Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B, Eckmann, ZOrich
361
John W. Schutz Monash University, Clayton, Victoria/Australia
Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time
Springer-Verlag Berlin.Heidelberg • New York 1973
A M S Subject Classifications (1970): Primary: 70 A05, 83 A05, 83 F05 Secondary: 50-00, 5 0 A 0 5 , 50 A 10, 50C05, 50D20, 53C70 I S B N 3-540-06591-1 Springer-Verlag B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-06591-1 Springer-Verlag N e w Y o r k - H e i d e l b e r g • B e r l i n This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag Berlin • Heidelberg 1973. Library of Congress Catalog Card Number 73-20806. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
To
Amina
IX
CHAPTER 9.1 9.2 9.3 9.4 9.5 9.6 9.7
9.
THREE-DIMENSIONAL
250
KINEMATICS
Each 3-SPRAY is a 3-Dimensional H y p e r b o l i c Space Transformations of Homogeneous Coordinates in T h r e e - D i m e n s i o n a l Hyperbolic Space Space-Time Coordinates Within the Light Cone Properties of Position Space Existence of Coordinate Frames Homogeneous Transformations of Space-Time Coordinates Minkowski Space-Time
CHAPTER i0.
CONCLUDING
APPENDIX
CHARACTERISATION
i.
APPENDIX 2. H O M O G E N E O U S E U C L I D E A N SPACES
BIBLIOGRAPHY
256 263 271 278 286 290 300
REMARKS
OF THE ELEMENTARY
COORDINATES
251
SPACES
302
IN H Y P E R B O L I C AND 309
312
PREFACE
The aim of this monograph is to give an axiomatic development of Einstein's theory of special relativity from axioms which describe intuitive concepts
concerning
the kinematic behaviour of inertial particles
and light
signals.
I am grateful to Professor G. Szekeres and Dr. E.D. Faekerell for their encouragement and constructive
suggestions
during the preparation of this
monograph.
John W. Schutz Monash University
TABLE OF CONTENTS
CHAPTER i.
INTRODUCTION
CHAPTER 2.
KINEMATIC AXIOMS
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13
3.7
SPACE-TIME
Primitive Notions Existence of Signal Functions The Temporal Order Relation The Triangle Inequality Signal Functions are O r d e r - P r e s e r v i n g The Coincidence Relation. Events Optical Lines A x i o m of the Intermediate Particle The Isotropy of SPRAYs The A x i o m of Dimension The A x i o m of Incidence The A x i o m of Connectedness Compactness of Bounded sub-SPRAYs
CHAPTER 3.1 3.2 3.3 3.4 3.5 3.6
FOR MINKOWSKI
3.
CONDITIONALLY
COMPLETE
PARTICLES
7 8 9 12 13 17 24 25 33 35 36 38
42
Conditional Completion of a Particle 42 Properties of E x t e n d e d Signal Relations and Functions44 G e n e r a l i s e d Triangle Inequalities 47 Particles Do Not Have First or Last Instants 48 Events at Which Distinct Particles Coincide 50 G e n e r a l i s e d Temporal Order. Relations on the Set of Events. Observers. 52 Each Particle is Dense in Itself 57
VIII CHAPTER 4.1 4.2 4.3 4.4
6.4
8.4
59 62
5.
COLLINEAR
SUB-SPRAYS
AFTER
6.
7.
66 69
COINCIDENCE
71
Coincidence
COLLINEAR PARTICLES
THEORY
8.
103 i19 of
OF P A R A L L E L S
0NE-DIMENSIONAL
72 77 84 95 1O0
103
D i v e r g e n t and C o n v e r g e n t P a r a l l e l s The P a r a l l e l R e l a t i o n s are E q u i v a l e n c e R e l a t i o n s C o o r d i n a t e s on a C o l l i n e a r Set I s o m o r p h i s m s of a C o l l i n e a r Set of P a r t i c l e s L i n e a r i t y of M o d i f i e d S i g n a l F u n c t i o n s
CHAPTER 8.1 8.2 8.3
59
Basic T h e o r e m s The C r o s s i n g T h e o r e m C o l l i n e a r i t y of T h r e e P a r t i c l e s . Properties Collinear sub-SPRAYs. P r o p e r t i e s of C o l l i n e a r Sets of P a r t i c l e s
CHAPTER 7.1 7.2 7.3 7.4 V.5
OF C O L L I N E A R I T Y
C o l l i n e a r i t y of the Limit P a r t i c l e The Set of I n t e r m e d i a t e P a r t i c l e s M i d - W a y and R e f l e c t e d P a r t i c l e s A l l I n s t a n t s are O r d i n a r y I n s t a n t s P r o p e r t i e s of C o l l i n e a r S u b - S P R A Y s A f t e r
CHAPTER 6.1 6.2 6.3
IMPLICATIONS
Collinearity. The Two Sides of an E v e n t The I n t e r m e d i a t e I n s t a n t T h e o r e m M o d i f i e d S i g n a l F u n c t i o n s and M o d i f i e d R e c o r d Functions B e t w e e n n e s s R e l a t i o n for n Particles
CHAPTER 5.1 5.2 5.3 5.4 5.5
4.
KINEMATICS
R a p i d i t y is a N a t u r a l M e a s u r e for S p e e d C o n g r u e n c e of a C o l l i n e a r Set of P a r t i c l e s P a r t i t i o n i n g a C o l l i n e a r Set of P a r t i c l e s i n t o Synchronous Equivalence Classes C o o r d i n a t e F r a m e s in a C o l l i n e a r Set.
123 132
147 148 164 172 191 210
233 233 239 243 246
§o.o] CHAPTER
0
S UMMARY
Minkowski elements
space-time
called
the "signal "inertial
"particles"
relation".
particles"
"light
signals".
Walker
[1948].
Particles
similar
in content
axioms,
four concern
are eleven
(i) between
any two distinct
(ii) each SPRAY
which
Of the remaining coincide
SPRAYs
particles
of a SPRAY,
there
from both,
is isotropic,
(iv) each bounded content
infinite
particles, sub-SPRAY
of the remaining
can be "connected" particles
is a third distinct
at any
We postulate:
is a SPRAY which has a maximal
any two distinct
to
The first five are
which
is distinct
SPRAY of four distinct
space-time
corresponds
[1948].
sets of particles
is a particle
The essential
axioms.
are called
to
is similar to that of
to those of Walker
one given event and which
relation,
physically
relation
basis
of undefined
undefined
correspond
and the signal
there
in terms
and a single
The undefined
Altogether
(iii) there
is developed
particle
is compact.
two axioms
coincide
which
sub-
and
by particles;
which
symmetric
is that:
and that,
given
at some event,
forms the third
there
side of a
"triangle". The ensuing
discussion
falls naturally
into two parts;
the
§0.0]
XII
development similar
of rectilinear
to the geometry
and the extension established hyperbolic homogeneous
showing
will now be described
particles
in the theory is applied
are analogous
of absolute
to particles
a given
particles event.
kinematics
geometry;
which
is
a correspondence coordinates.
between These
ideas
to properties
geometry.
and,
which
of maximal
eollinear
sets of
and it is found that they have many
faced with the possibilities distinct
in absolute
are shown to exist and their properties
Then the existence
which
is in many ways
in more detail.
is demonstrated
properties
subsets
and space-time
sub-SPRAYs
are discussed.
which
that each SPRAY is a three-dimensional
and then extending
coordinates
Collinear
of eoplanar
to three-dimensional
by first space
kinematics,
of coplanar
The concept
as in absolute
are parallel
It is shown that there
which may or may not be distinct,
of parallelism
geometry,
of there being none~
subsets
we are
one~
or two
to a given particle are two types
and that both types
through
of parallels, of parallels
lead to equivalence
relations
of parallelism.
events
eollinear
set can then be "coordinatised"
in a maximal
with respect relations
to any equivalence
of parallelism
to reflection it is possible
parallelism
class of parallels.
turn out to be invariant
mappings.
By composing
to generate
and time translations.
The set of all
"pseudo-rotations",
It transpires
is a theorem,
several
which
Both
with respect
reflection
space translations
that the uniqueness
is a marked
mappings
contrast
of
with the
§0.0]
XIII
theory of absolute
geometry!
This remarkable
that each particle
moves with uniform
It is shown that each SPRAy bolic
space,
relative
with particles
velocity
in three-dimensional coordinates
"within
correspondence coordinate
position-space space.
Associated
which
and with coordinates
to space-time
The extension
of this
gives rise to the concept with each coordinate
of homogeneous
to "homogeneous
the "inhomogeneous
Homogeneous
hyper-
of a
frame
is shown to be a three-dimensional
Transformations
correspond
to "points"
space correspond
a light cone".
to all events
frame.
is a three-dimensional
function.
hyperbolic
implies
velocity.
corresponding
as a metric
finding
Lorentz
Lorentz
coordinate
transformations"
transformations"
is a euclidean
systems
then
from which
are derived.
§O.1]
XIV
§0.L
GLOSSARY
This listing
OF DEFINITIONS
contains
order of appearance
AND NOTATION
definitions
within the text.
and symbols Numbers
in their
on the left
refer to the n u m b e r of the section in which the definition appears.
The symbol [] indicates
Section
the end of a proof.
Definition
Q,R,S,T,U,V,W
21
particles
21
set of particles
21
instants
21
set of instants
21
signal relation
22
signal function §3.2,
Qa, R1,Sx,... ~
(see also
(see also
§3.1)
§3.2,
(from Q to R) f ~ RQ
§3.6)
(see also
§3.6)
2.3
record function
2.3
distinct
instants
2.3
temporal
(order)
also
and Notation
§3.2,
(of R relative
relation(s)
to Q)
f o f QR ~Q
<,>,~,~
(see
§3.6)
23
before-after
<,>,¢,5
24
direct
24
indirect
26
coincidence
26
event
26
set of events
27
o p t i c a l line,
signal signal
[
(see also
relation ]
in optical
§3.2,
§3.6)
line
I
,
,
>
§0.1]
XlX
Section
Definition
9.3
origin
9.3
coordinate
9.3
within the light cone
9.3
light cone
9.3
vertex
9.3
position
9.4
parallel position
9.5
coordinate
9.5
time coordinate
9.6
homogeneous
A.I
topology
A.I
open sets
A.I
points
A.I
closed set
A.I
neighbourhood
A.I
closure
A.I
connected
A.I
open cover
A.I
subcover
A.I
compact
space
A.I
locally
compact
A.I
metric
A.I
distance
A.I
diameter
A.I
bounded
and Notation
in space time i d e n t i f i c a t i o n
(upper,
mapping
lower)
space spaces
frame
space
transformation
Lorentz
transformation
space
§o.1]
XV
Section
Definition
2.7
exterior
2.8
permanently
2.8
distinct
2.9
between
2.9
betweenness
relation
(also
§4.4)
and Notation
to coincident
particles
=
particles
§3.2,
for particles
2.9
SPRAY
SPR[
]
2.9
spray
spr[
]
2.9
isotropy
2 .i0
symmetric
sub-SPRAY
2.12
connected
(set of instants)
2.13
bounded
sub-SPRAY
2.13
cluster
particle
3.1
first
3.1
last
3.!
cut
3.1
gap
3.1
conditionally
3.1
ordinary
3.1
ideal
3.1
instant
3.1
set of conditionally
3.1
set
3.2
extended
mapping
<
,
,
e,~,~
(instant) (instant)
complete
Qx, Rl,...
Qo, RI, ....
of instants
Q,...
Qx, RI,'''
instant
instant
particle
~
(see also complete
particles
(see also
signal relation
§2.1)
§2.1)
(see also
§2.1,
§3.6)
>
50.i]
XVl
Section 3.2
Definition and Notation extended temporal order relation (see also §2.3, §3.6)
3.2
extended signal function
3.2
extended coincidence relation
3.2
ideal event
3.2
extended relation of "in optical line"
(see also §2.2, §3.6)
(see also §2.7, §3.6) 3.2
extended betweenness relation
3.6
generalised temporal order relation (see also §2.3, §3.2)
3.6
generalised signal relation (see also §2.1, §3.1)
3.6
generalised signal functions
3.6
generalised relation of "in optical line"
(see also §2.2, §3.2)
(see also §2.7, 53.2) 3.6
observer
3.7
dense
4.1
oollinear particles
4.1
side (left, right)
(see also §4.2, §6.2)
4.2
side (left, right)
(see also §4.1, §6.2)
4.2
right optical line, left optical line
4.2
to the right of, to the left of
4.3
modified record function
f o f QR RQJ
4.3
modified signal functions
f+,f-
~,~,~,. ....
[Q,R,S,...]
(see also §4.4)
§0. i]
XVII
Definition and Notation
Section 4.4
betweenness <
4.4
,
,
relation after
> after
[
(before) an event
]
collinear after (before) an event
[ , , ] after
(see also §4.1) 5.1
limit particle
5.3
mid-way between
5.3
reflection
5.3
reflected observer
kT
5.5
collinear sub-ZPRAY
CSP[
]
5.5
collinear sub-spray
csp[
]
6.2
side (left, right)
6.2
cross
6.4
collinear set (of particles)
6.4
collinear
7.1
parallel
7.1
diverge f r o m
7.1
(see also §7.1)
(see also §7.1) (see also §7.1)
(see also §4.1,
§4.2)
(verb)
set (of events)
COL[
col[
] ]
(divergent or convergent)
converge to
7.1
reflection
(see also §5.3)
7.1
reflected particle
7.1
reflected event
7.1
mid-way parallel
7.3
dyadic numbers, parallels,
7.3
indexed class of parallels
7.3
time scale
(see also §5.3)
instants
(divergent, convergent)
[
]
§0.i]
XVIII Definition and Notation
Section 7.4
pseudo-rotation
7.4
spacelike translation (see also §7.5)
7.4
time translation
7.5
natural time scale (see also §7.3)
7.5
space displacement mapping (see also §7.4)
7.5
time displacement mapping (see also §7.4)
7.5
time reversed
8.1
constant of the motion
8.1
rapidity (directed, relative)
8.2
congruent particles
8.2
distance
8.3
synchronous particles
8.4
position-time coordinates
8.4
coordinate frame (in coll)
8.4
origin in position-time
8.4
origin in position
8.4
velocity
9,1
3-SPRAY
9.2
origin of homogeneous coordinate system
9.2
homogeneous coordinates
9.3
space-time coordinates
9.3
origin in space-time
9.3
time coordinate
9.3
space coordinates
(see also §7.5)
(directed)
3SP[
]
(see also §A.2)
~0.1]
XX
Section
Definition
A.I
curve
A.I
length
A.I
arcwise
A.1
intrinsic
A.I
motion
A.I
doubly transitive
A.I
isotropie
A. 2
point
A.2
projective
A.2
special
A.2
change
and Notation
of curve connected metric
space
coordinates
projective of basis
coordinates
§i.0]
CHAPTER i INTRODUCTION Following Einstein's relativity
(Einstein
been proposed
formulation
[1905]),
for Minkowski
reasons
for developing
lines.
One reason,
several axiomatic
space-time.
There are several
which is not always made explicit,
more widely accepted.
in clarifying
that some m o d i f i c a t i o n
concepts
Lobaehevsky
theory of physical
subsequently
of physical
reader is encouraged
unknown,
interest;
[~ 300 B.C.] proposed Bolyai
priately modified
[1832] and
axiom of p a r a l l e l i s m
Similarly,
we might expect
versions
is the de-Sitter universe. to Axiom I (§2.2)).
space-time
and possibly non-Riemannian,
a possibility
to keep in mind!
but Riemannian,
lead
Such was the
of an axiom system for Minkowski
could lead to a previously
Minkowskian,
geometry!
a
it is conceivable
interest.
[1829] altered the Euclidean
and discovered hyperbolic
space-time
and exhibiting
to one or more of the axioms might
geometry:
that m o d i f i c a t i o n
and
if an axiomatic
based assumptions,
case with the axiom system which Euclid for elementary
is a
may be better understood
Another reason is that,
small number of intuitively
to an alternative
systems have
a physical theory along axiomatic
desire that special relativity
system is successful
of the theory of special
space-time
which the
(An example
of a non-
which satisfies
of the axioms given
appro-
here,
The principal m o d i f i c a t i o n
made is
§i.o] Prior to the present
treatment,
have already been formulated. of a coordinate
frame;
[1954,
approach
1959] and Noll
on the assumption
Some authors
in particular,
atised the conventional Suppes
several axiomatic
Bunge
systems
assume the concept [1967] has axiom-
due to Einstein
[1905], while
[1964] have based their systems
of the invariance
of the quadratic
form
Ax§ + Ax~ + Ax~ - c2At 2
with respect to transformations Zeeman
between coordinate
[1964] has shown that the inhomogeneous
is the largest group of automorphisms Robb
[1936] formulated
relation
an axiomatic
(before-after)
called "events",
Walker
grounds.
[1948,
relativistic
1959], who suggested cosmology
order".
sufficiently restricted succeeded
on mathematical
simplicity rather than
foundations
of "particles",
Walker's
"light signals"
and
axiom system was not developed
to sets of relatively
space-time
light signal",
The undefined
and was, in fact,
stationary particles,
in clarifying many kinematic
those of "particle",
for
in terms of an undefined basis which
to describe Minkowski
"collinearity".
elements which he
A system of axioms has been proposed by
involved the concepts "temporal
space-time.
system in terms of a single
aim of m a t h e m a t i c a l
is achieved by selecting his axioms physical
Lorentz group
of Minkowski
between the undefined
however Robb's
frames.
concepts,
"optical
especially
line"
basis of Szekeres
but it
and
[1968] bears
§l.O]
some r e s e m b l a n c e to that of Walker, a l t h o u g h Szekeres regards both particles
and light signals as objects whereas W a l k e r
regards particles
as objects and light signals as p a r t i c u l a r
instances of a binary "signal relation". axiomatic systems,
only that of Szekeres
describing M i n k o w s k i
Of these three [1968] succeeds in
space-time in terms of assumptions related
to what one m i g h t describe as either "kinematic experience" or p h y s i c a l intuition.
Our intention is to describe M i n k o w s k i space-time in terms of u n d e f i n e d elements called "particles", r e l a t i o n called the "signal relation"
a single u n d e f i n e d
and eleven axioms w h i c h
are intended to be in accordance with the reader's physical intuition.
In a subjective
sense,
a particle
corresponds to a
freely moving observer who is capable of d i s t i n g u i s h i n g between "local" events;
the concept of a particle
regarded as a more frame"
can therefore be
basic concept than that of a "coordinate
(which d i s t i n g u i s h e s between d i f f e r e n t events by
assigning d i f f e r e n t sets of coordinates to them):
the u n d e f i n e d
signal r e l a t i o n corresponds p h y s i c a l l y to "light signals". This u n d e f i n e d basis is similar to that of Szekeres a p h y s i c a l sense and to that of Walker sense, a l t h o u g h whereas Walker
[1948, 1969] in a formal
[1948, 1959] used
relations,
the "signal relation"
relation",
the present treatment has only
the "signal relation".
[1968] in
two
undefined
and the "temporal order one
u n d e f i n e d relation,
The notion of time ordering is implicit
§l.o]
in the concept of the signal r e l a t i o n and so temporal order can be defined in terms of the signal relation.
Apart from
this change and certain other m o d i f i c a t i o n s w h i c h result in a w e a k e r set of assumptions,
the first five axioms of the present
system, t o g e t h e r with their e l e m e n t a r y consequences,
bear a
strong resemblance to the excellent analysis of the concepts of "particles",
"light signals"
Walker
The subsequent
[1948].
different from those of W a l k e r
and "collinearity"
given by
six axioms are e s s e n t i a l l y [1948,
1959] and are b e l i e v e d
to be original in their a p p l i c a t i o n to special relativity. Four of these axioms refer to sets of p a r t i c l e s which r e p r e s e n t "velocity space"; they resemble axioms which have been used in the study of metric geometry by authors such as Busemann
[1955].
Before stating the axioms it may be as well to point out that there are several assumptions which we do not make.
In
particular, we do not assume the concept of a coordinate frame, we do not assume that the set of instants of each particle be ordered by the real numbers,
can
nor do we assume that p a r t i c l e s
and light signals move with constant speed.
In the present
axiomatic system, these p r o p o s i t i o n s turn out to be theorems. Three properties of Minkowski
space-time are of central
importance to the subsequent development. kinematics
0ne-dimensional
is in m a n y ways analogous to plane absolute geometry,
for it transpires that the concept of p a r a l l e l i s m can be
§z.o]
applied to particles and, furthermore,
the corresponding
question of uniqueness of parallelism is closely related to the uniform motion of particles. Szekeres
Both Robb [1936] and
[1968] observed that uniform motion implies uniqueness
of parallelism but, in the present treatment, we are able to prove the uniqueness of parallelism and then to show that this implies the uniform motion of (freely-moving)
particles,
so
that we need not assume Newton's first law of motion explicitly. The second important property is that, in contrast to the euclidean velocity space of Newtonian kinematics, space associated with Minkowski
the velocity
space-time is hyperbolic,
a
property which is established in the present treatment by making use of a recent characterisation by Tits [1952, 1955].
of the elementary spaces
The third important property is that
space-time coordinates are related to homogeneous in a three-dimensional hyperbolic
space.
is an isomorphism between homogeneous and transformations
of homogeneous
coordinates
Consequently there
Lorentz transformations
coordinates in hyperbolic
space. Our primary aim is to clarify the foundations of special relativity so that the theory becomes as acceptable and familiar as euclidean geometry.
Accordingly,
the question of independence
of the axioms is of secondary importance and is briefly discussed in Chapter i0.
Consistency of the axioms can be easily verified
§i.0]
by considering the usual model of Minkowski
space-time.
Since
much of the t e r m i n o l o g y and n o t a t i o n is new, a listing of definitions and n o t a t i o n has been i n c l u d e d before the m a i n text.
§2.1] CHAPTER
§2.1
KINEMATIC
AXIOMS
Primitive
Notions
A model terms
(i)
a set
FOR M I N K O W S K I
of M i n k o w s k i
of the following
2
SPACE-TIME
space-time
primitive
~
whose
elements
particle
being
a set whose
will
be d e s c r i b e d
in
notions:
are called
particles,
elements
each
instants;
are called
and
relation o
(it) a binary
Particles W,
....
particle
are denoted
Instants symbol
The binary r e l a t i o n such as
on the
by the
belonging
together
QI" Qe" Qa" Qx e Q.
expression
defined
set of all instants.
symbols
Q, R, S,
to a particle
with
some
subscript,
Qx o Ry
the
U,
are denoted
The set of all instants o is called
T,
by the
for example, is denoted
signal relation.
is to be read as
V,
"a
by
~ .
An
signal goes
from Qx to Ry tt or "a signal leaves Qx and arrives at Ry If In this notions Walker's
system,
is an a d a p t a t i o n undefined
the u n d e f i n e d and the
axiomatic
of the basis
elements
relations
the u n d e f i n e d
are
are the
temporal order relation.
basis
of p r i m i t i v e
used by W a l k e r
instants
and
[1948].
particles,
and
signal correspondence relation In the present
treatment,
§~.2] the signal relation is analogous to Walker's signal correspondence relation but the temporal order relation is defined in terms of the signal relation
That is, the present
(§2.3).
system makes use of only one undefined binary relation,
whereas
the system of Walker is expressed in terms of two undefined binary relations.
In physical terms, particles correspond to
"inertial particles" §2.2
and signals correspond to "light signals".
Existence of Signal Functions
AXIOM I
(SIGNAL AXIOM)
Given particles Q,R and an instant R y e
R,
there is a unique instant Qx ~ Q such that Qx ~ Ry, and there is a unique instant Qz e Q such that Ry ~ Qz" This axiom is used in the proof of Theorems i (§2.4), 2 (§2.6),
3 (§2.7),
13 (§3.6), 16 (§4.1),
17 (§4.2),
32 (§6.4)
and 61 (§9.5).
Q
Fig. i.
R
In all diagrams, particles are represented by solid
lines and signal relations are indicated by broken lines between instants which are represented by dots.
§2.3] The Signal Axiom implies the existence of a bijection from Q to
R, which will be called a
be denoted by the symbol
Qx,-~ Ry
fRQ
signal function
and will
where
if and only if
Qx a Ry.
Thus the Signal Axiom (Axiom I) is equivalent to the Signal Axiom of Walker [1948, Axiom SI, P322]. treatment,
In the present
the signal functions are equivalent to the
"signal correspondences"
and "signal mappings" of Walker
[1948] and Walker [1959], respectively. Connectedness of Szekeres
The Axiom of
[1968, Axiom A4, P138] has a similar
"physical content". Given any two particles Q and S, the composition of signal functions
f Qs
of sQ
is a mapping from Q to Q which is related to the "motion of S relative to Q" and is called the
record function
(of S
relative to Q). §2.3
The Temporal Order Relation In the present system, the "temporal
is defined in terms of the o-relation.
(order) relation"
This is
a departure
from the system of Walker [1948, P321], in which "temporal
§2.3] order" this
an i n d e p e n d e n t
difference
Walker ment
was
[1948,
in approach,
Axiom
as a t h e o r e m
Given
Qx / Qz
write
(Theorem
same
y
appears
i,
instant,
and say that
As a result
of
of an axiom of
in the present
treat-
§2.5).
Qx" Qz ~ 9"
Q and two instants
If Qx # Qz and if there R
relation.
the content
$3, P322]
a particle
and Qz are the
undefined
we write
Qx
Qz
and
exists
Qx = Qz;
if
Qx
otherwise
we
distinct instants.
are
a particle
R with
an instant
£ R such that ~
Qx ~ R Y we write
Qx"
Qx
< Qz and say that
The r e l a t i o n
(see the previous strict
and
temporal
< is called Fig.
I).
relation
that the previous
R
Qx
~ Qz
Y
before Qz
is
the
after
(strict) temporal relation
The next
axiom
is a n t i s y m m e t r i c ;
definition
and Qz is
is i n d e p e n d e n t
states that
that the
is,
it ensures
of the choice
of
R.
AXIOM
II
(FIRST A X I O M
OF T E M P O R A L
ORDER)
Given a particle Q and two distinct instants Qx" Qz e 9" then either Qx < Qz" or Qz < Qx" but not both. That
is, the t e m p o r a l
two instants sense that,
relation
is antisymmetric,
from the same p a r t i c l e given
sive possibilities:
Qx, Qz e Q, (i)
there
Qx = Qz
10
are
"comparable"
are three m u t u a l l y or
(ii)
Qx
< Qz or
and any in the exclu-
§2.3] (iii)
Qz < Qx"
1 (§2.5)
and ii
The
symbol
This
is used
in the proof
of T h e o r e m s
(§3.4).
~ will be used
tt
can be w r i t t e n
axiom
Qx
=
Qz
concisely
so that
statements
of the form:
tf
or
Qx < Qz "
as
"Qx ~ Qz" We define
the symbols
has the same m e a n i n g ment
> and ~ so that the as the statement
"Qx ~ Qz" has the same m e a n i n g
AX I O M
III
(SECOND A X I O M
"Qz
statement < Qx""
and the
as the statement
OF T E M P O R A L
Qz"
"Qx >
state-
"Qz ~
Qx"
ORDER)
Given a particle Q and instants Qx" Qy" Qz ~ ~; if Qx < Qy and Qy < Qz" then Qx < Qz"
That is, the t e m p o r a l the
conclusion
pa r t i c l e
of the axiom implies
R and an instant
Qx ~ R This
relation
axiom is used
An immediate
R
Y
Qx ~ Qy
and
that there
Furthermore,
exists
a
e R such that
and
in Theorems
corollary
is transitive.
R
Y
~ Qz"
i (§2.5)
to this
and ]0
axiom
is the proposition:
Qy ~ Qz ~ @x ~ @z"
11
(§3.3).
§2.4] The previous two axioms imply that the temporal
relation
is a simple ordering
on each particle,
order
so we shall
also call it the temporal order relation. §2.4
The T r i a n g l e I n e q u a l i t y Composite statements of the form Qx e Ry and Ry q S z are
sometimes c o m b i n e d for the sake of brevity to Qx q Ry q S z. (Note that q is not a transitive r e l a t i o n and so
Qx q Ry q S z =/~ Qx q S z ).
A statement containing one
a - r e l a t i o n is called a direct
signal,
and a composite state-
ment involving two or more a-relations
signal
(for example,
is called an indirect
Qx q Ry is a direct signal and
Qx q Ry ~ S z is an indirect signal).
A X I O M IV
(TRIANGLE INEQUALITY)
Let Q, R, S be particles
with instants
Qw e Q, R y e
R and
Sx, S z ~ S. If Qw q Sx and Qw ~ Ry ~ Sz, then S x <. Sz. That is, "of all signals leaving Qx and arriving at S, no signal arrives before the direct signal". equivalent to an axiom of W a l k e r
[1948, A x i o m $2,
A x i o m IV is used in the proof of Theorems 3 (§2.7),
5 (§2.9),
6 (§2.9),
This axiom is
8 (§3.2),
12
1 (§2.5),
P322]. 2 (§2.6),
28 (§6.1), and 57 (§9.1).
§2.5]
§2.5
Signal Functions
THEOREM i
are Order-Preserving
(Signal Functions
Let S,T be p a r t i c l e s Tx, T z E T such
with instants
~ T X
< S X
then T Z
and
S
X
o
of Theorems
T
Z
Z
< T . X
Z
This theorem is a consequence II (§2.3),
Sx, S z ~ S a n d i n s t a n t s
that
S
If S
are Order-Preserving)
of Axioms
III (§2.3) and IV (§2.4); 3 (§2.7),
5 (§2.9),
and is used in the proof
8 (§3.2),
and 60 (§9.4).
PROOF
(see Fig.
2).
R
S
11. f
f
T T T
R
~ f ~
1 1 I~
Fig 2
X3
I (§2.2),
Y X
17 (§4.2)
§2.6] By t h e
First Axiom
is a p a r t i c l e
of T e m p o r a l
R with
Order
an i n s t a n t
R
~
By the
Signal
( A x i o m II,
§2.3)
s R such that
S
y
Axiom
(Axiom
I,
T
e T such that R o T . y ~ y Y ( A x i o m IV, §2.4),
there
o R x
§2.2)
there
o S y
. Z
is an i n s t a n t
By the T r i a n g l e
Inequality
S x o T x and S x o Ry o Ty ==~ T x ( T y , By 0 Ty a n d Ry (7 S z (7 T a n d by t h e
Second
T
By the
Signal
The [1948,
§2.6
Axiom
The
$3,
a relation
of this P322]
.< T
y
I,
~
T
y
Order
T
§2.2),
(Axiom
llI,
§2.3),
.< T . z
x T
~ Tz,
x
# Tz,
appeared
so T
< T . z
x
in W a l k e r ' s
system
as an axiom.
Relation.
the n o t i o n
element
Events.
of an
of M i n k o w s k i o n the
as e q u i v a l e n c e
instants
~
z
theorem
coincidence
defined
Given
T
(Axiom
defining
of
of T e m p o r a l
and
y
Coincidence
undefined
are t h e n
.< T
Axiom
content
Before usual
x
Axiom
z
Qx a n d R
"event",
which
space-time,
set o f i n s t a n t s . classes
such that
of
is the
we d e f i n e Events
coincident
instants.
both
Y Qx a R
and R Y
we
say that
~ @x" Y
Qx and Ry are coincident
14
instants
and w r i t e
Qx -- Ry.
§2.6] THEOREM 2
The coincidence
relation
is symmetric
and transitive.
This t h e o r e m is a consequence of A x i o m I (§2.2) and A x i o m IV (§2.4). event
It is applied in the d e f i n i t i o n of an
(this section and also §2.12) and in the proof of
the Corollary to T h e o r e m 7 (§2.12).
PROOF.
By d e f i n i t i o n the coincidence r e l a t i o n is symmetric.
By the d e s c r i p t i o n of our model, each instant belongs to some particle,
so we consider any three particles Q,R,S
and any three instants Qx e Q, R y e
R,~ S z e S~ such that
Qx ~ Ry and Ry ~ S z. We will show that Qx = Sz"
By the Signal A x i o m
(Axiom I,
§2.2) there is an instant S u e S~ such that Qx ~ S u and since
Qx ~ Ry ~ Sz, the Triangle Inequality (Axiom IV, §2.4) implies that S u ~ S z. S
z
~ S . ~
Similarly Ry o Sz and Ry o Qx o S u imply that
Consequently S
z
= S
u
and therefore Qx q S
similar argument shows that S z o Qx"
z"
A
[]
At this stage we can not prove that the signal r e l a t i o n is reflexive.
However the symmetry and t r a n s i t i v i t y of the
coincidence r e l a t i o n allows us to deduce the following sufficient condition:
15
§2.6]
Given a particle
COROLLARY.
there is a particle
Q and an instant Qx e Q, if
R~ and an instant R y e R such that
Qx ~ Ry, then Qx a Qx and hence Qx ~ Qx"
This corollary
is used in the proof of Theorem
7 (§2.12)
and its corollary where we show that the signal relation and the coincidence
PROOF.
relation
are reflexive.
By transitivity.
[]
Qx' we call the equivalence
Now, given any instant of coincident [Qx]
an event
class
instants d~f
(see also
{Qx}
O
{R
§2.12).
Y
:
R
=
Y
Qx"
R
Y
e R, ~
}
R E
The set of all events
is called
event space and is denoted by the symbol ~ . Given particles
Qx e Q, Ry s R such that
@,R and instants
Qx = By, we say that the particles
@,R coincide
at the
event [Qx ], which is equivalent to saying that the particles Q,R coincide
at the event
[Ry].
The coincidence
has been defined and discussed previously although he did not define events coincident
instants.
16
relation
by Walker
as equivalence
[1948, P$23],
classes of
§2.7] §2.7
Optical
Given
Lines
instants
Qx,Ry,Sz
such
Qx ~ Ry ~ S z we
say that
which
the
we d e n o t e
of n i n s t a n t s only
set of i n s t a n t s
if,
IQx,Ry,Sz>.
by
that
and
{Qx,Ry,Sz)
is
in optical line,
Similarly,
we
say that
2 n } is {Q(II) "Q2"'''Q(n)
for all
a,b,c
with
Qx ~ Sz
in optical line
the
set
if and
I ~ a ~ b ~ c ~ n,
c
Note
that
this
relation
instead
of s a y i n g
use
alternative
the
that
is
not
Qx'Ry "Sz
expression:
an o r d e r are
relation.
in o p t i c a l
line,
S z is in o p t i c a l
and R
. This c o n c e p t is i n t e n d e d to be a n a l o g o u s Y g e o m e t r i c a l c o n c e p t of c o l l i n e a r i t y .
17
Sometimes, we may
line w i t h to the
Qx
§2.7] THEOREM
Q
Let Q,R,S,T,U
3.
e Q, R 1
~
e R, S 2
be particles with instants
e S, T 3
~
e T, and U ~
~
e U. 5
Then
~
(i)
IQI,R2,T4> and IR2,S3, T > imply
IQI,R2,Ss,T > , and
(ii)
IR2,S3, T > and IR2,T ,Us> imply
IR2,S3, T4, Us>.
This IV
theorem
(§2.4)
Theorems
PROOF.
and
is a c o n s e q u e n c e
Theorem
4 (§2.7),
(i)
i (§2.5);
16
(§4.1)
Q
o R 1
(2)
and
q S
20
Signal Axiom
in the p r o o f
T
o T
IV,
#
and R 4
(Axiom
o T . 2
I,
§2.4),
e T, w i t h X
~
S
§2.2) there
and t h e T r i a n g l e is an i n s t a n t
S
e S and
3
o S I
since
§2.5),
T
X
( A x i o m IV, Therefore
signal
functions
( T .
But Q
4
§2.4) by t h e
implies Signal
Q
(S)
1
o S
~
( S , such that X
Q
and
of
o T , and 1
X
an i n s t a n t
and
(§4.4).
and Q 4
3
2
(Axiom
o T 2
R
Inequality
and is u s e d
I (§2.2)
By d a t a ,
(i)
By t h e
of A x i o m s
(~ T X
tC
are o r d e r - p r e s e r v i n g
o T
4
that Axiom
and
T
so the
(Theorem
Triangle
~ Tx, w h e n c e T
(Axiom
I,
o T , a n d by
IQ~,s ,T > 3
18
§2.2)
(i)
and
i,
Inequality = T . (2),
§2.7]
Also,
from
(i),
(2) and
(3),
(4)
]Q
,R ,s > 1
The data, together with
2
(3) and
3
(4) are equivalent
to
IQ ,R ,S ,T >. 1
2
3
(ii)
The proof of (ii) is similar.
Let Q,R,S,T,U be particles with instants Q
COROLLARY. R
e R, S 2
~
Then
e S, T 3
(i)
~
e T, U 4
2
3
1
(i)
3
~ U.
h.
and IS ,T ,U > ~
5
(ii) IQ ,s ,T ,U > This corollary
3
t+
1
and IQ ,R ,S > ~
5
1
2
2
3
3
1
2
5
3
4
5
4 (§2.7).
IR2"S 3"T'Us >
IQI,S3,Us > and
IS3,T~,Us > ~
IQI"S3"T4"Us >
IQI,R2,S3 > and
IQI,S3,T~> ~
IQI"R2"S3"T4 >
and the data we have
IR ,T ,Us> , IQz,T ,Us> , IQI,R2,Us>, to
~
IQ ,R ,S ,T ,U >.
is used in the proof of Theorem
From these relations
relations
IQ ,R ,S ,T ,U >, and
5
IR2,S3,Us> and IS~'T4"U s> ~
are equivalent
e Q,
5
IQ ,R ,S ,U > 1
PROOF.
[]
IQI,R2, T4, Us>.
between quadruples
IQI,R2,T > which together
Now the data and the four
of instants
are equivalent
to
IQI"R2"S3"T4"Us >" (ii)
If
The proof is similar.
[]
Qx, Rx, Sx are non-coincident instants such that either
19
§2.7]
IQx,Rx, Sx > or ISx, Qx, Rx> or IRx, Qx, Sx > or ISx, Rx, Qx >, we say that the instant S
x
is exterior
to the (pair of)
instants Qx and R x.
AXIOM V
(UNIQUENESS OF E X T E N S I O N OF OPTICAL LINES)
Let Qx and R x be any two non-coincident Qx ~ Rx"
instants such that
If S x and T x are any two instants exterior to Qx
and Rx, then Qx and R x are in optical line with Sx and Tx.
This axiom is used in the proof of Theorems
4 (§2.7),
16 (§4.1) and 20 (§4.4). An equivalent,
though apparently weaker,
statement is:
"If S x and T x are any two instants exterior to Qx and Rx, then at least one of Qx and Rx is in optical line with Sx and T " x
The d e m o n s t r a t i o n of logical e q u i v a l e n c e involves
rate simple procf for each possible The A x i o m of Uniqueness
a sepa-
arrangement of instants.
of E x t e n s i o n of Optical Lines is
analogous to the axiom of U n i q u e n e s s of P r o l o n g a t i o n of Busemann [1955,
§8.1] and is w e a k e r than the c o r r e s p o n d i n g
axiom of Walker
[1948, A x i o m S.4., P324].
Thus the axioms which have been stated so far do not allow us to conclude,
as in the t r e a t m e n t of Walker
T h e o r e m 6.1, P324], that: with two n o n - c o i n c i d e n t
"All instants
[1948,
collinear optically
instants are in one optical line"~
20
§2.7]
since at this stage it is conceivable that we could have a situation analogous to multiple geodesics between antipodal points on a sphere.
THEOREM 4.
H o w e v e r we can prove the w e a k e r theorem:
(Existence of an Optical Line)
Given particles
Q,R,S with instants
QI e Q, ~ R 2 e R, ~ S 3 e S~
such that
•IQ1,R2,S3> then all instants
which
and QI ~ R 2 ~ S
are in optical
J
3
line with Q
and R , 1
or with R
and S , are in optical 2
3
A maximal set of instants, line, is called an optical that an optical
distinct
2
line.
line.
line i8 uniquely
instants
all of which are in optical Thus, this t h e o r e m states
determined
which are in optical
by any three
line.
This t h e o r e m is a consequence of A x i o m V (§2.7) and Theorem 3 (§2.7), together with its corollary; the proof of Theorems
22 (§5.2),
26 (§5.5),
and is used in
27 (§6.1),
Corollary 1 to T h e o r e m 30 (§6.3), and Theorem 32 (§6.4).
PROOF.
We consider any two instants Tx, Uy such that either
(i)
ITx, QI,R2 > or (ii)
IQI,Tx, R2 > or (iii)
IQI,R2,Tx>
or
(iv)
ITx'R2"S3 > or
IR2,Tx, Sa > or (vi)
IR2,S3, Tx>
,
(v)
and either
21
§2.7]
(i)
IUy,QI,R2 >
(4)
IUy,R2,S3 > or (5)
or (2)
IQz,Uy,R2 > or (3)
IQ1,R2,Uy>
IRz,Uy,S3> or
IR2,S3, Uy> .
