
+
W2 dY21
+ d
Y2).
the location of the
singularities (they
must
before proper time 2/(ki + k2), where ki, k2 denote the principal curvatures of the initial hypersurface). This class of solution may still occur
12
Here
we assume
irrelevant
that in
our case
the violation of the
genericity condition is
424
9.
Singularity theorems
manageable and can give us singularity theorems really give singularities. be
an
important clue
as
to
whether the
evidence for the existence of
physical
Corrections for the first printing of Spacetime
I would like to thank those readers of the first edition who gave me feedback regarding related literature and errors in the text. I am especially thankful to Nico Giulini who found a serious error in the presentation of the fundamental theorem in affine geometry and who provided me with an extensive list of other errors and suggestions. All errors known to me have been corrected in the second printing of the book. This document has been prepared for owners of the first printing. It contains corrections of those errors which are not merely obvious typos and in particular a replacement for the text of Section 1.1.2 up to (and including) the proof of the fundamental theorem in affine geometry, Theorem 1.1.1. Finally I would like to point to a review article on the material covered in chapter 8 and 9. This article provides many very illuminating examples of spacetimes as well as discussions which reinforce our sceptical approach towards the physical interpretation of singularity theorems: Senovilla, J. M. M. (1998). Singularity Theorems and Their Consequences. Gen. Rel. Grav. 30, 701–848. Hannover, 15th October 2001
M. Kriele
2
Corrections
Location
Error
Correction
Table of Contents: 2.7.1 and later
Levi-Civit`a
Levi-Civita
P. 19, line 6 from below
literarily
literally
P. 21, line 2 from below
intertial
inertial
Figure 1.4.7
ell0
`0
Definition 2.1.2 (iii)
In the text the condition U ∪ V = M is missing
A topological space (M, τ ) is connected if U, V ∈ τ with U ∩ V = ∅ and U ∪ V = M are necessarily of the form U = M , V = ∅ or V = M , U = ∅.
Definition 2.1.2 (v)
V U¯
V = U¯
P.46-p.47
ˆ , Ui , Uj , V, Wi some U, U ˆ Ui , Uj , V, W should read U, U, continued on next page
4
continued from previous page
Location
Error
Correction We assume k ≥ 1 since we speak of C k -diffeomorphisms. It is also possible to define C 0 -manifolds where ϕi ◦ ϕ−1 j are homeomorphisms. (This would be the approach taken in topology — but notice that C 0 -manifolds are much more general then C k -manifolds.)
Definition 2.1.3
Proof of Lemma 2.3.5
ηlk
nlk
Theorem 2.4.1.
f : J × U˜ × V → Kn , (t, x, y) 7→ f (t, x, y)
f: J × U × V → Kn , (t, x, y) 7→ f (t, x, y)
P. 118
Incomplete box
p. 79 ↓ [↓ p. 121]
P. 124
Incomplete box
p. 155 ↓ [↓ p. 125]
∂htxa ∂ 2 xh ∂xh ∂ x ˆb ∂ x ˆc
a = Γˆbc
∂x ˆ a ∂ 2 xh . ∂xh ∂ x ˆb ∂ x ˆc
P. 155, line 9
a = Γˆbc
P. 266, line 12 from below
Astronomical observations seem to imply that |Λ| is very small
I have been told that according to recent observations |Λ| is not negligible. Note that the constant Λ is retained in most of the book.
P. 293
Hubble discovered that distant galaxies are moving away from us
Hubble merely discovered the the cosmological relation of red-shift and distance. Others have interpreted this observations as being due to cosmological expansion. continued on next page
5
continued from previous page
Location
Error
P. 313
Correction Figure 7.2.1 should be replaced by the figure on the cover of the book.
P. 422, line 13 from below.
Chruschiel
Chru´sciel
10 .10 .20 Replacement for Section 1.1.2 up to (and including) the proof of the fundamental theorem in affine geometry In this section we present some results of affine geometry which will be needed in the proof of Theorem 1.4.1. This section is very technical and should be omitted on first reading.