We must show that for any instants
T
(6) and U
x
(i) - (vi) and
(i) - (6), respectively,
QI'R2"S3"Tx'Uy
are in optical
know that
2
(ii) (iii)
the instants
line in some order.
By data we
of Optical
Lines
Tx'Q I"R z,S 3> QI,Tx, Rz,S 3 > QI,Rz,Tx, S 3 > ,Tx, R2,S 3 >
(iv) ~i
(Axiom V, §2.7) and Theorem
3 (§2.7)
implications: (by Axiom V), (by Theorem or
IQ
R ,S ,T > iJ
or
2
5
3),
(by Axiom V),
X
]Tx,Q i ,R 2 ~ S 3 >
(by Axiom V),
(v)
QI,R2,Tx,S 3 >
(by Theorem
(vi)
Q ,R ,S ,T >
(by A x i o m V),
1
2
of
3
to obtain the following (i)
satisfying
IQ ,B ,S > and we apply the Axiom of Uniqueness 1
Extension
y
or
3
3),
CC
and
(1) (2)
UY'QI "R2 'S 3> IQI, Uy,R2,S3>
(by A x i o m V), (by Theorem
3),
(4)
=~ IQI,R2, Uy,S3> or ]Qz,R2,S3, Uy> IQI,Uy,R2,S3> or IUy'QI"R2"S3 >
(by A x i o m V),
(5)
=~ IQz'R2"Uy'S3 >
(by Theorem
(6)
=~ IQI,R2,S3,Uy>
(by Axiom V).
(3)
22
(by Axiom V),
3), and
§2.7]
Now each of the
36 cases
can be c o n s i d e r e d corollary similar
Case
first
18 cases,
all
to c o n s i d e r
can be p r o v e d
cases
IQI,Tx, R ,S3>
(ii)(3)
IQI,Tx, R2,S
in w h i c h
Case
(iii)(2)
that
and
(n)(M)
the first
18 cases.
using A x i o m V,
the c o r o l l a r y
U > (by A x i o m
to
IQI,R2,Uy,S3 > ~
and
IQI"Uy'R2"Tx'S3 >
or
(]Ql" Uy,R2,S 3) or 1Uy'Q,'R2" S 3>) or IUy,QI,Rz,S3,Tx>
The r e m a i n i n g
first
14 cases
A x i o m V, and are not i n c l u d e d
and
to T h e o r e m
IQI,R2,S 3,T x >)
to T h e o r e m 3) 3).
and
IQI,Uy,R2,Tx, S
~
ISy,Q ,R2,Tx, S 3> or IQI,Uy,R2,S3, Tx> 3),
IQ~'Tx'R2"Uy'Ss >"
(by the c o r o l l a r y
(by the c o r o l l a r y
(IQ1,R2,Tx, S 3>
3)
V).
3" y
IQI,Uy,R2,S3,Tx> (iii)(4)
to T h e o r e m
(IQ,'R2"Tx'Ss > or IQI,R2,S3,Tx>)
IQI"Uy'R2"S3 > ~
Theorem
be a p p a r e n t
(IQI"R2"Uy'S 3> or IQ 1"R2,S3, Uy >)
and
(by the c o r o l l a r y
IQI'Tx'R2"S3 >
(ii)(5)
Case
it should
3 is also applied:
Case
or
A x i o m V and the
apply to cases
so it is s u f f i c i e n t
==> IQI"Tx'Rz'Uy'S3 > or
will
for the f o l l o w i n g
Theorem
by applying
3 ; however,
considerations
Of these except
separately
to T h e o r e m
(6-n)(6-M),
(i)(1) .... (i)(6), .... ,(vi)(1) .... (vi)(6)
> or
(by the c o r o l l a r y
to
(by A x i o m V).
are all simple here
25
applications
of
for the sake of brevity.
[]
§2.8] This t h e o r e m which
does not exclude
are not e x t e r i o r
to Q
and S 1
optical
the p o s s i b i l i t y
and w h i c h may not be in 3
line with the pair of instants
Q
and R 1
pair R
and S 2
.
A stronger
of instants
result
, or with the 2
is o b t a i n e d
in C o r o l l a r y
of T h e o r e m uniquely
33
(~6.4),
determined
where we show that an optical by any two n o n - c o i n c i d e n t
line is
signal-related
instants.
§2.8
Axiom
of the I n t e r m e d i a t e
Two p a r t i c l e s each
Particle
Q,R are permanently coincident if, for
Qx e Q,~ there is some Ry e R such that Qx = Ry.
denote
1
3
permanent
By T h e o r e m symmetric
coincidence of Q and R by w r i t i n g
2 (§2.6)
permanent
and t r a n s i t i v e
coincidence
relation.
Q = R.
of p a r t i c l e s
Q,R,S and an instant
R
e R such that
If-1(Rx),R ,f (R )> and If-1(Rx),Rx, f (R )>, RQ x SR x RS QR x
24
is a
We say that particles
Q,R are distinct if Q ~ R. Given particles
We
§2.9] we say that the instants R
is between the particles Q and S. X
If, for all R
~
e R, the instant X
R
~
is between
the particle R is between the particles
Q and S, we say that Q a n d S;
and we denote
this by w r i t i n g
AXIOM
(INTERMEDIATE
PARTICLE)
Vl
the p a r t i c l e s
X
.
Given distinct particles Q,S and instants Qc ~ 9" S c s
S such
that Qc = Sc" there exists a particle R such that a n d Q /: R # S.
That
is, there
distinct
from both.
Theorems
22 (§5.2)
§2.9
The Isotropy
is a p a r t i c l e This and
is used
Q and S, w h i c h
in the p r o o f
is
of
28 (§6.1).
of SPRAYs
Any set of p a r t i c l e s given event
axiom
between
is called
which
a SPRAY.
c o i n ci d e
simultaneously
at a
We define
Sef SPR[Q c] ~
That is, SPR[Qc]
{R: f o f (Qc) = Qc, R E ~ }. ~ QR RQ
is the set of particles which coincide
at the event [Qc] (see Fig.
3).
A subset
a sub-SPRAY.
25
of a SPRAY
(with 9)
is called
§2.9
[Qe]
Fig.
3.
In M i n k o w s k i space-time,
SPR[Qc]
is the set of
"inertial particles whose paths are contained within the light cone whose vertex is the event
[Qc ]".
In this and
subsequent diagrams, events are r e p r e s e n t e d by dots.
The set of instants belonging to the particles of a SPRAY is called a
spray.
We define
spr[Qa] = {Rx: Rx e R,~ R~ e SPR[Qc]} A spray r e s t r i c t e d to a sub-SPRAY is called a
sub-spray.
next axiom states that all SPRAYs are isotropic. sent treatment,
The
In the pre-
it is this axiom which expresses the " P r i n c i p l e
of Relativity"
of E i n s t e i n
dean geometry,
a stronger analogue of this axiom has been used
by P o g o r o l e v
[1905,
§2].
[1966, A x i o m III7, Ch. II,
axiom an "axiom of motion".
26
In the theory of eucli-
§3] who called his
§2.9] AXIOM VII
(ISOTROPY OF SPRAYS)
Let Q,R,S be distinct
particles
with instants
Qc e Q, R c e R,
S c c S~ such that Qc = Rc ~ S c . If, for some instant
Qx E Q with Qx / Qc"
f o f (Qx) = f o f (Qx), QR RQ QS SQ then there is an injection
~ from
spr[Q c] to spr[Q c] such
that:
(i)
{ E sp~[Qc]
(ii)
for all particles
~
~(2)
f o f (Tx) = T ~ TU UT z
E SPE[Q c] ,
T,U e SPR[Q c] f ~(T)~(U)
o
(iii) for all Qx e Q, ~(Qx ) ~ Qx" (iv)
~(R)
,
f (~(Tx)) ~(U)~(T)
= ~(Tz),
and
= S.
It follows immediately
that:
(Q) = Q.
This axiom is used in the proof of Theorems 6 (§2.9), 2 4 (§5.3), The mapping
42(§7.3)
and 57 (§9.1).
~ is called an isotropy
statements mean that:(i)
~ maps particles
5 (§2.9),
onto particles;
27
mapping.
The previous
§2.9]
(it)
~ is a homomorphism.
A stronger p r o p e r t y than (it),
which is more o b v i o u s l y a homomorphism,
(it') "For all particles T x s T, U
T,U~ ~ S SPR[Qc]
is the following:
and for any instants
s U,
T x ~ Uy
¢(T x ) ~ ¢(fy)" ,
h o w e v e r in the present axiomatic system it is sufficient to assume
(it);
(iii) each instant of Q is m a p p e d onto an instant coincident with itself.
This is a weaker statement than:
(iii') "each instant of Q is invariant",
which is not assumed
in the present axiomatic system;
(iv)
R is m a p p e d onto a particle which is p e r m a n e n t l y coincident with S.
This is a weaker statement than:
(iv') "R is mapped onto S", which also is not assumed in
the
present a x i o m a t i c system; and finally the statement following the axiom means that: Q is mapped onto a particle which is p e r m a n e n t l y
coinci-
dent with 9' which is a weaker statement than:
"Q is invariant", w h i c h can not be proved in this system.
It may be worth noting that statements
(it'),
(iii'),
(iv')
likewise can not be proved in the present axiomatic system,
28
§2.9] since many p a r t i c l e s
can be p e r m a n e n t l y coincident or
"indistinct" as " o b s e r v e d by other particles".
This is a
consequence of choosing instants, rather than events, as the f u n d a m e n t a l u n d e f i n e d elements.
THEOREM 5.
Let Q,R,S~ ~ be particles
in SPR[Qc] , as in the
p r e c e d i n g axiom, and let T be any p a r t i c l e in SPR[Qc].
(i)
f o f = f o f , and QR RQ QS SQ
(ii)
f o f = f QT TQ Q¢(T)
o
Then
f ~(T)Q
This t h e o r e m is a consequence of Axioms iV (§2.q) and VII
(§2.9) and Theorem 1 (§2.5).
Theorem 6 (§2.9), Theorems PROOF.
It is used in the proof of
C o r o l l a r y 2 of T h e o r e m 22 (§5.2) and
23 (§5.3) and 30 (§6.3). (i)
By the Triangle Inequality
f Q¢(R)
o
f g f o f ¢(R)Q QS S¢(R)
But by the preceding axiom,
¢(R) ~
o
(Axiom IV,
f of ¢(R)S SQ
= S, so ~
§2.4)
f S¢CR)
o
f ~(R)S
is an identity m a p p i n g and therefore
f o f ~ f o f . Q~CR) ~(R)Q Q§ SQ
The opposite inequality is proved in a similar manner.
29
§2.9]
(ii)
Qx e Q there is an instant Qz e
For each instant
such that
f o f (Qx) = QT TQ
(1)
and by part
(2)
(ii) of the preceding
f
~(Q)¢(T)
o
f
¢(T)¢(Q)
Qz "
axiom,
(~(Qx))
Also by part (iii) of the preceding Inequality
(3)
(Axiom IV,
= ¢(Qz).
axiom and the Triangle
§2.4),
f o f (Qx) ~ f o f Q¢CT) ¢(T)Q Q#(Q) ¢(Q)¢(T)
=
f
Q¢(Q)
=
f Q~(Q}
o
f
o
f ¢(T)¢(Q)
o
¢(Q)¢(T)
f
¢(T)¢(Q)
(~CQz})
= Qz"
Now if
(4)
f o f (Qx) = Qy Q¢(T} ¢CT)Q
then, as above,
30
<
Qz,
o
f (Qx) ¢(Q)Q
(~(Qx))
§2.9] (5)
f
o
f
(~CQx)) <.
f
o
¢(Q)Q =
f
o
¢(Q)Q
=
f
o
Q¢(T) f
f
o
Q¢(T)
o
¢(T)Q f
f
(%(Qx))
Q¢(Q) (Qx)
~(TJQ
f (Qy) = t(Qy) ¢(Q)Q
< ~(Qz ), by Theorem i (§2.5), which is a contradiction
of (2).
That is,
(4) should be
replaced by (6)
f
o
Q~(T) which,
together with
(I), completes
The next result, two axioms,
¢({)Q(Qx ) = Qz" the proof of (ii).
which is a consequence
is equivalent
[]
of the previous
to an axiom of Szekeres
[1988, Axiom
A.3, P137]. THEOREM
6
(Permanently
Two distinct particles
Coincident
~oinoide at no more than one event.
This theorem is a consequence (§2.9) and Theorem 8 (§3.2),
5 (§2.9).
ii (§3.4),
of 30 (§6.3),
Particles)
IV ( § 2 . 4 )
and VII
It is used in the proof of Theorems
21 (§5.1),
33 (§6.4),
of Axioms
25 (§5.4),
34 (§6.4),
31
29 (§6.2),
Corollary
36 (§7.1) and 46 (§7.5).
1
§2.9] PROOF. S ,S 0
Given p a r t i c l e s
Q,S and instants Q o , Q I
Q ~ and
e S such that i
~
Q
=S
#S
0
=Q,
0
1
1
we will show that Q = S. ~ Suppose the contrary; that is, suppose that Q ~ S. A x i o m of the Intermediate Particle
(Axiom VI,
By the
§2.8), there is
a particle R such that
(I)
and Q / R ~ S.
Since Q0 = S0 "QI = $I and ~ ~ ~ of "betweenness"
it follows from the d e f i n i t i o n
that there are instants R jR 0
Q
= R 0
-~ S 0
and Q 0
= R 1
e R such that I
~
= S , 1
1
or equivalently,
f QR
o f (Q ) = f RQ I QS
o f (Q ) = SQ i
Q
. I
By the A x i o m of Isotropy of SPRAYs (Axiom VII,
§2.9), but with
Q0 taking the place of Qo' there is an i s o t r o p y m a p p i n g that ¢ ( R )
= S,
(2)
But statement
~ such
and by the previous theorem,
f o f = f o f . QR RQ QS SQ (i) and the d e f i n i t i o n of "betweennesss"
imply that, for at least one instant Qz e Qj
32
(§2.8)
§2.10 ] (3)
f o f (QX) < f o f (Qx) , QR RQ QS SQ
which contradicts
(2).
Thus the s u p p o s i t i o n that Q ~ S is
false, which completes the proof.
§2.10
[]
The A x i o m of Dimension
In the theory of absolute geometry, we can specify the dimension n by the axiom:
"there is a set of n+1 equidistant
points, but there is no set of n+2 e q u i d i s t a n t points". the case of 3-dimensional absolute geometry,
In
the set of four
e q u i d i s t a n t points is, of course, a tetrahedron.
An axiom of
this form implies that the set of points is non-empty;
that
is, an axiom of d i m e n s i o n is also an axiom of existence (for n ~ 0).
In the present treatment we do not have a measure of "distance"; particles functions.
however, we can compare the relative motions of
from the same
SPRAY by comparing their
record
This leads to the following d e f i n i t i o n which is
analogous to the concept of a set of e q u i d i s t a n t points in a space with a n o n - s y m m e t r i c metric.
35
§2.10 ]
Given an event
{R(i): i=l,...,n;
[Q ], the sub-SPRAY of n particles,
R (i) e SpR[Qc] } is a symmetric sub-SPRAY.
if, for all i e {1,...,n}, R x(i)~
there exists R (i) ~ R (i) with x ~
[Qc] such that, for all j,k e {1,''',n},
f j ji [ x where f
ik
ki
j # i, k # i,
"
is the signal f u n c t i o n which sends R (i) onto R (j).
By
ji T h e o r e m 5 (§2.9) it follows that a symmetric
sub-SPRAY has the
stronger property:
j # i, k ~ i,
for all i,j,k c {l,..,n},
f
of
ij
A X I O M VIII
ji
=f
of
ik
ki
(DIMENSION)
There is a SPRAY which has a maximal symmetric sub-SPRAY of four distinct particles. This axiom states that particles the "velocity space" of each SPRAY (see §9.1 and §9.5). Theorems and
7 (§2.12),
exist and ensures that
is t h r e e - d i m e n s i o n a l
The A x i o m is used in the proof of ii (§3.4),
15 (§3.7), 25 (§5.4)
61 (§9.5).
34
§2.11] §2.11
The A x i o m of Incidence
AXIOM IX
(INCIDENCE)
Let @ and R be distinct particles with instants Qz ~ Q and R z e R~ such that Qz ~ Rz" Given any instant Qx ~ Q with Qx ~ Qz" there is some particle S~ with instants Sx, Sy c S~ and an instant Ry ~ R~ with Ry ~ Rz, such that
QX ~ SX and Ry ~ Sy. (see Fig. 4)
s
Q
R
Z
\x S
Fig.
4
Note that o r d e r relations have not been s p e c i f i e d b e t w e e n the pairs of instants Qx and Qz' Ry and Rz, or S x and Sy. axiom is used in the proof of Theorems 25 (§5.4),
28 (§6.1) and 32 (§6.4).
35
7 (§2.12),
The
15 (§3.7),
§2.12] §2.12
The A x i o m of Connectedness
AXIOM X
(CONNECTEDNESS)
The set of instants is connected in the following sense: given any two instants Qx and T z there are particles R and with instants Rx, R y e Qx
Rx
R and Sy,S z e S such that Ry ~ s Y
and
and
Sz ~ Tz
(see Fig. 5).
Q
s
Qx ~
R
Rx
Fig.
Sz ~
z
5
This axiom has some resemblance Euclid.
T
to the first axiom of
It is independent of the r e m a i n i n g set of axioms,
since a model can be c o n s t r u c t e d from a 4+l-dimensional Minkowski space by considering two 3 + l - d i m e n s i o n a l M i n k o w s k i having no event in common. Theorems
sub-spaces
The axiom is used in the proof of
7 (§2.12), i0 (§2.10), 15 (§3.7),
26 (§5.5),
Corollary 2 of T h e o r e m 33 (§6.4) and T h e o r e m 61 (§9.5). A p h y s i c a l i n t e r p r e t a t i o n of the next theorem is that "no particle
can move as fast as a signal".
36
§2.12] THEOREM
7.
(The Signal
Given any particle
Relation
is Reflexive)
Q and any instant
Qx ~ ~"
Qx ~ Qx" That is, ~ is an identity QQ This t h e o r e m IX(§2.11), (§2.6).
PROOF.
In order to apply
there
such that
§2.11)
to show that there
§2.12)
Qc = Re.
asserts
However
that there
2 (§2.6)
16
to T h e o r e m
2 (§2.8) S
with
By the A x i o m of Connected-
R and instants
is a p a r t i c l e
that Qc = Qx" (Axiom IX,
S ~ Q and an instant
Now the conditions []
(Axiom VIII,
Qc e Q and R c e R
we can not assert
37
2
(§4.1).
is some p a r t i c l e
Qx
are satisfied.
(§2.10),
to T h e o r e m
Qx # Qc" the A x i o m of Incidence
S z E S such that Qx = Sx" to T h e o r e m
VIII
and the A x i o m of D i m e n s i o n
is a p a r t i c l e
For any instant
of T h e o r e m
the c o r o l l a r y
S z e S such that S z
(Axiom X,
of Axioms
and the C o r o l l a r y
It is used in the proof
an instant
§2.10),
is a c o n s e q u e n c e
and X (§2.12)
it is n e c e s s a r y
ness
mapping.
of the C o r o l l a r y
§2.13] COROLLARY.
(i)
The coincidence relation
is an equivalence relation.
(ii)
For any instant Qx" the corresponding event [Qx ] is
@iven by
[Qx ] = {Ry: Ry ~ Qx" Ry PROOF. Part
Part
(i) is a consequence of T h e o r e m 2 (§2.6).
(ii) is an a p p l i c a t i o n of
event
(§2.6).
§2.13
(i) to the d e f i n i t i o n of an
[]
Compactness of Bounded sub-SPRAYs.
Before stating the final axiom, which involves a p r o p e r t y of compactness,
it is n e c e s s a r y to define the concepts of "a
bounded sub-SPRAY" and "a cluster particle".
In several
treatises on g e o m e t r y an analogous property,
with points
taking the place of particles,
has been taken as an axiom;
the p r o p e r t y has been called "finite compactness" by Busemann [1955,
§2.6], and is a stronger a s s u m p t i o n than the axiom of
"continuity" of absolute geometry.
38
§2.13]
Given [Qc]~
a particle
9' an i n s t a n t
Qc ~
S P R [ Q c ] ~ we say that ~ [ Q c ]
if, for some R ~ S P R [ Q o ] , t h e r e
f (Qc) RQ
Q,
is a b o u n d e d
are i n s t a n t s
< R
i
and a s u b - S P R A Y
< R
2
R ~ R
,
such that
~
[Qc] ~__- {S:
(see Fig.
f RS
o f (R) SR I
~ R , S e SPR[Qc]} 2 ~
6).
R
Fig.
6
39
sub-SPRAY
E R w~th
§2.Z3 ]
Given a particle ~[Qc] ~[Qe],
~
Q, an instant
Qc c Q and a sub-SPRAY
SPR[Qc] , we say that S is a cluster if (i)
S e SPR[Qa],
(ii)
there exists
some T~ e ~ [ Q c ] S
SI,S 2 e S with
< S
i
< S
,
2
such that
(see Fig.
and
for any instants
f (Qc) SQ
ST
particle
of(S)<S TS I
2
7).
T
S
S2
SI
Fig.
7
40
of
§2.13] By applying Theorem i (§2.5), condition replaced by the equivalent condition "relative to Q" (ii')
(ii') which is specified
and is:
given QI,Q2 T ~ ,
(ii) could be
e Q with Qc < Q i < Q 2 " there exists some
such that
Q
< f-1 SQ
i
o f o f o f (Q) ST TS SQ I
< Q . 2
Note that the conditions
CQ c) < S
< S
2
or
Qc < Q
l
< Q
2
weaken the assumption of compactness so that it applies only "after coincidence".
Without these
(equivalent)
restrictions,
the following axiom would also apply "before coincidence": consequently the results of Chapter 5 would apply not only "after coincidence" but also "before coincidence"
and
Theorem 30 (§6.3) would be a simple corollary of Theorem 29 (§6.2). AXIOM XI
Every
(COMPACTNESS)
bounded
infinite
sub-SPRAY
This is the last axiom.
has a c l u s t e r
It is a stronger axiom than
the "Axiom of Continuity" of absolute geometry, similar results.
and yields
It specifies properties of "velocity space",
and also implies the "conditional completeness" regarded as a set of instants proof of Theorems
particle.
(see §5.4).
21 (§5.1) and 57 (§9.1).
41
of each particle
It is used in the
§3.i] CHAPTER
CONDITIONALLY
In this ditional
section,
completion",
particle"
This
"conditionally
of W a l k e r
of
each p a r t i c l e
later d i s c u s s i o n s
which
shows that
"completion" however,
not categorical,
so his
"completion"
§3.1
that it is trivial,
Conditional
ordered
order
the first an element
sets X(1),
in the arguments
which
sets of instants.
are by
treatment.
apply to any linearly
since p a r t i c l e s
An element
the
are
a e X is called similarly,
last element of X if, for all
If X can be d e c o m p o s e d
X(2 ) such that
X(, ) U X(2 ) = X
systems
can not be j u s t i f i e d
element of X if, for all x e X, a ~ x.
x s X, b z x.
(i)
some notions
b e X is called
are
of a Particle
set X, and to any particle,
linearly
and is justified
axiom
as in the present
Completion
We will define
W a l ke r ' s
sequence
of the given axioms.
is e m p l o y e d
1959];
"con-
completed
all p a r t i c l e s
as a c o n s e q u e n c e
[1948,
showing
in its
in the " c o n d i t i o n a l l y
simplifies
complete"
A similar process
PARTICLES
so that every bounded m o n o t o n e
has a "limit"
25 (§5.4),
COMPLETE
we embed
of instants
by T h e o r e m
3
and
42
into two n o n - e m p t y
sub-
§3.1]
(ii)
for all
x c X( 1
the d e c o m p o s i t i o n
x e X(a), x 2
< x , then 2
If
X(~) has
X(2 ) has no first element, we say that
[X(1),X(2 )] is a gap (in X).
conditionally
1
[X(1),X(2 )] is called a cut.
no last element and the cut
and for all 1)
An ordered set X is
complete if every bounded subset has
an infimum
and a supremum.
Any linearly ordered set X can be embedded in a conditionally complete linearly ordered set X, as is shown by M a c N e i l l e [1937,
§ii, T h e o r e m 11.7] and also,
in a form closer to the
present treatment, by B i r k h o f f [1967, Chapter V, §9].
The
adjoined elements correspond to those cuts which are gaps in X.
The order r e l a t i o n on ~ is defined:
for any y e X
[X(1),X(2)] , [X(3),X(~ )
and any gaps
y ~ x ( ~ ) ¢=>y < [ X ( ~ ) , x ( 2 ) ]
,
,
y ~ z ( 2 ) ~-> y > [ X ( ~ ) , X ( 2 ) ] "
x(1 ) c Z(3 ) ¢~ [x(,),X(2 )] < [x(3),X(4)]. The c o n d i t i o n a l c o m p l e t i o n of a p a r t i c l e Q is denoted by and is called a
conditionally
complete particle.
of the given p a r t i c l e Q are called the a d j o i n e d elements are called
Instants
ordinary instants,
ideal instants;
while
all elements
of the c o n d i t i o n a l l y complete particle Q will from now on be called
instants.
Instants of Q are denoted by the symbol
together with subscripts,
for example, QI" ~a" Qa" Qx s ~. If
~x e Q, and if it is known that Qx is an ordinary instant, we
43
§3.2]
shall write Qx instead of Qx; thus the statement that Q
is an o r d i n a r y instant of Q (and of Q), and the stateX
ment
"Qx s ~" means
~
Qx = Qx e ~" means that Qz is an ordinary instant of Q. "--
The set of all conditionally complete particles is denoted by .
The set of all instants
(ordinary and ideal) is denoted
by Y .
§3.2
Properties of Extended Signal Relations and Functions
Since ideal instants correspond to gaps, the Signal A x i o m (Axiom I, §2.2) induces a binary r e l a t i o n ~ on ~ x ~ .
The
extended signal relation ~ is a unique e x t e n s i o n of o on x~
, and so we shall abbreviate the symbol ~ to o.
there are extended signal functions,
relation,
Similarly,
an extended coincidence
ideal events, an extended relation of "in optical
line", and an extended betweenness relation, all of which will be denoted by the previous symbols.
The (extended) signal
functions are bijections which send ordinary instants onto ordinary instants and ideal instants onto ideal instants. following theorem is e s s e n t i a l l y the same as a result of Walker [1948, P322].
THEOREM 8
Let !'~'~ (ii)
f
SQ
(Extended Signal Functions)
e ~ $ f
SR
.
Then (i)
f is order-preserving and SQ
o f .
RQ
44
The
§3.2]
This theorem is a consequence of Axiom IV (§2.4) and Theorem 1 (§2.5).
It is used in the proof of Theorems 9 (§3.2),
11(§3.4), 12(§3.5), 14(§3.6) and Corollary i of Theorem 22
(§5.2). PROOF (in outline) It is only necessary to prove these results for ideal instants. (i)
Consider an ideal instant Qy e Q~ and any other instant of
Q.
Apply the (ordinary) signal function f to the definition SQ of the order relation for a conditionally completed particle (see the previous section). (it)
Given an ideal instant Qy e Q, apply the Triangle
Inequality (Axiom IV, §2.4) to all ordinary instants of the cut (in Q) which corresponds to Qy.
[]
COROLLARY (Walker [1948], Theorem 5.5, P323)
Given ideal particles QjR and instants Q xjQz c ~ and ~y e ~, if Qx ~ ~y ~ Qz'then Qx = ~
PROOF.
Y
= Qz = Qx or Qx < Qz"
For ordinary instants, this is simply the definition
of the temporal relation (§2.3).
If Qz is an ideal instant
the definition of the temporal relation can be applied to all ordinary instants in the cut corresponding to Q . x Qx ~ Qz"
Thus
If Qx = Qz" then Qx ~ ~y o Qx q ~y" that is ~x = ~y"
which completes the proof.
D
45
§3.2]
Statements similar to those contained in the following theorem have been made by Walker [1948, P322, P325].
The
theorem is not stated in terms of the concept of a "limit", since we have not yet shown that a particle is a H a u s d o r f f set (of instants).
It would have been possible to insert T h e o r e m
15 (of §3.7) at an earlier stage
and thus simplify the
statement of the next theorem; but it is not n e c e s s a r y to do so, and the present approach is independent of the A x i o m of Incidence
THEOREM 9
(Axiom IX,
§2.11).
(Monotone Sequence Theorem)
Let Q,R s ~ and let Q' be a subset of Q. If Q' is bounded above,
it has a supremum and
sup{fRQ (~t): Qt ~ Q'} ~ fRQ (sup{Qt: Qt ~ Q'}) ~ ~ if Q' is bounded below,
it has an infimum and
inf{f (~t): ~t ~ Q'} = f (inf{Qt: Qt c Q']) RQ ~ RQ In particular, increasing
sequence,
"
if {Qn:n=l,2,..o;
.
Qn ~ ~] is a bounded
then
sup{f (Qn)] = f (sup{Qn]) RQ RQ If {Qn:n~l,2,..;
~n ~ ~} is a bounded inf {Q(Qn )
decreasing
= ~Q(inf{Qn}).
46
sequence,
then
§3.3]
This t h e o r e m is a consequence of T h e o r e m 8 (§3.2).
It is
used in the proof of Theorems 12 (§3.5), 17 (§4.2), Corollary 2 of Theorem 22 (§5.2), and Theorems 41 (§7.3),
PROOF.
32 (§6.4),
36 (§?.I),
42 (§7.3) and 49 (§7.5).
By the p r e v i o u s l y quoted t h e o r e m of M a c N e i l l e
Q' has a supremum or an infimum, (extended)
as the case may be.
signal functions are one-to-one
ing, by the previous theorem,
[1937], Since
and o r d e r - p r e s e r v -
f if an o r d e r - i s o m o r p h i s m RQ
between Q and R, which leads to the stated equalities.
§3.3
G e n e r a l i s e d Triangle
Inequalities
The next result is due to Walker
THEOREM i0.
O
[1948, T h e o r e m 5.2, P323].
Let ~,~ ...... ~ be a finite set of conditionally
complete particles with instants Qx, Qy s ~; Rx, Ry s R; .... ; and Tx, Ty c 3"
Then a ~
then
~fQx
(ii)
If Qx a ~x a ... ° ~x and Qy o ~x" then Qx ~ Qy"
y
~ ... ~ ~
and ~x ~ ~ ,
(i)
y
~
x
~ ~
y
This theorem is a consequence of Theorems i (§2.5) and 8 (§3.2).
It is used in the proof of Theorems 13
and 17 (§4.2).
47
(§3.6)
§3.4]
PROOF.
Part
(i) is a consequence of Theorem 8 (§3.2) by
i n d u c t i o n on the number of particles. part
§3.4
(i) by Theorem 8(i).
Part
(ii) follows from
[]
Particles Do Not Have First or Last Instants
The next theorem applies to ideal as well as to ordinary instants.
The first p a r a g r a p h of the proof resembles a
theorem of Walker [1948, T h e o r e m 5.1, P323].
THEOREM ii.
Particles
do not have first
or last instants.
This theorem is a consequence of Axioms VIII
(§2.10) and Theorems
II ( § 2 . 3 )
6 (§2.9) and 8 (§3.2).
It is used
in the proof of T h e o r e m 25 (§S.4) and the Corollary to T h e o r e m 40 (§7.3).
48
and
§3.4]
PROOF.
By Theorem 8 (§3.2), if (conditionally complete)
p a r t i c l e s had first at their first
(or last) instants,
(or last) instants.
o r d e r - p r e s e r v i n g bijections,
they w o u l d all coincide
Signal functions are
so if one particle has a first
(or last) ordinary instant, then all particles have a first (or last) o r d i n a r y instant, at their first
and all p a r t i c l e s would coincide
(or last) o r d i n a r y instants.
By the A x i o m of D i m e n s i o n
(Axiom VIII,
two distinct particles Q ( 1 ) , Q ( 2 )
which we shall denote by S , T
(for ease of writing) with instants S
e S, T s T such that 0
S
= T . 0
0
~
.
~
Since S / T, there is an instant S
0
Sm # S
§2.10), there are
e S X
By the First A x i o m of T e m p o r a l Order
with
~
(Axiom II,
§2.3),
0
there is a particle (i)
if S
< Sx,
f SU
o f (S) US o
= Sx,
> Sx"
f SU
o f (S) US x
= S
o
(it)
if S
U such that
o
or
. o
Thus, in either case, respectively, (i) (it)
(f SU
o f )-I (S ) < S US o
f o f (S) SU US o
and o
> So,
which shows that S instant of S.
is n e i t h e r the first instant nor the last 0 If S had a first or last (ordinary) instant,
then Theorem 6(§2.9) would imply that S ~ T, w h i c h would be a contradiction.
Therefore no particle has a first last
(ordinary)
instant.
(ordinary)
By definition,
49
instant or a
gaps can not correspond
§3.5] to first or
last instants,
last ideal instant.
§3.5
Events
THEOREM 12.
T
O
so no particle
has a first or
[]
at Which Distinct
Particles
Coincide
Given distinct particles Q,T and an instant
e T, we define for all integers n, ~
T n If sup {Tn} e ~, let ~ ~_ d~f inf{Tn}.
d~f(f o f )n(T ). TQ QT o d~f sup{Tn}, and if inf{T n} e ~, let
Then
(i)
if ~
(ii)
if 7_= exists, ~ coincides with ~ at [7_~], and
(iii)
exists, Q coincides with ~ at [T ],
for all Tx e n=-=U{~y:Tn ~ ~y ~ Tn+1"~y e ~}~, Q~ does not
coincide with ~~ at [~x ] (see Fig. 8). This theorem is a consequence
of Theorems
9 (§3.2).
It is used in the proof of Theorems
25 (§5.4),
40 (§7.3),
46 (§7.5)
and 50 (§8.1).
50
8 (§3.2) and i5 (§3.?),
§3.5]
T3 T
Fig.
8.
{ and T~ coincide at the event
PROOF.
sup{T n} s -T, and the particles
In this illustration,
[L].
We define a function
q def f
o f ;
TQ
QT
and then T
=
qn(T ).
n
o
Consider an instant ~
s ~ with X
T
<~ 0
then,
~ T X
; I
since q is a strictly monotone
51
increasing
function by
§3.6] Theorem 8 (§3.2),
T
= q(T ) < q ( L ) 1
and so
0
L
< q(~x ) .
That is, for any instants Qy e Q and Tz e ~ with T ~
Similar
considerations
apply for each semi-closed
fn < Tx ~ Tn+I , which proves (§3.2)
(iii).
If ~
< ~ 0
exists,
g T X
1
interval Theorem
9
implies that
q(To~) = q ( s u p { T n } ) which proves
(i).
= sup{q(Tn)
} = 8up{Tn+ 1 } =-T~o ,
The proof of (ii) is similar.
[]
This theorem and its proof are based on a theorem of Walker
[1948, Theorem
8.1, P325].
result in the following words:
Walker has paraphrased
the
"any instant of T, at which T
does not coincide with Q, lies within an interval of T at every instant of which T does not coincide with Q".
§3.6
Generalised Events.
Temporal
Relations
on the Set of
Observers.
An intuitively is proved
Order.
obvious property
in the following
the signal relation
theorem.
and related
of the coincidence
relation
This allows us to generalise
concepts to apply to the set
52
§3.6]
of events.
Walker
[1948, Theorem 5.h, P323] proved a special
case of the following theorem,
and generalised
the temporal
order relation to apply to instants of different particles; however he did not relate instants and events in either of his papers
(Walker [1948, 1959]).
THEOREM 13
(Substitution
Let
be c o n d i t i o n a l l y
Q,R,S,T e Q,R
-Q1
~
If Q
e R,S 2
a -R 1
~
2
e S, T 3
~ ~
Property of the Coincidence
~
with
instants
"
= ~ , then Q
2
2
~ ~ 1
This theorem is a consequence Theorem i0 (§3.3).
particles
e -T ~.
and -R 3
complete
Relation).
~ -T • 2
3
of Axiom I (§2.2) and
It is used in the proofs of Theorems
14
(§3.6) and 16 (§4.1). PROOF
By the Signal Axiom
Qx £ ~ and T ~
y
(Axiom I, §2.2) there are instants
£ T such that
~xS~
2
~.Y
So we have Q1
~
Qx ~ ~
2 2
~ ~ ~ ~
which, by the Generalised
2 2
and Q and Q
Triangle
X 1
~ ~ ~ R
2 2
and ,
Inequalities
(Theorem 10
§3.3), imply that ~i $ Qz and Qx ~ QI" respectively. ~x = QI 7 that is, Q1
2
53
So
§3.6] Similarly
R
0 ~ z
and
~
o ~ 2
imply that ~
COROLLARY
= ~
(extended)
, s
~
g
y
2
o 2
3
signal functions
we can now define
temporal
need not belong
that £
and 3
; that is, ~
(extended)
8, §3.2),
complete
a ~ 2
o ~ 2
3
and ~
y
If R = S, then f o f = f o f . QR RQ QS sQ
Since (Theorem
Y
o ~ 2
order relation
are order-preserving
a generalisation
for pairs of instants
to the same particle.
particles is before
Q,R with instants ~
Y or
and ~
Y
of the which
Given two conditionally Qx ~ ~ and ~x ~ ~" we say
is after Qx if:
(i)
f (Qx) < % , RQ
(ii)
f (Qx) = R and Qx ~ ~ " RQ Y Y
IfV
isbefore
we write < and is after X y y x Y we write Ry > Qx' as in §2.3. Note that there could be pairs of instants
between which none of the relations
The above definition
is a paraphrased
definition
[1948,
of Walker
A particular relation
occurrs
particles
case of the for instants
(generalised) which
instants
order
line.
the instants
the instant
Sz is after the instants is after Qx and before
54
~z"
R
Y
Given
Qx a Q, By e ~,
IQx, Ry,Sz> ,
Qx is before
%
temporal
are in optical
the instant
instant
of a similar
P323].
Q,R,S and non-coincident
~z ¢ ~ such that
version
<, =, or > hold.
and Sz'
Qx and Ry, and the
§3.6]
THEOREM
Let Q,R,S be conditionally
14.
instants Qx ~ Q" R y ~
Qx ~ ~y ~
(ii)
~ and Sz E S. R,
Qx ~ ~z"
(iv)
~x ~ ~y and ~y = ~z ~
~x ~ ~z"
(v)
~X = ~
and ~
This theorem Walker
[1948,
y
8 (§3.2)
Theorem
32 (§6.4).
PROOF.
By Theorem
~
z
~
< ~ .
x
z
8 (§3.2)
of
It is a consequence
of
13.
D
of the coincidence
the o-relation
and hence
temporal order relation,
and is used in the proof of
and Theorem
property
§3.6) permits
as to apply to events
the same as a theorem
5.6, P323].
and 13 (§3.5),
The substitution 13,
< ~
is essentially
Theorem
Theorems
(Theorem
Then
Qx ~ ~y and Qx ~ ~y"
(iii) ~x ~ ~ y and ~ y < ~ z ~
y
complete particles with
relation
to be generalised
the si@nal funotions,
(i)
[Qx],E~y],[~ z] we define:
[Qx ] o [Ry] if and only if there ~x ~ [Qx ] and ~y ~
(ii)
the
and the concept of optical lines can
all be extended to apply to the set of events. Given events
so
[Ry]
are instants
such that L
o L
;
f [Qx ] = [Ry] if and only if [Qx ] o [Ry] RQ
;
55
§3.6] (iii)
l[Qz],[~],[Sz]> L
e
[Qx], ~y
if and only if there are instants
e [Ry], and
VZ e
[~z ] such that
I~x, Uy, Vz> ; and
(iv)
[Qx] < [Ry] if and only if there are instants x
e [Qx ] and ~
It is a consequence order
relation
e [3 ] such that y
of the previous
is t r a n s i t i v e
An important any composition
consequence
provided
is unaltered;
R ~- S ~
.°.
< ~ . y
x
theorem that the t e m p o r a l
the set
of events.
is unaltered
permanently
another particle"; particle
coincident with
that the domain and range of the
f o f QR RT
{R:
.......
f o f QS ST
a corresponding R = Q,R
e~}
...
observer:
.