Let o, x1 , . . . , xk ∈ An and α1 , . . . , αk ∈ R such that the barycentre with masses α1 , . . . , αk , α1 x1 +α2 x2 + . . . +αk xk := o +
k X
Pk
i=1
αi = 1. Then
αi (xi −o),
i=1
is independent of o and therefore an affine invariant. The symbol + is defined via the right hand side and can only be applied to “linear combinations” where the real factors add to 1. An¯ P affine subspace B of k An is a set of points {x = α1 x1 +α2 x2 + . . . +αk xk ¯ i=1 αi = 1}, where x1 , . . . , xk are pairwise different, fixed points. The affine dimension of B is k − 1. It follows that an affine subspace is an affine space. An affine subspace of dimension 1 is called an affine line. We call points lying on a single line collinear. Observe that lines are the smallest sets which are invariant under parallel transport. Lemma 10 .10 .10 . Let x, y, z ∈ An . Then x, y, z lie on an affine line if and only if there exists a λ ∈ R such that x = y + λ(z−y). Proof. x lies on the line generated by y, z if and only if there exists an β ∈ R with x = βy+(1−β)z = y+β(y−y)+(1−β)(z−y) = y+(1−β)(z−y).
Definition 10 .10 .10 . An affine map is a map f : An → An , f (x) = A(x−o) + b, where A is a linear map, o ∈ An , and b ∈ Rn . If A is bijective then f is called an affine transformation. A collineation is a bijection f : An → An which maps any three collinear points into collinear points.
6
Consider a line l and three points x1 , x2 , x3 on l. Then the number λ given by x3 −x1 = λ(x2 −x1 ) is denoted by x3 −x1 . x2 −x1 The following lemma is the classical theorem of Thales. It will be used in the proof of the fundamental theorem in affine geometry (Theorem 10 .10 .10 below). Lemma 10 .10 .20 . Let H1 , H2 , H3 ⊂ Rn be parallel hypersurfaces and l be a line which intersects these hypersurfaces. Let xi (l) = Hi ∩ l. Then x3 (l)−x1 (l) . x2 (l)−x1 (l) does not depend on l. ~ the subspace of Rn which is the associated vector Proof. Denote by H space to the affine space H1 (and since H1 , H2 , H3 are parallel also to ~ defined by H2 , H3 ). We consider the quotient space An /H x∼y
if and only if
~ y − x ∈ H.
This space has a natural affine structure with associated vector space ~ given by π(x)−π(z) = ~π (x−z) where π, ~π denote the projections R n /H to the equivalence classes. We have π(x3 (l))−π(x1 (l)) = ~π (x3 (l)−x1 (l)) ¶ µ x3 (l)−x1 (l) (x2 (l)−x1 (l)) = ~π x2 (l)−x1 (l) x3 (l)−x1 (l) = ~π (x2 (l)−x1 (l)) x2 (l)−x1 (l) x3 (l)−x1 (l) = (π(x2 (l))−π(x1 (l))) x2 (l)−x1 (l) which implies that x3 (l)−x1 (l) π(x3 (l))−π(x1 (l)) = x2 (l)−x1 (l) π(x2 (l))−π(x1 (l)) only depends on the projected values. Now it is sufficient to observe that π(xi (l)) is independent of l since all points in Hi are equivalent: x, y ∈ Hi ⇒ π(x) = π(y). It is easy to see that all bijective, affine maps are collineations. Conversely, the fundamental theorem in affine geometry asserts that any collineation must be affine:
7
Theorem 10 .10 .10 . Let An be an affine space over R with n ≥ 2 and fix o ∈ A. Let f : An → An be a bijection which takes each three collinear points into collinear points. Then there exists a point b ∈ An and an invertible linear map f~ such that f (x) = f~(x−o) + b for all x ∈ An . The proof is elementary but lengthy and requires some preparatory lemmas. We will follow (Berger 1987, p. 52–55) where one can also find a version of this theorem which holds in the complex case. Observe that the following proof makes heavy use of the assumption n ≥ 2. The theorem does not hold for n = 1 since in this case any map maps collinear points into collinear points. Lemma 10 .10 .30 . Let o, x1 , . . . , xk ∈ An , f be a collineation, λ1 , . . . , λk ∈ R, and k X λi (xi −o) ∈ An . x=o+ i=1
Then there exist µ , . . . , µ ∈ R such that 1
k
f (x) = f (o) +
k X
µi (f (xi )−f (o)) .