We see from the above remarks that particles to the same observer
by changing
that is,
Q we define d~f
T
of Theorem 13 is that
to a particle
the given particle,
To each particle
on
of signal functions
any given particle
composition
y
belonging
"appear to be the same" as "seen by any that is, if R,S e Q then for any
T ,
f of =f of TR RT TS ST
Observers have been defined as equivalence particles,
which is analogous
classes of coincident
to the definition
56
of events
as
§3.7]
equivalence
classes
"conditionally such
complete
a concept.
particles
are
previous
of c o i n c i d e n t
(In
Several extended
observers"
to o b s e r v e r s :
~
show
complete,
We
do not d e f i n e
we h a v e
that
no use
for
all o r d i n a r y
which
means
that
the
is t r i v i a l ) .
definitions
RQ
since
§5.4 we w i l l
conditionally
completion
instants.
which
apply
to p a r t i c l e s
can n o w be
for example:
~
if and only
[Qx ]~-~ [By]
if[Q}
o[R
]. Y
(ii)
[Q,R,S] ~ ~ ~ all
(iii)
~
for all
V e S , IT, U, V]
all
~=~ for
definitions
which
extended
§3.7
all
line"
Each
are
the
Particle
e X with 2
z
dense that
and
for
for all
U e R,
and
for
with
relation
to the
the p r e v i o u s
and the
definitions
relation
"in
set of events.
in I t s e l f
ordered
set.
< x , there 1
< y < x , we i
U~ ~ R, ~
.
is Dense
x
all
consistent
Let X be a l i n e a r l y all x ,x
T e Q,
signal
to a p p l y
for
.
[ s S ,
These
optical
T~ e Q, ~
If Y ~ X and if,
exists
some
y e Y such
for that
2
say that
Y is a d e n s e
subset
of X.
If X is a
2
subset
of X, we
X is a d e n s e
say that
X is d e n s e
set.
57
in i t s e l f ,
or s i m p l y
§3.7] THEOREM 15
(Each Particle
is Dense in Itself)
Given a particle Q and ordinary instants Qa, Qc e Q with Qa < Qc" there i8 an instant Qb e Q such that Qa < Qb < Q " ~
This theorem is a consequence IX (§2.11)
and X (§2.12)
in the proof of Theorems PROOF.
distinct
Vlll
It is used
23 (§5.3) and 40 (§7.3). (Axiom VIII,
(Axiom X, §2.12),
§2.10) and the
there is some particle
[Qc], the Axiom of Incidence
that there is some particle
coincides with Q at [Qc ].
COROLLARY.
S, distinct
§2.11)
from Q, which
By Theorem 12 (§3.5),
Each conditionally
Moreover,
If
(Axiom IX,
Qa < Qb def f o f (Qa) < Qc " QS SQ
itself.
(§2.10),
from Q, which coincides with Q at some event.
this event is not implies
of Axioms
and Theorem 12 (§3.5).
By the Axiom of Dimension
Axiom of Connectedness
C
[]
complete particle is dense in
each particle is a dense subset of its
conditional completion. PROOF
Let Q be a conditionally
complete particle with instants
Qa,Qc e ~ such that Qa < Qc" Case i. If Qa or Qc (or both)
are ideal,
then by §3.1, there is
some ordinary instant Qb s @ with Qa < Qb < Qe" Case 2. If both Qa and Qc are ordinary theorem applies.
[]
58
instants,
the above
§4.1] CHAPTER
IMPLICATIONS
Most of the results given by Walker differs
OF COLLINEARITY
contained
in this chapter have been
[1948] but since the present
from Walker's,
the sake of logical
§4.1
4
Co!!inearity.
axiom system
proofs have been given in detail for
completeness.
The Two Sides of an Event. X is collinear
A set of particles
if, for all particles
@ e ~ and for each instant Qx e Q, either: (i)
there are two distinct optical
lines,
Qx and one instant from each particle (ii)
all particles
We shall indicate enclosing
of ~ \ {Q}, or
of ~ coincide with Q at [Qx ].
that a set of particles
the particles
means that {Q,R,S,T}
each containing
in square brackets;
is collinear.
to denote an arbitrary
is collinear by
eollinear
Before establishing
thus
The symbol
[Q,R,S,2]
E will be used
set of particles.
the main result we prove the following:
59
§4.1]
PROPOSITION
(Walker [1948], Theorem 7.2, P324)
Let Q,S,T e Z and let Sy
S.
If [f-1(Sy),Sy,f (S )> , then If-:(S ),S ,f (Sy)> SQ TS ~ ST Y Y QS That i8, the instant S
is between Q and T and by Theorem 13
(§3.6), the event [S ] i8
between Q and T.
Qx d~f f-1(Sy), T d~f f-1(Sy), Qz d~f f (S) and SQ x ST QS Y TZ def= fTs(Sy)" We must show that IQx,Sy,Tz> implies ITx,Sy, Qz>.
PROOF.
Let
Consider the optical line which contains the instants T
and X
S
If T
y
Tx / Sy, T
x
and S
x
= S , there is nothing further to prove. y
then the instant of Q which is in optical line with is either:
y
(i)
Qx
which implies
IQx,Tx, Sy >
(ii)
Qz
which implies
ITx,Sy, Qz >.
Now
IQx,Tx,Sy>,
IQx, Tx, Sy, T z> so
Tx = Sy,
or
ITx,Qx, S Y >,
and by the Signal A x i o m
which is a contradiction.
§2.7) imply that so
or
the data, and Theorem 3 (§2.7) imply (Axiom I, §2.2),
Tx = Tz
ITx,Qx, Sy>
and the
Also
A x i o m of U n i q u e n e s s of E x t e n s i o n of Optical Lines
Tx = Tz,
If
ITx,Qx,Sy,Tz>
Tx = Sy,
(Axiom V,
and by Theorem 7 (§2.12),
which is another contradiction.
The only
r e m a i n i n g p o s s i b i l i t y is (ii) above, which was the result to be proved.
Q
60
§4.1] The p r o p o s i t i o n
THEOREM
16
can n o w be e x t e n d e d :
(Walker
[1948],
Theorem
7.3,
P324)
Given a particle S E ~ and an instant S
~ S, each particle 2
can be p l a c e d i n one o f t h r e e Is ], ~ [S ], ~ IS ] C 2
2
~.
disjoint
subsets
Particles
in ~ [S ] coincide
2
2
with S at [S ]; the event [S ] is between any particle ~
2
of
2
[S ] and any particle
of ~ IS ], but not between any two
2
2
particles
of ~ [S ] or of ~ [ S 2
and
of
~
~[S
].
The sets of p a r t i c l e s ~ [ S
]
2
2
] are called the left side
(of [S ] in ~) and the
2
2
right side
(of [S ] in ~), respectively. 2
This V (§2.7) is u s e d
PROOF.
and
is a c o n s e q u e n c e
Theorems
in the
proof
3 (§2.7),
to prove.
T does
not
of A x i o m s
7 (§2.12)
of T h e o r e m
If all p a r t i c l e s
further that
theorem
17
coincide
there
with
and
(§2.2) 13
and
(§3.6).
It
(§4.2).
in Z c o i n c i d e
Otherwise
I
S at
at
[S ], t h e r e 2
is n o t h i n g
is a p a r t i c l e
T e ~ such
[S
Signal
].
By the
Axiom
2
(Axiom T
I,
~ S 0
§2.2)
o T .
there Again
are
instants
by the
T 0 , T ~ e T such
Signal
Axiom,
for any p a r t i c l e s
2
Q, U e F. there
are
Q ,Q 1
E Q such
U , U 1
instants:
3
E 3
that
Q
~
U ~
o S 1
such
that
U
61
o Q 2
o 1
S
and 3
cr 2
that
U 3
~4.2]
We s p e c i f y that T s ~ [ S 2]
, so by the p r e v i o u s
proposition:
for any ~ Q e ~ such that
IQI,S2,T ~ > and Q i ~ Q 3 , Q~ e ~
for any U e Z such that
IS2,U3,T > or
IS2,T~,U3>
[S 2] "•
and U I ~ U 3
e ~ IS 2 ] ; and all o t h e r p a r t i c l e s Having
specified
I[S2]
part of the t h e o r e m proposition
§4.2
in Z are in
The next t h e o r e m In the p r e s e n t
Instant
Theorem"
PROPOSITION
O
Theorem
[1948,
it is c a l l e d
of its r e s e m b l a n c e
of r e a l v a r i a b l e
theory.
Theorem the
7.4,
"Intermediate
to the
"Intermediate
Before proving
this
the f o l l o w i n g :
(Walker
such
of ~.
treatment
because
e ~ and
be d e f i n e d
Instant
of the p r e v i o u s
is due to W a l k e r
P324].
r e s u l t we e s t a b l i s h
~ [ $ 2 ] , the r e m a i n i n g
is a c o n s e q u e n c e
The I n t e r m e d i a t e
Let T , U , V
, ~ [ $ 2 ] , and
and the d e f i n i t i o n
Value T h e o r e m "
~ [ S 2]
[1948,
Lemma,
let T a e T .
that,
Let
P325]).
the f u n c t i o n
for each i n s t a n t
T
g:
~ ~, X
g ( L ) de=f mini f o f (~x) , f o f (-Tx) } TU UT TV VT If U and ~
with
V are on the same side
of [T ] then,
~
T
a
~ T a
~ g ( T a) x
~
for all T
e T x
U and
V are on the same side of [T x]
~
~
~
°
62
§4.2]
PROOF
(See Fig.
9)
By the Signal A x i o m
(Axiom I, §2.2) there is an instant
U b s U such that T a ~ U b.
We assume, without loss of general-
ity, that [U b] is between T and V, or that U coincides with at [Vb].
Then by the Signal A x i o m
previous theorem,
(Axiom I, §2.2) and the
there are instants
Va, V b e V and Tb,T c e
such that
ITa, Ub, Yb> and
IYa, Ub, gb > and
Yb ~ T c
By T h e o r e m i0 (§3.3),
U b ~ V b ~ T c and
Ub ~ T b ~
T b ~ TcJ so g(T a) = T b.
T
U
V
Vx
Fig.
9
We now suppose the contrary to the proposition; we suppose that for some instant T
e T with X
Ta < T x ~ T b ,
63
that is,
§4.2] U and
V are on o p p o s i t e
sides
of
[T ]; t h a t
~
is, w e
suppose
x
that there
are
instants
U
x
e U and ~
V
s V such that ~
x
IV x, T x, Ux> and we
shall
By T h e o r e m
deduce
i0
a contradiction.
(§3.3),
T a o U b a n d T x o Ux and
Ta
Va ~ U b a n d
Vx o U x and
U b < U x -=> V a < Yx,
Vx o T x and
Va o T b and Tx
which
< Tx ~
Ub < Ux"
<, T b ~
is a c o n t r a d i c t i o n .
so but
Vx "< Va"
Q
We c a n n o w prove:
THEOREM
Let If
17
T. , U , V . e Z .a n d U and
V are
on opposite at
(Intermediate
some
Theorems used
sides
instant
This
on
let .
Ta, T d ~ T.
the
same
of
T at
between
theorem i (§2.5),
in t h e p r o o f
Instant
side
Theorem)
of T at
[Td] , then
Ta and
of T h e o r e m s
Corollary
i to T h e o r e m
36 (§7.1)
and
30
U and
~~ or 7~ c o i n c i d e s
V are ~
with
T d.
is a c o n s e q u e n c e 9 (§3.2),
ETa] , a n d
i0
of A x i o m
(§3.3)
26
and
(§5.5),
(§6.3),
38 (§7.2)°
64
28
I (§2.2) 16
(§4.1).
(§6.1),
and Theorems
and
33
29
It is (§6.2),
(§6.4),
§4.2]
PROOF.
(Walker
In the
following
case
g is d e f i n e d
been
P325])
argument
we
assume
T a > T d) can be t r e a t e d
(with
integer
[1948,
as
in the p r e c e d i n g
more
generally,
[Tn,Tn+1].
interval
U
interval
Thus
[Tn, Tn+~],
T a < T d ; the o t h e r
similarly.
U,V~ are
on the
so for all
could
to each
same
side
< T d.
n, ~
function
For any p o s i t i v e
The p r o p o s i t i o n
so as to a p p l y
and
The
proposition.
def n - g ( Ta) . n , we let T n
stated
that
have
closed of T in the By the M o n o -
n=0
tonic
~
Sequence
Theorem
d~f sup{Tn } s ~
(Theorem
and g ( ~
9,
§3.2)
there
is an i n s t a n t
) = T , so ~ c o i n c i d e s
with
~ or
n
. at
[~ ]..
Thus,
then
T d > -T > T a.
If T d < Ta,
Thus, icle could
from
either
be a n u l l
previous
coincidence,
sides
side
in
interval,
the o t h e r
and
set of all
left
instants
with
side
In the
are
two
sides
after
6 (§6.4) and r i g h t of)
each
of T at T d > Ta,
distinct
~.
can be o a l l e d the
right
the
side
after
the be
event
of
sub-
of c o i n c i d e n c e .
It
sets
of p a r t i c l e s
can be d e f i n e d
particle.
left side
(c.f.
the
which
in the
eollinear
sides
65
interval,
of any p a r t i c l e
the e v e n t
a part-
c h a p t e r we w i l l
collinear
that
until
In this
following
which
so the
Chapter
T in E are
one
sides
Q
can be c a l l e d
sub-SPRAYs
and that
of
coincides
can be w e l l - d e f i n e d
is s h o w n exist
is similar.
section).
considering
SPRAY
the p r o o f
the two
(of T) and
U,V are . on o p p o s i t e
.if
for
(the
§4.3] An o p t i c a l
line
containing
S x and Ty, w h e r e
instants
Ty s T~ g ~[Sx] , s u c h t h a t S x o Ty, is c a l l e d a right optical line ( t h r o u g h Sx). There is a s i m i l a r d e f i n i t i o n for a left optical line ( t h r o u g h S ).
In o r d e r
x
through
through
Ty, we can d e f i n e the sides of IT ] in ~ such that: U e Z having
be a r i g h t
(left)
(left)
line
x
should
a right
optical
for any p a r t i c l e
S
that
Y U
an i n s t a n t
g U such z
ISx, Ty, U z> we d e f i n e U~ g ~ [ T y ] .
optical
Similarly,
line
that
~
we
can d e f i n e
[Ty]. If U e ~ [Ty] similarly, If the
we
say that
U~ is
to the right of [Ty]; and
~ [Ty] we say that U is to the left of [Ty].
if U g
Sx,Ty,U z are on a r i g h t o p t i c a l
instants
line
such
that
ISx, Ty, Uz> y is to the right of S x and U z is to the right of S x and Ty; also Ty is to the left of U z and Sx is to the left
we
say that
of T
and
y
a left
U . z
optical
relations
§4.3
T
We m a k e line.
Signal
We n o w d e r i v e
defined
in later the
"modified
definitions
for the
instants
on
Along any given optical line, the
"to the right of" and "to the left of" are transitive.
Modified
proofs
similar
some
results
sections.
"modified
signal
Functions
which
Walker
record
functions"
and M o d i f i e d
are
[1948,
function";
used
to
§9, P326] we d e f i n e
in an a n a l o g o u s
66
Record
way.
Functions
simplify has p r e v i o u s l y two k i n d s
of
§4.3]
Given
two p a r t i c l e s
record function
o
(Qx)
QR
-
, if R is to the r i g h t
Qx
, if R c o i n c i d e s
(Qx),
record
function
on w h e t h e r
We d e f i n e
the
if R is to the
indicates
If o RQ f j QR depending
modified
the
° fRQ CQx)
R ° ~ Q1_ ~
Q
The m o d i f i e d
set,
Q
0R
o
in a c o l l i n e a r is defined:
• R
f
{,R
relative
of
with
Q at
left
of
position,
<>
R~ is to the r i g h t ,
or left,
[Qx ~.
of
modified signal function f+ ,
which
is
RQ related
to r i g h t
optical
I~Q
(Qx )
f+(Qx) d~f ~ Qx BQ If (Q) [QR x
as follows:
f (Qx) is to the r i g h t RQ , if R c o i n c i d e s w i t h Q at " if
-i
Similarly,
lines,
"
fQR (Qx) -i
if
we d e f i n e
the
. is to the
Qx'
of
[Qx ],
left
or
Qx"
of
modified signal function f-, RQ
which
is
f--(Qx) d~f
related
f (Qx) RQ Qx
to
J
left if
•
optical
f (Qx) RQ
is to the
, if R c o i n c i d e s
RQ
f-1(Q x) QR
if
f-1(Q x) QR
lines, as follows:
with
left
of
Q at ~
[Qx ],
is to the r i g h t
67
Qx'
of
or
Qx"
for
Qx" [Qx 3, Qx"
§4.3] THEOREM 18.
Let
(i)
=
Q
(ii)
f+ sR
e Z.
=
,
f
,
Q~
f- o f- = fSR RQ sQ
=f-o QR
Q
Then
,
QR
o f+ = f+ RQ SQ
f o QR
(iii)
Q,R,S
and
RQ
This theorem is a consequence
of the previous definitions
and is used in the proof of Theorems
33 (§6.4),
43 (§7.4), 45 (§7.4), 48 (§7.5), 49 (§7.5), 52 (§8.2),
PROOF.
53 (§8.2),
Results
41 (§7.3),
50 (§8.1),
51 (§8.1),
54 (§8.3) and 55 (§8.3).
(i) and
(ii) are consequences of the previous
definitions. To e s t a b l i s h
(iii), we consider separately the p o s s i b i l i t i e s
of R being to the right of Q, coincident with Q (which is not shown since it is trivial),
and to the left of Q.
apply the previous definitions.
{
L°
IL ° {J =
We
Thus
Q O
-1
QR
Q
fR
RQ
{I51°' -l
-I
o
Q
R
O
68
§4.4] THEOREM 19
(Walker
[1948, Theorem
Let @,R,{ g E and let Qx g Q" [f+(Qx)], RQ
8.2, P826]).
The order of the events
[f+(Qx )] on the right optical SQ
line through [Qx ] is
the same as the order of the instants f o f Qs sQ
(Qx) and
f o QR
This theorem is a consequence It is used in the proof of Theorems 23 (§S.3),
24 (§5.8),
25 (§5.4),
29 (§6.2),
30 (§6.3),
Corollary
36 (§7.1),
37 (§7.2),
the Corollary
41 (§7.3), the Corollary PROOF.
From the previous
COROLLARY.
If
f o QR
then R and S coincide
§4.4
Betweenness
THEOREM (i)
20.
28 (§5.5),
Relation
and ~
,
22 (§5.2),
3 to 32 (§6.4),
28 (§6.1),
33 (56.4),
to 39 (§7.3), 43 (§7.4),
40 (§7.3), and 46 (§7.5).
[]
f o f QS SQ
at [f+(Qx)]. R@
definitions.
27 (§6.i),
(Qx)
,
[]
for n Particles
Let Q,R,S,T be distinct particles.
where ments:
21 (§5.1),
definitions.
(Qx) =
~
of the preceding
to 41 (57.3),
Q
(Qx) in Q . Q
Then
, and
is a concise expression for the four state,
and .
69
§4.4]
This Theorem 21
theorem
3 (§2.7).
(§5.1),
22
At this
"
and
T could
It is u s e d
(§5.2)
REMARK.
and
is a c o n s e q u e n c e
stage
"cross
Theorem
6 (§2.9)
PROOF.
Proposition
Proposition
(ii)
of E x t e n s i o n
We
shall
represent so,
linearly
we
rationals,
can not p r o v e
not
(i)
of T h e o r e m s
or ",
at an i d e a l
because
event
and
S
so
apply.
is a c o n s e q u e n c e Lines
brackets for
V,
the set
[]
> to
any n u m b e r
3 (§2.7).
of U n i q u e n e s s
§2.7).
<
eR(1),...R(n),R (n+1) a,b,c w i t h
of T h e o r e m
of the A x i o m
(Axiom
relations
indexing
and
the p r o p o s i t i o n :
is a c o n s e q u e n c e
use the
V (§2.7)
(§5.3).
other"
can e x t e n d
ordered
in the p r o o f
would
also
integers
Similarly,
~
of O p t i c a l
for e x a m p l e ,
24
we
each
betweenness
positive
and
of A x i o m
concisely of p a r t i c l e s ;
..> m e a n s
that,
for all
0 ~ a ~ b ~ c, . definition such
or the reals.
70
to a p p l y
as the
to any
integers,
the
§5.o]
CHAPTER 5
C O L L I N E A R SUB-SPRAYS A F T E R COINCIDENCE
In this
chapter we will show that there are
sub-SPRAYS
which are "collinear after the event of coincidence" which contain a " r e f l e c t i o n of each particle particle".
and
in each other
In T h e o r e m 25 (§5.4) we will show that the condi-
tional completion of Chapter
3 is trivial,
by showing that
all instants are ordinary instants.
Since the A x i o m of Compactness to bounded sub-SPRAYs
(Axiom XI,§2.13)
"after the event of coincidence",
useful to modify some previous definitions "after a certain event".
~ ~ ~ after
Thus,
[Re ] means that,
If-l(Rx ), R x, ~R(Rx)> ~Q
applies
and
it is
so that they apply
the statement: for all Rx e R~ with Rx >
Rc"
If-i(Rx ), Rx, f (RxJ>. Rs
QR
There is a similar d e f i n i t i o n for the statement: [Q,R,S] ~fter [R ].
Both of these d e f i n i t i o n s
to apply to any number of particles,
71
can be e x t e n d e d
as in the previous section.
§S.1] §5.i
Collinearity of the Limit Particle
THEOREM 21
(Collinearity of Limit Particle)
Let Q be a particle with an instant Qc E Q and let {R(n) : n=1,2,...; R (n) ~ SPR[Qc]} be a bounded sub-SPRAY of SPR[Qc]. If after [Qc ], there is a unique particle S ~ SPR[Qc] such that: (i)
(±i)
for any instant Qx ~ Q with Qx > Qc" f o f (Qx)= QS SQ
, ...S>~ after [Qc ]
supl f o f (Qx)} ~ (n) R(n) Q
f-z ° f-1 (Qx) = infl f-1
SQ
QS
~R(n) Q
o
and
and
f-1 (Qx) } QR(n)
We call S the limit particle of the sequence of particles (R(n)). This t h e o r e m is a consequence of Axioms IV (§2.4), XI (52.13) and Theorems
6 (§2.9),
19 (§4.3) and 20 (§4.4).
It is used in the proof of Theorems 25 (§S.4) and 36 (§7.1).
72
22 (§5.2),
23 (§5.3),
§s.1]
Q
R (m)
//
R (n) S
f
Q1
Q2
QI
Fig.
PROOF (i)
(see Fig.
I0
i0).
The set of particles
{R(n): n=1,2,...} is an infinite
bounded set and so by the A x i o m of Compactness it has a cluster p a r t i c l e S;
(Axiom XI,
§2.13)
that is, for any instants
QI,Q2 ~ Q with
subset N of the positive
73
integers
such
§5.1] that,
for all n e N,
f o f o f (Q ) < f (Q ). 1 2 SR (n) R(n) s SQ SQ
The Triangle Inequality e
(Axiom IV,
§2.4) implies that,
for
all
n
N ,
(i)
f o f (Q ) <. f o f o f (Q ) < f (Q ). SR(n ) R(n) Q i SR (n) R(n) s SQ I SQ z
Given any positive
integer m, there is an integer n e N with
n ~ m, and so
after
and the Triangle
(Axiom IV,
Inequality
f o f (Q ) <. f sR(m ) R(m) Q i SR(n) =
[Qc].
o
f
o
SR(n )
52.4),
f R(n) R(m) f
o
(Q
R(n) Q
< f (Q), SQ 2
Now Q
Thus, by collinearity
f (Q ) R(m) Q I
)
i
by (i).
>Q,
was arbitrary apart from the r e q u i r e m e n t that Q R
2
1
so for all positive integers m,
f o f (Q ) <. f (Q ). SR(m ) R(m) Q l SQ l By the Triangle Inequality and the previous equation, we see that for all positive integers m,
(2)
f o f (Q ) = f (Q ). SR(m ) R(m) Q l SQ i
74
§5.z] NOW let QI'
def - f QS
o f (QI)" Qz' d~f - f o f (Q2) . sQ Qs SQ
second of the inequalities
The
(i), which applies for all
n e N , can now be written in the a l t e r n a t i v e form
-I(
(3)
,
f o f o f QI ) .< f o f o f o f-1(Q~) < Q2" QR (n) R(n) S QS QS SR (n) R(n) S 6~S
As before,
for any p o s i t i v e integer m there is some integer
n e N with n ~ m
and so by collinearity,
the Triangle Inequality,
and the above inequality,
f o f o f-1 (QI') <. f o f o f o f-~(Q~) QR (m) R(m) s QS QR (m) R(m) R (n) R(n) s QS =
f
o
QR (n)
t o f-1 (QI)
f
R (n) S
QS
I
<
t
Now Q2 was arbitrary,
Q2
apart from the r e q u i r e m e n t that Q2r
r >
Q1
~
so for all p o s i t i v e integers m,
f o f o f-1(Q;) QR (m) R(m) s QS and, by
(4)
Equations
!
~ Qt
using the Triangle Inequality,
f o f o f-1 (Q~) = QIf QR (m) R(m) s QS (2) and
(4) apply to any cluster p a r t i c l e ~ for any
instants QI,Q;s Q~ with QI > Qc and QI' > Qc" particle S and for all p o s i t i v e integers m,
75
so for any cluster
§5.l]
after
[Qc ].
The proof of (i) is completed by applying Theorem 20(i)
(ii)
f o f (Q ): n=l,2,...] QR(n ) R(n) Q I
Since
ing sequence,
[
the A x i o m of Compactness
the Triangle Inequality
(Axiom IV,
(§4.4).
is an inereas-
(Axiom XI,
§2.3) and
§2.4) imply that for any
e Q~ with Qc < Q 0 < Q I" there is an integer M such
instant Q
that, for all n > M, -I
Qp(n)
R(n) Q
i
SR (n)
qS >f QS
R(n) s
SQ
i
of(Q). SQ o
Consequently
( sup i
f
and since Q
f
°
QR (n)
(QI): n=1,2,... ~] ~ f
R(n)Q
QS
@0 is arbitrary,
o f (Qo) "
SQ
subject to the c o n d i t i o n
< Q , we see that 0
i
f (Q ): n=1,2,''' ~ >~ f R(n) Q i )
8up I f QR (n)
Qs
Theorem 19 (§4.3), t o g e t h e r with part
o f (Q ). I sQ
(i) of this theorem,
implies the opposite inequality, which establishes e q u a t i o n of (ii). Q
e Q with Q 2
~
In order to derive the second equation,
< Q 0
the first
< Q . I
let
Then, by (i), there is an integer
2
M such that, for all n > M ,
76
55.2]
f-1 o f-1 (Q) R(n) Q QR(n ) i
<, f-1 o SQ
I
f o f SR (n) R(n) s
)+i
i ) o f - (01 QS
< f-l o f-1(Q ). SQ QS 2 From this stage on, the proof is similar to the proof of the first equation.
By Theorem 19 (§4.3) and Theorem 6 (§2.9), there is only one
(distinct)
§5.2
cluster particle S.
[]
The Set of Intermediate Particles
We now d e m o n s t r a t e the existence of a sub-SPRAY which is collinear after the event of coincidence and "has no gaps". THEOREM 22
(see Fig. ii)
Let Q,U be distinct particles
which coincide at the event
[Qc ] and let Qx > Qc" Given any instant ~y ~ ~ with Qx < Qy < f Qu
o f (Qx) uQ
there is a particle S e SPR[Qc] such that: (i)
~S
0
sQf (Qx) = ~y = Qy
(ii) REMARK. event
and
after [Qc ].
S e ~* which is a linearly ordered subspray after the
[Qc] .
77
§S.2] This theorem is a consequence Theorems
4 (§2.7),
19 (§4.3),
of Axiom VI (§2.8) and
20 (§4.4) and 21 (§5.1).
is used in the proof of Theorems
23 (§5.3),
It
2S (§5.4),
26 (§5.5) and 57 (§9.1).
Q
s
QY
u
Q
S
T
U
"
Qx [Qa]
PROOF
Fig.
ii
(see Fig.
12)
Fig.
By the Axiom of the Intermediate there is a particle and Q ~ w ~ u.
W ~
SPR[Qc]
Particle
12
(Axiom VI,
such that
§2.8),
after [Qc ],
In this proof we shall consider particles
the set
78
from
§s.2]
E d~f {R: <@,R,W> a f t e r [Qo ], R e SPR[Qc]} V
O{R: <W,R,U> after [Qc ], R ~ SPR[Qc]} By Theorem 4 (§2.7) E is a c o l l i n e a r set of particles, Theorem 20 (§4.4),
E is a p a r t i a l l y ordered set.
Hausdorff maximal principle, the A x i o m of Choice maximal set E*.
and by
Thus, by the
which is logically equivalent to
(see Rubin and
Rubin [1963]),
(with nespeet to set inclusion)
E has a
linearly ordered sub-
By T h e o r e m 19 (~4.3) and the previous theorem, there
is a particle S ~ E* such that
~l) f
as
of
sQ
c~x) = ~uplf
~QR
d~f
o f C~x): f RQ
QR
o f CQx) ~ ~ , RQ
Q 1
Let
E' d~f
~: ~R
RQf (Qx) ~
let ~x def_ f-1(~y)QU and let Uz def_ fuQ (Qx)
R ~ E*
;
"
For any particle R ~ Z'
<@,R,U> after [Qc ], so -1 o f-1 o f (Qx) = f o f (Qx) >. ~y , f of uR uQ QR RQ QU RU whence
Let
f-1 o f-1(U ) >. RU UR z
x
-U2def inf{f -I o f-1(U ): R E 2'} >. -U RU UR z ~ x
79
R ~ z*} ~
§5.23 By Theorem particle
19 (§4.3)
theorem,
there
is a
T ~ Z* such that
f-1 TU and since
and the previous
o f-1(Uz) = U = ~ , UT z 2 after [Qc],
-I o f-I (~y)
fQ
= f-i
QT
of
UQ
of UT
-1(Qy)
of TU
QU
= f-1 o f o f (-U ) UQ UT TU x f-1 o f o (~), UQ UT TU 2 -I o f-1(~
~Q
) ~ f-l(U
QT
Y
UQ
z
SO
) = Qx
and therefore O
By equations
(i) and
after [Qc].
the Intermediate
and by Theorem equation equations
(Axiom VI,
between
19 ( §4.3),
this contradicts of Z*.
(2) imply that
O
which completes
the proof.
§2.8) implies
S and T and distinct
(2), or the maximality (I) and
with Theorem
20 (§4.4),
If S ~ T, the Axiom of Existence
Particle
tence of a particle
(2) together
O
80
of
the exis-
from both;
equation
(i),
Thus S = T, and
§5.2]
Let Q,U be distinct particles which coincide at
COROLLARY
i
the event
[Qc ].
L e t Qo > Qc and, f o r each i n t e g e r
Qnd~f [fQu o ~QJn(Qo). [Qn, Qn+1] a
=
is a consequence
{Qx: Qn~Qx~Qn+l'Qx of Theorem
and Theorem
PROOF.
[Q ,Q ] C
By (i) of the preceding
theorem,
0
signal functions
and map ordinary
instants
are one-to-one,
onto ordinary
e
8 (§3.2)
used in the proofs of the next corollary
(extended)
let
Then
9" where [Qn, Qn+1]
This corollary
n,
l
Q}
.
and is
25 (§5.4).
Q.
Since
~
order-preserving,
instants
(Theorem
8
§3.2), (~R o ~Q)n:
[Q0 ,Q i] ~
[Qn, Qn+l] C
from which the stated set containment
Q,
follows.
[]
The proof of the next theorem is based on a proof of the Intermediate Fulks
Value Theorem of real variable
(as in
[1961]).
COROLLARY
2.
Let Q,R,S be distinct particles which coincide at
the event [Qc ].
Let Qo > Qc and, for each integer n, let -
If
theory
f QR
)
f o f (Q ) , QS SQ o +~ then, for all Qx e ~] [Qn, Qn+1] , o
f (Q ) RQ o
e
<
f o f (Qx) < f o f (Qx) . QR RQ QS SQ
81
§5°2] This
corollary
9 (§3.2)
and t h e
of
Theorems
Theorem
is
a consequence
previous
23 ( § 5 . 3 )
corollary,
and 24 ( § 5 , 3 )
of
Theorems It
is
and t h e
5 (~2.9)
used in Corollary
and
the
proofs
to
25 (§5.4).
PROOF.
We suppose the contrary; that is, we suppose there is +~ an instant Qu ~ U [Qn,Qn+l] s u c h t h a t
f o f (Qu) ~ f o f (Qu) , QR RQ QS SQ and deduce a contradiction.
Case i.
Qu < Q
and f o
QR
o f > f o f (Qu) RQ QS SQ
Let
Qw def sup{Qt: f o f (Qt) > f o f (Qt), Qu .< Qt < Q , QR RQ QS SQ o
Qt ~ ~},
and let
QR
o f (Qt) < f o f (Qt) , RQ QS SQ Qu < Qt < Qw • Qt e Q}]
whence by Theorem 9 (§3.2),
Qu ~ Qv < Qw < Qo" Then,
supl f o f (Qt) : Qv < Qt < Qw" Qt E Q} " "QR RQ >~ supl f o f (Qt): Vv < Qt < :w" Qt ~ Q} ~QS SQ and
82
;
§5.2]
inf~f( o f (Qt) : Qw < Qt < Q " Qt ~ ~qR RQ o inf{f
S
o f (Qt) : ~w < Qt < Q , SQ o
so by Theorem 9
Qt ~ Q};
(§3.2),
f o f (Qw) ~ f o f (Qw) QR RQ QS SQ respectively,
~}
and f
QR
o f (~w) ~ f o f (~w) RQ qS SQ
f
o f
whence f
o f
QR
RQ
QS
SQ
By the previous corollary, Qw is an o r d i n a r y instant and so by Theorem 5 ( 2.9),
f
of
QR
= f
RQ
of
QS
,
SQ
which is a contradiction.
Case 2.
Qu > Qx and f o f (Qu) > f o f (Qu) QR RQ QS SQ
The proof is similar to the proof of Case i.
Case 3.
f
QR
o f
(Qu)
RQ
= f
QS
o f
(Qu)
.
SQ
By Theorem 5 (§2.9),
f of = f of QR ~Q Qs SQ which is a contradiction.
[]
83
§s.3]
§5.3
Mid-Way
and Reflected Particles
Let Q,S,U be particles [Qc].
If
after
which coincide
at the event
[Qc] and if
f of =f of , SQ QS SU US
we say that S is mid-way transpires
between
after
[Qc ] ~
Q
s
(Existence
Let Q,U be distinct Then
.
u
Fig.
23
It
in §6.3 that
THEOREM
Q and U (see Fig. 13).
13
of Mid-Way Particle)
particles
there is a particle
which
S mid-way
84
coincide between
at the event
Q and U.
[Qc ]
§5.3] This t h e o r e m is a consequence of Theorems 15 (§3.7), 19 (§4.3), T h e o r e m 22. 37 (§7.2),
PROOF.
21(§5.1),
22 (§5.2) and Corollary
It is used in the proof of Theorems 39 (§7.3), 44 (§7.4) and 47 (§7.5).
(with respect to set inclusion)
collinear sub-SPRAY ~* which contains between them.
Qx
Let
Qz d~f f
and let
o
By T h e o r e m o
21
=
c o m p l e t e l y ordered
Q and U and p a r t i c l e s
e @ be an instant such that
Qx > Qc
uQ
(§5.1)
f (Qx) SQ
there exists
f (Qx).
Qu
f QS
2 of
24 (§5.3),
As in the proof of the previous theorem,
a maximal
(i)
5 (§2.9),
there
is
a particle
(
I
sup f o f (Qx): f ~QR RQ QR
S e Z* s u c h o
;Q
(Qx)
that
¢ Q~, R e Z*
}
de=f Q 1
Case i.
(see Fig.
Suppose
IfQS
o
f SQ
14)
]2(Qx)
<
Qz
Then Theorem 15 (§3.7) and Theorem 22 (§5.2) imply the existence of an i n s t a n t
Q
e Q and a particle
V e E* s u c h
9
Q 1 < I fQV ° f I-I (Qz) = which means that ordered subset of E.
that
~
3
< I fQS ° f
after [Qc 2,
since Z* is a linearly
By T h e o r e m 22 (55.2) there is a p a r t i c l e
T ~ Z* such that
f o f (Qx) = minIQ , f o f (Qx)l > Q QT TQ 3 QV VQ
85
~5.3] and, since Q,S,T,V
~ ~*, w h i c h is linearly
after
[T c]
ordered,
and # # T .
But now, by T h e o r e m 19 (§4.3),
f
o
QT
Q
(Qx) .< f
QT
o f (Q ) TQ
3
{V o fVQ (Q3) = Qz " which contradicts
{
[
v
Fig.
Case
2. (see Fig.
Suppose
I
f
QS
(I), and shows that the supposition is false.
o f
sQ
u
Q
14
v
Fig.
s
u
15
15)
1 ~ (Qx J
> Qz
Then there is an instant Q 3 E Q~ and a particle
~ > I~v o ~ ] ~~z~ ~
V E 2* such that
> I~ o ~ 1~ ~0z
86
§5,3]
which
means
that
there
is a p a r t i c l e
after
[Qe].
T e E* such
By T h e o r e m
22
(§5.2)
that
f o f (Qx) = max(Q , f o f (Qx)} < QI QT TQ 3 Qv vQ and
after
I
f o QT
[Qc ]
and
contradicts
Having
(i) and
eliminated
[
~
Sz
~
f
o
QV
shows
f
VQ
that
the p r e v i o u s
]2
f o f QS SQ Also
after
But n o w
(Qx) >~ f o f (Q ) QT TQ 3
Q
>~ which
.
T # S
(Q
3
the
two
)
=
Qz
"
supposition
cases,
we
is false.
conclude
that
(Qx) = Qz = f o f (Qx) " QU UQ
[Qc ], so l e t t i n g
~
S x
d~f f (Qx) SQ
and
def i - f- (Qz), QS f o f (Sx) = Sz = f o f (Sx) SQ QS SU us
By T h e o r e m
.
5 (52.9)
f of =f of , SQ QS SU US which
completes
the proof.
If a p a r t i c l e we say that of Q in
U.
[]
U is m i d - w a y
Q is a reflection In the next
theorem
between
of we
87
two p a r t i c l e s
W in U and show
that
Q and
W is a r e f l e c t i o n
W,
§5.3] for any two d i s t i n c t particles Q and U which coincide at some event~ there is at least one r e f l e c t i o n of Q in U, and all reflections of Q in U are p e r m a n e n t l y
coincident;
that is,
there is a unique o b s e r v e r w h i c h we denote by the symbol ~U (see §3.8).
Before proving this existence theorem,
we
establish the following:
PROPOSITION
(see Fig.
16)
Let @,R,S coincide at the event
[Qc ].
zf f o f = f o f RQ QR RS SR
f o QP
and
=f Q
of QS
, sQ
then R is mid-way between Q and S.
Q
R
S
[Qc I / \ Fig. PROOF.
16
Since f o f o f o f = f o f • it follows that QR RQ QR RQ QS SQ f of of of = f of. QR RS SR RQ QS SQ
88
§5.3]
By the Triangle
f
Inequality
o f
QR
~ f
RS
f
and
QS
o f
SR
= f
RQ
24
and
SQ
~ f
RQ
f
o f
= f
RS
(§5.2)
19
of R e f l e c t e d
Observer)
which coincide
is a c o n s e q u e n c e
(§4.3),
and T h e o r e m
Theorems
26 (§5.5),
(see Fig.
20 24
,
QS
Then there is a unique observer
Theorems
, whence
SQ
[]
(Existence
This t h e o r e m
PROOF
§2.4),
o f
QR
Let Q,U be distinct particles [Qc ].
f
SR
wh i c h was to be proved.
THEOREM
(Axiom IV,
(§4.4), (§5.3).
36
(§7.1)
~U"
of A x i o m VII
Corollary
42
in the proof
(§?.3).
17 )
Ro
R 1
R2
R3
R~
q
R
S
T
U
Fig.
Rs
B6
R7
R 8
~u
17
89
(§2.9)
2 to T h e o r e m
It is used and
at some event
and 22 of
§5.3] Applying the previous theorem successively, there are particles: i such that S is mid-way between Q and U; (i)
such that R is mid-way between Q and S; and such that T is mid-way between S and U.
By Theorem 20
(§4.4),
(2)
a f t e r
[Qc ].
We will now show that S is mid-way between R and T.
If,
for some instant S x s S with S x > Qc"
f
(3)
o f (Sx) > f
SR
RS
o f (Sx)
ST
TS
then, by Theorem 23 (§5.2), f SR
The statements
I f
SR
o
o S
(S) x
>
f ST
o
(Sx). S
(i) imply that
II 2 S
--f SQ
of
=f QS
of SU
which shows that (3) is false.
= US
o T
S
Similarly, we can show that
the opposite inequality is false, and since S
was arbitrary, X
f SR
of
= f RS
ST
of TS
This together with (2), shows that S is mid-way between R and T.
90
§5.3] In order
to simplify
particle
symbols
and fnm will
the proof,
we shall
now substitute
the
R°,RI,Rz,B3,R ~ for @,R,S,T,U r e s p e c t i v e l y for fRnR m
be s u b s t i t u t e d
represent
the proposition:
(4)
For n=l, 2, • • .m,
We now let P(m)
(i)
after [Qc ]
(ii)
f n(n-1)
(iii)
for all R n-l > Qc
o
f = f (n-1)n n(n+1)
o
f (n+1)n
and
X
(n-l)n The r e s u l t
n(n-1)
(n-t)(n+l)
of the preceding
paragraph,
(2) can be summarised
as:
(5)
P(3) is true. The A x i o m
that,
of Isotropy
for any i s o t r o p y
of SPRAYs
mapping
¢(P(m)) is true where ¢(P(m)) (6 )
Let
f
de=f
(J) t (k) (i)
is the
f
f
o
for all
(Axiom VII,
(1)
§2.9)
is true,
and
implies
then
statement:
For n=l,2,. . .,m,
f
after =
¢(n-IJ~(n)
¢(R n-l) > R X
do(n-1)¢(n)
with
~ (Rj) ¢ (R k)
¢(nJ¢(n-1) (iii)
together
¢, if P(m)
<¢(R°),~(RI),...¢(Rm+I)>
(ii)
(n+l)(n-1)
0
[Qc] , f
¢(nJ¢(n+1)
J
¢(n)¢(n-1)
[~(n-IJ¢(n+l)
@(n+1)¢fn-1)
91
o
f
¢(n+1)¢(n)
and
§5.3] Since
P(3)
§2.9)
and the
there
is an i s o t r o p y
is true,
the A x i o m
first
of e q u a t i o n s mapping
(4),
% such
of SPRAYs
with
(Axiom
n = 3, i m p l y
VII,
that
that
~(R 3) = R 3 and ~(R 2) = R ~.
(7)
By the
ordering
Theorem
20
relations
(§4.4)
of
(4) and
and T h e o r e m
Let R G-n de=f ¢(Rn),
for
19
n=0,1.
(6),
together
with
(§4.3),
We o b s e r v e
that
~ o t ( R I) ~- R I
~ o ¢ ( R °) ~- R °, so
and
R 6-n -- ~(Rn),
(9)
By (4),
(5),
(6),
(8) and
By a p r o c e d u r e can
similar
that:
is true.
to that
of the p r e c e d i n g
paragraph,
show that
PC9)
Now by a few
successive
f 40
That
for n = O , l , ' ' ' 6 .
(9) it f o l l o w s
P(5)
(i0)
we
of I s o t r o p y
is,
o f 04
is true.
applications
: f 48
R 8 is a r e f l e c t i o n
o f
of e q u a t i o n s
(after
(4),
[Qc]).
8~
of R ° in R ~ , so B 8 is a r e f l e c t i o n
92
§5,3] of @ in U.
By Theorem 19 (§4.3), ~U is unique.
the proof.
[]
COROLLARY
(see Fig. 18)
Let
S ° , S I be
[S~]. such Sm X
distinct
There that,
is
for
E Sm with
where
f mn
particles
a sub-SPRAY all
which
coincide
at
{S n : n=O, t l , ± 2 , . ..;
integers
m,
n and
for
any
This completes
the
event
S~ n e S P R [ S ~ ] }
instant
Sm > o X SOY
n
m
d£f -
f s m s n"
m(m+1)
(m+1)m
This corollary is used in the proofs of Theorem 25 (§5.4), the Corollary to Theorem 39 (§7.3) and Theorem 57 (§9.1).
S-~
S-3
S-2
S-I
So
SI
Fig. 18
93
S2
S 3
S ~
§5.3] PROOF
(by i n d u c t i o n )
For n > 1, let S n+1 e ~n+1 w h e r e reflection
of ~n-1
For n < 0,
let S n-l e ~n-1 w h e r e
reflection
of ~n+1 in ~n.
Case
~n+1 is d e f i n e d
as the u n i q u e
~n-1 is d e f i n e d
as the u n i q u e
in i n.
i. m < n
As an i n d u c t i o n
hypothesis,
suppose
for all S k e S k w i t h
that
X
Sk > S O X
and
for all
k,~ w i t h
1 ¢ £ - k ~ n
~
,
O
kZ Then. since
~k
k(k+1)
(k+1)k
x
<sk,sk+I,SZ>~ ~ after [S$] ,
f o f (Sx) = f o f o f of (Sx) k(£+I) (£+I)k k(k+1) (k+i)(£+I) (£+I)(k+I) (k+1)k
=
k(k+1)
(k+1) (k+2)
k(k+1)
(k 1)k
[
o
f
k(k+1) So if the
induction
hypothesis
all
k,£ w i t h
£ - k ¢ n,
for
all
k,Z w i t h
~-k ~ n+l.
(trivially)
true
k(k+1)
(k+1)k
(k+1)k
f i£+I-k (sk). (k+l)k
for
is
(k+2) (k+1)
is true
it is a l s o Since
for n = 1, the
complete.
94
the
true induction
proof
of this
hypothesis case
is
§s.4]
Case
2. m > n
In accordance
with the d e f i n i t i o n
the proof
is analogous
functions
instead
of m o d i f i e d
record
to that of Case 1 but with
functions,
inverse
of functions.
Case 3. m = n This trivial
§5.4
All
THEOREM
case
Instants
completes
Are
Ordinary
All instants
25.
This t h e o r e m
the proof.
Instants.
are ordinary instants.
is a consequence
of Axioms
IX
(§2.11)
and X (§2.12)
and Theorems
12
(§3.5),
19
(§5.1),
(§4.3),
Theorem
22
(§5.2)
is used
in the proof
Theorem
26
(§5.5)
21
D
22
and the Corollary of the C o r o l l a r y
and the C o r o l l a r y
95
VIII
6 (§2.9),
(§5.2),
ii
(§2.10), (§3.4),
Corollary
to T h e o r e m to T h e o r e m to T h e o r e m
1,2 of
24 (§5.3). 25 (§5.4), 40
(§7.3).
It
§5.4]
Qy -4® Q1
Q-1
Qx
Uo
c
Fig.
PROOF
19
(see Fig. 18)
Theorem ii (§3.4) implies the existence of instants before and after Q0 •
Let Qc be an instant of Q ~ with Qc < Q 0 "
By the A x i o m of Incidence Connectedness (Axiom VIII,
(of Q)
(Axiom X,
(Axiom IX, §2.11), the A x i o m of
§2.12) and the A x i o m of D i m e n s i o n
§2.10), there is a particle
coincides with Q at [Qc].
96
U # Q such that U
§5.4]
Qn d~f [f o ~ In (Q o) "
For each integer n, let Corollary
1 of
Theorem
22
(§5.2)
and
By
Q
Qu
Theorem
12
(§3.5),
+~
t.) [Qn, Qn+,] C
(i)
{Qx: Qx > Qc" Qx ~ ~}
We now show that these two subsets are equal, two exceptional
by excluding
cases.
Case i. Suppose there is an ordinary
instant
Qy e Q such that,
for all integers n, Qy > Qn"
By the corollary
theorem,
{un: n=0,1,2,''; ~Un ~ SPRAY[Qc]}
there is a sub-SPRAY
which is collinear
for which
[Qc],
after
sup
o f cQ
n
nO
21 (§5.1),
such that,
for all positive
(2)
such that Q = U~ U~ = ~Ui
= s p{f
By Theorem
~On
integers
~
~
"
and
< Q. 0
there is a limit particle S e
SPRAY[Q o]
n,
after [Qc ] ~
to the previous
and
~
~
= Q= - f o f (Q ). QS SQ o
Also,
~ = sup Qn = sup Qn+1 = 8up{fn o fno(Qi)} < QY and as before,
there is a limit particle
for all positive
integers
and
~
1 of Theorem 22,
U n~-oo
such that,
n ,
(3) ~ after [Qa] By Corollary
T sSPRAY[Q c]
[Qn, Qn+1] ~
97
~ ,
= Q~ = fT
o
~m~f(Qi)
§5.4]
so T h e o r e m
6(§2.9), T h e o r e m 19 (§4.3) and Corollary
2 of
T h e o r e m 22 (§5.2) imply that
after[inf{Qn}] T ~ which contradicts
(2).
and
S
before[Q
,
Hence there is no o r d i n a r y instant
Qy such that, for all integers
n ,
Qy > Qn
Case 2. Suppose there is an ordinary instant
Qx > Qc" such that for all (§2.9),
] , and
integers
n, Qx < @n" [Qx ].
U does not coincide with Q at
(§3.5), for all negative
@x e Q with By T h e o r e m 6
By Theorem 12
integers n,
Q Qu
Q
As in Case i, there is a set of particles
~Q 0
{un: n=0,1,'''}
such that
o
i IIfn O oI I 0} °x"
From this stage on, the proof is similar to the proof for Case i;
so there is no ordinary instant
all integers n, Qc <
Qx e Q such that, for
Qx < Qn; whence Q_oo = Qc"
98
§5.4] We have now shown that oo
(4)
U
n--o
[Qc'Qn I -- {Qx: Qx ~ Qc'Qx ~ Q~"
But Qc was arbitrary, (ordinary)
and since for any Qw e Q there
Qv e Q such that Qv < Qw' it follows
instant
taking Qc = @v' that ~w ~ @" the proof.
is some
Thus ~
C
Q, which
by
completes
[]
An immediate is the following
consequence stronger
of (the proof of) this theorem
version
of Corollary
2 of Theorem
22
(§5.2): COROLLARY.
some event
Let Q,R,S be distinct particles [Qc ].
which coincide at
If, for some instant Qo c Q with Qo > Qc"
f o f (Q) QR RQ o
< f o f (Q) QS SQ o
then, for all Qx ~ ~ with Qx > Qc
"
f o f (Qx) < f o f (Qx). QR RQ QS SQ This corollary
is a consequence
of Corollary
Theorem
22, and it is used in the proof of Theorem
PROOF.
By Corollary
2 to Theorem
99
22 (§5.2).
[]
2 of 26 (§5°5).
§5.s]
§5.5
P r o p e r t i e s of C o l l i n e a r Sub-SPRAYs A f t e r C o i n c i d e n c e
Given two distinct particles Q,T which coincide at some event [ Q c ], we define the c o l l i n e a r
{R:
CSP
d ~ f
[Q,T,R] a f t e r
asp
~ ~ d ~ f {[Rx]:
sub-SPRAY
[Qc], R g S P R E Q c ] }
,
and
Rx
> Qc"
R~ e O S P < Q~, T >~} .
(csp)
is a set of events,
Note that a c o l l i n e a r
sub-SPRAY
whereas a spray
is a set of instants.
(spr)
d i m e n s i o n a l M i n k o w s k i space-time,
In a 1+I
-
a collinear sub-spray is
"a set of events c o n t a i n e d w i t h i n the upper half of a light cone".
Some of the previous results can now be stated concisely in the following:
u
v
Q
2t7_ ~ [Qc ]
Fig.
100
20
§5.5] THEOREM
26
(Collinear
subSPRAYs
after coincidence)
Let Q,R and {,T be pairs of distinct particles which coincide at the event [Qc]. Then aSP is a set of particles which is collinear
(i)
after [Qc] and CSP<~,R> = {~: <S,Q,R> after [Qc], after [Qc], or after [~c]; S e SPB[Qc]}, (ii)
S,T e CSP ~
CSP<@,{>=~T,~SC CSP<9,{>,
(iii) {,[ ~ (iv)
CSP = CSP<S,T>,
the relation
"to the right of" between particles in
CSP after [Qc ] is a linear ordering, (V)
and
for any instants Qx, Qy e Q with Qc < Qx < Qy" there are particles
U,V ~ CSP such that
after [Qc] and
f o f (Qx) = f o f (Qx) = Qy QU UQ QV VQ This theorem 17 (§4.2),
19 (§4.3),
its corollary. 28 (§6.i), Theorem
is a consequence 22 (§5.2),
after [Qc]
and
(see Fig. 20) •
of Theorems 24 (§5.3),
4 (§2.7),
25 (§5.4),
It is used in the proof of Theorems
29 (§6.2),
32 (§6.4),
30 (§6.3),
and Theorem
31 (§6.3),
33 (§6.4).
101
and
27 (§6.1),
Corollary
3 to
§s.s] PROOF.
~R C
(i)
By Theorem
24 (§5.3) there is an observer
CSP such that <~R,~,~> after
By Theorem 4 (§2.7) optical
[Qe].
lines containing
instants
Qc ) from both Q and R, are uniquely characterised. proves the first proposition; consequence
of the previous
This
the second p r o p o s i t i o n
c o r o l l a r ~ Theorem
(after
is a
19 (§4.3)
and Theorem 17 (§4.2). (ii)
By (i),
CSP is a collinear set of particles
after [Qo], and is uniquely particles
contained
specified
in it.
(iii)
As with
(iv)
As with the second proposition
sequence (v)
by any two distinct
(ii), this is a consequence
of the previous
corollary
As in the proof of Theorem
of (i).
of (i), this is a con-
and Theorem 25 (§5.4)
19 (§4.3).
(see equation
(4)),
if we let R ° ~ Q and R I ~ R, there is some R n E CSP such that
o f [Qx ] > [Qy].
f On
Now Theorem 0
nO
22 (§5.2) implies the existence
after [Qc]
and
o
Now take any particle
g ~
e
~Q.
[]
102
of
V
such that
§6.1] CHAPTER
COLLINEAR
In this c h a p t e r particles
there
w h i c h have many p r o p e r t i e s
are described
more
in absolute
are c o l l i n e a r
analogous
geometry.
sets of
to those
These
of
properties
fully in §6.4.
Basic Theorems
The main result theorem.
The p r o o f
of this of some
if the A x i o m of Compactness so as to apply dence": would
PARTICLES
we show that
coplanar sets of lines
§6.1
6
the
"before
Second
be a trivial
(Theorem trivial
27,
would
(Axiom XI,
coincidence"
Collinearity
similarly
of Theorem
is c o n t a i n e d
theorems
extension
§6.1)~
extension
chapter
have been
§2.13)
simplified
had been
as well as "after
Theorem
of the First Theorem
31
26 (§5.5).
103
in the next
(Theorem
30,
Collinearity (§6.4)
would
stated coinci§6.3) Theorem be a
§6.1]
THEOREM 27
(First C o l l i n e a r i t y Theorem)
Let Q,B,S be
(distinct)
particles
with instants
Qa, Qb ~ Q;
Rb,R c ~ R; Sa, S c ~ S (see Fig. 21). If
Sa = Qa < Qb ~ Rb < Be = Sc
csp<6,{> c
osp<{,~> c
R
, then
csp<{,[>.
s
Q
\ R C
S
7 Fig.
21
This t h e o r e m is a consequence of Theorems 4(§2.7), 19 (§4.3) and 28 (§5.5). 28 (§6.1),
29 (§6.2),
It is used in the proof of Theorems
30 (§6.3) and C o r o l l a r y 1 to 30 (§6.3),
and Theorem 32 (§6.4).
104
§6.1]
PROOF.
By the preceding
optical line containing
theorem and Theorem 4 (§2.7), two events from a collinear
(after the event of coincidence) since R c
particular,
is uniquely
any
sub-spray
determined.
Sc,
=
[Sol = [s c] E osp
~
osp<~,{>.
Therefore: (I)
any event which is in optical
line with either the
[R c] and [f (Rc)] , or with the events [R ] QR c and [f-1(Rc)] , is contained in both
events
RQ
osp<~,~>
and
osp<~,~>.
We now choose any particle
T e CSP such that
after [Qb ] and Q ~ T.
By the previous
Theorem 4 (§2.7),
(2)
If we can show that
{[Tw]: Tw > Qb" T
e T} ~
csp,
it will follow that
o~p<~,{> Let T
d~f f-1(R ), T i
RT
By proposition
(S)
~ o~p<{,{>
c
c
osp<{,{>.
d~f f (R ). 2
TR
a
(i) and the definition
[T ], 1
[T ] e csp. 2
~
105
~
of T,
theorem and
In
§6.1]
U
R
S
Q
S
Fig.
In this
22
for b o t h
cases
i and
on the p a r t i c u l a r
Case
i.
T
w
> T
2
Let T w = T z > T 2 . s CSP<@,R>
diagram,
such
2.
(see
same
Tz
o f QU
particle
Fig.
U depends
respectively.
22)
theorem, a£ter
o f (T ) = UT i
o f UR
U is d e p i c t e d
the p a r t i c l e
and T x ,
By the p r e v i o u s that
Qa
a
In g e n e r a l ,
instants
f TU
f TQ
the
T
o f RQ
T
,
and
[Qb]
so
z
o f (T) QT ~
106
there
=
T z
is a p a r t i c l e
§S.l]
Therefore
IT , f - l ( R c ) , I RQ
and by p r o p o s i t i o n
f UR
f
o
RQ
of(T)> QT
(I)
[f o f UR RQ
Also,
RO"
o f (T)] QT I
e osp. ~ ~
trivially
[f QU
Since
o f o f o f (T ) ] e c s p < Q , S > . UR RQ QT
[T ] is in o p t i c a l
line w i t h the d i s t i n c t
events
Z
[f o f UR RQ
it f o l l o w s
and
o f (T)] QT i
Let T
< TW < T I
Qb =
W
such that
,
] ~ csp. Z
2.
o f (T)] QT i
that
[T
Case
If o f o f QU UR RQ
T .
~
(see Fig.
As in ease
22)
i, there
is a p a r t i c l e
X
~
after
[Qb ] and
f o f (Tx) TU UT
f-l QT For r e a s o n s
U £ CSP
o f-1 UQ
similar
o f-1 RU
= T ,
SO
z
o f- I o f- i (T2) QR TQ
to those
= T
g i v e n in ease i,
[mx] ~ osp.
107
x
~
~
§6.1]
(The p a r t i c l e case
i.
U is n o t n e c e s s a r i l y
Actually
there
instant
Tw).
different
R
S
the
is a d i f f e r e n t
V
particle
Q
~ % ~
U as in
U for each
T
...~Rc
same particle
"~
T2
Ty
T1
Case
3.
T
< T 1
Let T
w
< T W
= T ; then y
is a p a r t i c l e
Fig.
23
(see
Fig.
23)
2
T
i
< T
< T . 2
Y
[ e CSP
such
By t h e p r e v i o u s that
and f
TV
o f
(T ) = T
VT l
108
y
theorem,
after [Qb]
there
§6.1]
Since
after
[Qb ], and -I
f o f o f TQ QR VR
after
[Qb],
1
o fRV
o f (T ) = T RT i y
Therefore
ITI, f-I RV
o f (T), RT I
Re> ,
and since both
[T ], [R ] E esp it follows by (i) that [f-i o f ( T ) ] RV RT i Also,
e csp, ~ ~
trivially [f o f-1 QR VR
o f-i o f ( T ) ] RV RT i
e csp<@,S>.
Now either
after
[Qb ] , or a f t e r
and in either case [T ] is in optical Y [{ -1 ° RT f (T i
in
some
order,
and
IQ'
[Qb ] ,
line with the two events
o f-I VR o RV f-I o fRT (T i
so
109
O
bJ
/
~ ////"
~m
§6.1]
Now proposition implies
(4)
(2) applied to (3) and to cases i, 2, 3
that
o~p<~,C> = o~p ~
o~p.
We w i l l n o w s h o w t h a t
(5)
sy > So,
{[sy]:
d~f f ( S ) . z
RS
At t h i s
stage we do not k n o w w h e t h e r
or .w h e t h e r R and Q are on the same side of
[Sc] ; the f o r m e r
previous
let R x def ) and - f-1(Sy SR
Y
. . a f t e r. [So], . S after
asp. ~ ~
Sy e S~ w i t h Sy > so,
For any i n s t a n t let R
S Y e S} ~ ~
theorem,
such that < T , R , U >
case is d e p i c t e d
t h e r e are p a r t i c l e s
after
in Fig.
24.
T,U e CSP
[R b] and
f o f (R ) = f o f (Rx) = f o f (Rx) = Rz. RT TR x RU UR RS SR By (4),
[f (Rx)] , [f (Rx)] , [S ] c esp, TR UR Y ~ ~ so by T h e o r e m
19
(§4.3),
either
[f (R )] = [S ] TR x y in e i t h e r
case
[Sy]
e csp,
or [f (R )] = [S ]; UR x y which
establishes
and h e n c e
(6) This,
~p together
with
c
o~p.
(4), c o m p l e t e s
111
the proof.
[]
(5),
By the
§6.1]
The A x i o m ficiently the
strong
stronger
THEOREM
of Incidence
28
statement
result
(Ordered
such
§2.11)
for some purposes
contained
Let R,S be d i s t i n c t ~
(Axiom IX,
so we e s t a b l i s h
in the following:
Incidence
particles
is not a suf-
- see Fig.
with
25)
instants
~
R
e R, S C
that R
c S C
~
~ S . C
C
Given an i n s t a n t S a with i n s t a n t s
c S~ with S a
< Sc,
there is a p a r t i c l e
- - o"Q, Q a e xQ a n d there is an i n s t a n t
that Sa
~- Qa < Qb ~- Rb < Rc = Sc
R
S
C
/
Q
\ Q
Fig.
25
112
Q
R.D ~ R such
§6.1]
This t h e o r e m is a consequence of Axioms IV (§2.4), IX (§2.11) and Theorems 17 (§4.2), 27 (§6.1).
19 (§4.3),
Vl (§2.8),
26 (§5.5) and
It is used in the proof of T h e o r e m 30 (§6.3),
Corollary 1 to T h e o r e m 30 (§6.3), and Theorems
32 (§6.4) and
33 ( § 6 . 4 ) . PROOF.
By the A x i o m of Incidence
(Axiom IX,
§2.11), there is
Wa, W d e W and there is an instant
a particle W with instants
R d e R such that
W a = Sa Case i •
Wa < W d < S a
(see Fig.
In this trivial case we and R b d~f Rd Case 2.
and W d -- R d # Rc. 25)
let Q d~f W, Qa d~f Wa • Qb d~f Wd
.
W d < Wa
(see Fig.
26)
Consider an instant S b e S with S a < S b < S c. (§5.5), there are particles
<W,S,T> . .
after [S a] . . .
(]_)
T,U E CSP<W,S>
and .
By T h e o r e m 26
such that
a f t e r [S a ]
and
f o f (Sb) = f o f (Sb) = f o f (Sb). ST PS SE RS SS US
By the previous theorem,
[f (Sb)], TS By e q u a t i o n
[f (Sb)] US
e csp<W,S>
~
csp<W,R>.
(i) and Theorem 19 (§4.3), either T or U coincides
with R at [f (Sb)] ; so, accordingly, we define Q to be either RS ~ or U so that Q coincides with R at [f (Sb)]. We then define RS ~
113
§6.1]
Rb d ~ f f (Sb) and Qb d ~ f f (Rb)" RS QR
w
R
and so Qb ~ Rb"
s
Q
Rb
-
S 7a
f Fig.
Case
3.
S c < Wd
26
(see Fig.
We first find a particle
27)
T which has properties
in the p a r a g r a p h containing
Case 3(i).
Rd
as stated
(4).
~ f - * ( S c ) , we let W be the SW ~ particle T;~ that is we define T~ d ~ f W , T a d -£ f W a , T 3 d e - f Wd and R
If f o f o f (S) WS SR RS a
d~f Rd
.
3
Case 3(ii).
If f o f o f (S) WS SR RS a
> f - 1 ( S c ) , by T h e o r e m 26 SW
(§5.5) there is a particle T e C S P < W , S > <W,{,S>
~fter
[Sa],
or < W , S , T >
after
114
with either
[Sa],
such that
,
§6.1]
(2)
f o f o f-1 (Sc) WT TW SW
Since
R coincides
with
S at
~
by
o f (Sa) RS
O
a
° f ( S ) < S RS a
c "
(2),
f o f o f-1(S ) < f o f WT TW SW c WS SW
and
therefore
<W,T,S>~ ~ ~
after
f o f-1(S WT ST
and
T
d~f i
W at
0
(see
[S a]
o f-1(S ) SW c
Fig.
) = f o f o f WT TS SR
27).
by
(2),
o f (S ) a RS
<W,{,{> [Wd] ;
after so
f-1(S ) = f o f c TS SR
ST
by
[Sa] ; R c o i n c i d e s Theorem
17
(§4.2)
o f (S ) . RS a
with there
S at is
an
~
such
Hence,
so
(3)
Now
> f-1 (Se) SW
[S ] > IS ] ,
~
f SR
whence
= f o f WS SR
IS c] and instant
with T
g T 3
that
R coincides
with
T at the
event
~
[T ] 3
Sc
3 <W d
.
Let R
def 3
f RT
(T ) -- T ~
3
and let T a
de=f f (S ) ~- S TS a a
115
~
~j
Oq
\ \ \ \
~i /
•
/
l/
//
//
o
o
I-J
t~
§6.i]
So for both cases T and instants
Ta, T
~
3(i) and
e T and R 3
T
= S
a
3(ii):
~
a
-- R
O
is a p a r t i c l e
that (see Fig.
e R such 3
< S
there
28),
~
< R
C
3
= T
and
3
f o f o f (S ) ~ f - ~ ( S ) de=f T . TS SR RS a ST c i
Then,
by the Triangle
(4)
Inequality
f (Sa) ~ f-1 RS SR
(Axiom IV,
o f-t o f-1(Sc) TS ST
§2.4),
~ f-t(T ) TR i
By the Axiom of the Intermediate
Particle
there is a particle
with
T # U # R. ~
~
R
Let U
28).
§2.8),
and
d~f f ( U ) 2
RU
(see Fig.
)
and T
d ~ f f-t(Rc) 2
~
d~f f-1(U 2
U g CSP
(Axiom VI,
and let
2
TU
Then
2
UR
(5)
ITl" U 2 ,R c >
By Theorem
26 (§5.5),
such that ~ ~ (6)
there
after [S a ]
f o f (T)= TV VT i
and
IR2,U2,T
is a particle
2
> V e CSP
and
f TU
o
f
UT
(T)
=
i
T
.
2
de f f-1(Sc) we have .ITt,V2,Sc > and, from (5), SV ITt,U2,Sc >. Thus by Theorem 19 (§4.3) and (6), U -- V
Letting
V
2
2
by (5),
IR ,V ,T >; that is 2
2
2
[~ 2] ~ c ~ p < { , p
117
= c~p<~,s>.
2 ~
SO
~J
r~ co
~q
o
\
\ \
\
/
!
L~
o~ ~J
§6.2]
Since
T
> T , it follows 2
S
~- T a
By T h e o r e m
26 (§5.5),
19
there
f-1(T ). TR 2
o
Q e CSP
is a p a r ti c l e
after [T ]
f-1 QT By T h e o r e m
(4) that
~ f-1 o f-1 (2 ) < f -I RT TR I RT
a
such that
from
1
and
o f-1(T ) = f-1 o f-1(T ) TQ 2 RT TR 2
(§4.3),
Q coincides
~
let R b d~f R , Qb d~f f
QR
2
§6.2
The Crossing
and T c c T such that into two
by specifying
on one p a r t i c u l a r at no more
particles
that
events
called after
side of @.
@,{ with
instants
Qc e
with
left and right sides,
[Qo] and c o i n c i d e n t with T are
Since
(Theorem
distinct 6,
the c o n c l u s i o n s
"events
after
§2.9),
(right)
say that
"{ is on the
left
csp.
Sometimes
if the
the words
"in
particles this
of §4.1 and
coincide
definition §4.2.
Instead
[Qa] and c o i n c i d e n t with T are on
side of Q in osp"~
the left
shall delete
[]
Qe = Tc" we p a r t i t i o n the events of
than one event
is in accordance of saying
(Rb) ; so that Qb = Rb"
classes,
that
2
Theorem
Given two distinct
csp
We now
with R at [R ].
(right)
simply
side of Q after
context
csp"
119
we shall
is u n a m b i g u o u s , At this
[Qc ] in we
stage we
§6.2]
do not define the sides of Q before
[Qc]; they are specified
completely by the corollary to Theorem
33 (§8.4).
Let U be a particle with an instant
[Ud].
such that U coincides with Q at
U d e U, with U d > Qc"
If
after [Qc ],
(i)
[@,T,U]
(it)
U is on the left
(right)
side of @ before
[U d] in
csp and (iii) U is on the right
we say that that U
(left)
side of Q after
[U d] in
U crosses Q at [U d] in csp, or simply
crosses Q at [Ud] if the context of the statement is
unambiguous
(see Fig.
T
2 9).
Q
Fig.
U
29
120
§6.2]
T H E O R E M 29
(Crossing Theorem)
Let Q,R,S be distinct
particles
with instants
Qa, Qb e Q;
Rb,R c c R; Sa,S c e S such that Sa ~- Qa < Qb ~- Rb < Rc ~- Sc. Then R crosses
S~ at [S c] in csp.~ ~
This t h e o r e m is a consequence of Theorems 17 (§4.2), 19 (§4.3),
26 (§5.5) and 27 (§6.1).
6 (§2.9), It is used in
the proofs of Theorem 30 (§6.3), C o r o l l a r y i to Theorem 30 (§6.3) and Theorems PROOF.
32 (§6.4) and 33 (§6.4).
(see Fig. 30 )
S
Sd
T
R
"
Sc
Fig.
30
121
§6.2] Let the right side of S (in
csp be
that Q is on the right side of S after
defined such
[Sa].
By Theorem 27 ( 6 . 1 ) ,
[Q,R,S]
[Rb].
after
Let us suppose that R does not cross
S at [S ]
~
and we
d
If R does not cross S at [Se] ,
shall deduce a contradiction.
it follows that R \ {R c} is on the right side of S after [Rb]. Consider any instant S d E S such that S d > S e and
f o f (Sd) < f o f (Sd). SR RS SQ QS (This is possible
since otherwise Q would coincide with S at [S ] ~
and so by
O
Theorem 6 (§2.9), Q = S, which is a contradiction).
By Theorem 26 (§5.5), there is a p a r t i c l e T e CSP such that <S,T,Q>
after[Qa ] and f ST
o
f (Sd) TS
=
f SR
f (Sd) , RS
o
whence by Theorem 19 (§4.3), T coincides with R at [f (Sd)].
RS The p a r t i c l e
R is
on t h e
left
side
of
T at
the
R coincides
[Ra ] (and after [Rb]). with
T at
two distinct
Theorem 6 (§2.9).
with
T at
[R ]
[R b]
so by
some event
before
and it is on the right side of T at the event Theorem 17 ( § 4 . 2 ) ,
event
But now we have shown that R~ coincides events,
which
is
a contradiction
by
We conclude that R crosses S at [S ]. ~
122
O
[]
§6.3] §6.3
C o l l i n e a r i t y of Three Particles.
Properties
of
Collinear Sub-SPRAYs.
In absolute geometry any three lines are coplanar if they intersect in any three distinct points.
We now prove the
analogous:
THEOREM 30
(Second C o l l i n e a r i t y Theorem)
Any three particles which coincide events are (permanently)
(in pairs) at three distinct
collinear.
This theorem is a consequence of Theorems 19 (§4.3),
26 (§5.5),
is used in the proofs
PROOF.
(see Fig.
27 (§6.1),
5 (§2.9),
28 (§6.1) and 29 (§6.2).
of Corollaries
i and 2 of Theorem
31)
Let Q,R,S be distinct particles with instants Qa, Qb E Q;
Rb, R c e R; Sa, S c ~ S such that
Sa
Take
any instant
S
Qa < Qb
y
e S
~
with
[~,~,#]
Rb < Rc
S
y
< S
a
.
after [sy]
123
Sc
We
.
will show that
It
32 (§6.4).
0~
L~
cJ~ crJ
§6.3]
S
Let
d~f f-* o f-1(Sy),
x take a n y t w o
RS instants
SR S ,S 0
S
let
d~f f-1 o f-1(S )and Y
QS
SQ
Y
e S such that 2
So < $2 < min{Sx'Sy}" By T h e o r e m T ,T 2
28
(§6.1),
£ T and there 3
there
is an i n s t a n t
Q 3
=T
2
Again
by T h e o r e m
U ,U
~ U and there I
--Q
2
28
3
(§6.1),
instants
~
a
there
is a p a r t i c l e
is an i n s t a n t
T
U with
instants
e T such that 1
~
S
-- U 0
By T h e o r e m
T with
e Q such that
~
S
0
is a p a r t i c l e
27
< U 0
-- T 1
< T 1
-- S 2
. 2
(§6.1),
csp <S, T>
osp<{,{> osp<9,{> D csp. We d e f i n e which
the r i g h t
contains
side of S in
U after
[S
].
csp<S,U>
By T h e o r e m
to be the 26
(§5.5)
side there
are
0
particles
(2)
e CSP<S,T> with
V on the
and
W e CSP<S,U>
left
side
of S a f t e r ~
with
W o n the
left
side
of S a f t e r
[S ] [S ] 0
125
and
2
such
that
§6.3]
(3)
f
o f
SV
(s ) = f
VS
Y
By Theorem
Y
" at the event
26 (§5.5) and Theorem
27 (§6 i)
" csp<S,U>
(4)
(s ) -- sy
WS
V and W coineide
Thus by Theorem 19 (§4.3),
[f ( S ) ] . VS ~
o f
SW
~
osp<[,W>.
Let W
def f-1(S ), let W def f (Sa) ; then by Theorem 26 (§5.5), SW a s WS there is a particle Q* ¢ CSP with Q* on the right side of W after
If (Sy)] such that
VS (5)
f
o
f
(w
)
=
w
. 5
WQ* By repeated
application
(T,U),
crossing of the pairs and
Q*W of the previous
(T,S),
(T,Q),
(S,Q),
(V,W), . . . .(W,Q*), we see that after coincidence
the first particle second particle. and Q* coincide
(of each pair)
9" and
(R,S)
(of each pair),
is on the left side of the the particles
at [Sa] ; that is
]
and Q* is on the right side of S after ~ of
(R,Q),
So by (5) and Theorem 19 (§4.3),
Q,Q* ~ SPR[S
itions
theorem to the
Sy,
and
f
SQ
Theorem
of
=f
QS
5 (§2.9),
of
SQ*
126
[S a ].
Q*S
By (3), the defin-
§6.3]
Since both Q and Q* are on the right side of S after [Sa] , Theorem 19 (§4.3) implies that Q* = Q.
Hence
~
{[Qz]: Qz
(6)
> Sy, Qz ~ Q}~ ~
By a similar p r o c e d u r e paragraphs),
(7)
(based on the third and subsequent
we can show that
{[Rz]:
Thus from
> Sx, R z e R} ~ ~
Rz
(6) and
was arbitrary;
COROLLARY i.
at the event
csp<S,U>. ~ ~
(7)
[Q,R,S] ~f#er ButS
csp<S,U>.~ ~
hence
[S X]
[Q,R,S],
which completes the proof.
Let Q,R,S be distinct particles
which
coincide
[Qc ].
If after
[Qc ], then
.
This corollary is a consequence of Theorems 6 (§2.9), 17 (54.2),
27 (§6.1)~
4 (§2.7),
28 (§6.1), and 29 (§6.2).
It
is used in the proof of Theorem 31 (§6.3). PROOF.
The previous m e t h o d of proof applies here, but with the
first set of order and coincidence relations being r e p l a c e d by
So
Qc = Qc
= Rc = Rc
= Sc
after [Qc];
127
and
[]
§6.3]
the s e c o n d relations required
statement
(i) of the p r e v i o u s
<@,{,R>
or (iii)
17
(§4.2)
proof.
or
changes
are
6 (§2.9)
either
<S,Q,R> before [Qc ],
(ii)
before [Qc].
is true by s h o w i n g
last t h r e e of
No f u r t h e r
and T h e o r e m
before [Qc ],
to j u s t i f y the
[Q,R,S].
to show that
By T h e o r e m (i)
is r e q u i r e d
We w i l l
that e a c h of cases
show that case
(i) and
(iii)
(ii) lead to
contradictions.
before
Case (i)
~
(see Fig.
QI s Q w i t h
is a p a r t i c l e
Qc"
Qi <
T and i n s t a n t s
(see Fig.
Q
1
-- T
< T 1
= R
it f o l l o w s
Theorem
17
(§4.2)
E T and R 3
3
from Theorem there
0
[Q,R,T]
that
4 (§2.7)
is an i n s t a n t
T coincides
w i t h S at [T ].
~
~
T
that
and since
[Q,R,S,T].
s T with T ~
By T h e o r e m
By
< T 1
< T 2
27 (§6.i)
2
o~p<{,{>
=
o~p<{,<>
~
o~p<{,{>,
o~p<{,{>
~
o~p<{,{>
~
osp<{,6>,
o~p<{,{>
=
o~p<~,{>
~
o~p<{,{>.
Thus by T h e o r e m
29
and
(§6.2),
{ crosses
@ at [Qc] in
csp
crosses
@ at [Qo] in
°sp<@,{>,
{ crosses
{ at [Qe] in
asp;
which
e R 3
°
2
such t h a t
~
(§8.1)
= Q .
< R
3
f r o m the above t h e o r e m
28
32)
1
It f o l l o w s
By T h e o r e m
T ,T
~
s u c h that
32)
C
Take any i n s t a n t there
[Q ]
~
is a c o n t r a d i c t i o n .
128
and whence
<@,R,S>
before [Qo ]
3
§6.3]
T
Q
R
S
R3 T3
R
S
Q Fig.
Case (ii)
32
<S,Q,R> before [Qc ].
Similarly, by i n t e r c h a n g i n g the symbols Q and S w h e r e v e r they occur
in the
above paragraph, we deduce a contradiction.
(Note that Fig.
32 does not apply to this case, even with Q
and S interchanged).
The only remaining p o s s i b i l i t y is case
before [Qc]. COROLLARY
2.
If
(iii); that is
O
S ~ ~R then .
(This is a stronger
statement than the d e f i n i t i o n of §5.3).
This corollary is a consequence
of the previous
corollary.
It is used in the proofs of T h e o r e m 31 (§6.3) and Corollary 2 of T h e o r e m 32 (§6.4).
129
§6.3] PROOF.
If the particles Q,R,S coincide at some event [Qc],
then the previous
corollary applied to the d e f i n i t i o n of
§5.3 shows that <@,R,S>
(both before and after [Qc]).
D
The previous two corollaries give rise to the following theorem, which is a stronger version of Theorem more general statement of part
26 (§5.5); a
(v) of Theorem 26 (§5.5) appears
as a special case of T h e o r e m 33 (§6.4) and is not stated here.
T H E O R E M 31
(Properties of Collinear sub-SPRAYs)
Let Q,R and S,T be pairs of distinct particles coinciding at the event [Qc].
(i)
Then
CSP is a collinear set of particles, CSP -- {S:
<S,Q,R>,
, or ; s
(ii)
S,T
s
CSP
(iii)
S,{
s
CSP =
(iv)
CSP is a simply ordered set.
~
and
e
S~R[Qc]}
,
CSP,
CSP<S,T>
=
~T'~S C
CSP, and
This theorem is a consequence of T h e o r e m 26 (§5.5) and Corollaries
i and 2 of Theorem 30 (§6.3).
proof of T h e o r e m 32 (§6.4).
130
It is used in the
§6.3]
PROOF.
Part
(i) is a consequence of the p r e v i o u s two
corollaries applied to Theorem 26 (§5.5). Part
(ii) and part (iii) are the same p r o p o s i t i o n s as (ii)
and (iii) of T h e o r e m 26 (§5.5). In the case of part right of" after
(iv) we define an order r e l a t i o n "to the
[Qc],
which is specified by d e f i n i n g the
[Qc ] in [Qc ], the
particle S to be on the right side of Q after
csp.
Before the event of coincidence
particles have the same ordering,
but the ordering does not
correspond to a r e l a t i o n of "to the right of", but rather to a r e l a t i o n of "to the left of" 33 ( § 6 . 4 ) ) .
(see the corollary to T h e o r e m
[]
131
§6.4] §6.4
Properties of Collinear Sets of Particles
We now prove theorems w h i c h are analogues of the following p r o p o s i t i o n s which occur in absolute geometry: (i)
given any two distinct points in a given plane, there is a line in the plane containing both points (Theorem 33, §6.4),
(it)
a line and a point not on the line determine a plane (Corollary 2 of T h e o r e m 33,
(iii) any line which intersects
§6.4) and
a plane in two distinct points
is contained in the plane
(Theorem 34, §6.4).