i=1
Proof. For k = 1 the claim is clear by the definition of a collineation. Assume now, the assertion is true for all m ∈ {1, . . . , k − 1}. For x=o+
m+1 X
λi (xi −o)
let
x0 = o +
i=1
m X
λi (xi −o) .
i=1
Then we have x = x0 + λm+1 (xm+1 −o)
(10 .10 .1) 1
and by induction there are real numbers µ0 , . . . , µ0 Pmhypothesis 0 0i f (x ) − f (o) = i=1 µ (f (xi ) − f (o)). We define also y = o + λm+1 (xm+1 −o), 1 1 z = y+ x0 . 2 2
m
with
(10 .10 .2) (10 .10 .3)
The triples {z, x0 , y}, {y, o, xm+1 }, and {z, o, x} consist each of collinear points. This is clear for the first triple and follows from Lemma 10 .10 .1 for the second triple. To see this for the third triple observe that y−o = x−x0 . z = 21 y+ 12 x0 is the centre of the parallelogram defined by o, y, x, x0 and therefore the intersection of the line connecting y with x0 and the line connecting o with x. Since each of these three triples consists of collinear points there exist α, β, γ such that
8
f (z) = αf (x0 )+(1 − α)f (y), f (x) = βf (o)+(1 − β)f (z), f (y) = f (o)+γ(f (xm+1 )−f (o)). This implies f (x) = βf (o)+(1 − β)f (z) = β(f (o)−f (o)) + (1 − β)(f (z)−f (o)) + f (o) = (1 − β)((αf (x0 )+(1 − α)f (y))−f (o)) + f (o) = (1 − β)(α(f (x0 )−f (o)) + (1 − α)(f (y))) + f (o) m X i µ0 (f (xi )−f (o)) = (1 − β)(α i=1
+ (1 − α)γ(f (xm+1 ) − f (o))) + f (o) =
m+1 X
µi (f (xi )−f (o)) + f (o).
i=1
Lemma 10 .10 .40 . Let o, x1 , . . . , xn ∈ An such that {x1 −o, . . . , xn −o} is a basis of Rn . If f is a collineation then {f (x1 )−f (o), . . . , f (xn )−f (o)} is also a basis of Rn . Proof. Let x ˜ ∈ An be any point and let x = f −1 (˜ x). Since {x P1n−o, . . . , xn − o} is a basis of Rn there exist ξ i ∈ R such that x−o = i=1 ξ i (xi −o). Lemma 10 .10 .30 implies that there exist µ1 , . . . , µn ∈ R such that x ˜ − f (o) = f (x) − f (o) =
k X
µi (f (xi )−f (o)) .
i=1
Since x ˜ was arbitrary the assertion follows. Lemma 10 .10 .50 . A bijection f is a collineation if and only if it maps affine lines onto affine lines. Proof. Let x, y ∈ An and denote by l the line spanned by these points. Let zˆ be a point on the line spanned by f (x), f (y). We have to show that z = f −1 (ˆ z ) ∈ l. If this was not true than the vectors z−x, y−x would be linearly independent. But then Lemma 10 .10 .40 would imply that f (z)−f (x), f (y)−f (x) where linearly independent as well. Contradiction to the construction of zˆ = f (z)
9
Lemma 10 .10 .60 . Let f be a collineation. Then f maps parallel lines into parallel lines. Proof. Let l, ˜l be two parallel lines (which do not coincide — otherwise there would be nothing to prove). Since they are parallel they span a plane P rather than a 3-dimensional subspace of An . This plane is mapped into a plane P 0 . In order to see this consider a line ˆl such that the lines l, ˆl intersect and span P . It is clear that any line which intersects both l and ˆl is contained in P . Moreover, any point y ∈ P lies on a line ¯l which intersects both l and ˆl. Let P 0 be the plane generated by the (intersecting) lines f (l) and f (ˆl). f (y) lies on the line f (¯l) which intersects f (l) and f (ˆl). Hence f (¯l) (and therefore f (y)) lies in P 0 . Having established that f (P ) is a subset of a plane we only have to show that f (l) ∩ f (˜l) = ∅. If there was a point z ∈ f (l) ∩ f (˜l) then f −1 (z) would lie in both l and l0 which is impossible since both lines are parallel.