For any pair of distinct particles Q,S which coincide at some event
[Qc], we define
COL[Q,S] d~f that is,
{~: [R,Q,S], R e ~
};
COL[Q,S] is the set of particles which are collinear
with Q and S.
We also define
col[Q,S] ~
d~f {[R ]: R
~
X
e R,R ~ COL[Q,S]}. X
.
.
.
.
We note as a result of the previous t h e o r e m that:
CSP
COL[Q,S]
~
SPR[Q c] C
csp = col[Q,S]
~
{[Rx]:
=
COL[Q,S] and
Rx>Qc,Rx e spr[Qc]};
(observe that any col and any asp are sets of events, whereas any spr is a set of instants).
132
§6.4]
THEOREM
32
(EXISTENCE OF COLLINEAR
SETS OF PARTICLES)
~ ~ Let Q,S be distinct particles with instants Qc e Q, Sc
such that Qc ~- Sc" particles.
e
~
S
Then COL[Q,S] is a collinear set of
That is, for any particles
Q,S,T,U with Q and S
as above, [Q,S,T] and [Q,S,U] ~
This theorem is a consequence IX (§2.11), 27 (§6.1),
and Theorems 28 (§6.1),
4 (§2.7),
previous
I (§2.2) and
9 (§3.2),
14 (§3.6), It is implicitly
theorems.
By the A x i o m of Incidence theorem,
of Axioms
29 (§6.2) and 31 (§6.3).
used in many of the following PROOF.
[Q,S,T,U]
(Axiom IX,
there are particles
not coincide with Q and S at [Qc];
§2.11)
in COL[Q,S]
and the
which do
so we must show that,
each instant Qz e Q, there are two distinct optical each containing
COL[Q,S]
\
Qz and one instant
{Q}, in accordance
lines,
from each particle
with the definition
for
of
preceding
this theorem.
Case i.
Qz # Qc
By Theorem 31 (§6.3), T e 9S C
CSP
there is a particle
such that .
(52.7) there are two distinct Qz and one instant
optical
from each particle
133
By Theorem lines,
4
each containing
of COL[Q,S] ~
{Q}.
u~
~.~
~ ~
~ ~~
\
I
~'~
II
III
I
/
II
\ ~ ~ ~
\\\\ \
09
§6.43
Case
3.
Qz = Q C
By T h e o r e m
28
(see
(§6.1)
if
Fig.
33)
we c o n s i d e r
an
instant
Q
e Q with 1
QI < Ca" S
there
is a p a r t i c l e
~T w i t h
instants
= S
S
~
TI,T2
E T~
and
e S such that 2
=
~I and t h e r e
T
l
is a p a r t i c l e
<
T
2
U with
2
<
=
°
Qc;
Uz, U~e U~ and
instants
Q3 e Q~
such t h a t
S
= T 2
By T h e o r e m
27
= U 2
< U 2
= Q 3
< Q 3
~- S d
. °
(§6.1),
osp C
osp<{d>
C
o~p c
asp<{,{> C
osp
and
Now by T h e o r e m csp
such
T is on the
29
left
if we let
Ud d~f f (Qc)" UQ
(1)
if we
define
the
sides
of Q in
that: of Q a f t e r
U is on the r i g h t Thus,
(§6.2),
osp
of Q a f t e r
[Q
]
[Q3 ]
in in
asp,
then
csp. ~
Tb d~f f-i(Qc)" Td d~f f (Qc) and Ub d~f f-i(Qc)" QT TQ QU we have
ITb, Qc, Ud> and
IUb,Qc, Td>.
135
§6.4] For each particle
Vb d~f f-1(Qc) Qv
and V d
r
e
COL[Q,S] ~ CSP,
d~f f (Qe) ; so y b < Yd" vQ
we let Since n e i t h e r
T nor V coincides with Q at the event [Q ], each of the four ~
instants Tb,Td,Vb,V d appears in one
(but not two) of the Now Vb < Qc < Td
optical lines d e s c r i b e d in the above Case i. and T b < Qc < Vd
and since signal functions are one-to-one
(by the Signal A x i o m
(Axiom I, §2.2)), the only p o s s i b l e
combinations of signal relations between the four instants
Tb, Td, Vb, V d are: (i)
T b q V b and T d a V d whence
ITb, Vb,Qc > and
IQc, Td,~d > ,
(ii)
V b q T b and T d o V d whence
IVb,Tb,Qc>
and
IQc, Td, Vd> ,
ITb, Vb,Qc > and
IQc, Vd, Td > ,
(iii) T b ~ Vb and V d o T d whence (iv)
V b o T b and V d o T d whence
IVb,Tb,Qc > and IQc, Vd, Td > ,
(v)
T b ~ V d and Vb ~ T d whence
ITb,Qc, Vi> and
By (I) and Theorem 4 (§2.7), the r e l a t i o n s
IVb, Qc, Td> .
(i)-(v) imply
Vb,V d belong (one-to-one) to the u n i q u e l y
that the instants
d e t e r m i n e d optical lines containing Tb, Qc, U d and Ub, Qc, T d. Now [ was any particle in COL[Q,S]
\ CSP,
so
there are two distinct optical lines, each containing Qc and one instant from each p a r t i c l e of COL[Q,S]
\ CSP.
By Theorem 14 (§3.6), there are two distinct optical lines, each containing qo and one instant from each particle of
COL[Q,S] \ {Q};
the proof is now complete.
The relations
(i)-(iv)
can not occur, as is easily shown
by a continuity argument involving Theorem 9 (§3.2),
136
so the
§6.4] remaining relations
(v) apply; that is
IT b, Qc" Vd>
If Q,R,S are three distinct
COROLLARY i.
coincide
IV b, Qc" Td>"
and
at three distinct
events,
COLEQ, R]
=
col[Q,~]
= col[Q,S]
COL[Q,S]
[]
particles
which
then =
COL[R,S]
= col[R,S]
and
.
This corollary is a consequence of the above theorem and Theorem 30 (§6.3). (§6.4)
It is used in the proof of T h e o r e m 33
and Corollary 2 of T h e o r e m 33 (§6.4). For any T s OOL[Q,R]
PROOF.
[Q,R,T].
it follows by d e f i n i t i o n that
By Theorem 30 (§6.3) we know that [Q,R,S]
above theorem implies that [Q,R,#,T].
s COLE{,#] COL[Q,R]
~
and so the
That is,
a~d T E COLER,#],
COL[Q,S]
and COL[Q,R]
~
COL[R,S].
By cyclic interchange we obtain the other containment relations, from which the c o n c l u s i o n follows.
COROLLARY
coincide
2.
[]
Let Q,R,S be three distinct
at some event.
If e{,R,#>
after
particles [Qc ], then
coLEQ,R]
= COLEQ,S]
: COL[R,S]
ool[Q,R]
= col[Q,S]
= coI[R,S]
137
which
and
.
§6.4]
This corollary is a consequence of the above theorem and Corollary i of Theorem 30 (§6.3).
It is used in the proofs
of the next corollary and Theorem 33 (§6.4).
PROOF.
The method of proof is the same as for the previous
corollary,
but with Corollary i to Theorem 30 (§6.3) taking
the place of Theorem 30.
COROLLARY
[]
3.
COL[Q,S] = (U:
U coincides with two distinct events in csp, U ~ ~
This corollary is a consequence
of the above theorem,
the previous corollary and Theorems 19 (§4.3), 30 (§6.3). PROOF.
).
26 (§5.5) and
It is used in the proof of Theorem 34 (§6.4).
By T h e o r e m 30 (§6.3) any particle which coincides with
two events of csp
is either a particle of CSP,
or is collinear with two distinct particles
of
CSP and therefore with Q and S, by the previous corollary.
That is, the right side of the e q u a t i o n
(above)
is contained in the left side.
In order to demonstrate the opposite consider any particle R e COL[Q,S].
containment,
we
For any instant Qx e
with Qx > Qe" there is by Theorem 26 (§5.5) some particle
T ~ CSP
(C
COL[Q,S])
such that
138
§s.4]
f
o
(Qx) =
QT By T h e o r e m
Similarly
19
(§4.3),
(Qx) . Q
R coincides
with
T at
[f+ (Qx )] ~ csp TQ ~ ~
C
by taking
instant
Qy ~ Q with Qc < Qy
another
event of csp;
completes
THEOREM
o
QR
any other
we find that R coincides this
f
Q
with
the proof.
33 (Existence
[]
of Particles
Let 9,3 be distinct particles such that Qc = Sc;
col[Q,S].
in a C o l l i n e a r
with instants
Set)
Qc e Q, S c e S
let the right side of Q in col[Q,S]
be such that S is on the right side of Q after [Qc ], Given any four instants
Qw, Qx, Qy, Q z e Q with Qw < Qy and
Qx < Qz" there is a particle
I f
QR
(see Fig.
REMARK. theorem
o
Q
R ~ OOL[Q,S]
(Qw) = Qx and
[ f
QR
o
Q
such that (Qy) = Qz
34)
The p a r t i c l e so that
the
S is i n c l u d e d
in the statement
sides of Q in col[Q,S]
139
of this
can be specified.
Qx"
§6.4]
Q
R
Q Z
Qy
Qw Qx
Fig.
34
In this illustration
This theorem
Qw > Qz and Qy < Qz "
is a consequence
17 (§4.2),
18 (§4.3),
19 (§4.3),
29 (§6.2),
and Corollaries
PROOF
45 (§7.4)
(see Figs.
Take any instants
(i) By Theorem
26 (§5.5),
34 (§6.4),
6 (§2.9),
28 (§6.1)~
1 and 2 of Theorem
is used in the proof of Theorems 36 (§7.1),
of Theorems
32 (§6.4).
It
35 (§7.1),
and 48 (§7.5).
35 and 38) Q0"Q2
e Q with
Qo < Q 2 < min{Qc, Qw, Qx, Qy,Qz} . 28 (§6.1)
there
are particles
140
T,U and instants S
e S;
§6.4]
TI,T
,T 3 e T and U ,U 2
Qo m
0
~
e U such that I
Uo < U 1 -- T i < T 2 -- Q 2 and Q 2 = T 2 < T 3 -- S 3 < Sc -- Qc"
By Corollary i to the previous theorem,
(2)
COL[{,[]
= COL[{,{]
= COL[@,{]
col[Q,u]
= eol[Q,T]
= col[Q,S].
and
Since S is on the right side of @ after [Qo] that is in co/[{,T], @
T h e o r e m 29 ( 6 . 2 )
at [Qe] in col[Q,[],
implies that S crosses
that is in coil@,{].
(§2.9), S can only cross Q at one event, (§4.2),
By T h e o r e m 6
so by T h e o r e m 17
S is on the left side of @ before
Now the sides of Q in col[Q,S]
in coil@,{],
[Qc] in coil{,{].
are completely specified.
T h e o r e m 26 (§5.5) implies the existence of particles
V s CSP
(3)
and W s CSP
I
f o f QV VQ
1
(Qw) = Qx and
By Theorem 19 (§4.3)
[
f o QW
(Qw) = Qx" Q
V and W coincide at the event ~
Now Q , V , W
such that
~
[f+(Qw)]. VQ
satisfy the conditions of C o r o l l a r y i to the previous
t h e o r e m and Q , U , W
satisfy the conditions of Corollary 2 to the
previous theorem,
so by
(4)
OOL[V,W]
(2),
= OOL[Q,W]
= COL[Q,U]
141
= COL[Q,s].
CO 0~
Oq
hl
CO
Oq
0~
§s.4]
Now let V
Y
d~f f+(Q ) and let V d~f f-(Qz) " VQ Y z VQ
By Theorem 26 (§5.5) there is a particle R ~ CSP such that
o
R
(V
V
)
=
V z,
Y
and by Theorem 18 (§4.3) and the above definitions
it follows
that
I
(s)
f o QR
Since
R
e
CSP,
Q
(Qy) = Qz"
equation (3) and the remarks following
it imply that
(~)
fQR °
and e q u a t i o n
Q
(Qw) = Qx"
(4) implies that
R e COL[Q,S],
(7)
which,
together w i t h
(5) and
(6), completes the proof.
COROLLARY i. Let Q,S be distinct
at the event
[Qc]~
particles
Then S crosses
which
Q at the event
[]
coincide [Qc] in
col[Q,S]. This corollary is used in the proof of Theorems 36 (§7.1),
PROOF.
35 (§7.1)
37 (§7.2) and Corollary 2 of T h e o r e m 58 (§9.3).
See the first p a r a g r a p h of the previous proof.
143
[]
§6.4]
COROLLARY 2.
Given a particle S and an event [Qy] which
does not coincide with S, there is a unique col which contains
[Qy] and all events coincident with S .
This corollary is a consequence of A x i o m X (§2.12) and Corollary i of T h e o r e m 32 (§6.4). of Theorems
PROOF.
60 (§9.4) and
61 (§9.5).
Take an instant S
~ S such that X
Sx
(i)
It is used in the proof
~
< f-l(Qy) QS
By the Axiom of Connectedness
(Axiom X, §2.12), there are
particles T,U with instants Ty, T z e T and Ux, U z s U such that
Sx = U x
(2)
and
Uz = T z
and
By (i) and (2) and the above theorem,
s COL[T,U] [S ]
T y ~ Qy there is a particle
such that R coincides with both events
[Qy] and
Now
X
[Qy] e col[R,S] and S e COL[R,S] Furthermore,
col[R,S]
Corollary i of T h e o r e m 32 (§6.4) implies that
is independent of the instant S x.
144
[]
§6.4]
COROLLARY
3
( C h a r a o t e r i s a t i o n of Optical Lines)
If Sx and Qy are n o n - c o i n c i d e n t S x o Qy , then
there
is a unique
instants
optical
such
that
line c o n t a i n i n g
S x and Qy. This corollary is used in many of the subsequent theorems.
PROOF.
Since S does not coincide with Q at [Qy]
of the previous corollary are satisfied.
, the conditions
D
This corollary is a much stronger result than T h e o r e m 4 (§2.7).
THEOREM
34
(Containment Theorem)
Let Q,S be d i s t i n c t [Qc ] .
particles
which
coincide
at some
event
Then
COL[Q,S]
= {U: U coinoides
with
two distinct
events
of col[Q,s], u E ~ }. REMARK.
Any two distinct instants of a p a r t i c l e must be
temporally ordered,
so there can be no particle coincident
with two unordered events.
This theorem is a consequence of T h e o r e m 6 (§2.9), Corollary
3 to T h e o r e m 32 (§6.4) and T h e o r e m 33 (§6.4).
used implicitly in many of the subsequent theorems.
145
It is
§6.4]
PROOF.
By Corollary 3 to Theorem 32 (§6.4),
COL[Q,S] is
contained in the right side of the above equation. to demonstrate the opposite U having two instants ~
containment,
In order
consider a particle
UI,U 2 ~ U~ with UI < U 2 such that
[u~], [u 2] ~ co~[@,s]. Let
Qw def f+(U1)" let Qx def f-CU1)" let Qy def f+(U2) and let Qu
Qs
Qu
Qz def f-(U2) ; the conditions of the previous t h e o r e m are now Qu satisfied so there is a particle
R ~ COL[Q,S] such that R
coincides with [ at the two distinct events By T h e o r e m 6 (§2.9),
U
~- R
which completes the proof.
e
COL[Q,S], []
146
[U I] and [U 2]
§7.0]
CHAPTER 7
THEORY OF PARALLELS
In previous
sections we have d e m o n s t r a t e d the existence
of collinear sets of particles,
and we have seen that collinear
sets of p a r t i c l e s have some properties
analogous to eoplanar
sets of lines in the theory of absolute geometry, "parallel postulate"
is r e q u i r e d to d i s t i n g u i s h b e t w e e n the
E u c l i d e a n and B o l y a i - L o b a c h e v s k i a n geometries, "parallel postulate" However,
Whereas a
no special
is r e q u i r e d in the present treatment.
until we prove the t h e o r e m which I take the liberty
of naming the "Euclidean" Parallel Theorem
(Theorem 48,J§7.5),
we must consider the p o s s i b i l i t y of there being two different types of parallels.
Having shown that there is only one type of parallel,
it
follows that each particle has a "natural time-scale" which is d e t e r m i n e d to within an arbitrary i n c r e a s i n g linear transformation.
Then it is not difficult to show that m o d i f i e d
signal functions are linear, one-dimensional kinematics
and the ensueing
discussion of
is taken up in the next chapter.
In most of the subsequent proofs, questions of collinearity are trivial due to the results of §6.4. any (maximal)
collinear set of particles,
represent the c o r r e s p o n d i n g
We let COL r e p r e s e n t and we let col
set of instants.
147
Sometimes we
§7.1]
shall
abbreviate
the
statements
of t h e o r e m s
by not m e n t i o n i n g
COL or col e x p l i c i t l y .
§7.1
Divergent
and
Convergent
Q
R
Parallels
Q
T
T
R
/ [U
/ Fig.
Given say t h a t (see
Fig.
Q,T~ s COL
particles
Fig.
and
Q~ A (T,[U ]) _ y
38
[Qy] ~ col, we
an event
Q is a divergent parallel to { t h r o u g h 37)
and we w r i t e
(i)
Q coincides ~
~i)
Q does
(iii)
Q V (T, EUy ])
37
Y
not
for each
with
the
coincide
particle
the
[Uy]
event
(T,[Uy]) if:
Q V
event with
[U y ] ,
T at any event,
and
R e SPR[U ] such that Y
after R coincides ~
Sometimes
with
we m e r e l y
T at
[U
some
say that
] and
event
R
~ Q
before
,
[U ] y
Q~ diverges from T~ t h r o u g h
148
[U ] . y
§7.1]
Similarly, we say that Q is a convergent parallel through the event
A
to
[Uy] (see Fig. 38) and we write
(f, [Uy]) if: [Uy]
(i)
~ coincides with the event
~i)
Q does not coincide with T at any event~
(iii) for each particle R e SPR[Uy]
before
and
such that
[V ] and R # Q ,
R coincides with T at some event after [U ] Sometimes we merely say that Q converges
to T through [U ]
In the r e m a i n d e r of this section we will often use the symbol
II to r e p r e s e n t either
V
or
that the substitution is consistent
A
, where it is implied
in any statement or
proof. We now show that parallelism
is a relation between particles
by proving the following:
T H E O R E M 35
( T r a n s m i s s i b i l i t y of Parallelism)
If Q II (T,[U ]) and Q ~
~
y
C
~ Q ~
J
then Q II (T,[Qc]) ~
~
That is, Q is parallel to T and we write Q V T or Q A T~ , as the case may be; or simply Q II T with the above convention. This t h e o r e m is a consequence Corollary i of T h e o r e m 33 (§6.4). of Theorems
36 (§7.1),
37 (§7.2),
and 42 (§7.3).
149
of T h e o r e m 33 (§6.4) and It is used in the proof 38 (§7.2), 40 (§7.3)
§7.1]
~s
T
Q
R
S
Q
[uy]
u9 ]
Q
S Fig.
PROOF.
T
(a)
RT
Q
39
Fig.
Transmissibility
Suppose the contrary;
SR 40
of Divergent Parallelism
that is, suppose there is a particle R
w h i c h coincides w i t h Q at [Qc ] such that before and R ~ Q . (see Fig.
T
[Qc ]
Take an instant R b a R with R b > Qc if Qc > Uy
39) or with R b < Qc if Qc < U
Y
(see Fig.
40).
By
T h e o r e m 33 (§6.4) there is a particle S which coincides with
150
§7.1]
Q at
[Uy]
and with R at
[Rb].
By Corollary i of T h e o r e m 33
(§6.4) <@,S,R,T>
before min{[Uy],[Rb]}
so S does not coincide with any event of T before
[U9] ~
but
this contradicts the third r e q u i r e m e n t in the d e f i n i t i o n of Q.
(b) Transmissibility of Convergent Parallelism The proof is similar to the proof of (a) with the expressions "before" and "min" changed to "after" and "max", respectively.
If we want to specify that Q diverges from T, or that Q converges to T, or that Q is parallel to T, we use the more concise expressions Q V T, Q A T, and Q II T, respectively. At this stage we have not shown that p a r a l l e l particles exist or that the r e l a t i o n s of p a r a l l e l i s m are symmetric.
For con-
venience we define both relations of p a r a l l e l i s m to be reflexive, itself,
so that each particle
is (trivially) p a r a l l e l to
in both the divergent and the convergent
sense.
We
extend the definitions of p a r a l l e l i s m to apply to observers, that:
Q
so
II ~ ~=> for all R e Q and for all S E T, R II s.
We now extend the d e f i n i t i o n of mid-way and r e f l e c t e d particles (§5.3) so as to apply to particles,
such as p a r a l l e l particles,
which need not coincide at any event.
As a result of Theorem
31 (§6.3) we can extend the definitions of §5.3 to become: if Q,S,U are particles
~
~
~
such that
and
f SQ
o f QS
151
= f SU
o f US
,
[]
§7.1] we say that S is mid-way
between Q and U, and we say that Q is
a reflection of U in S, and that U is a reflection of Q in S. We also define reflected events,
so that if [T x] and [V ] are Y sides of S in col and if there are instants
events on opposite
Sw,S z e S such that IS w] ~ IT x] ~ IS z] and IS w] o IVy] ~ [Sz] , we say that IT x] and [V ] are reflections y and we write
of each other in S, ~
[Tx ] = [Vy]s and [Vy] = [Tx] S.
It follows from
Theorem 33 (§6.4) and Theorem 19 (§4.3), that each event
col has a unique r e f l e c t i o n in each particle (of COL).
(of We
will now demonstrate the existence of p a r a l l e l particles of both types
and their reflections.
T H E O R E M 36
(Existence of Parallels and their Reflections)
Let S~ be a particle in COL, and let IVo] be an event in col. There are particles Q,U e COL such that ~
(i)
~U II (S,[Vo ]) and Q~ II (S,[Wo]
(ii)
~ e ~S and Q e ~S"
and
This t h e o r e m is a consequence of Theorems 6 (§2.9), 17 (§4.2), 19 (§4.3),
21 (§5.1),
24 (§5.3),
Corollary i of 33 (§6.4), and 35 (§7.1). proof of Theorems
37 (§7.2),
39 (§7.3),
33 (§6.4) and
It is used in the 40 (§7.3), 41(§7.3)
and its corollary, T h e o r e m 43 (§7.3) and its corollary, Theorems
46 (§7.5) and 61 (§9.5).
152
9 (§8.2),
and
§7.1]
~(~)
S
T
Fig.
PROOF.
a
R(I+I)R (~)
R (7)
W (7)
41
We define the sides of S so that [V ] is on the right ~
0
side of S in col.
Case
(a)
Divergent Parallels
(i)
We first show that there is a particle
(S~: ~=1,2,''';
U V (S,[V ]).
S~ e S)~ be an u n b o u n d e d decreasing sequence of
instants with S
< V . I
By Theorem 33 (§6.4) for each positive
0
integer ~ there is a particle R (h) which coincides with the events Fig.
Let
[V0 ] and [Sh] , and which is contained in COL
41).
153
(see
§7.1]
(i) u
s
R (la)
[s
R
Z
Y
/ Fig.
By the corollary to Theorem
<S,R(~+I),R(I)> {R(1): ~=1,2,...}
42
33 (§6.4),
after [V ].
for each
By Theorem
I ,
21 (§5.1) the set
has a limit particle
U e CSPeR(1),R(2)>
c
COL such that, for any instant
R (I) e R (IJ with R (I) > V , Z
~
Z
(1)
f o
R(1) v
f
UR(I )
for each positive integer
0
(R(lJ ) = supt z integer
f
o
I ~R(1)R(k) I, .
I, the limit particle
f
(R(1) )).~
and
R(k)R(1) So for each positive
U is to the right of the particle
154
§7.1] R (x)
before
before
IV ] ,
so
U does
not
coincide
with
S at
any
event
[V ]. 0
Next we show that U does not coincide with S at any event after IV ].
Suppose the contrary;
that is, suppose that
0
~U coincides with S~ at some event
[Sy] > [V0 ] (where Sy e S).
By Corollary 1 of (see Fig.
T h e o r e m 33 (§6.4), U crosses S at IS ] Y 42) so by Theorem 19 (§4.3), for any instant
R(1)z ~ R(1)~ with R(1)z { Sy, f R(i) U By equation
o
f (R (I)) > f o f (R (I)) . UR(I ) z R(1) S SR(I ) z
(i), for any p a r t i c u l a r instant R (1) z' there is a
particle R (~) such that
f R(1)B(~)
o
f (R (I)) R(V)R(1) z
By Theorem 19 (§4.3) the events
>
f o f (R(1)). R ( I)S SR ( i) z
[V ] and o
[ f (R (I)) ] R(~)R(1) z
are on opposite sides of S, so by Theorem 17 (§4.2) the particle
R (~) coincides with S at some event between [V ] and 0
[ f (R (I)) ]. R(~)R(~) z
But R (u) also coincides with S at
~
~
[S ] < IV0 ]' so by Theorem 6 (§2.9) R(~)~
= S,~ which is a con-
tradiction.
In order to show that the third condition of the definition (§7.1) is satisfied,
consider any R e SPR[V o] such that <S,U,R>
after [V ] and R ~ U (see Fig. 43); then there is some positive 0
~
155
}7.1]
integer
p
such
that
<S,U,R(P),R> after .
with
S at some
event
.
.
after
~
shown
[V ] , and
.
IS
] and b e f o r e p
]
We have
now
that
])
u
Fig.
By T h e o r e m
{R( x ) "
I=1,2,
Let Q be the then
[Y 0
u v (s, [ v
~S
so R c o i n c i d e s
0
24
"'"
(§5.3)
} such
limit
there that
particle
R (p)
R
43
is a set of p a r t i c l e s
RS(X) is a r e f l e c t i o n of the
sequence
as above,
Q V (s,[v ~
]T) 0
156
.
_(I)
•S
of R (l) in S.
: I=1,2,...};
§7.1]
S
U
R (n)
R
Sy i f
Sx J
Sn S1
I s
/I"
f
I[VO
(ii)
We w i l l
other
in S.
R
(i)
now We
Consider
show
will any
Fig.
44
that
Q and
use
the
instant
symbol S
(see
2)
3)
(the the
(4)
Fig.
reflections
R as
E S with ~
y define
U are
of
each
an a b b r e v i a t i o n S
> V . Y
We
for
now
o
44):
S' def f-1 o f-1 (Sy) S def i i x QS SQ " x - f-us o SU f- (Sy), S' de_f f-1 _ n _(n)_ ~S ~ set
of
sequence
o
instants of
-i (n) SRs f
(S
)
=
Y
d£f -
S
n
f- i R(n) S
o
f-1 ( S ) SR(n ) y
{S : n=1,2,... } s h o u l d n o t be c o n f u s e d (SI: I=1,2,''')
instants
R d e f f - 1 (S ) I
SR
Y
157
of p a r t
(i)
above),
with
~7.1] (5)
d~f
R
f
n
o
f
(n)
R
(6)
By equation
x
def -
f RU
1
RX
=
(R ) . i
UR
s u p ~ R n") ~
Since < S , U , R ( n ) , R > a f t e r .
of
(i) (of the proof of (i) above)
(7)
.
and
(R ),
R(n)R
RR
.
"
[V ]
.
it follows that for instants
0
after V 0 ~
(8)
f
=
SR
(9)
f-I
f
o
SU
=
RS
Thus
f
f-1 US
o
f-1
o f-1 SU
-I
= f-1 US
=
o f
o f(R
(R
UR
f-i
o
R(n) s
SR
),
and
R(n) R
f-1
=
RU
US
US
f
o
SR(n)
Sx = f-1
= f
f
=
UR
RR (n)
),
by
(2)
and
(4),
I
by
(8),
I
o f-1(Rx),by
(6),
RU
f-1(Rx) ,
by
(9),
BS
-I
= f (sup{Rn)), RS n = sup
(R n)
158
by
(7),
, by Theorem
9 (§3.2),
§7.1]
= sup{f-1 o f o f (R)} by (5) RS RR(n ) R(n) R i " ' =
8up~f -If n ~ RS
f
o
f o f-1(Sy)~, ] ) R(n)R SR
RR (n)
= sup~ f-1
o
n ~R(n)s
by (4),
f-1 (Sy)~,) by (8) and (9), SR (n)
= sup{Sn} , by (3). n Now equations
analogous
to equations
(4)-(9)
can be defined
Q,B~n) in place of U,R(n) which leads in the
for the particles same way to (io)
S'x = suP{Sn} = sup{Sn} = Sx" by (3). n n
That is, for any S
> V , Y
o
f-1 o f-1(Sy ) = f-1 o f-1(Sy), US SU QS SQ or equivalently,
for any S x ~ V , 0
f o f (S ) = f o f (S ). SU US x SQ QS x
(ii)
Similarly
if we take any instant
U0 e U~ with U0 < V0"
then as in the proof above there are sets of particles
{w~X):
~=1,2,...} and {W~I): I=1,2,...} which have limit
particles which are parallel parallels
to S (see Fig.
coincide with the events
similar to equation
159
These
[U ] and [U ]S respectively, 0
and satisfy an equation
41). 0
(ii) for all
§7.1]
instants S x e S w i t h S x ~ U . ~
By the previous theorem,
the
0
p a r a l l e l which coincides with [U ] must be U, and by the 0
~
equation analogous to (ii), the other parallel must be Q. That is, for any S
e S with S X
(12)
f o f (S) SU US x
But U
~ U 0 J
X
is arbitrary,
= f o f (Sx). SQ QS
so (12) applies for all S
0
e S; that is, X
Q and U are reflections of each other in S.
~
This completes
the proof for the case of divergent parallels.
Case
(b)
Convergent
Parallels
(i)
We first show that there is a particle T A
(S, [Y ]). ~
{~ : w=1,2,...;
~
instants with ~
E S}~ be an u n b o u n d e d increasing sequence of > V .
i
the sequence represent
Let
X
(The bars are intended to d i s t i n g u i s h
0
(~w) from the previous
ideal instants as in Chapter
for the case of divergent parallels
(S h) and do not
sequence 3).
As in the proof
(see Fig.
41), for each
positive integer ~ there is a particle ~(~) which coincides with the events
[V ] and [~w].
So for all p o s i t i v e integers
O
I and ~, < ~ ( 1 ) , R ( W ) , R ( ~ ) , R ( I J > has a limit particle where
and the set {~(~) : ~ = 1 , 2 , ' ' ' ~
T such that < ~ ( 1 ) , ~ ( ~ ) , T , U , R ( h ) , R
(I)>
U is the dive~gent p a r a l l e l of the previous Case a.
Hence
T can not coincide with S at any event after [V ] and by the above ordering
(with respect to U and R(1)),
with S at any event before ~
[V ].
T can not coincide
In order to show that the
0
third condition of the definition
( § 7.1) is satisfied, we
could use an argument analogous to the c o r r e s p o n d i n g argument
160
§7.1] for d i v e r g e n t
parallels
we have no further
above.
use for the
parallel
U of the above
sent the
convergent
that
is, we define
In the r e m a i n d e r symbol
Case a.
parallel
So we let the symbol
as in the p r e v i o u s
so far been
~
~(~)
case,
(s,[v
there
] ].
~
s
o
T (n)
.
S
u
~(a)
i
\1
Fig.
called
T;
])
such that
Q^
U repre-
U so that
u A (S,[V Similarly,
T or for the divergent
which has
the p a r t i c l e
of this p r o o f
45
161
exists
a particle
Q
§7.1]
(ii)
We will now
other
in S.
is shown proof
show that
The proof
in outline
are similar
Case
a, but here
instant ~
(see Fig.
to those
such that
> f
~
o f
for this
used for the proof meanings.
Consider
is some p o s i t i v e
-
(<)
n
(17)
d~f
~ Y
f
(~x
) •
f o f (~(a)) • ~(a)~(n) ~(n)~(a) l
Therefore
by T h e o r e m
(18) ~(a)
=
By t h e
and
instants
S
of
n
~(a),
e S and Z
9 ( 3.2)
sup{~(a)]. n
choice
integer
d~f f o f ( S ) , SU US x
~(~) d~f f o f (~(~)) Y ~(~)8 8Z(a ) I
Y
any
We now define:
Z(a) s (16) ~ ( W
of
f o f (~x) = ~n d~f f o f (<), SR-(sn) =(n)~S ~ s~(n) R(n)s
d£f
(15) ~ ) i
used
a, and
US
d~f f o f (<), SQ QS
~'n d~f
(14)
of Case
The symbols
different
of each
0
SU (13)
45).
{ V ; then there X
~
to the proof
w h i c h were
they have
~
U are r e f l e c t i o n s
is similar
e S with ~ X
Q and
~
for
U
integer
n
> a and
e U with Z
x
any
< S
~
z
< ~
y
and
162
V
o
< U
z
< Sy,
for
any
§7.1] (19) f
(S ) =
us
z
f
o
U_~(n)
f
o
f
_Snj_K(a )
_6(a)S
(S z ) =
f
o
U_~(n)
f (Sz) , -6(n) S
and (20) f
su So ~
Y
( U z)
=
f
o
s-~(~)
f
o
f
-#(aJ-~(n)
g(n) w
( S z)
=
f
o
S-~(n)
f
-~(n) U
(U). z
= f o f (F) SU US x
=
r
o
s-~(a) =
f
r
o
r
-~(a)w
o
r
u-~(a)
(sup{-R(a)}),
C~x)
-~(a)s
by (15), (17), (18),
= supl<S_~(~ f ) (-~(a) n ) } , by Theorem 9 (§3.2),
] ~ sup~
f
o
tS-ff((~)
o f
f -~(a)-~Cn)
-~(n)-~(a)
o f
(Fx) ~
-~(a) S
by (15) and (16),
S
n)
-#(n)S
= sup{~n} , by (14). The remainder of the argument is similar to the argument for divergent parallels.
[]
163
§7.2] §7.2
The Parallel Relations are Equivalence Relations
We now show that both relations of p a r a l l e l i s m are equivalence relations. particles,
It then follows that in any c o l l i n e a r set of
there are equivalence
classes of p a r a l l e l particles,
of both the divergent and the convergent type.
THEOREM 37
Let
Q,S
e
(Symmetry of Parallelism)
COL.
I f Q rr s then S fJ Q.
This theorem is a consequence of Theorems 19 (§4.3), 23 (§5.3),
Corollary i of Theorem 33 (§6.4) and Theorems
and 36 (§7.1).
It is used in the proof of Theorem
35 (§7.1)
38 (§7.2)
and implicitly in many subsequent theorems. PROOF.
We have already specified that the relation(s)
p a r a l l e l i s m are reflexive (§7.1)),
Case
(a)
of
(in the remarks following T h e o r e m 35
so we consider distinct particles Q and S.
Divergent
Parallels
(See Fig.
46)
We suppose that S ~ Q; that is, we suppose that for some S o e S, there is a particle
(i)
U such that
S V (Q,[S ]) and U % S.
By T h e o r e m 23 (§5.3) there is a particle and U
and a r e f l e c t i o n mapping
¢(T)
~- T,
t(s)
T m i d - w a y between S
¢ such that
--u,
164
t(u)
~- s.
OQ
~ ~a>
~i:u>
e~
/
I
\
I
\
\
-e-
/
\
\
I
I
~>
-e-
~-
l.~b>
LIJ
§7.2] Now
<S,T,U,Q>
before [S ] so
(2)
Q v
By the p r e v i o u s reflection
Q v
S,
theorem,there
Q v U.
is an o b s e r v e r
~(Q)
which
is a
of Q in T, and
Oc~) v }.
(3) Since
0 is a r e f l e c t i o n
f
o
TO(Q) and
and
T,
since
f
= f
O(Q)T
U v Q
mapping
(by
f
o
TO(U)
19
(§4.3).
f
= f
O(U)T
O(Q) and O(U)
Let Qb d~f f (Sc) ; then
and
QT
(i)),
[So] , by T h e o r e m
before
o f
TQ
in T,
o f •
TS
UT
(= S) do not cross
Hence
before IS ].
by the
above
ordering
and
(3),
Qs (4)
0rQ) v rk,[~rQb)])._ We n o w
namely,
that
show
that
there
S V ¢(Q).
is a p a r t i c l e
If we
suppose
R such
that
V (9(Q),[S ]; and then,
as above,
for a n y
T
g T with X
f o f (Tx) TR RT f TS
o
<
R
T X
~ S,
< Sc,
f o f (T) Tt(Q) O(Q)T x
f (Tx) < f o f (Tx) , ST TR RT
166
the
and
contrary;
§7.2]
whence
f o f (T) TQ QT x
f TR
f T#(Q)
o
f (T) ¢(Q)T x
o f (Tx) = f RT T¢(R)
o
f (T x) ¢(R)T
f o f (T) TS ST x
=
= f o f (Tx) , since U e ¢(S), TU UT ~
SO
which contradicts false.
[S c]
before
#
~,
(i) and shows that one of the suppositions
If the first supposition
further to prove;
and~(~)
if the second
is false there is nothing supposition
is false, we have
shown that
Consider a reflection mapping
e(s)
By the preceding
theorem and
(6)
e o cr~)v
e such that
~- s
(4),
(k,[e
We now show that S v 8 o ~(QJ.
o CrQb)]).
If we suppose
the contrary,
namely that there is a particle R' such that ~
R' V
and R'~ ~ S,
(8 o ~(Q),[Sc])~
then, by an argument
similar to that of the preceding
167
is
§7.2] paragraph,
for any instant f S@°t(Q)
>
o
f
so S~ ~
f (S ) = f eot(Q)s x st(Q)
o
SR'
Sx ~ S~ and ,ex < Sc"
f
(S
R'S
t(Q);~ but this
(7)
)
=
f
x
contradicts
d~f f - St(Q)
where
T
e d
By (I) and follows
and
~
T
=
S
=
f Tt(Q)
after
19 (§4.3)
0 and
f o f (Sc) , SOot(Q ) Oo~(Q)S
o
f (To) ~(Q)T
[Sc],
and since
.
.
.
C
it follows
are on the same side of IS c] in col. and therefore
I[So],[Qb],e Since both Q and 8 o t(Q)
U e t(S)~ it
[S ].
after
.
t are reflections,
S a' < Ta < Sa,
J
that
.
Since
=
C
(2), < U , T , S , Q >
(8)
]).
.
d
from Theorem
,
and
dkf _ f o f ( T C) TQ QT T
)
x
C
f (S) ~(Q)S a
S' de=f f o f (S ) a SQ QS c
Ta
(S
(5), whence
~
o
f
@(R')S
~ v (0 o ¢ ( ~ ) , [ s
Let S a
f (S ) t(Q)S x
o
SO(R')
~
o
By (i), S # T,
by Theorem
o t[Qb]> diverge
168
and
that Q and
0 o t(Q)
so
19 (§4.3), e
o t(~)_ # Q.
from S, Theorem
35 (§7.1)
§7.2]
implies that there is no event at which Q and e o ¢(Q) can coincide. Thus by (8)
and
Corollary 1 of
<¢(Q),~',~',U,Q,e
o
T h e o r e m 33 (§6.4),
¢(~)> before
[ S ].
This is a c o n t r a d i c t i o n of (7) since by (i), S ~ U. conclude that the supposition
We
(i) was false, which completes
the proof for the case of divergent parallels.
Case b. Convergent P a r a l l e l s . A similar proof can be based on a figure which is a r e f l e c t i o n of Fig.
46 in a "horizontal" line.
THEOREM
38
Let
[]
(Transitivity of Parallelism) If Q II R and R II S, then Q El S.
Q,R,S~~ a COL.
~
~
~
~
This t h e o r e m is a consequence of Theorems 35 (§7.1) and 37 (§7.2).
17 (§4.2),
It is used implicitly in many
subsequent theorems.
PROOF.
This proof is analogous to the proof of the c o r r e s p o n d i n g
theorem
of absolute geometry.
Case
(a)
Divergent
The result
Parallels
is trivial unless Q ~ R ~ S ~ Q which is assumed from ~
now on.
We define
~
the right side of Q to be the side which ~
contains R.
Now Q ~ S, so Theorem
35 ([7.1) implies that there
~
is no event at which Q and S can coincide.
169
§7.2]
Case (a)(i)
(see Fig.
47)
Take any instant Qc a Q, and any particle T ¢ SPR[Q o]
[Qc ].
such that { is on the right side of Q before @ V ~, T coincides with
COL
Since
(and crosses) R at some event [R b]
where R b ¢ R and [R b] < [Qc ].
Similarly,
is on the right side of R before some event [S a ] where S a
~
since R V { and {
[Rb] , T coincides with S at
a S and [S a ] < [Rb].
Since Q~ and S~
coincide at no event, we conclude that Q v S.
T
Q
R
(
S
T
Q
S
[R~R;)]
]
F~ a ]
[s a ]
Fig.