Lemma 10 .10 .70 . Let k : R → R an automorphism, i.e., k(αβ) = k(α)k(β) and k(α + β) = k(α) + k(β) for all real numbers α, β. If k 6= 0 then k = id Proof. k(0) = k(0+0) = k(0)+k(0) implies k(0) = 0. Assume, there is an α 6= 0 with k(α) = 0. Then k(β) = k(α)k(β/α) = 0 for all β and k must vanish. Hence k(α) 6= 0 ∀α 6= 0. k(1) = k(1 · 1) = k(1)k(1) implies k(1) = 1. By induction we obtain k(n) = n for all natural numbers. k(−n) = k(0 − n) = k(0) − k(n) = −k(n). Similarly, we have k(1/n) = 1/k(n) = 1/n. For n, m ∈ Z we have now k(n/m) = n/m and the lemma is proved for all rational numbers. α ≤ β implies k(α) ≤ k(β) since for any positive number γ 2 we have k(γ 2 ) = k(γ)k(γ) ≥ 0. Let now γ be any number. Then there exists a monotonically increasing sequence αi → γ of rational numbers and likewise a monotonically decreasing sequence of rational numbers βi → γ. Hence αi = k(αi ) ≤ k(γ) ≤ k(βi ) = βi which implies k(γ) = γ. Observe that this lemma would be false if we had replaced R by C as z 7→ z¯ would be a counter example. This is why theorem 10 .10 .10 (as stated above) is not true for affine spaces over the field C. Proof of Theorem 10 .10 .10 . Let f~ : Rn → Rn , v 7→ f~(v) = f (o + v)−f (o). The idea of proof is to construct an automorphism k : R → R such that f~(λv + µw) = k(λ)f~(v) + k(µ)f~(w) holds for all λ, µ ∈ R and v, w ∈ Rn . We will use constructions based on parallel lines in order to represent vectors such as v + w, (λ + µ)v, λµv. Since f maps parallel lines into parallel lines (Lemma 10 .10 .60 ) these constructions will be preserved by f
10
and can therefore be used in order to prove linearity and multiplicativity of f~, k. We will first show that f~ is additive. f PSfrag replacements f (o + w) o+v+w
o+w
f (o + v + w) f (o)
f (o + v)
o+v o Fig. 10 .10 .10 . Additivity of f
Let v, w ∈ Rn and consider the lines lv , lw spanned by o, o + v and o, o + w. The point o + v + w is the intersection of the parallel translation of lw that contains o+v and of lv that contains o+w (cf. Figure 10 .10 .10 ). Since parallel lines are mapped into parallel lines we know that f (o + v + w) is constructed analogously from f (o), f (o + v), f (o + w). Hence f~(v +w) = f (o+v +w)−f (o) = f (o+v +w)−f (o+v)+f (o+v)−f (o) = f (o + w)−f (o) + f (o + v)−f (o) = f~(w) + f~(v). Here we have used the fact that the vectors connecting f (o) with f (o + w) and f (o + v) with f (o + v + w) are identical since they correspond to opposite sides of a parallelogram in a plane. Now we show that there is a well defined automorphism k : R → R such that f~(λv) = k(λ)f~(v) for all v ∈ Rn and λ ∈ R. We first fix a vector v and consider the line l through o spanned by v. Denote by gl : l → R the map o + λv 7→ λ and by gf (l) the map f (o) + µf~(v) 7→ µ. Since f maps the line through o which is spanned by v into the line through f (o) which is spanned by f (o + v)−f (o) the map k : R → R is well defined through the relationship f~(λv) = k(λ)f~(v). From f (o) + k(λ)f~(v) = f (o) + f~(λv) = f (o + λv) = f (gl−1 (λ)) we