R
47
Fig.
170
48
§7.2]
Case (a)(ii)
(See Fig.
48)
Take any instant Qc e Q and any particle T e SPR[Qc] such that T is on the right side of Q before
[Qc].
n
COL
Since
V R, the p a r t i c l e T coincides w i t h R at some event [R a] where
R a e R~ and IRa ] < [Qc ]"
By T h e o r e m 17 (§4.2) the particle
coincides with S at some event
[R ] < IS b] < [Qc].
T
[S b] where S b g S and
Since Q and S coincide at no event we
a
~
conclude that Q V S.
Case
(a)(iii)
<S,@,R>
The previous t h e o r e m implies that S V R and R v Q so interchanging the symbols that S V Q.
"Q" and "S" in case
(a)
A g a i n by the previous theorem,
(it) Q V S.
, we find This
completes the proof for divergent parallels.
Case (b)
Convergent Parallels
A similar proof applies w i t h the word
"before" and the symbols
< and V r e p l a c e d by "after" and > and A,
171
respectively.
D
§7.3]
§7.3
Coordinates
on a C o l l i n e a r
By c o n s i d e r i n g we can attach
Before consider
any e q u i v a l e n c e
"coordinates"
discussing
those
Set
of p a r a l l e l
of p a r a l l e l s
of col.
particles
which
a dyadic
we first
can be "indexed"
number
numbers
the form
n/2 m, where n is any integer and m is any non-
THEOREM
(Existence
Let Q,U be distinct
theorem
36 (§7.1). and
47
PROOF. Theorem
of M i d - W a y
particles
S which is mid-way
This
is a number
of
integer).
39
particle
that
in COL,
by dyadic
negative
(recall
of p a r a l l e l s
to the events
classes
subclasses
class
with Q II U.
between
is a c o n s e q u e n c e
It is used
Parallel)
There
a
Q and U and parallel
of Theorems
in the proof
exists
23
of Theorems
(§5.3) 44
to both. and
(§7.4)
(§7.5).
The proof 23 (§5.3)
of Theorems
is e s s e n t i a l l y except
21 (§5.1),
Theorem
22
(§5.1);
Theorem
5 (§2.9).
that
and
the
Theorem
22(§5.1)
and T h e o r e m
36
[]
172
same as the proof 36
(§7.1)
takes
and C o r o l l a r y
(§7.1)
takes
of
the place
2 to
the place
of
§7.3]
COROLLARY.
There
Let S°,S I be distinct
is a collinear
class
particles
of parallel
with S o [[ S I
particles
{sd: d is a dyadic number} such that,
for any integers
I
f o f sdms dn sdnsdm
and for any dyadic
=
m and n,
f sdmsd(m+1)
numbers
and
the
proof
corollary
Corollary
of T h e o r e m
PROOF.
to T h e o r e m 41
between
<sa, sb, sc>
is a c o n s e q u e n c e
l)sdm
I
.
of T h e o r e m
24 (~5.3).
19
It is u s e d
(§4.3) in the
(§7.3).
By the a b o v e
S ½ mid-way
sd(m
a,b,c
a < b < c ~
This
o
theorem
there
S o and S I and
is a p a r t i c l e
by i n d u c t i o n
there
is a
particle
(1)
S 2-(m+I)
We d e f i n e
the
right
In the r e m a i n d e r
mid-way
side
of this
between
of S O to be the proof
p d e f
we
I
f
shall
o
sas b
where
the
As in the
superscripts Corollary
S o and S 2
a and
f sbs a
-m
side w h i c h use the
24
173
S I.
notation:
1
b are not n e c e s s a r i l y
to T h e o r e m
contains
(§5.3),
for
each
numbers. integer
p the
§7.3]
set of particles
{sn/2P:
n=O, +_I, +_2, • .. }
has the property:
p<m/2P;n/2P>
(2)
= pn-m<m/2P; (m+l)/2P>
.
Now by (I),
p2
= p
,
= p
.
and so by induction,
o~q Therefore,
for any integer n,
p n.2 q <0;1/2 (p+q) > = p n < o ; 1 / 2 P >
,
and by (2),
p
= p
.
Now by Theorem 19 (§4.3), for all integers n and for any non-negative integers p and q,
s n . 2 q / 2 (p+q)
~ sn/2p
.
that is, the set of particles
{sn'2q/2(P+q):
q=0,1,2,...}
174
§7.3]
is an e q u i v a l e n c e class of p e r m a n e n t l y Equation proved;
coincident particles.
(2) is e q u i v a l e n t to the e q u a t i o n w h i c h was to be the ordering p r o p e r t y is trivial.
[]
A subclass of (convergent or divergent)
parallels
indexed by dyadic numbers will be called a dyadic class of
parallels.
We can define a dyadic
class of instants of the
particle S o , by taking any p a r t i c u l a r instant of S o and giving it the index S~, and then letting
sodef[ 2p -
for each dyadic number
f o f SOS p sPs ° p.
]*
o
(So) •
If it is clear from the context
that we are referring to a p a r t i c u l a r class of parallels and instants we shall simply call them dyadic parallels
dyadic instants,
and
respectively.
A further consequence of the p r e c e d i n g c, .-ollary is that the dyadic subscripts
(of the subset of dyadic instants of
the particle S o ) are ordered in accordance with the ordering of the instants they represent;
that is, for any dyadic
numbers a and b,
a
SO<
a
175
Sbo "
§7.3]
THEOREM
Any dyadic
40.
is a countable
class of instants
of any particle
dense subset of (the set of instants
of) the
particle. This 15
theorem
is a c o n s e q u e n c e
(§3.7),
19
(§4.3),
to T h e o r e m
39
(§7.3).
41
(§7.3)
PROOF. dyadic
and
46
has
(§7.1),
It
(§7.1)
and
12
(§3.5),
the
Corollary
in the p r o o f
of T h e o r e m s
with
subset
(§7.5).
instants,
sequence
36
is u s e d
Let S ° be a g i v e n
Tn : This
35
of T h e o r e m s
and
let
particle T e 30 .
~ sO(2_i/2 n ) :
Tn
is b o u n d e d
and
a given
Define
a sequence
n=O,l, • ..; T
strictly
n
of
of i n s t a n t s
T)
increasing
and t h e r e f o r e
a supremum
TO __ S o def sup~S o : n=0,1,2,..} co ~ ~ (2_1/2 n)
(i)
We w i l l
first
show t h a t
S ° = S °. 2
Let SI-I def_ f-1 (S~) and let S 11 def_ f CS°o) S°S i SIS o By the p r e v i o u s
corollary, 2n
1
(2)
$IS(1-I/2 n) By T h e o r e m divergent
36
(§7.1)
as the
S(1-1/2n)$1 there
case m a y
is a p a r a l l e l be)
such
176
that
Q (convergent
or
§7.3]
(3)
and
f o f {s~) = s o s°Q Qs ° sOe = sup~SOn [ 2-I/2 n ~ $20}, it f o l l o w s
since
positive
integers
So by e q u a t i o n all p o s i t i v e
that
for all
n,
(2) a n d
integers
Theorem
19
(§4.3),
it f o l l o w s
that
for
n,
S Q
o
f QS 1
s Q
Qs I
($1_1) $I "
whence
and
so by T h e o r e m
the
event
12
(§3.5)
the p a r t i c l e s
SO
(4)
The r e m a i n i n g a theorem
instant
at
Q -- S 1 and so by (3),
Therefore
instants
Q and S 1 coincide
part
of W a l k e r
0 = S2 •
of t h i s
[1948,
proof
Theorem
is b a s e d
13.1,
on the p r o o f
P330].
Given
any two
S x" O S Oz e sO w i t h S xO < S zO " we w i l l f i n d a d y a d i c S O such that S O < S O < S O . y
x
y
z
177
Theorem
15
(§3.7)
of
and
§7.3]
Theorem
36 (§7.1)
or convergent
imply that there is a parallel
U (divergent
as the case may be) to the right of S O such
that
SO <
(5)
-I (S 0) < S O ,
u
X
where
u d~f
By Theorem by equation
Z
Z
f o f sou US 0 35 (§7.1)
U can not coincide with S0 at [S~] and
(i) there is some positive
integer m such that
S0 > u -I (S~) (2_2 -m+2 )
(6)
•
We define the set of particles
{R(n):
R (n) --S 2-n, . n=1,2,3, . . .
}
and so
-I 0 r n ($2) =
(7)
SO (2_2 -n+z)
where
r
d~f n
By (6) a n d
f sOR(n)
o
SO
>
(2_2-n+2)
f R(n)sO
(7),
rml (S ~) > u-I(S02 ) and since parallels
can not cross
r
-1
m
>
u
-1
178
(by Theorem
35 (§7.i))
§7.3]
Thus from (S), (8)
S O < r - l (s O) x m z
Since R
(m)
< SO z
does not coincide with
increasing and decreasing
SO
at any event,
the
sequences
Irn(S02): n=O, 1,2j'' I and Irmn(S02): n=0,1, 3,..I are unbounded,
S0
so (the set of instants of)
is covered by
the set of semi-closed intervals
{[rnm'~O' n+1(S02)]:
"" "I
Therefore there is some integer p such that
rP(sO) < sO<'z rP+I(s02 ) m
(9)
Now from (8) and
sO
(9),
r-lm (SO "< rml o rP+I (s02 = rp (S02) < SOz
and by definition,
r~(S~)
is a dyadic instant.
[]
The set of instants of each particle is orderisomorphic to the set of real numbers.
COROLLARY.
This corollary is a consequence of Theorems ii (~3.4) and 25 (§5.4).
It is used implicitly in many of the following
theorems.
179
§7.3]
PROOF.
By T h e o r e m
ii
(§3.4)
last instants.
By the above
countable
subset
dense
set of instants §3.1). these
theorem
of instants.
of each particle
Sierpinski conditions
particles
[1965,
~,
ordered
set to be o r d e r - i s o m o r p h i c
THEOREM
41
(Indexed
Class
has
By T h e o r e m
(§5.4)
Theorem
and
first or
each p a r t i c l e
has no gaps
§i0,
are n e c e s s a r y
do not have
25
(defined
i] has
sufficient
a the
in
shown that
for a linearly-
to the reals.
[]
of Parallels)
A class of parallels and the instants belonging
to them can
be indexed by the real numbers such that, for any real numbers a s bj c Sc
(i)
cb
a
a-b+c
cb
a
a+b-c "
whence
a-2b+2c " where we have i n t r o d u c e d the (see Fig.
for any real numbers
S b and an instant S b c S b. this way,
f d~f f ab sas b
49).
Furthermore, ~
notation:
a
~
a and b, there is a parallel
A class of parallels,
is called an indexed class of parallels.
indexed in The subscript
indices of any p a r a l l e l are said to constitute a divergent or convergent parallels
time scale,
according as to whether the class of
is a divergent or convergent class.
180
§7.3]
Sb
S C
Sb
a-2b+2c
I
Sa_b+c
J f
f J f f J J
Sb a
d
Sa+b-c
Fig.
49
This t h e o r e m is a consequence of Theorems 18 (§4.3),
19 (§4.3),
9 (§3.2),
36 (§7.1), the C o r o l l a r y to Theorem
39 (§7.3) and Theorem 40 (§7.3). Theorems 45 (§7.4), 46 (§7.5),
It is used in the proof of
47 (§7.5),
48 (~7.5),
49 (§7.5)
and the Corollary to Theorem 56 (§8.4). PROOF.
We have already indexed the dyadic instants of the
particle S O by letting
(1)
SO 2y
d~f
o
y
f yO
181
~7.3] We will first show that,
for any dyadic numbers a and 6,
o
(2)
(S a) =
2a+2~
"
o
Let d be a dyadic number
(with d ~ 1) such that there are
integers m and n for which
a = dm Thus, by the d e f i n i t i o n
o
B
(S a)
=
0
~ = dn
and
.
(i) and the C o r o l l a r y to T h e o r e m 39, (§7.3),
f
o
0 dn
f
o
f
dn 0
= Ifd ° fdO)n °
o
f
0 dm
(S
)
dm 0
Ifd ° fdolm(s~ )
= [fd o fdolm+n(s~ )
=
I
f
f
o
0 d(m+n)
d(m+n)
0
1
(S
)
0
=
S2d(m+n )
=
SO 2a+26
which establishes e q u a t i o n
"
(2).
We now extend this result to apply to all instants of all parallels.
By the previous theorem,
any instant of S O specifies
a (Dedekind) cut in the dense subset of dyadic instants of S O .
182
§7.3]
We
index
each
also
by the M o n o t o n i c
each
real
for e a c h
of S O by the
instant
number real
there
number
(3)
Sequence
real
number
Theorem
(Theorem
is a c o r r e s p o n d i n g
a and for any
~ < a < ~
dyadic
9,
§3.1),
for
instant
of sO;
numbers
a and ~,
S O < S O < S~
~>
so o b t a i n e d ;
thus
.
a
Furthermore, set,
and
by T h e o r e m
parallels
there
as before, in the
any class
by the r e a l 19
(§4.3),
is some
subset number
(§4.3),
of d y a d i c
any
parallel.
specifies
two
a (Dedekind)
also
real
the
same
way
cut
(see
§3.1)
We i n d e x
each
parallel
by T h e o r e m
number
ordered
distinct
In m u c h
parallels.
so o b t a i n e d ; for each
is a p e r m a n e n t l y
between
dyadic
any p a r a l l e l
dense
Theorem
19
of p a r a l l e l s
b there
36
(§7.1)
and
is a p a r a l l e l
S b such that
o
thus,
as above,
numbers
(4)
for
(S ) = S2b
b
0
each
real
g < b < g <-->
by e q u a t i o n s
for all
b,
and for
all d y a d i c
B and-~,
o S
Thus,
number
;
dyadic
(2) and
numbers
$2~+2 B <
<
o
0
<
b (3),
for any
e and ~ such
O
that
(S~a)
183
o
0
0 given
real
number
~ < a < [ ,
< $2-~+26 •
a and
§7.3]
whence
o
Similarly,
by the above
b and for all dyadic
= $2a+2 B
~0 equation
numbers
2a+26 < b
and
(4), for any real number
6 and p such that
o f ] (S a ) < bO)
6 < b < ~ ,
2a+26 "
whence
(S)
2a+2b We can now index
for all real numbers
the instants
a
Now with the equation
instants
of the parallels 18
so
(7)
a÷c
by T h e o r e m
=
cb
(8)
(6) and
18
(§4.3)
indexed
imply
in this way,
that
f (s)C -
Oc (§4.3)
and
o
cO
)
a-c
cO
(5) and T h e o r e m
Furthermore,
fcb
0
=
fcO
o
(7),
f + ( S b) cb
=
by defining,
a and c,
sC def f+ (S O
(6)
so by
of each p a r t i c l e
Sc a - b + c and
f-(Sba) = S c a+b-c cb
184
"
§7.3]
whence,
again by Theorem 18 (§4.3),
(9)
a-2b+2c
This, together with equations COROLLARY.
Let Q~ ~ COL a n d
of p a r a l l e l
to Q~ t h r o u g h
distinct parallel
This corollary 36 (§7.1). PROOF.
the proof.
[U a ] E col.
[U a ] coincide,
is a consequence
I f the two types
then
each event
[]
there
is only
one
of col.
of Theorems
19 (§4.3) and
It is used in the proof of Theorem 46 (§7.5).
(see Fig.
Let {R~: a real,
50). R ~ E COL}
classes of divergent Q ~ R 0 ~ S 0.
real numbers,
a n d {SB:
B real,
S fl ~ COL} be
and convergent parallels
Then through
convergent parallels
functions
(8), completes
let
to Q t h r o u g h
"
[U a ] there are divergent
R k and S l, respectively,
such that R k = S1;
and applying
such that
Theorem
where
so by considering
and k and 1 are record
19 (§4.3), we find that,
integer n, R nk ~_ S nl
185
for each
§7.3]
Q = Ro ~ So
Sb
Ra
R mk
~ S ml
Ro Z
11/Jf/~ iiiJTl
Ro X
Fig.
Now
take
50
any
In
this
event
ROx d ~ f
f+
diagram,
0 < z-x
[Vy] e aol
and
(Vy)
R Oz d ~ f
and
[V
such and
y
] is
that we
to m
take
left
of
Q, ~
< z-x
< O;
if
[V
integer
m
f-
(Vy)
ROv
the
an
.
let
ROv If
< m
z < x and
we
take
an
integer
m
] is t o t h e r i g h t o f Q, Z > x Y s u c h t h a t 0 < z-x < m (see Fig. 50).
186
§7.3]
By Theorem 36 (§7.1),
there are parallels Ra, s b such that
Ra V (RO,[v
(I)
~
])
Sb A
and
y
~
(sO, Iv ])~" ~
y
Since the relations of p a r a l l e l i s m are equivalence relations,
(2)
R a V R 0 and R 0 V R mk ~ sbA
Now
R a V R mk and
S O and sO A S ml and R mk ~ S ml ~
S b A R mk
(I) and (2) imply that
~
after
~
[V ] and y
and hence R a ~ S b.
~
~
after
[V ]
~
y
That is, there is only one particle which
is p a r a l l e l to Q, in both the divergent and convergent senses, ~
which coincides with the event
[V ]. Y
[]
This corollary and its proof are analogous to a p r o p o s i t i o n of absolute geometry.
The corollary is used in the proof of
the "Euclidean" Parallel Theorem
( T h e o r e m 46 (§7.5))
The next t h e o r e m shows that for any given particle,
all
time scales of the same type are d e t e r m i n e d to within an arbitrary strictly increasing
linear transformation.
axiomatic system of Szekeres,
this p r o p e r t y was regarded as an
axiom and was called the "Axiom of Standard Time" 1988, A x i o m A.8, P142]. is a consequence (Axiom VII,
In the
[Szekeres,
In the present system, the p r o p e r t y
of the A x i o m of Isotropy of Sprays
§2.9).
187
§7.3]
THEOREM 42
(Affine Invariance of Divergent and Convergent
Time Scales) Let
COL
let
{Sa:
(I) and COL ~ real,
be c l a s s e s (i)
(2) be c o l l i n e a r
S a ~ COL
of p a r a l l e l s
If sa ~ ub,
then
(I)} and
sets
{US:
of the same
of p a r t i c l e s ,
& real,
U ~ s COL
type i n d e x e d
there are real c o n s t a n t s
and (2)}
by the reals.
c , d , k such
that
for all real x,
S ac+kx (ii) Also,
if COL
(1) = COL
(2),
safky c+kx the u p p e r or lower sign b e i n g the
left side
of S a is the
~ U~+ x then for all real x a n d y,
~ ub+Y d+x " chosen
left,
according
or right,
side
This theorem is a consequence of A x i o m VII Theorems
9 (§3.2),
24 (§5.3) and 35 (§7.1).
the proof of Theorems 47 (§7.5),
PROOF.
(i)
as
to w h e t h e r
of U b.
(§2.9) and
It is used in
48 (§7.5) and 49 (§7.5).
If S a = Ub, we consider any instant S a s sa;
b
then there is some instant U 8 e (i)
{b
such that
Sa b a -_ UB
For any p a r a l l e l sC such that {c ~ that there is a p a r a l l e l
sas c
sCs a
sa,
T h e o r e m 36 (§7.1) shows
U d ~ U b such that
ubu d
188
udu b
§7.3] In §3.6 we saw that signal functions could specify mappings between the sets of events coincident with particles, well as between the p a r t i c l e s themselves.
as
With this interpre-
tation,
(2)
f
o
sas c
f
=
sCs a
f
o
f
Ub U d
udu b
the proof being analogous to the proof of T h e o r e m 36 (part (ii)) (§7.1) with r e f l e c t i o n mappings being r e p l a c e d by the more general isotropy mappings.
By (i) and
(2) we see that,
for any integer n,
Ub
By considering parallels m i d - w a y between S a and S c, and mid~
way between U b and U d, it follows that
I
f
o
sas(a+c)/2
)2 I
f
=
f
o
ubu(b+d)/2
s(a+c)/2sa
f
]2
u(b+d)/2U b
Since these record functions are continuous functions of a real variable,
the intermediate value t h e o r e m
example,
[1961, Theorem 3.3a]) applies
Fulks
instants S a ~ S a and U b ~ Ub such that x ~ y ~ fs~]
=
[ub] Y
189
(see, for and so there are
§7.3] and
f o f [S a ] = f o f [Ub ] . sas(a+o)/2 s(a+c)/2sa x ubu(b+c)/2 u(b+c)/2ub y Then,
as with
(2) above, for any integer n it follows that
[sa+2n(c-a)/21 Proceding
= IU~+2n(b-d)/21
by i n d u c t i o n we see that,
for all p o s i t i v e integers
m and for all integers n,
Isa+2n (c-a )/2m] = [ub6+2n(d_b)/2ml Thus,
Ub
letting
k def (c-a)/(d-b)
and noting that the indices of
Sa
are mapped onto the indices of
by some strictly m o n o t o n i c
increasing function we see that, for all real x, 8b
(ii) left 19
side (§4.3)
If of
COL (1) = COL ( 2 ) , S a is the
implies
that,
left, for
or all
then
according
right, real
side
of
as Ub ,
to
whether
Theorem
y,
sa+ky ~_ uD+Y and so by (4) and the relations given by the previous theorem it follows that,
for all real x and y,
rSa+-k [ub+Y l ~ a+kYx] = L 6+x]
190
the
§7.4]
§7.4
Isomorphisms o f a
Collinear Set of Particles
We will now apply a r e f l e c t i o n operation to any collinear set of particles.
By composing these r e f l e c t i o n operations we
can obtain mappings w h i c h are like space translations and pseudo-rotations.
Finally by composing four of these mappings,
we can generate time t r a n s l a t i o n mappings.
We first show that any its members
COL
can be reflected in any of
and that this a u t o m o r p h i s m preserves both relations
of parallelism.
THEOREM 43
Let
COL be a c o l l i n e a r
distinct mid-way
particles between
(i)
there
(ii)
R
(iii)
R A
For
(Invarianee of Parallelism)
V
are
set
R and
R and
of p a r t i c l e s
V,
together
V such
instants
that
containing
with
a particle
the T
either:
R c ~ R~ and
V c ~ ~V such
that R c ~ Vc,
V , or
V .
any p a r t i c l e
S e COL such
that
S ~ R,
there
is an o b s e r v e r
~
~T C
COL a n d ~T
II
T"
This t h e o r e m is a consequence of Theorems 19 (§4.3) and 36 (§7.1).
18 (§4.3),
It is used in the proof of T h e o r e m
44 (§7.4).
191
or
~rJ ha.
H"
~ _ _~_ . _/ _ _' _ ,- - -?- - T . , ~ ',#
_
~
~"
4=L-a
§7.4]
PROOF.
There are six eases to be considered,
could be either:
since R and V
(i) members of a SPRAY, or (ii) divergent
parallels,
or (iii) convergent parallels;
be either:
(A) c o n v e r g e n t p a r a l l e l s or (B) d i v e r g e n t parallels.
Case
R and V members of a SPRAY
(i).
and S and R could
We define the sides of T such that R is on the right side of T after [R ]. c Case
(i)(A).
Convergent Parallels
Firstly we consider those convergent parallels
on the left side of
R, since these parallels n e c e s s a r i l y cross T at some event and this simplifies the proof. Case
(i)(A)(a).
S on the left of R (see Fig. 51). ~
There is some instant T 1 ~ T~ with T 1 > Rc with T at IT1] ; that is, S A
U A
there is a particle
(V,[TI]).
definition of p a r a l l e l i s m
Since S A (R,[TI])
(2)
f
TS Similarly, inequality;
(3)
o f
ST
(T
(R,[T1]).
such that S~ coincides
By Theorem 36 (§7.1)
By Theorem 19 (§4.3) and the
(§7.1) it follows that
it follows that for all instants T x s T,
X
)
~
f
o
TU T
U A (V,[TI]) , ~
f
(T
UTT
X
) = f
TU
UT
(T
X
)
from which we obtain the opposite
therefore
f TS
o f
o
f ST
=
f TU
193
o
f UT
§7.4]
That is,
and by d e f i n i t i o n S A R and U A V, which completes the proof of Case Case
(i)(A)(a).
(i)(A)(b).
S on the right side of B (see Fig.
52).
By Theorem 36 (§7.1), there is a particle Q which is a r e f l e c t i o n of S in R, so then Q is on the left Case
Take any instant R 2 e R
(i)(A)(a).
completely a r b i t r a r y even though in Fig.
R2
>
Rc).
of R as in
(note that R 2 is 52 it appears that
We now define:
(5)
-
R3
S
o
f
(R2)
SR
(S)
T2 d~f
f+(R2) TR
(7)
T3 d~f
f+ (R3) • TR
(8)
T4 d~f
f-(R2) TR
(9)
T5 d~f
f- (R3) TR
=
f
RS
o
SR
(R2)
,
•
and
In the r e m a i n d e r of this proof we shall r e p e a t e d l y use the results of Theorem 18 (§4.3). then
(7) and
By equations
(6) and
(9), we have
o
(i0)
R (ii)
T o
R
(T 2) = f- o (T 2) = T 4 and TR RT (T3)
=
T 5
.
T
194
(8), and
§7.4]
Also (12) Q
o f QT
1
(2 s) = f- o f+(T s) TQ QT =
f- o f- o f+ o f+(T3) TR RQ QR RT f-o TR
Q
of
f-o
{[{
TR
o QR
S
f- o
RT
o f I*}-I o f + ( T 3 ) ~ s i n c e SR RT
o f
TR
S
Q e ~R"
, by equation (7),
SR
f (R 2) , by equation (5) , TR T 4 , by equation (8).
Also
o f S
ST
1 (T2) =
f- o f+ (T 2) TS ST f- o f- o f+ o f+(Tv) TR RS SR RT f- o TR
S
o f SR
f o TR
S
o f SR
o
(T2) RT
(R 2) , by equation
~R(R3) • by equation (5)~ T 5 , by equation
195
(9),
(6),
§7.4]
= [{R o { IT
: [{~ o
(T3) " bY equati°n
(ii)'
{~]~o {I{~ o~ ]I-~~ ,
by equation
*
(12),
(T2)
T
by equation
(i0), and since T 2 was arbitrary,
(13)
By Theorem 36 (§7.1) there are particles U e kT By the previous procedure,
We~V
and
"
case U A V and so W A V .
but with f-,f+,{(f
U and W such that
Then by a similar
o f),}-I taking the place of
f+,f-,{(f o f)*}
respectively,
it follows
that
But [ ~ ~T and V ~ $~T • so e q u a t i o n s
(13)
That is (lS)
= ~T
and
~ = ~T '
196
and ( 1 4 )
imply that
§7.4]
and by definition S A R and W A V , w h i c h completes the proof of Case Case
(i)(A)(b),
(i)(B).
and hence the proof of Case
(i)(A).
Divergent Parallels
First we consider those d i v e r g e n t p a r a l l e l s on the right side of R, since these parallels n e c e s s a r i l y
cross T at some event.
The proof is analogous to the proof of Case symbols w,U,Q,S
(i)(A) with the
being r e p l a c e d by the symbols U,W~S,Q
respectively, with the inequalities in the c o r r e s p o n d i n g version of e q u a t i o n
(i) being reversed,
(13) and (14) being w r i t t e n so that S
and with equations o f ST
and
o W
T
are d i s p l a y e d explicitly. Case
(ii).
B V V .
We define the sides of T such that R is on the right side of T . If there is only one class of parallels parallels
(that is, if divergent
are also convergent parallels)
then all particles
under c o n s i d e r a t i o n are members of the same equivalence of p a r a l l e l s and there is nothing further to prove. are two classes of parallels,
divergent p a r a l l e l i s m is transitive
parallels
If there
then a proof is only required
for the case of convergent parallels,
parallels is trivial.
class
since the relation of
and so the case of divergent
The proof for the case of convergent
is similar to the proof of case
(i)(A), the only
differences being that both references to R
197
are omitted.
§7.4]
Case
(iii).
The proof except
R ^
V
for this
case
that the words
interchanged,
and all
is similar
"divergent"
to the proof and
for case
"convergent"
signal
functions
Q
QR
should
(ii)
should be
be r e p l a c e d
as
follows:
QR
QR
and so on. case
(iii)
A simpler is similar
reversed".
COROLLARY.
RQJ
way to imagine to case
R
this
(ii) with
Q
is to consider "the d i r e c t i o n
that of time
[]
Let COL be a collinear set of particles
containing
T and let (Qa: a real, Qa ~ COL} be a class of parallel particles.
Then there is a class of parallel observers. {~(~e): ~ real,
where
~ denotes a reflection
~(~a) C
in T.
corollary
in the proof
PROOF.
Take any instant
of T h e o r e m
T
44
e T. X
is a p a r t i c l e
COL.)
is a c o n s e q u e n c e
is used
,
(Also for each particle
S~ e COL, there is an observer _ST C This
COL}
of Theorem
36
(§7.1)
and
(§7.4).
By T h e o r e m
36
(§7.i)
there
~
R ~ COL such that R coincides w i t h T~ at [T x]
and R II QO and so, for all real
~, R II Qe.
Now by the above
~
theorem,
for any two p a r t i c l e s
Qa and Qb there are observers ~
198
§7.4]
~(~a)
~(~b)
and
and s o
such that
~(~a) II ~c~b)
either
if we compose coincide detail
THEOREM
compositions
spacelike
in two m u t u a l l y
44
parallel
following
if we compose
particles;
reflections
(Mapping
in two distinct These
mappings
section
§7.5.
of an Indexed
{WS:
~ real,
set of events.
such
Class
(i)
V coincides
(it)
V V Qa(= wd), or
(iii)
V A Qa(=
and let col be the Qa ~ COL} and
of divergent
for some Qa and W d ,
such that either
with Qa(~ W d) at some event,
W d) ,
Then there is a mapping ¢: col + col such that ~(Qa)
~_ ~(W d) = V
and the indexed classes of parallels
199
in more
of Parallels)
Qa = W d
Let V e COL be a particle
which
are d i s c u s s e d
Let {Q~: ~ real,
that,
lead
reflections
particles
W ~ e COL) be indexed classes
convergent parallels
which
or to " p s e u d o - r o t a t i o n s "
Let COL be a collinear set of particles, corresponding
~c~)
~(~b)II
of r e f l e c t i o n s
translations
at some event.
in the
and
u
We now consider to:
~(~a)II ~c~)
are mapped
or
and
§7.4]
into indexed classes of parallels of the same type. the indexed class of divergent parallels
That is,
{Qa: ~ real, Qa e COL}
is mapped onto an indexed class of divergent parallels {US: ~ real, U s ~ COL} and, furthermore,
A similar result applies to the indexed class of convergent parallels
{WE: ~ real, W E e COL}.
A m a p p i n g c o r r e s p o n d i n g to case
(i) is called a "pseudo-
rotation", while mappings c o r r e s p o n d i n g to cases are called spacelike
(ii) and
translations.
This theorem is a consequence of Theorems
23 (§5.3),
39 (§7.3) and 43 (§7.4) t o g e t h e r with its Corollary. t h e o r e m is used in the proof of Theorems
PROOF
(See Fig.
By Theorems
(iii)
The
45 (§7.4) and 48 (§7.5).
53)
23 (§5.3) and 39 (§7.3), there is a particle T mid-
way between Qa(= W d) and V.
As in the previous corollary, we
use the symbol ~ to denote a r e f l e c t i o n mapping in T, so that for any particle S e COL,
(I)
of
S
=
ST
f
T~(S)
o
f
~(S)T
(In this and all subsequent equations,
=
o
T~(S)
f-
~(S)T
record functions
represent mappings between observers as e x p l a i n e d in §3.6).
200
II
u....l
°°
t~
c~
Co
M
~-h 0
po
f~
H~ 0
0
c, ~3
¢
c~ ~3
,°
('D
0
0
M
c~
0 r+
O'q
H1
0
~. Oq
..----.
/
\
/
/
/
\
q'
\
/
\ \
7"
'%
~
§7.4]
which
clearly
has
so f r o m
n o w on we
instant
T
e T,
the p r o p e r t y
shall
let
T
X
and
by this for
use
for a n y S s COL,
the
be d e f i n e d
symbol
~.
by e q u a t i o n
Given
any
(i) so that
Z
[Tz] Then
only
that
= f- o f + [ T x ] TS ST
equation
~ny T
and
=
f+ T~(S)
(2) we
see
o
f- [T ] . ~(S)T x
that,
for any
S s COL,
e T , Z
(3)
~ : f-[fz]
--
ST since
T
f+ ~(S)T
and S w e r e
It can
arbitrary.
be
seen
from
(2) and
(3)
X
that ~ is a b i j e e t i o n its own = ~-1
inverse). and
(4)
f- = f+ TS T~(S) (2),
that
So i n v e r t i n g
interchanging
By e q u a t i o n s ^
and
(2) and
S and ~(S)
o ~
(3) a n d
~ is i d e m p o t e n t
and
(3),
(that
noting
is,
~ is
that
, we h a v e
f+ = fTS T~(S)
(4) we see t h a t
o
for a n y
observers
^
R, S ~
COL,
(5) f+ = f+ o f+ = ~ o fRs RT TS ~(R)T and
(6)
o
fT~(S)
o ~ = ~ o
similarly
f- = ~ o f+ o RS ~ (R) ~ (S)
202
f~(R)~(S)
o
§7.4]
Letting R and S be ~o and ~b respectively,
(7)
f+ = ~ o f@bQC @(Qb)@(@c)
o ~
and
we have
f- = ~ o f+ QbQC @(@b)@(@c)
o
Right and left signal functions have been i n t e r c h a n g e d because is a r e f l e c t i o n mapping. functions
In order that right and left signal
should map onto right and left signal functions,
respectively,
we shall compose two r e f l e c t i o n mappings.
we define a r e f l e c t i o n m a p p i n g
~(~a)
Thus
8, w h i c h is a r e f l e c t i o n in
(= V) so that for any particle S e COL and for any
instant V
s V , X
~
col -~ aol
(8) 8:
S}(@ a)
8 (S) ~(@a)
S~(Qa)
which is analogous to the definition
(2).
Consequently,
similar arguments, we find that
s~(@ a)
8 (s) ~(@a)
SgcQa)
and
(i0) f+ = 8 o
RS
f-
o 8
and
f- = 8 o
e(R)e(s)
Rs
f+ o 8 e(R)e(s)
If we now define (ii)
and combine equations
¢ d~f 8 o (7) and
(i0), we obtain
203
by
§7.4]
(12)
f+ = ~-1 o f+ QbQC ~(Qb)~(QC)
since ~-1 =
(8o4)-1
that {~(Qa)}
= ~-1 o
= 4-1 o 8 -1 = ~ o 0 .
f~(Qb)~(Qc)
o ¢ ,
We have already noted
of the same type as {~e}
{~(Q~)} is a class of parallels
of the same type
For each real a we take a particle
U a £ ~(Q ), which
is possible
by the Axiom of Choice,and
this set of particles
such that,
[uX]
(13)
Accordingly,
equations
[U~+ b c] -
^a
index the instants
of
¢[Qx]
=
Y
(12) imply that
f+ [U c]
=
,
for all real x and y,
Y
(14)
f-
QbQc
is a class of parallels
and similarly as {Q~}
o ~ and
ubuc x
and
[U b ] x-b+c
=
f- [U c] ubuC x "
and hence (15)
Ub U c Equations
(13),
(14), and
U c Ub
(15) show that with the indexing
specified by (13), the set {U s} is an indexed and by (2) and
(8), ~c~
A similar result gent parallels
= ~
= ~.
can be obtained
{W ~} by replacing
with ~c and ~b ~ respectively, similar lines.
class of parallels,
for the class of conver-
~c and ~b of equations
and then proceeding
[]
204
along
(7)
§7.4]
THEOREM
Let
45
(Time Translation)
COL be a c o l l i n e a r
{Qa:
~ real,
classes
set of p a r t i c l e s
Q~ ~ COL} and
of d i v e r g e n t
{R~:
and
~ real,
and c o n v e r g e n t
let
R B ~ COL} be i n d e x e d
parallels,
respectively,
with Q O -_ R 0 0 QO Let Qa E ~ with
0 0 QO ~- RO "
and
0 0 0 QO < Qa < Q1 " T:
There
col-~
is a b i j e c t i o n
col
[~km ] [Qy] ~-~ ~a+ky
[R~] ~ where
k,~,b
Furthermore,
are p o s i t i v e
[Rb+~z]
real numbers.
for any p o s i t i v e
integer
,
In p a r t i c u l a r
n,
QO = R0 a ( 1 + k + . . + k n) b ( l + ~ + . . + ~ n) The m a p p i n g
This
T is called
theorem
33 (§6.4), of Theorems
translation
is a consequence
41 (§7.3) 46
a time
(§7.5)
and 44 and 48
of Theorems
(57.4). (§7.5).
205
mapping.
It is used
18
(§4.3),
in the proof
§7.4]
PROOF
(see
Fig.
54)
o Qo o Since QO < a < Q1 " it f o l l o w s that
Qil < Q1 a+l so by T h e o r e m
33
(§6.4)
f
(i)
(2)
there
f
o
QI T
We w i l l
parallels)
from
(I) a n d
Qa-1 I =
and
so by T h e o r e m
an i n d e x e d we
Qa-1
COL
such
that
and
class
choose
of p a r a l l e l s
divergent
1 def f+ 1 = (Qa+l) SIQ 1
and
$1
;
(2),
( 7.3)
and
QI1 =
and
Theorem
f- (S~) , QISJ 18
(4.3)
f+ (S~I) = QO2 q~S I
and
f+ (S~) = qa0 QOsI
f - (SII)- = Qa0 QOs1
and
fQ2SI
206
(S~)
{S a}
or c o n v e r g e n t
S1 s T ,
f- (S!I) QISl 41
=
T e
(Qa+1) =
S1 def f+ (Q~ -1 = 1) SIQ 1 whence
1
Q1 "
is a p a r t i c l e
(Qll)
whether
such that
<
TQ
now define
(it is i m m a t e r i a l
1
Qa-1
TQ I
o Q 2
and
= QO2 ;
and
"M
/
"k
/ ~
/
~
~b
/
\
T
w
"i¢'~
.'-
/
Q~
i....1
§7.4]
these four relations imply that, if we define particles
sO,s 2 s {S a} in accordance with Theorem 41 (§7.3), then S O coincides with Q0 at [QO] (= [$0]) and S 2 coincides with Q2 at
[Q2O]
(: [S~])
.
By the previous theorem, there are mappings
~1,~2,~3,~4: col ~ col which send classes of parallels onto classes of parallels of the same type such that
(~2: ~2_~ ~2
and
[Q~]-~ [S~] = [Q~]
~3:~2 ~0
and
[S~]-~ [S~]
$4:~0 + ~0
and
[S O ] -~ [QO] = [S~]
,
and
We now define the time translation mapping
d~f ¢~
o ~s o ¢2 o ~I
and so by the above relations, T: k O
kO
and
[Q~]-~ [QO]
Also, by the previous theorem,
However we can not assert that T sends each parallel onto itself; we can only assert that T sends the right side of QO
208
§7.4] onto itself (and the left side of QO onto itself). Accordingly,
there are real positive numbers
k and Z such
that
so by Theorem
Since Qx0 =
42
(ii)
(§7.3),
f+ (Q~) and R 0 = f+ (R~) , fo fo QOQx/2 Qx/ZQo Y ~oRy/2 Ry/~po
it follows from the previous theorem that there is a positive real number b such that T: [Q~] ~
[Q~+kx ] , [R~] ~ [R~+~y] .
By Theorem 41 (§7.3) and the previous theorem,
T: [Qx+a ~ ] = f+ [Q~] ~ [Qa+kx+k~] k~ f+ [Q~+kx ] = Qk~QO QaQO and similar considerations
apply to convergent parallels,
T: [Q~] ~ [_k~ ~a+kx ] " [R y~ 3 ~
so
~ ] . [Rb+~y
Since QO0 ~ RO0 " it follows that for any positive integer n,
n O :
~[Q~ [Qo ]
=
[R
] ~
]
(1+k+..+k n)
209
[?0 =
b(1+£+..+~n)
] .
[]
§7.5]
COROLLARY.
The
time
T
translation
-I
aol~
T has
an
inverse
CO1
~-a/k + y/k ]
[R~] ~
-b/~ + z/z ]
This corollary is used in the proof of the following theorem (Theorem 46 (§7.5)). PROOF.
The mapping
-I
is clearly a bijection and it is
easily verified that the composition of T and - 1 , order,
§7.5
is the identity mapping.
Linearity of Modified
in either
D
Signal Functions
The culmination of the present theory of parallelism is in the next theorem where we show that there is only one type of parallel.
The axiom which implies the uniqueness
parallelism is the Signal Axiom (Axiom I, §2.2).
of
It turns
out that the only space-time satisfying our axioms is the Minkowski
space-time, which shares the property of uniqueness
of parallelism with the Euclidean geometry. which is the other absolute geometry, property of uniqueness of parallelism;
Hyperbolic geometry,
does not have the likewise the de Sitter
space-time, which does not satisfy the Signal Axiom (Axiom I, §2.2) but which does satisfy appropriately modified versions of
210
~7.5]
the remaining axioms, also does not have the property of uniqueness of parallelism.
The property of uniqueness space and time translation
of parallelism allows the
mappings of the previous
section
to be expressed relative to any indexed class of parallels (Theorems 47 and 48).
The final theorem of this section
involves the linearity of modified signal functions, which is the stage at which the geometrical discussion
ceases temporarily
and the kinematics begins. THEOREM 46
("Euclidean" Parallel Theorem)
Let COL be a collinear set of particles c o r r e s p o n d i n g set of events.
and let col be the
Let Q,S s COL and let [W ] e col. ~
X
Then S V (Q,[W ]) ~
That is,
given any particle
one p a r a l l e l
S ^ (Q,[W ])
and any event,
to the given particle
there is exactly
through the given event
This theorem is a consequence of Theorems 12 (§3.5), 19 (§4.3), its corollary, Theorems
6(§2.9),
38 (§7.1), 40 (§7.3), 41 (§7.3) and
and 45 (§7.4).
It is used in the proof of
47 (§7.5), 49 (§7.5) and 60 (§9.4).
211
.
[7.5]
PROOF.
{Qa} a n d {R ~}
Let
be c l a s s e s
gent parallels
as in the p r e v i o u s
a n d Qoo = R o°
Since
previous
theorem)
following (i)
k = 1, £ < I,
k < 1 and
it is s u f f i c i e n t diverging
the r e a l p o s i t i v e
a r e as y e t
(i')
k > 1 and
(iii')
theorem
unknown,
with
numbers we
shall
and
Q = Q0 k,~
conver= R0
(of the
consider
the
cases:
£ = I,
(ii')
of d i v e r g e n t
(ii)
k < I and
~ > I,
(iii)
k > I and
£ > I,
£ < I . to s h o w
f r o m Q and one
By the that
corollary
there
converging
are
to T h e o r e m
two parallels,
to Q, w h i c h
41
([7.3)
one
are p e r m a n e n t l y
coincident.
By the p r e v i o u s
theorem,
o k # 1 =~ T[Qa/(l_k)]
(i)
# i ~ That
is,
T [ R ~ / ( I _ ~ )] = [R~/(~_~)]
the e v e n t ( s )
[ 0
Qa/(l_k)]
are f i x e d w i t h 0
o ] = [Qa/(l-k) " and
respect
[R~/(1-2) ]
and
to the
time
translation
~.
Since
0
Qa = Rb " a n d
QOQa/2 Qa/~QO it
follows
coincide
by at
Theorem°19
the
RORb/~ Rb/iRO
a ([4.3)
that
event
212
the
p a r t i c l e s Qa/2
= Rb " and
Rb/2
§7.5]
Also by
(3)
T h e o r e m 41 (§7.3)
[
f
QOQka/2
o
f]*(QOa) = Qa+ka 0 Qka/2QO
and
f o f (Rb) = ROR~b/2 R£b/2R 0 b+~b We shall now discuss each ease separately.
Case
(i)
~ = 1
By the previous theorem,
[QO+ka] =
T[QO] =
TIRe] = [R~b]
If k = I , T h e o r e m 19 (§4.3) and equations
(3) imply that the
particles Qa/2 and Rb~2 coincide at the event [ + (QO)] . Qaf2QO But these two particles also coincide at the event
[ ~/+ (Q00)] , Qa 2QO
and so by T h e o r e m 6 (§2.9)
Qa/2 -_ Rb/2 If k # I , then since
after
213
§7.s]
Qo Ro Rb/~ Qka/~ Qa/~
111
Fig.
it follows
that
k < 1 (see Fig.
is a fixed
event.
Applying
55
55).
Since
0
k # 1, [Qa/(1_k)]
the time t r a n s l a t i o n
mapping
T
n times,
n [Q~] Thus
[Rb]. 0
[~c1_~+1)/cl
for every p o s i t i v e
integer
n
R0 ~ QO b(n+l) a(l_kn+l)/(l_k) But by T h e o r e m {R 6} coincide
12
(§3.5)
this would
at some event
before
diction.
214
Ro
, 0 < Qa/(1-k)
imply
that all members
of
0 [@a/(l_k)], which is a contra-
§7.5]
Case (i')
k = 1
The p r o o f
is s i m i l a r
Case (ii)
k < I and
Applying
the
time
to the p r o o f
of case
(i).
~ > 1
translation
mapping
T
n times
QO Thus,
for
Ro
all i n t e g e r s
n
,
R0 < 0 b(l+£+. • .+~n) Qa/(1-k) which
as in case
is a c o n t r a d i c t i o n
since
the
"
sequence
(1+~+. • •+£n) Case (ii') The p r o o f
k > I and is s i m i l a r
Case (iii) By
(1)
will
first
Theorem
36
to
k > 1 and
there
are
show
that
events
these
there
I
f o f QOQc QcQO
equation
the
f
o
(ii)
above.
o [Qa/(l_k)] and
events
are
is a p a r t i c l e
1 [Qa/(l_k)]
theorem,
onto
QOQkc
case
~ > I
fixed
(§7.1)
By the p r e v i o u s
~ < I
the
[R
identical. Qo
such
f
o
translation
]
215
By
= [R~/(l_~)]
time
[Qa/(l-k)
; we
that
T maps
equation
QkcQO
/(1_£)]
=
o ] [Rb/(1-g)
this
(i),
§7.5]
and hence but
by Theorem
k ~ 1 , whence
19
(§4.3)
of b o t h class
classes
QO = ~
r
The
time
expressed {S a}
and
of p a r a l l e l s
and
also,
class TO
"
{S~:
cs~)
translation more {T B}
transformation by d e f i n i n g
a real,
of p a r a l l e l s
r
kc = c ,
(1_~)]
{T~:
o ~ To=
mapping
simply
with
on the
and
S real,
an i n d e x e d
0 0 " TO = R b / ( 1 - ~ )
r
o
to the
divergent
T fl a C O L }
T has p r o p e r t i e s
respect
indexing
an i n d e x e d
S ~ s COL}
R0 0 = QO ~ ~ " SO a/(1-k)
o
that
[R~/ =
an a f f i n e
of parallels
convergent SO
define
conclude
e = 0 , so t h a t o [Qa/(l_k)]
We n o w
we
"
f
that
and
cT~)
which
indexed
such
can be
classes
. Thus
since
S 01 ~ T 1O " 0 0 S k = T~
Given
any positive
can be a p p l i e d
integer
n times
and
n
, the
time
translation
mapping
so
• n'°:. . ,~n. " ~' " ~''' " [~]" + [~"n ]" [ 4 " (4)
216
[%]
T
§7.5]
and
S O ~ TO kn £n By the corollary to T h e o r e m 40 (§7.3) the set of instants of Q is o r d e r - i s o m o r p h i c to the sets of real numbers which of S O and T O
index the instants
so there is a strictly
m o n o t o n i c bijection g from the real indices of S O to the real indices of T 0
(5)
g:
that is,
SO
TO
y ~
z if and only if S O ~ T O y z
We will now define a function h so that
(6)
h(y) de=f g(y+l)
g(y)
We will show that h is an u n b o u n d e d function by showing that, for each real n u m b e r 6 > I, S I/2 crosses TB; for, if S I/2 crosses T h(y)/2
T h e o r e m 19
at the event
[ f (sOy)1 then by $1/2S 0
(§4.3) and the Indexing T h e o r e m
f sosl/~ f TOTh(Y)/2
o
f (S~) sJ/2sO
=
(Theorem 41,
S y+l O
§7.3),
and
f (T~(y)) = T O o Th(y)/2T 0 g(y+l)
Suppose the contrary;
that is, suppose that S 1/2 does not cross
all T 8 (with 8 > I).
Then there is a smallest real number y > I
217
~j ~.
y / /
~rJ~
LmJ
§7.5]
such that S 1/2 does not cross T Y (see Fig.
56).
Take any
instant T Y e T Y and let W be any particle w h i c h crosses TY at [T~] and w h i c h is to the left of TY after [T~].
Then,
for each real n u m b e r ~ < y, W crosses the c o r r e s p o n d i n g particle TS; and hence W crosses $ 1 / 2 Thus T Y A
S I/2
$1/2A
at some event after [T~].
and so
TY
,
TY A
TO
TO
= SO
SO V
S I/2
whence S 1/2A
Consequently,
and
S0
SOv
S I/2
by the corollary to T h e o r e m 41 (§7.3), each
divergent parallel
is a convergent p a r a l l e l and vice versa,
in which case there is nothing further to prove, unbounded.
Furthermore,
increasing function,
or h is
h must be a strictly m o n o t o n i c
since
otherwise
S 1/2 would
cross some
convergent p a r a l l e l at two distinct events, which would lead to a c o n t r a d i c t i o n by T h e o r e m 6 (§2.9).
Thus,
in the case
where h is unbounded,
there is some integer n a such that:
(7)
real
for
all
y > n a , h(y)
Since k > I , the two sequences (km+1-km:
m=1,2,...)
Ckm:
> 2 .
m=1,2,''')
and
are both u n b o u n d e d so there is some
integer n b such that: (8)
for
all
integers
m > n b , k m > n a and k m + l
219
- km > 2
§7.5]
Let
n
d =e f max{na, n b} .
integer
such
K(m)
for
any i n t e g e r
<
2
(4) a n d
the
largest
non-negative
(S),
k m+l
-
km
.
m > n ,
km+l-km
(i0)
By
be
that
(9) Then
K(m)
Let
< K(m)
<
km+l
_ km
for m > n ,
~m+l
= g(km+l)
> g(km+K(m))
by
(9),
since
and f r o m
5tm+1
Also
g is a s t r i c t l y
increasing
function,
(6)
> h(km+K(m)-l)
(7) and
monotonic
(8) i m p l y
> 2K(m)
> km+l
Therefore,
+ h(km+K(m)-2)
increasing
~m+l
for a n y
since
+ ... + h ( K m)
+ g(k m)
> 2 a n d h is a s t r i c t l y
so
+ g ( k m)
_ k m + ~m
integer
k > I and
h(k m)
that
function,
~m+l
and
monotonic
by
(4),
m > n ,
_ ~m > km+l
£ > 1 ,
220
_ km
(5) a n d
(i0)
.
§7.5]
>~ which implies
that
(Ii)
k < We now establish
an inequality which is opposite
by a similar procedure. convergent
parallels
We define classes
{U s} and
of divergent
{V 8} , respectively,
not indexed classes
of parallels
because
of {U s} and {V B} are time
the indices
to (ii) and
which are
in the sense of Theorem
reversed;
41 (§7.3), that
is, for all a and 8 , s us b < c <=~ U b > The relations
corresponding
and
V
> V 0~
to the relations
of Theorem 41 (§7.3)
are :
f+ (U b) = U c ucub ~ a+b-c
and
fuCu b
(U b)
c = Ua_b+ c
,
whence
o
ubuc
and similar relations parallels
{V ~}
ucub
a+2b-2c
apply for the convergent
As before the classes
and {V 8} are defined
such that:
221
class of
of parallels
{U a}
~7.5] uo ~
Qo
vo
-- ~
"
f-i UI/2u 0
o
As before,
Ro "- ~
o
"
UO
o ~-
o
Qa/(l-k)
o
" Wo -- R b / ( 1 - £ )
and
f-I o f-I (V~) f-I (U~) = U 10 ~ V 01 = uOu1/2 Vl/2V 0 vOvl/2
the time
reversal
mapping
T can be applied
n times
and so
(4')
Tn:
U~
U kna
V~
p~nB
U
U nz
y and
z now apply
and
U 0 -_ V 0 kn ~n Note
that the subscript
indices
of convergent
respectively, way. (5')
G:
(6')
before
there
V0 ~ y
Similarly
and divergent
whereas
As before,
symbols
they
classes
applied
def G(y+l)
- G(y)
As b e f o r e
k m+l
=
in the opposite
U0 if
we define
H(y)
of parallels,
is a f u n c t i o n
~* z if a n d only
G(£ m+l)
222
•
to the
V0 ~ U 0 y z
§7.5]
and
so on, w h e n c e
(ii')
which
Case
£ < k ,
is a c o n t r a d i c t i o n
(iii')
The p r o o f
k < I and
is s i m i l a r
We h a v e k and (i)
(ii).
Z < 1
to the p r o o f
seen that
the
of
case
(iii).
only permissible
combinations
of
~ are: k = ~ = I , (iii)
< 1 ; and parallels, have
of
k > 1 and
in e a c h p e r m i s s i b l e which
are b o t h the
An
immediate
consequence
42
is t h a t
relative
(§7.3)
to a c l a s s
is d e t e r m i n e d transformation
"uniqueness
a time
and
(iii')
is o n l y
scale
223
one
class
Thus
theorem
the
strictly
given
we
and defined
particle,
increasing
time scale.
of
O
o f any p a r t i c l e ,
containing
a natural
k < 1 and
convergent.
of parallelism".
an a r b i t r a r y
is c a l l e d
there
of t h e p r e v i o u s
of p a r a l l e l s
to w i t h i n and
case
divergent
demonstrated
Theorem
~ > I , and
linear
§7.5]
THEOREM
Let
47
(Space
Displacement
COL be a collinear
{Q~: ~ real,
Mapping)
set of particles.
Q~ ~ COL} be an indexed
any real numbers
Let
class of parallels.
Given
a and b, there is a bijection 6:
col
~
col
and for all R ~ COL,
6c~)
IJ ~ •
Furthermore,
for any indexed class of parallels
{Ua: a real,
U s e COL},
there are real
constants
c and d such
that 6:
The m a p p i n g
This 3g
theorem
(57.3),
the p r o o f
6 is c a l l e d
41
displacement
is a c o n s e q u e n c e
mapping.
of T h e o r e m s
22
It is u s e d
(§7.3)
and
46
(§7.5).
of T h e o r e m s
48
(§7.5)
and
4g
(§7.5).
a # b.
This
and we
a space
LFUc+xld+t ~
42
The
we
~
(§7.3),
PROOF.
which
[U~]
case
a = b is trivial,
proof
shall
is b a s e d
constantly
so f r o m n o w on we a s s u m e
on the p r o o f
refer.
let T d~f Q(a+b)/2.
224
(§5.3),
of T h e o r e m
Accordingly,
we
44
let
in
that
(§?.4)
Y d~f Qb
to
§7.53
Thus
~: [Q~]
t - a+b-x.j LQ
~_.
and
e:
whence,
[Q~] d£f
if we define
-
~b-x~
~'~ L Q t
¢ d~f
J
,
e o ~ ,
6- [ < ]
(i)
Given any particle R ~
COL,
if a # b,
(i) implies that
there is no event at which R and 6(2) coincide, previous theorem,
6(R)II
~.
If a = b , the space d i s p l a c e m e n t
is trivial and 6(R) = R , so 6(R) that for all R c
so by the
II ~ trivially.
We conclude
COL,
(2)
6r.~) II ~ . T h e o r e m 41 (§7.3) implies that for any real numbers
t,x,y
with x < y ,
[u xt]
(3)
c~ [u U,- x +.y ]
and
[UYt ]
~ [
t+x_y]
Ux
These relations c o r r e s p o n d to right and left m o d i f i e d signal functions respectively.
(4)
(aEut])
We will now show that
o (s[uYt-x+y])
and
225
(a[uYt])
(~
X
(a[Ut+x_y])
§7.s]
Given particular A,B,C,D
real numbers
such that [Q ] : [Utx]
(5) since
there are real numbers
t,x,y
{Q~} is an indexed
and
[Q~] : [ U ~ _ x + y ]
class of
parallels.
,
Now (3) implies
that A [QB]
[QDC]
from which (6)
D = B - A + C .
Also by ( 1 ) , 6: [QA]
(7)
~
Consequently and
[~B~A-a+b~, ]
[Q;]
-+ [ Q C - a + b ]
.
with
by Theorem
4l (§7.3)
together
(6[Q~])
~ (6[Q~])
,
(7),
and by (5) this is equivalent
to the first relation
second relation
similarly.
can be proved
The relations Theorem {V~:
(6)
41 (§7.3)
a real
(4) are in accordance
so we can define
, V ~ e COL}
dj Evil
226
with the results
an indexed
such that
of (4):
the
of
class of parallels
§7.s] By
(2),
there
is some
V ~ such that
U 0 ~ VB ,
and
so by T h e o r e m
that
for a l l
real
44
(§7.3)
~
can c h o o s e
t =
is a c o n t r a d i c t i o n
mapping,
by
so k = I ; w h e n c e
48
Let
COL
be
Let
{Ua:
o (8[U~]) (l),
with
a collinear
instants
,
c,d,k
set
Us E
U bc" U~
OOL}
since
, ~ is a s p a c e
displacement
~ c+xLUd+tJ
=
[]
Mapping)
of particles. be
E Ub such
an
indexed
that
class
Ub < U
.
of parallels
There
is
a bijection
C
• : col
~
[U~] "-+ The m a p p i n g
This 33
(§6.4),
47
(§7.5).
and
T is c a l l e d
theorem 41
such
(8) b e c o m e s
(Time T r a n s l a t i o n
~ real
constants
, then
(Icl-d)/(k-1)
~[U~] THEOREM
real
d+kt"
([U~]) which
are
t a n d x,
(8) If k ~ I we
there
a time
col [U~_c+d]
displacement
is a c o n s e q u e n c e
(§7.3),
It is u s e d
42
.
(§7.3),
44
in the p r o o f s
61 (§9.5).
227
mapping.
of T h e o r e m s (§7.4),
45
18
(§4.3),
(~7.4)
of T h e o r e m s
49
and
(§7.5)
§7.5]
PROOF.
This proof is based on the proof of Theorem 45 (§7.4).
We define an indexed class of parallels
{Qa: ~ real , Qa e COL} such that,
for some real number a with 0 < a < 1 ,
gO def ub -
0 d~f U b
" QO
0 def
a " Qa
-
U~
.
By Theorem 42 (§7.3), ~ b+~x
~
[Q ] = [Uc+ktJ
(i)
The space displacement
where k =
mappings
d-e a
'1 and '3 (of the proof of
Theorem 45 (§7.4)) are such that
'1: [QO]0 so by the previous
~ [Q~]
and
*3:
_~+x. [~t j and
*3:
[Q~]
"
[Q~]
[Q~]
~
- -~+x. [Qt+a j
theorem,
x *1: [Qt ] ~
We now define a time displacement
mapping T ~ d~f *3 o '1 "
and so
* : [Q ~ ] whence
from
~-~ [Q~+a ]
(i),
T*:
b+kx. [Uc+ktJ
~
228
~ b+kx LUd+kt]
;
§7.5]
w h i c h is equivalent to
• ~: [u yz] -- [uz~- c+d] , which is the r e q u i r e d mapping.
THEOREM 49
(Linearity of M o d i f i e d Signal Functions)
If Q a n d R are p a r t i c l e s
f+(Rt) QR where
D
in COL with n a t u r a l
= Qat+b
a , b , c , d are c o n s t a n t s
and
f-(Rt) QR
time scales,
then
= Qct+d "
a n d both a a n d c are p o s i t i v e .
Furthermore
f+(Qt ) = R(t_b)/a RQ
and
f-(Qt ) = Q(t-d)/c RQ
This t h e o r e m is a consequence of Theorems 18 (§4.3), 48 (§7.5).
41 (§7.3),
42 (§7.3),
46 (57.5),
"
9 (§3.2),
47 (§7.5) and
It is used in the proof of Theorems
50 (§8.1),
51 (§8.1) and 57 (§9.1). PROOF.
If Q and R coincide at no event, or if they are p e r m a n e n t l y
coincident, they are p a r a l l e l and the result is a special case of Theorem 42 (§7.3). Theorem 46
Otherwise Q and R coincide at some event by
(§7.5).
We now define indexed classes of p a r a l l e l s {S~:
~
real , S ~ e COL} and
{US:
229
~ real,
U ~ e COL]
§7.5] such
that
sO
For any real event
number
by T h e o r e m
and
c such
Let
6 and
= Q
46
•
U0 = B
a, { a ~
[0 • so S a and
~U0 e o i n c i d e
(§7.5);
that
are r e a l
T be
space
and
time
two t h e o r e m s ,
8:
IS
]
Consequently
we
can d e f i n e
~
numbers
b
respectively,
as in
that
and
[S
] -~ [St+ b ]
a mapping
6
o
T
:
T
o
6
that
(2)
[~t+b ] •
Since
• is a c o m p o s i t i o n
composition
(3)
there
translations,
such
[S~ +a]
d~f
47
is,
at some
that
the p r e c e d i n g
such
0 0 SO = QO "
and
(§7.5)
of t h r e e
and
of two
displacement
displacement
mappings,
(1),
I:
[U~] --[U~+c]
230
.
mappings,
I is a
so by T h e o r e m
§7.5] Since 6 and T are bijections,
~ is a b i j e e t i o n and so, for any
integer n,
sna] Now a was arbitrary, substitute positive
a/2 m
so if we choose any p o s i t i v e integer m and
for a w h e r e v e r
integer
0
a appears, we find that for any
m and for any integer n, U0
=
L bn/2mJ that
is,
for
any
dyadic
number
[S~]
(4)
p,
= [U~p]
C o n s e q u e n t l y by T h e o r e m 41 (§7.3)
Lsg sOuO and,
and p- p
cp
-LS p+ap
=
sOuO
since signal functions are continuous by Theorem 9 (§3.2),
it follows that for all real t,
f+ [S~] = sou 0
[U 0ct/(b_a)]
and
f
sou 0
[S@] = [U~t/(b+a)]
That is, the m o d i f i e d signal functions
f+ sou 0
and
f-
•
are linear
sOu 0
strictly increasing functions and therefore Theorem 42 (§7.8) implies that
f+ QR
and f- are linear strictly increasing functions QR
w h i c h can be w r i t t e n in the general form
231
§7.5]
f+(Rt) QR where
A,B,C,D are
Theorem
18
=
QAt+B
constants
and
f-(Rt) QR
and A a n d
=
QCt+D "
C are p o s i t i v e .
By
(§4.3)
RQf+(Qt) = R(t-B)/A
and
232
f-(Qt) RQ
=
R (t_Dj/C
.
rn
§8.1]
CHAPTER 8
O N E - D I M E N S I O N A L KINEMATICS
In this chapter all p a r t i c l e s have natural time scales and m o d i f i e d signal functions are linear.
We will often delete the
particle symbol where there is no chance of ambiguity; example,
in the next theorem,
f o f Qs sQ
for
instead of writing
Qx = QM ( x - q ) + q sQ
"
we shall write
f o f Qs sQ
§8.1
(x) = M s Q ( X - q ) + q
R a p i d i t y is a Natural Measure for Speed
In this section we define sional measure of speed.
"rapidity" which is a non-dimen-
For collinear sets of particles,
directed rapidities are composed by simple arithmetic addition, which means that r a p i d i t y is a natural measure for speed. name "rapidity"
is due to Robb [1921] who introduced this
concept in a d i f f e r e n t way.
233
The
§8.1]
THEOREM
Let Q,S,T
50.
e COL.
If S ~ Q , there is a positive
(i)
"constant
of the motion"
MQS and a real number q such that
[
f o f {s sQ
where
1
(x) = MQs(X-q)+ q ,
the real number q is such that S coincide8
with
Q at [Qq]. -I
(ii)
MSQ
(iii)
If R II Q and T II
S
then
,
MRT REMARK.
= (MQS)
= MQS
•
The constant of the m o t i o n M is invariant with respect
to affine t r a n s f o r m a t i o n s of natural time scales, by part
This theorem is a consequence of Theorems and the previous theorem. Theorems PROOF.
12 (§3.5), 18 (§4.3)
It is used in the proof of
51 (§8.2) and 56 (§8.4). (i)
By the previous theorem,
both f+ and f- are linear
sQ functions,
so their c o m p o s i t i o n
I
f
Qs
If Q ~ S
(iii).
o f
sQ
]"
QS
is a linear function.
, there is some instant Qq e Q such that S coincides
w i t h Q at [Qq],
and so the record f u n c t i o n is of the form
f o f Qs sq
(x) = MQS(X-q)+ q ,
234
§s.1]
where
MQS > 0 , by T h e o r e m
12
(ii)
The p r e v i o u s
implies
Q , S~
~
s COL,
there
theorem
are
constants
f-(X)Qs = ~QSx + q~ so by T h e o r e m
18
that,
for any
two p a r t i c l e s
8@s, qs,aSQ,S q+ such that SQf+(x) = asQx + s +q
and
(54.3),
-I -I fSQ- (x) = ~QS x - BQsqs
that
(§3.5).
f+
-I -I + QS (x) = asQX - asQSq "•
and
is -I
-I
8SQ = BQS
aQS = (~SQ "
and
whence
MQ s = ~SQaSQ = (~QSeSQ)-I = MSQ-I . (iii)
The p r e v i o u s -
theorem
implies
that
there
are
+
aQR, BRQ, rq, qr such that
+
f-(x) RQ and
since
= flRQX + rq Q II R
and
f+ (x) = aQRX + qr "
QR
,
~QRSRQ
=
1
SO
-1 + rq f - (x) = (XQRX RQ
and
f+(x) QR
235
= aQRX + qr+ "
constants
§8.1]
C o n s e q u e n t l y by Theorem 18 (§4.3),
[l o r l~x~ : r S
SR )
o r +~x)
RS
SR
= f-
o f-
o f+
RQ
Qs
sQ
: f
o
RQ
f
QR
o
o f+(x)
QS
= MQsX
o f+(x)
Q
QR
+ a-QIR(MQs(q;-q)+q)+rq
•
Thus we have shown that
MRS = MQS and similarly,
since T II S ,
MTR = MSR and so by
•
•
(ii), -1 MRT =
-I
(MTR)
Given any particles
=
(MsR)
Q,S
~ COL
= MRS
= MQS
.
[3
such that S II Q,
Theorem 42 (§7.3) shows that
I
f o f Qs SQ
1
where d is a real constant.
(x) = x + 2d ,
The constant of the m o t i o n M s Q i S
and is therefore not shown explicitly. p r e c e d i n g theorem
The results of the
(with the e x c e p t i o n of (i)) apply t r i v i a l l y
to the case where S II Q.
236
I ,
§8.1]
Q,S e COL we define the d i r e c t e d
G i v e n any two p a r t i c l e s
rapidity
of S r e l a t i v e
to Q to be
rSQ
dgf -
½ log e MSQ
•
Since M > O ,
_co
In the case of p a r a l l e l
The next t h e o r e m was defined unbounded
Also,
transformations
time
THEOREM
51
Given
Q,S,T
co
is in no way
rapidities of n a t u r a l
M = I and hence
surprising,
by simple
are u n a l t e r e d time
scales,
with respect
for speed which
Law for D i r e c te d
by a r b i t r a r y
is a natural
Rapidity)
•
measure
237
affine
so the following
~ COL,
"rapidity
is
arithmetic
scales.
(Addition
to Q.
since r a p i d i t y
to t r a n s f o r m a t i o n s
rQT : rQS + rST That is,
r = O.
of S~ with respect
have a m e a s u r e
is composed
is i n v a r i a n t
natural
<
rapidity
so that we would
and w h i c h
addition.
r
particles,
IrsQ I the relative
We call
result
<
for speed".
of
§8.1]
This 49
theorem
(§7.5)
Theorems
PROOF.
and 52
50
is a c o n s e q u e n c e (§8.1).
(§8.2)
By T h e o r e m
flQS'qs;aSQ'S~;flST'S
and
49
of Theorems
It is used
18
in the proof of
57 (§9.1).
(§7.5)
there are real c o n s t a n t s
t;~TS" t+s such that
fs-(x) = BQSx + qs
and
f+ (x) = asQx + s q+ SQ
f-(x) = 6ST x + s t ST
and
f+ (x) = aTS x + t+s TS
By T h e o r e m
18
(§4.3),
f-(x) QT
= flQSBSTX + BQSSt- + qs
f+(X)TQ = ~TSasQx whence,
as in part
+ aTSS~
= ~QS~SQ6STaTS = MQsMsT logarithms
of
• both sides,
rQT = rQS + rST
238
and
+ t÷s "
(ii) of the previous
MQT = flQSBSTaTSaSQ
and taking
(§4.3),
theorem,
and
§s.2]
COROLLARY
(Urquhart's
Theorem
: see
Szekeres
[1968])
Given Q,S e COL,
rQS = -rSQ PROOF.
Put
If are
two
that
{Qa: a real, classes
there
parallels
§8.2
T = Q in the
of p a r a l l e l s ,
relative
Congruence
Given
theorem.
[]
Qe e COL} and {S~:
is a u n i q u e {Qa}
above
it f o l l o w s
directed to the
of a C o l l i n e a r
any two p a r t i c l e s
B real,
rapidity class
from
S ~ ~ COL}
Theorem
of the
class
of p a r a l l e l s
{S ~}
we
say t h a t
Q and
S are
Q,S ~ COL, there
f-(x) QS
and
congruent
if
~sQ = ~Qs " which
is e q u i v a l e n t
to the
condition
aQs = ~SQ • since
-1 ~QS = ~SQ
and
(iii),
of .
Set of P a r t i c l e s
are
constants
+ q~ such that ~SQ, BQS'Sq,
f+ (x) = asQX + s + SQ q
50
-1 BSQ = BQS .
239
= ~QS x + q-s ;
§8.2]
The word "congruent" has also been used by Milne different
sense:
the following
[1948] in a
here, we use the work "synchronous"
(see
§8.3) where Milne used the work "congruent".
We
have departed in t e r m i n o l o g y because the word congruent is very descriptive of the idea of equality of time durations.
T H E O R E M 52
Congruence
is an e q u i v a l e n c e
relation
on a c o l l i n e a r
set of
particles. This theorem is a consequence of Theorem 18 (§4.3) and is used in the proof of T h e o r e m 53 (§8.2).
PROOF.
By definition,
congruence
is a reflexive and symmetric
relation.
In order to show that congruence we consider three particles Q,R,S
is a transitive relation,
e COL such that Q is congruent
to R and R is congruent to S; that is,
~RQ = 6QR
and
~SR = 6RS
Then by Theorem 18 (§4.3),
~SQ
= aSR~RQ
: ~RS~QR : ~QR~RS
which shows that Q is congruent to S.
240
= ~QS []
"
§8,2]
If Q and S are p a r t i c l e s we can define defined
a particle
COL w h i c h are not c o n g r u e n t
in
T s S whose
natural
time
scale
is
such that
f+(x) d~f (~QS/~SQ)½ x = f-(x) TS TS Then
~2Q
(P QsasQ)~
from w h i c h we see that natural within specify set,
time
Q and T are congruent.
scale of each particle
an a r b i t r a r y the time
by choosing
affine
scales
of p a r t i c l e s
that each other particle preceding
theorem,
congruent
to each other.
one c o l l i n e a r the t h e o r e m
Given
all particles Since
to two or more assume
in a p a r t i c u l a r
say
this t h e o r e m
distinct
only
be careful
collinear
the t r a n s i t i v i t y
collinear
Q s COL, and specifying
in the c o l l i n e a r
we must
to
we could further
COL is congruent to Q.
set of particles,
for this would collinear
in
the
is only d e t e r m i n e d
transformation,
a g i v e n particle,
Since
By the set are now applies
to
not to apply
sets of particles,
of congruence
for non-
particles.
two congruent
their r e c o r d
functions
particles
Q,S s COL such that Q ]I S,
are of the form:
241
§8.2]
(f o f ]~ (x) = x + 2dQs Qs sQ
I
f o f ]*(x) [sQ Qs where
distance of S relative
defined
to 9" and the directed distance of
to 4' respectively. in terms
of the time
they are not invariant natural
time
THEOREM
53
2dsQ
+
dQS and dSQ are called the directed
the constants
relative
=x
and
with
The d i r e c t e d scales respect
distances
are
of the particles, to t r a n s f o r m a t i o n s
and so of
scales.
(Additivity
of Directed
Let Q,S,T be congruent particles
Distances)
in a collinear set.
zf
Q II S II T, then (i)
dQT = dQs + dST , and
(ii)
dSQ = _ dQS •
REMARK.
It is important
analogous applies
property
in terms
This t h e o r e m 52
for eollinear
if Q,S,T are
are defined
to note
congruent,
that,
rapidities, since
of the time
in contrast
is a c o n s e q u e n c e
(§8.2).
242
this p r o p e r t y
the d i r e c t e d
scales
to the only
distances
of the particles.
of Theorems
18
(§4.3)
and
§8.3]
PROOF.
Part
(ii)
is a s p e c i a l
so it is o n l y n e c e s s a r y congruent,
there
Ai + ( x )
=
x
are
+
SQ f
case
to p r o v e
constants
y
(x)
=
x +
,
-F + ( x )
18
f+(x) TQ
(i).
Since
=
x
+
Ts ,
6
QS
By T h e o r e m
part
(i) w i t h
ySQ, YTS, 6QS, 6ST
SQ
QS
of p a r t
T = Q ,
Q,S,T
are
such that
¥
and
TS
f (x) ST
= x +
f-(x) QT
= x + ~ ÷ 6 QS ST
+6
+
8
ST
(§4.3),
= x + y + y TS SQ
and
Thus
f QT
o
(x) = x +
y SQ
Q
= x + 2dQs since
§8.3
¥ SQ
+
= 2dQs
6
and
QS
Partitioning
S
+ 2dsT
y + 6 TS ST
a Collinear
+6
QS
Set
=
,
2dsT
Into
ST
•
Synchronous
Equivalence
Classes.
Given
synchronous
any particles
E COL
we
say t h a t
Q and S are
if
f+(x) QS (One
Q,S
condition
=
f-(x) SQ
implies
and
f-(x) QS
the other,
243
= f+(x) SQ
by T h e o r e m
18
(§4.3)).
§8.3]
THEOREM 54
(Synchronous Parallel Particles)
The synchronous collinear
relation
i8 an equivalence
relation
on any
class of parallels.
This t h e o r e m is a consequence of Theorem 18 (§4.3).
PROOF.
By definition,
and symmetric. is transitive,
the synchronous r e l a t i o n is reflexive
In order to show that the synchronous r e l a t i o n we consider three particles Q, S j T e
COL such
that Q II S II T and such that the pairs Q,S and S,T are synchronous. distance,
It then follows from the d e f i n i t i o n of directed
that
f+(x)QS = f-(x)SQ = x + dQS
and
f+(X)sT = f-(X)fs = x + dsT .
Then, by Theorem 18 (§4.3),
f+(x) = f+ o f+(x) = x + dQS + dsT QT
QS
ST
= x + dsT + dQS = f-
o f-(x)
TS = f- (x)
SQ [3
TQ
This result also follows from T h e o r e m 43 (§7.3).
244
§8.3]
THEOREM The
55
(Synchronous
synchronous
collinear
i8
Sub-SPRAYs)
an e q u i v a l e n c e
relation
on any
sub-SPRAY.
This
PROOF.
relation
Collinear
theorem
is a c o n s e q u e n c e
By definition,
and symmetric.
the
s y n c h r o n o us
To show that
particles
belonging
particles
Q,S,T
such that
synchronous.
At the event
Q must
be the
same as the real
be the
same as the real
synchronous.
Therefore
18
relation
it is t r a n s i t i v e
to a collinear
~ OSP
of T h e o r e m
sub-SPRAY,
the pairs
(§4.3).
is r e f l e x i v e
on the set of we c o n s i d e r
Q,S and S,T
of coincidence,
the real
are
index of
index of S, w h i c h must
index of T, since both pairs the m o d i f i e d
signal
three
also are
functions
are of
the form:
By T h e o r e m
f+(x) QT
18
= e(x-a)
+ a , and
f + (x) = f - ( x ) ST TS
= 6(x-a)
+ a
relation
.
(§4.3),
= f+ o f+(x) QS ST
We have
these
f+ (x) = f - ( x ) QS SQ
= a~(x-a)
shown that the
on any class
are the only
+ a = f- o f-(x) TS SQ
synchronous
of p a r a l l e l s
subsets
relation
and on any
of a c o l l i n e a r
245
= f-(x) TQ
is an e q u i v a l e n c e
collinear
sub-SPRAY:
set of p a r t i c l e s
on w h i c h
§8.4] the
synchronous
relation
is not p o s s i b l e synchronous. could
is an e q u i v a l e n c e
for all the particles
However,
be synchronous,
all particles and each
class
synchronous
with that m e m b e r w h i c h
synchronous
collinear
§8.4
Let
Coordinate
{Sa: ~ real,
in COL.
Given
coefficients
reals,
is called
and the
of a c o l l i n e a r of parallels
is contained
in a Collinear
indexed
set to be sub-SPRAY
could be
in the given
class
of p a r a l l e l s
sx and any instant
S xt E S x , the
pair
(x;t) of reals
of the event
[S~].
by the c o r r e s p o n d i n g
a coordinate
frame in col;
the origin in p o s i t i o n - t i m e
set of events
it
Set
S a s COL} be an indexed
coordinates
in col,
is called
Frames
any particle
events
of a e o l l i n e a r
Thus
sub-SPRAY.
of the ordered
position-time
relation.
{(O,t):
p o s i t i o n of the coordinate
t real}
frame.
246
of the
are
The
called
set of all
ordered
pairs
the event coordinate
is called
the
of
(0,0) frame;
the origin in
§8.4]
THEOREM 56
(Some Useful Kinematic
Let {Q~:
~ real,
Q~ e COL} and
distinct
indexed
classes
Relations)
{SB:
B real,
of p a r a l l e l s
(i)
QO and S O are s y n c h r o n o u s ,
(ii)
QO0
=
0 and So •
(iii)
QO
~
S O"
Let r be the d i r e c t e d For any real x,
rapidity
let u,w,y be real n u m b e r s x
COL such
that:
of {Q~} w i t h r e s p e c t
0
and
f+ (u) = t , S t ~ QW" sXs 0 -
(see Figure
in
S fle COL} be two
such
to {S B}
that
f (t) = Y sOs x
57).
Then (i)
x/t = tanh r d ~ f v , where
v is the
"velocity"
to {SB} ,
of {Qa} with r e s p e c t r
(ii)
w
= e u = t sech r , and
(iii)
y
= te r sech r .
This theorem is a consequence is used in the proof of Theorems Corollary i to 58 (§9.3),
of Theorem
57 (§9.1),
and Theorems
and 62 (§9.6).
247
50 (§8.1).
It
58 (§9.3) and
59 (§9.4),
60 (§9.5)
§8.4]
Sx
s6
S o
/
h
SO Y
<~'~.
< Sx
//' SOu
QO
f/
4
/ Qg
7 Fig.
57
PROOF.
This
diagram
By T h e o r e m
rapidity
illustrates
50
(i)
f QOsO
o
(§8.1)
the
and
case w h e r e
the
S O and S x are
(2)
t
(3)
y =
=
f sOQ o
]*
(u) = e
2r
u
synchronous,
f+ (u) sXs 0
=
u
+ x , and
f- (t) = t + x = e 2 r u sos x
248
x > 0 .
definition
(§8.1),
(i)
Since
I
.
of d i r e c t e d
§8.4]
By equations (2) and (3), (4)
x/t
= tanh
r = v .
Since QO and S O are synchronous, equation (i) implies that (5)
f+
w =
(u)
= eru
, and
(W)
= erw
.
Q°S°
(6)
y =
By equations (7)
fSOJ
(3), (4) and (5),
W = e r u = e -rt ( 1 + t a n h
Equations
r)
= t seth
r
(4) and (7) correspond to parts (i) and (ii), and
part (iii) is obtained by combining equations COROLLARY The
set
time
lines
=
in
col
through
coordinates
x/t
the
which
lim(tanh
r)
which
origin
are
= I
are
related
and
respectively.
on
the
right,
or
in p o s i t i o n - t i m e ,
x/t
r~
PROOF.
[]
(Kinematics of Optical Lines)
of events
optical
(6) and (7).
by
=
the
lim
left, have
position-
equations:
(tanh
r)
= -1
,
r~_~
Thus,
signals
have
"infinite
rapidity"
By the Indexing Theorem (Theorem 41, §7.3), it follows
that x / t = ±1, for right and left optical lines, respectively. That is, optical lines have unit velocity, which corresponds to infinite rapidity.
[]
249
§9.o]
CHAPTER 9
T H R E E - D I M E N S I O N A L KINEMATICS
Whereas the previous
discussion had some similarity to
the theory of absolute geometry, this final chapter departs r a d i c a l l y from both absolute geometry and the more usual discussions of M i n k o w s k i importanoe;
space-time.