see that k is given by k(λ) = gf (l) ◦ f ◦ gl−1 (λ). In order to prove additivity of k we use the fact that (λ+µ)v = λv+µv can be constructed using parallel lines (cf. Figure 10 .10 .20 ) Let w ∈ Rn be linearly independent from v and consider the triangle defined by the points o, o + w, o + λv. This triangle can be parallely translated so that the point o is mapped into o + µv. (We simply parallely translate the lines generated by its sides as indicated in the figure). Since this translation preserves the vectors defined by the sides of the triangle we
11
PSfrag replacements f f (o + w)
o+w
λv o + λv + µv f (o) f (o + λv) o + µv f (o + µv) o + λv f (o + λv + µv) µv
o
Fig. 10 .10 .20 . Additivity of k
have obtained a geometric construction of the point o + λv + λw. Since this construction only employs intersection points and parallel lines it is preserved by the map f . Hence we obtain f~((λ + µ)v) = f~(λv) + f~(µv) = k(λ)f~(v) + k(µ)f~(v) and therefore k(λ + µ) = gf (l) ◦ f ◦ gl−1 (λ + µ) = gf (l) ◦ f (o + (λ + µ)v) = gf (l) (f (o) + f~((λ + µ)v)) = gf (l) (f (o) + k(λ)f~(v) + k(µ)f~(v)) = gf (l) (f (o) + (k(λ) + k(µ))f~(v)) = k(λ) + k(µ).
PSfrag replacements
f f (o + w)
f (o + λw)
o + λw f (o)
o+w
o + λµv o + µv
o
o + λv o+v
f (o + v) f (o + λµv) f (o + λv) f (o + µv)
Fig. 10 .10 .30 . Multiplicativity of k
The proof of multiplicativity is similar and employs a slightly different geometrical construction (cf. Figure 10 .10 .30 ) which is justified by Lemma 10 .10 .20 . The configuration in the first part of Figure 10 .10 .30 lies in a plane whence hypersurfaces are simply lines. Denote by H2 the line which connects o + v with o + w, by H1 its parallel translation through o, and by H3 its parallel translation through o + λv. Further denote the
12
line through o and o + v by l and the line which connects o with o + w by l0 . Using the notation of Lemma 10 .10 .20 we have λ=
x3 (l)−x1 (l) (o + λv)−o = . (o + v)−o x2 (l)−x1 (l)
Hence Lemma 10 .10 .20 implies that the intersection of H3 and l0 is really o + λw as depicted in the figure. We apply this lemma a second time where the three parallel hypersurfaces H20 , H10 , H30 are now given by the line connecting o + µv with o + w, its parallel translation through o, and its parallel translation through o + λw. It follows that the intersection of H30 with l is o + µ(λv) = o + λµv. Since this construction only employs intersections and parallel lines it is preserved by f and we obtain f~(λµv) = k(λ)k(µ)f~(v). This implies k(λµ) = gf (l) ◦ f (o + λµv) = gf (l) (f (o) + f~(λµv)) = gf (l) (f (o) + k(λ)k(µ)f~(v)) = k(λ)k(µ). Hence k is really an automorphism of the real line. One can geometrically show that this automorphism neither depends on v nor on o. However in our case this automorphism is trivially well defined since we already know that the only non-zero automorphism of R is the identity. This also implies f~(λv) = λf~(v) for all λ ∈ R, v ∈ Rn . Hence the theorem is proved. The text continues now with Definition 1.1.3 in the main text.