Two ideas are of central
namely, that the velocity space of M i n k o w s k i space-
time is hyperbolic,
in contrast to the e u c l i d e a n velocity
space of Newtonian kinematics, are related to homogeneous
and that space-time
coordinates
coordinates
in a t h r e e - d i m e n s i o n a l
h y p e r b o l i c space.
It is shown that each SPRAY is a t h r e e - d i m e n s i o n a l h y p e r b o l i c space with particles c o r r e s p o n d i n g to "points" with relative velocity as a metric function.
and
Homogeneous
coordinates in t h r e e - d i m e n s i o n a l h y p e r b o l i c space correspond to space-time coordinates of particles
in a SPRAY.
This
c o r r e s p o n d e n c e is eventually extended to all events and gives rise to the concept of a coordinate frame.
The position
space
a s s o c i a t e d with each coordinate frame is shown to be a threedimensional e u c l i d e a n space,
so the present axiomatic system
is also an axiom system for e u c l i d e a n geometry.
T r a n s f o r m a t i o n s between homogeneous correspond to h o m o g e n e o u s
coordinate
Lorentz transformations,
250
systems from which
§9.1] the inhomogeneous
Lorentz t r a n s f o r m a t i o n s
are derived.
In
conclusion we describe the trajectories of particles and optical lines relative to any coordinate
§9.1
Each
frame.
3-SPRAY is a 3 - D i m e n s i o n a l H y p e r b o l i c Space
It has been known for some time that the velocity space of special r e l a t i v i t y is hyperbolic; early references given by Pauli
see, for example,
[1921, p.74].
the
The p h e n o m e n a
of spherical aberration and Thomas p r e c e s s i o n are simple consequences of the v e l o c i t y space being h y p e r b o l i c and they have been discussed recently by Boyer [1965], Fock [1964] and Smorodinsky
[1965].
In the next theorem, we show that each SPRAY is a m e t r i c space with observers being the "points" of the space and w i t h relative rapidity as an i n t r i n s i c metric. ing definition:
a 3-SPRAY
(denoted 3SP[
We make the follow] is a SPRAY w h i c h
has a m a x i m a l symmetric sub-SPRAY of four distinct particles. The existence of at least one 3-SPRAY is p o s t u l a t e d in the A x i o m of D i m e n s i o n
(Axiom VIII,
§2.10).
In a following theorem
(Theorem 61, ~9.5) we will show that each SPRAY is a 3-SPRAY. THEOREM 5 7 .
Each 3-SPRAY is a h y p e r b o l i c space of three
dimensions with curvature of -1:
the "points" of the space
are the observers of the 3-SPRAY and relative rapidity is an intrinsic metric.
251
§9.1] This theorem is a consequence of Axioms IV (§2.4), VII (§2.9), XI (§2.13) and Theorem 22 (§5.2), Theorem 24 (§5.3), and
56 (§8.4).
and Theorems
49 (§7.5),
the Corollary to
51 (§8.1)
It is used in the proof of Theorems
58 (§9.3)
60 (§9.4) and 63 (§9.7). PROOF.
A characterisation of 3-dimensional euclidean and
hyperbolic
spaces is given in Appendix i.
We will first
show that all the conditions of this characterisation
are
satisfied by any 3-SPRAY, and then it will follow that each 3-SPRAY is either a 3-dimensional euclidean space or a 3-dimensional hyperbolic
space.
Definitions of concepts
which have not yet been defined will be found in Appendix i.
(SPR) is a metric space having
TO show that a given SPRAY
relative rapidity as an intrinsic metric, we consider any three particles Q,S,T e SPR. rapidity
By the definition of relative
(§8.1),
lrQSt = IrSQI and also
IrQs I = o if and only in Q = S.~
The triangle inequality for relative rapidity is a consequence of the Triangle Inequality
(1)
f
QT
o
f
TQ
(Axiom, IV, §2.4) which implies that
~ f
o
f
ST
QS
252
o
f
TS
o
f
SQ
§9.1] For any three particles
Q,S,T
~ SPR there are instants
Qa e Q, S b s 4" Tc s { such that
Qa -- Sb "- Tc and by Theorem 49 (§7.5),
the signal functions
after coincidence
have the form
Q(Qx and so on.
= Ty where y = c + aTQ(X-a)
,
As in the proof of Theorem 51 (§8.1),
IrQTl = ½1n(aQT
so by the inequality
• aTQ) •
(i),
rQT = ½1n(~QT ½1n(aQS
• ~TQ) " ~ST " aTS " aSQ)
IrQs I + IrSTI We have now shown that relative and, by the Addition
Law for Directed
it follows that relative
arcwise-connected
of Bounded
locally compact sub-SPRAYs
(by Theorem 22,
the Axiom of Isotropy of SPRAYs the conclusion
of Appendix
is either a euclidean
Rapidity
(as a consequence
to Theorem 24, §5.3),
Axiom of Compactness
is a metric function (Theorem 51, §8.1),
rapidity is an intrinsie metric.
Each SPRAY is unbounded Corollary
rapidity
of the (by the
(Axiom XI, §2.13)),
§5.2) and isotropic
(Axiom VII,
§2.9)).
i, it follows that each
or a hyperbolic
253
(by Now by
3-SPRAY
space of three dimensions.
§9.1]
We now consider a given 3 - S P R A Y Q,S
e
3SP.
Homogeneous
(3SP)
coordinates
and any particles
in t h r e e - d i m e n s i o n a l
e u c l i d e a n and h y p e r b o l i c spaces are d e s c r i b e d in A p p e n d i x 2, to w h i c h we will refer.
Let
relative to S in e o l [ S , Q ] in T h e o r e m 56 (§8.4).
(2)
(x;t)
=
(i) of this t h e o r e m is
ta~h
r =
v
,
which shows that t and x are homogeneous relative to S in either a hyperbolic, d i m e n s i o n a l sub-space of
one-
with t c o r r e s p o n d i n g to x 0
3SP;
2, according as to w h e t h e r
hyperbolic.
coordinates of Q
or a euclidean,
and x c o r r e s p o n d i n g to x I of equations Appendix
coordinates
of an event coincident with Q, as
Equation
x/t
be p o s i t i o n - t i m e
We will now show that
3SP 3SP
(i) or (3) of is e u c l i d e a n or is h y p e r b o l i c by
assuming the contrary and deducing a contradiction: if 3 S P
is a e u c l i d e a n space,
with equation
a comparison of equation
(3) of A p p e n d i x 2 shows that relative velocity
is an intrinsic m e t r i c so, for collinear particles
(3)
VQT
=
VQS
+
vST
,
=
rQS
+ rST
.
and by T h e o r e m 51(§8.1),
(4)
(2)
rQT
254
§9.1]
Now the general solution to Cauchy's functional e q u a t i o n
g(x+y)
=
g(x)
+
g(y)
(where g is a continuous f u n c t i o n for p o s i t i v e real variables as discussed by Aczel
g(x)
Equation equations
[1966]) is
=
~x
(2) implies that
,
~
v(r)
.
is a continuous
(3) and (4) show that
functional equation.
real
v(r)
function and
satisfies Cauchy's
C o n s e q u e n t l y there is a real constant
such that
V(r)
=
ar
which is a c o n t r a d i c t i o n of (2).
,
Thus
can not be a
3SP
e u c l i d e a n space and so we conclude that
3SP
is a h y p e r b o l i c
space.
Moreover,
if we now compare e q u a t i o n
of A p p e n d i x 2, we see that each 3-SPRAY, rapidity as an (intrinsic) metric, curvature of -1.
D
255
(2) with equation
(i)
equipped with relative
is a h y p e r b o l i c space with
§9.2]
§9.2
Transformations Dimensional
of Homogeneous
Hyperbolic
Transformations in the
3-dimensional
(see Appendix
2):
(1)
xi =
between
Coordinates
in Three-
Space sets of homogeneous
hyperbolic
space,
coordinates
H3, are of the form
3
[ aik x k k=O
(i=O, 1, 2, 3) ,
with det[aij] # 0 Hyperbolic ations~
distance
is independent
so for any two points x,y e
of coordinate
represent-
H3 ,
h(x,y) = ~(~,~) , whence,
by equation
(2) of Appendix
Arcosh{'~(x,Y)l[~(x,x)~(y,y)l-½}
2,
= Arcosh{l~(x,y),[~(x,x)~(y,y)l-½ }
and so
Thus
[-xOYo+xaY ~ ]2l-(aOjxj)2
+ (aljx j) 2 + (a2jxj)2 + (a~jxj)21 x
x [- (aOkYk) 2+
(alkYk) 2+
256
(a2kYk) 21
§9.2]
where r e p e a t e d indices imply a summation convention:
Latin
indices take the values 0,~,'2, 3 and Greek indices take the values
1,2.3.
E q u a t i n g coefficients of x~, ~0~0, of x ~ ) y ( ~ )
64ith no sum over ~), and of x 2(i) x2(fl)y2(i)Y2(~) with ~ = 1 , 2 , 3 and i = 0,1,2, 3 and fl # i(and with no sum over ~ and i).~ we find that~
since the argument of A r c o s h
must be real,
3 (2)
[
a=l
a2 ~0
-
2 ao0
-
2 ao~
=
-1
=
1
=
0
3
I G2~ 8
(~=1
(fl = 1,2,3)
and
(i,k=0,1,2,3
a n d i ~ k)
3 ~= 1
aaia&k
respectively. that
[aij]
- aoiaOk
An immediate consequence of these equations is
has an inverse
Co0 = CO0
•
[aij]
a o ~ = -aao
~
where
aao = - a o a
•
aag = afa
The m a t r i x [a..] r e p r e s e n t s the inverse coordinate transfor90
marion
and the
equations
corresponding
to
equations
(2)
above
are :
3 (3)
[ a2 O~ ~=1
-
~
ao0
=
-1
3 I
a=l
a 2
floi
_
2
aflo
1
(fl = 1, 2, 3)
and
0
(i,k=0,I,2,3
a n d i # k)
3
I ~=I
aiaak&
- aioako
=
257
.
§9.2]
[aij]
Since
and
[aij]
are
inverses,
det[aij][akZ]
=
it f o l l o w s
(det[aij])2
that
= I ,
whence
(4)
det[aij]
The
inverse
of the
= det[aij]
transformation
(S)
xi
= +1
(i)
is t h e r e f o r e
given
by
= aij xj
where
ao0 = aO0
(6)
We
can n o w
• a o e = -a 0 , a ~ o = - a o ~
verify
that
a00
by m a t r i c e s
having
[Aside:
§9.6 b e l o w
In
correspond
reversal.
form
a group,
non-singular
-
d~f
-
then
ao0
we will
we o b s e r v e
-
d~f
-
(2)
transformations
without
transformations
first
the
above
that
has
~
_ and
transformations
the
a unique
[aij] , we d e f i n e
• aoa
represented
to s h o w t h a t
so e a c h m a t r i x
equations
see t h a t t h e s e
of L o r e n t z
In o r d e r
if f o r a n y m a t r i x
ao0
transformations
•
> 0 f o r m a group.
group
to t h e
time
coordinate
• as8 = - a B ~
aao (3)
inverse.
a matrix
-
d~f
• aao
-
258
[aij]
_/i-[
can be w r i t t e n
forms
matrices
Secondly, such -
aoa in the
are
• aaB
that
d~f -
equivalent
-aBe ,
§9.2]
aijakj = 6kj and ajiajk = 6ik . It is now easily v e r i f i e d that, two such matrices,
if
[aij]
and
then the product m a t r i x ~..
= ~..~..
[~ij]
[cij],
are any where
,
satisfies equations w h i c h are equivalent to equations and (4).
(2),
It is now only n e c e s s a r y to show that the 00-terms
of the original m a t r i c e s are positive.
elk
=
aijbjk
COO
=
aoobo0 + ~ aoab~o
Since
then
>0
by the first of equations
A set of h o m o g e n e o u s
origin
(3)
(2) and (3).]
coordinates has the point x as
if x has the coordinates
x i = (xo,xl,x2, x 3) = (xo, O,O,O)
259
§9,2]
Relative to the same set of coordinates, we denote the
coordinates system
of any point z s H 3 with respect
having x as origin Z~
X
X
to a coordinate
by X
X
= (Zo, Zl, Z2, Z3)
and, in particular,
the coordinates of the origin x are denoted
by ~gX.
=
X
X
X
X
X
(Xo, Xl, x2,x 3 ) = (Xo, O, 0,0)
A t r a n s f o r m a t i o n of coordinates
.
can therefore be expressed
in the form:
(7 )
zx
a.
~O
z y.
z y.
J
~
a
*
X
.
z .
~J
J
We now derive some results which will be used in following sections.
A p a r t i c u l a r case of
(8)
y~
so by equations
(9)
(yl)
+
= aij
(7) is
Y
yj
=
Y2 )
Also, by equations
h(y,x)
y~
(2),
+ (y3) J
+ =
(i0)
aio
-I + ao0
(2) and (9),
Arc°sh l] ~ Yi" i ) = Araosh
ao0
260
+ a30
§9.2]
and so
sinh
(ii)
Thus, from equations
(12)
h(y,x) (9),
2
(I0), (II) we find that
yxo : yuo oo~h h(y,x)
(13)
½
= (-1 + ao0)
[ (y~)2 + ( y2) 2
,
x 2]½ = [yyo[ sinh h(y,x ) + (y3)
and
(14)
c~xI) 2 +
(~2~ x 2
Coordinate
+ cyx~ 2 ] ~l~Xot-:
= tanh
h(y,x)
which have ao0 = 1
transformations
leave the origin invariant,
by equation
having specified
say m, any choice of a system of
homogeneous
an origin,
coordinates
class of transformations
is arbitrary
(i0); so we see that
to within an equivalence
having
ao0 = 1 and also, by equations
(2) and (3),
a 0 = aoa = 0 so equations
(~ = 1,2,3)
(2) become
261
,
§9.23
3
C153
~ a~6 ax6 6=I
which are the conditions three-dimensional
6 y
=
(~,y = 1,2, s) ,
for orthogonal
euclidean
space.
transformations
The quadratic
in a
form
(z~)2 + (z~)z + (z~) 2 is therefore coordinate
invariant with respect to the subgroup
transformations
In the following mappings
between each
bolic space
(H3).
having
section,
ao0
=
I .
we will consider
To clarify the distinction
for corresponding
particles
~
3SP correspond
and, for any two particles
Q,S
IrQsr =
to q , s , u , w
e 3SP,
h C q , 8)
262
hyper-
between the
upper and lower and points.
for example, Q,S,U,W
isometric
3-SPRAY and three-dimensional
S-SPRAY and H3, we will use corresponding case symbols
of
e H3
Thus,
§9.3]
§9.3
Space-Time
Coordinates Within the Light Cone
In this section we will define space-time
coordinates by
e s t a b l i s h i n g a c o r r e s p o n d e n c e between the 0-component of homogeneous
coordinates and the t-component of p o s i t i o n - t i m e
coordinates which, we recall, apply only to the r e s t r i c t e d case of " o n e - d i m e n s i o n a l motion".
The reader may already have
noticed the similarity between the formulae of §9.2 and the Lorentz t r a n s f o r m a t i o n formulae.
T H E O R E M 58.
Each 3-SPRAY is a hyperbolic space and so any
particle in a given 3-SPRAY can be represented by a set of homogeneous
coordinates.
natural time scale, each particle
Now, given a particle S with a
we can define a mapping TS such that for
Q g $SP [So].
TS:
(i)
S
qo
_.~
t
=
S
qo
and then (2)
where
I(qsl)2 + (q2)2 + (q3)2] ½ = Ix' (x;t) are position-time
col[Q,S],
of any event
The quadruple
coordinates,
,
relative to S in
[Qw ] c o i n c i d e n t with 9"
[qSo,qSl,q~2,q3s] is called a set of
space-time coordinates
of the event
263
[Qw ] .
§9.3]
relative space
to a c o o r d i n a t e
s y s t e m having S as an o r i g i n
[S O ] as an o r i g i n
and
time
is called the
coordinate space
coordinates
are called
coordinate8
are d e t e r m i n e d
space
O-coordinate
The
and the r e m a i n i n g
coordinates.
to w i t h i n
three
the space
Thus
an a r b i t r a r y
orthogonal
transformation.
Furthermore, such
in s p a c e - t i m e .
in
that
events
for any
[@,S,U],
coincident
same c o o r d i n a t e as an o r i g i n
two p a r t i c l e s
any
two sets of s p a c e - t i m e
coordinates
with
Q and U, r e s p e c t i v e l y ,
relative
system
h a v i n g S as an o r i g i n
in s p a c e - t i m e , S
S
are r e l a t e d S
S
S
: u3
"
S
-8
In p a r t i c u l a r ,
if [qo, q l , q 2 , q 3 ] and
S
S
coordinates,
two e v e n t s
8
S
relative
coincident
to the and
-S
Q,
[S O ]
-S
[qo, ql,q2, q3 ]
are
the
to the same c o o r d i n a t e
with
of
by the set of e q u a t i o n s
S
ql
of any
in space
: Ul = q2 : u2 = q3
(3)
space-time
@,U ~ 3SP[So]
system,
then
s -8 s -8 s -s s -s qo : qo = ql : ql = q2 : q2 = q3 : q3
(4) REMARK.
An origin
equivalence
class
in space
is only d e t e r m i n e d
of p e r m a n e n t l y
coincident
to w i t h i n
an
synchronous
particles.
This and 57
theorem
(§9.1).
61 (§9.5)
and
is a consequence
It is used
of Theorems
in the proof
62 (§9.6).
264
56 (§8.4)
of Theorems
59 (§9.4),
§9.3] PROOF.
Let Q be any p a r t i c l e in 3 S P [ S O] and let {S e : ~ real.
S a e COL[Q.S]}
be an indexed class of parallels
such that for all real t,
0 S t ~- S t Now, by T h e o r e m 56 (§8.4),
{[QW ] : Qw
{[S:]
s Q} =
and we note in particular,
(5)
x/t
where r is the directed
: x/t
=
tanh
r = u}
that
= tanh
r = v
.
r a p i d i t y of Q and S.
It is important
to realise at this stage that Q and S are not n e c e s s a r i l y synchronous;
furthermore,
and the above equation
equation
(i) of T h e o r e m 56 (§8.4)
(5) make statements which are independent
of the time scale of Q. Equation
(6)
I (q~),
(14) of the p r e c e d i n g
+
(q~)2
+
8.2)½, 8-1 J 'qo I
(q3'
section can be written as
=
tanh
h(q.s)
There is an obvious analogy between equations leads us to make the c o o r d i n a t e
time
8
(7)
TS, Q
: qo
identification 8
~
t = qo
265
(5) and (6) w h i c h mapping.
§9.3]
Since
IrQs [ equations
(5) and
8
hlq, s)
(6) imply that
which is equivalent 8
=
to equation
(2).
The coordinates
8
ql,q2, q3 are thus determined only to within a class of coordinate equation
transformations
which
(i5) of the previous
orthogonal
transformations
leave
section,
(8) invariant.
this is the class of
in a three-dimensional
space, which will be identified
later in Theorem Q ~
euclidean 60 (§9.4).
3SP[So] ,
Similar considerations
apply for any particle
so we define
T S which is an obvious extension
a mapping
By
of
the mapping defined by (7). The collinearity
conditions
eollinearity
condition
of equations
(4) is a consequence
three-dimensional (see Appendix
2).
(3) are equivalent
stated in Appendix
hyperbolic
2.
O
266
The final set
of representing
space by classes
to the
points
in
of quadruples
§9.3]
COROLLARY i.
If,
furthermore,
Q and S are
synchronous
and if,
in col[Q,S], 0
X
Qw = St then
the m a p p i n g
T S implies
"
that
w = q~ where
q~ is the O - c o m p o n e n t This c o r o l l a r y
of the set of c o o r d i n a t e s
is a consequence of T h e o r e m 56 (§8.4).
It is used in the proof of Theorems
PROOF.
q qi
59 (§9.4) and 62 (§9.6).
By T h e o r e m 56 (ii) of §8.4,
w = t sech r and this is analogous to equation
(13) of the p r e c e d i n g
section which is
q
= qo sech
h(q,s)
whence
w = qq
[]
267
,
§9.3]
If we
let
[yO,Yl,Y2, Y3 ] be the s p a c e - t i m e
an e v e n t
relative
in space
and
the p r e v i o u s coordinates
to a c o o r d i n a t e
[S 0] as the o r i g i n theorem satisfy
applies the 2
say that
these
vertex is [So].
Events
YO > 0 are
said
to be w i t h i n
to e v e n t s
2<
Y3
we
whose
see
that
space-time
are within within
2
YO
the
;
the light cone w h o s e light
cone w h i c h
have
(or YO < O)
the upper
(or lower)
vertex is [S O ] .
268
of
S as an o r i g i n
condition 2+
events
having
in s p a c e - t i m e ,
only
Yl + Y2 we
system
coordinates
light cone w h o s e
§9.3]
COROLLARY
2
(Position Space)
Given a coordinate space,
(1)
a set of events
system which has S as an origin in represented
by
[t, Yl,Y2,y 3] : t2>yl+Y2+y 3 : yl,Y2,y 3 constant;
is the set of events, with some particle given a particle
within
the light cone,
which is parallel
which is paraZlel
which
sented by
coincide
coincide
to S; and conversely,
to S, there are real
numbers yl,Y2, y 3 such that the set of all events, light cone, which
t real
with this particle
within
the
can be repre-
(I).
Thus any set of events
(1) is the set of events,
the light cone,
of an observer which is parallel
shall represent
this set of events
within
to S.
We
and the corresponding
observer by the triple
(Yl,Y2, Y3 ) • For any given coordinate system, the corresponding positionspace is defined to be the set of all particles which are parallel to, and synchronous with, the origin in space. position-space
can be represented
as
{(yl,Y2, Y3 ) : yl,Y2, y 3 real)
269
Thus
~9.3]
REMARK.
Each triple
represents
permanently
coincident
synchronous
with,
This 33
(§6.4).
Theorem
61 (§9.S)
PROOF.
By equations
the given number
in some
1 of
of C o r o l l a r y
3 of
theorem,
the set of
col and by e q u a t i o n
space,
from S.
if we take
(2), Since any real
a with 2
is some p a r t i c l e
YJ
Theorem
the
are
Y3
such that Q coincides [a,yl,Y2,y3]
Now by equa-
= col[Q,S]
we see that the
cone,
.
with
theorem
of S in c o l [ Q , S ] light
-
(4) of the above
33 (§6.4),
side
2 > 0
Y2
Q e 3SP
col
and by equations
2 -
coordinates
(3) of the above
within
to, and
(§9.6).
3-SPRAY is a h y p e r b o l i c
the event whose
same
82
are at the same distance
-
tions
of Corollary
(3) of the above
a2
there
of
in space.
and T h e o r e m
events
class
are p a r a l l e l
It is used in the proof
(i) is contained
all of these
w h ic h
is a c o n s e q u e n c e
Theorem
events
particles
the origin
corollary
an e q u i v a l e n c e
theorem
and C o r o l l a r y
set of events
and is t h e r e f o r e
which
coincides
270
with
the
1 to
(i) is on the set of events,
some p a r a l l e l
to S
§9.4]
Conversely,
given any particle
p a r a l l e l to S and both U and S are
[Yo(W),Yl(W),Y2(w),Y3(w)] [Uw] , equations y1(w)
: yl(O)
If
of any event
(2) and (3) of the above t h e o r e m imply that
= Y2(w)
: Y2(O) = Y3(w)
: y3(O)
= +1 : I
the positive sign must apply
for all w, which completes the proof.
§9.4
U is
c o n t a i n e d in some COL.
are the coordinates
U can not cross S in col,
~nce
U in p o s i t i o n space,
[]
Properties of Position Space
Before e s t a b l i s h i n g the main result of this section, we prove the following important p r o p e r t y of the synchronous relation.
This p r o p e r t y is also applied in Theorem 63 (§9.7)
where we show that each coordinate frame "can be calibrated in the same p h y s i c a l units"
THEOREM 59.
relation
The synchronous
relation
on the set of particles
is an equivalence
of any SPRAY.
This t h e o r e m is a consequence of Theorems
56 (§8.4),
58 (§9.3) and Corollary 1 of Theorem 58 (§9.3). in the proof of Theorems
PROOF.
It is used
60 (§9.4) and 63 (§9.7).
Clearly the synchronous relation is reflexive and
symmetric,
so we only have To prove transitivity.
Let S be
any particle with a natural time scale and let Q,U e SPR[S 0]
271
§9.~]
be any two particles which are synchronous with S, though not n e c e s s a r i l y homogeneous
synchronous with each other.
coordinates
Then the
of any particle W e S P R [ S o ]
with respect to coordinate
systems having S , Q
are related by t r a n s f o r m a t i o n s
,
and U as origins,
having the same form as equations
(7) of §9.2:
Ws
w.
=
a..
wq
~j
J
and
~j
=
*
and
wu
J
=
8
a..
w.
~J
J
~
w u.
= b..
w q.
b .*.
~
and
w so
~J
J
Thus, in particular,
qiu =
b i*j
ajk
qq
= b*ij
7X
~
bjk
uk =
aij
ajo
qq
bjo
u0
,
and similarly, uq
~
=
aij
U
whence u
qo : qq
=
~q
"
uo
where a is a constant.
=
coo boo
-
aSo bBo
Now the pairs Q , S
s~f a
and U , S
are
synchronous so, as in T h e o r e m 58 (§9.3) and Corollary I to T h e o r e m 58 (§9.3), identified.
all terms in the above e q u a t i o n can be
The second of equations
(ii) of T h e o r e m 56 (§8.4)
implies that a signal leaving Q at [ Q u ]
arrives at U at [u w]
where w =
u
u0 =
a
-1
u
q
=
a-lt
=
272
ua
-I
e
r
cosh
r
§9.4]
and
similarly,
a signal
leaving
U at
[U u ] a r r i v e s
[Qw ],
at Q~ at
where O
w = q~ = a It f o l l o w s that
the
that
-I u qo = ~-I t = u~-le r cosh r
Q and
subscripts
U are
synchronous.
u and w h a v e
statement
of T h e o r e m
56
(§8.4):
particles
U or W or to sets
the
(It s h o u l d
same
they
meanings
do not
of h o m o g e n e o u s
be n o t e d
as in the
refer
to the
coordinates.)
We
~
have
now
THEOREM
(i)
shown
80
that
the
synchronous
relation
(Properties
of P o s i t i o n
Space)
Every p o s i t i o n space.
space
is a t h r e e - d i m e n s i o n a l
If a p o s i t i o n
space
is r e p r e s e n t e d
~ ( y l , Y 2 , Y 3 ) : y l , Y 2 , y 3 real~ then yl,Y2, y 3 are o r t h o g o n a l Every p o s i t i o n
(ii)
synchronous
This Corollary and
57
theorem
space
cartesian
is an e q u i v a l e n c e
[]
euclidean as
, coordinates. class
of p a r a l l e l
particles.
is a c o n s e q u e n c e
2 of T h e o r e m
(§9.1).
is t r a n s i t i v e .
33
It is u s e d
(§6.4)
of T h e o r e m
and T h e o r e m s
in the p r o o f
273
i (§2.5), 46
of T h e o r e m
(§7°5), 61
56(§8.4)
(§9.5).
§9.4]
PROOF.
Let there by a y-coordinate
system which has the
particle S as an origin in space and the event origin in space-time.
[S 0] as an
Let Q and U be any two particles
in
3SP[S 0] which are synchronous with 4' and let [I,0,0,0]
,
be the space-time events
[qo,ql,q2,q3 ] , coordinates
[Uo,Ul,U2, U 3]
in the y-coordinate
system of
[SI] , [Qw ], [U s ] such that a signal goes from [S 1] to
[Qw ] and a signal goes from [Qw ] to [Uz]. theorem and equation
w = 1. exp
(i) of Theorem
56 (§8.4),
f~Qsf
z = w.~xpfrs~ t and by equation
By the previous
xp[i Q1+Qs4]
(ii) of the same theorem,
qo = w aosh rQS = exp IrQs I cosh rQS
,
u 0 = z cosh rus = exp[lruQ] + IrQSI] cosh rus whence
(i)
UO/qO = exp IruQI cosh rus sech rQS
By Theorem
57 (§9.1) and equation
(2) of Appendix 2,
cosh rUQ = I~(u,q) l[~(u,u)~(q,q)] -½
274
§9.4]
and so
exp IruQI=
[Q(u,u)Q(q,q)]½[lQ(u,q)l+{Q2(u,q)-£(q,q)Q(u,u)} 2
2
2
2~-%
2
2
2
2
= lfl(q,s) I [~(q,q)~(s,s)
cosh rQS = qo[qo-ql-q2-q3 a
~] ,
]-~
and
oosh rus = uO[UO-Ul-u2-u 3] Substituting
these
multiplying,
we o b t a i n
-% = l~(u,s)I[~(u,u)~(s,s) ]-%
relations
in
(i),
simplifying
and
cross
Q(u,u)- I ~2(u,q)l = [Q2(u,q)-Q(q,q)Q(u,u)] ½ which
becomes
sides
by
the
light
after
squaring,
simplifying
and
dividing
2 2 2 2 (which is not zero since Uo-Ul-U2-U3 cone w h o s e
vertex
is
both
[U z ] is w i t h i n
[So]) ,
(Uo_qo)2 = (ul_ql)2 + (u2_q2)2 + (u3_q3)2 whence
(3a)
Uo = qo + [(ul-ql)2+(u2-q2)2+(u3-q3 )2]½
the p o s i t i v e
square r o o t being t a k e n ,
[u z] ~ and,
by T h e o r e m
that
u 0 ~ q0"
56 (§8.4) The
since
[%]
and T h e o r e m
ambiguity
"
i (§2.5),
of sign was
275
it f o l l o w s
introduced
by the
§9.4]
operation of squaring:
(3b)
the case with the negative
UO = qo - [(ul-ql)2+(u2-q2)2+(u3-q3
)2]½
square root,
"
[S 1] to [U ] and from
corresponds to a signal which goes from
z
[U z ] to [QW] It follows from equations
(3a) and (3b) that any set of
particles r e p r e s e n t e d by
(4)
{(yl,Y2, y3) : Yi = mqi ÷ (1-m)ui ; i=1,2,3 ; ~ real}
is collinear before to T h e o r e m
[S O ] and after [So].
By Corollary 2
33 (§6.4), this set of particles
is contained in some
collinear set and, since no two distinct members
coincide
at any event, they are p a r a l l e l by T h e o r e m 46 (§7.5). equations
(3a) and (3b), it follows that any two
particles
in p o s i t i o n - s p a c e
the proof of p r o p o s i t i o n
are synchronous:
(parallel)
this completes
(ii).
Since p o s i t i o n space is an equivalence p a r a l l e l particles,
Also, by
the distance between
(Ul, U2, U 3) is given by lq 0
UOl
(ql,q2, q3 ) and
It follows that equations
(3a) and (3b) are forms of Pythagoras'
T h e o r e m in a three-
dimensional e u c l i d e a n space with o r t h o g o n a l which establishes proposition(i).
276
class of synchronous
[]
cartesian coordinates,
§9.4]
COROLLARY
(Orthogonal T r a n s f o r m a t i o n s
in P o s i t i o n Space)
Given a fixed origin in position space, all transformations between orthogonal cartesian coordinate systems are of the form
where [a B] is any orthogonal
3×3-matrix.
This corollary is used in the proof of C o r o l l a r y
3 of
T h e o r e m 61 (§9.5).
PROOF.
By equations
(7) and (15) of §9.2.
The fixed natural
time scale of the origin excludes t r a n s f o r m a t i o n s
Ya where ~ # I .
=
~a ~ x~ ,
x
=
~
-I a ~
y~
of the form
,
[]
As a consequence of the previous theorem, we can now define an equivalence r e l a t i o n of p a r a l l e l i s m between p o s i t i o n spaces. We say that two
(or more) position spaces are parallel if all
their particles are parallel, w h i c h means that their relative velocity is zero.
277
§9.s]
§9.5
Existence
of Coordinate
In the f o l l o w i n g can be assigned
THEOREM
61
(Existence
space-time
coordinate an origin
theorem
we show that
of Coordinate
S with a natural
coordinates
system
having
time-scale,
for all events,
S as an origin
in space-time.
to this coordinate such
real numbers
Conversely,
and events
Furthermore,
set of events
we can
relative
in space
and [S O ] as
any ordered
given
between
will
any particle
there are constants coincident
ordered
be called
with
quadruples
a coordinate
T which
to the given
Two events [y~,y~,y~,y~] (2)
with
coordinate
, (Yo-Yo)2
=
, (Yl-Yl)2
frame.
if and only if
+ (Y2 -Y2
278
with S
T can be represented
[yO,yl,y2,Y3 ] and
are signal-related
frame.
coincides
: -~ < ~ < +~}
coordinates
of
a, Vo, Vl,V2, V3 such tha~ the
the particle
{[~VO-a, hv1,~v2,hv3]
with respect
quadruple
of some event relative
by
(i)
to a
system.
correspondence
at some event,
coordinates
Frame)
[yo,Yl,Y2,Y3 ] is the set of coordinates
Any
space-time
to all events.
Given a particle define
Frames
)2
+
(,y3_y 3.)2
§9,5]
This t h e o r e m is a consequence of Axioms I VIII
(§2.10) and X (§2.12),
and Theorems
36 (§7.1),
(§2.2),
Corollary 2 of T h e o r e m 33 (§8.4)
48 (§7.5),
58 (§9.3) and
It is used in the proof of Theorems
62 (§9.6),
60 (§9.4).
63 (§9.7),
64 (§9.7) and 85 (§9.7).
PROOF.
(i)
We begin by considering the case of a particle S
S0 e S
which has some instant (In part
such that
SPR[S 0]
is a 3-SPRAY.
(ii) of this proof, we will show that all particles
have this property).
By T h e o r e m 58 (§9.3), there is a
coordinate system which has S as an origin in space and
IS 0]
as an origin in space-time. We first show that any given event, say [U ] , coincides with some particle
in the p o s i t i o n - s p a c e whose origin is S.
The case of an event which coincides with the origin in space is trivial,
so we consider the case of an event w h i c h does not
coincide with the origin in space. 33 (§6.4),
By Corollary 2 of T h e o r e m
such an event and the origin in space are contained
in a unique c o l l i n e a r set. existence of a p a r t i c l e
T h e o r e m 36 (§7.1) implies the
V which coincides w i t h the given event
and is p a r a l l e l to the origin in space.
Now take a particle
W e V such that W is synchronous w i t h S.
In p o s i t i o n space, the p a r t i c l e
(yl,y2,y 3)
The Signal A x i o m
W has coordinates,
say
(Axiom I, §2.2) implies that,
coincident with the origin in space, there is an event
279
ISb]
§9.5]
with coordinates
[Xo, O,O,O]
such that
IS b] o [Uc].
define the space-time coordinates of [U ] to be O
[x 0 + (y2+y2+ 1 2 Y32,½ j "
Yl "
Y2 "
Y3
We now
:
]
and we observe that this d e f i n i t i o n corresponds w i t h the previous
definition of §9.3, within the light cone, by
equation
(3a) of the proof of the previous theorem.
Conversely,
given an ordered quadruple
there is a particle
[yo, yl,y2,y3 ] ,
W in p o s i t i o n space with coordinates
(yl,Y2, Y3 ) and, by the Signal A x i o m (Axiom I §2.2), there is an instant W c ~ W_ and some event
[S b] w h i c h is coincident with
the origin in space and which has coordinates 2
2
2½
[Yo - (YI+Y2+Y3)
, 0,0,0]
, such that
[S b ] a [Wc ~ Thus
[W c] is an event which has coordinates [yo, Yl,Y2,Y3 ] . It is worth noting that the 1,2,3-components
of each
event are the coordinates of a particle in p o s i t i o n space which coincides with the event;
the 0-component of the event
is equal to the n u m e r i c a l index of the c o r r e s p o n d i n g instant from the particle in p o s i t i o n space. (3b) of the proof of T h e o r e m 60
Thus equations
(§9.4) apply to all signal-
related pairs of events, w h i c h establishes this case
(3a) and
(i).
280
equations
(2) in
§9.s]
In order to e s t a b l i s h first show that, 3-SPRAY.
(i) for this case
(i) we will
for any instant S a e S~ , S P R [ S a ] is a
T h e o r e m 36 (§7.1) implies that,
for any particle
£ SPR[Sa] , there is a particle R s 3SP[S O] such that II Q and conversely,
given any particle R s 3SP[S O]
there is a particle Q e S P R [ S
a
] such that Q~ II R.
Equations
(4) of T h e o r e m 58 (§9.3) imply that there are constants
Vo,Vl,V2, V 3 such that the set of events coincident with the p a r t i c l e R can be r e p r e s e n t e d by
(3)
{[hVo, lV1, hV2, lV 3] : -~ < ~ < ~}
By T h e o r e m 48 (§7.5), there is a time
:
eol[Q,S]
~
displacement mapping
eol[Q,S]
such that
furthermore,
T translates
the 0-component and leaves the sum
of the squares of the 1 , 2 , 3 - c o m p o n e n t s and R are both contained in col[Q,S], are proportional,
by equations
invariant.
Since Q
their l ~ 2 , 3 - c o m p o n e n t s
(3) of T h e o r e m 58 (§9.3).
Also
Q II R , so the previous two conditions imply that the 1,2,3 components are invariant w i t h respect to the m a p p i n g T~ since otherwise Q and R would cross at some event. all real k,
281
Therefore,
for
§9.5]
(4)
T : [lVo, lV1,lv2, hv3]-~
Thus, for particles
(5)
x 0,
=
Xo-a
,
[lVO-a, lv1, lv2, lv 3]
contained in SPR[Sa] , the coordinates
x I'
=
x I
,
are h o m o g e n e o u s coordinates,
x 2'
=
x 2
,
x 3'
=
x3
and the m a p p i n g T can be extended
to a b i j e c t i o n between 3SP[S O] and SPR[Sa].
Therefore
SPR[S a ] is a 3-SPRAY. (ii)
We will now show that each SPRAY is a 3-SPRAY by
showing that every particle has the property assumed in (i). There is at least one 3-SPRAY as p o s t u l a t e d in the A x i o m of Dimension
(Axiom VIII,
§2.10).
Now the A x i o m of Connectedness
(Axiom X, §2.12) implies that any event can be "connected to" this
3-SPRAY by two particles.
The result
(ii) above,
applied
twice, implies that the SPRAY specified by the given event is a 3-SPRAY.
(iii) (i).
Thus each SPRAY satisfies the a s s u m p t i o n made in
The mapping
(4) implies
(1).
[]
We immediately have the following: COROLLARY i.
Each
Consequently,
SPRAY
is a 3-SPRAY.
the results
[]
of the previous
to any SPRAY.
282
theorems
apply
§9.s]
COROLLARY
2
(Time Coordinate
Given any coordinate coordinate frame
frame
frame which
and any real number
is related
by the coordinate
3
(Coordinate
Given any coordinate
of equations
Transformations
frame,
any quadruple
[bo, bl,b2,b 3] and any orthogonal a coordinate
frame,
on a parallel coordinate
(i)
parallel
having
position
,
(5) of the above
given
position
+ z0 + b0 ,
za
metric),
metric
defined
to the given
transformations
÷ b ~ + aa6 z B frames with synchronous
(that is, parallel
by the transformations of real numbers
[a 6] , there is
is related
any two coordinate
the same euclidean
quadruple
which
Space)
of real numbers
the same euclidean
space,
spaces
in Position
3x3-matrix
frame by the coordinate
A : z0
Conversely,
related
coordinate
H
COROLLARY
having
to the given
+ [zo-a, zl,z2, z 3]
This is a consequence
theorem.
a, there is a
transformation
T : [Zo, Zl,Z2, Z 3]
PROOF.
Transformation)
position
spaces
the two coordinate
(1), where
283
are
[bo,bl, b2,b 3] is some
and [a 6] is some orthogonal
3x3-matrix,
frames
§9.s]
This corollary is a consequence of Corollary
2 of
T h e o r e m 58 (§9.3) and the Corollary of T h e o r e m 60 (§9.4). It is used in the proof of Theorems
PROOF.
62 (§9.6) and 63 (§9.7).
By Corollary 2 to T h e o r e m 58 (§9.3), there is a
particle
in p o s i t i o n space with coordinates
By the above theorem, there is a coordinate
(-bl,-b2,-b
3)
frame which has
this particle as an origin in space and the event, whose coordinates are [ - b o , - b l , - b 2 , - b
3]
frame, as an origin in space-time.
in the given coordinate The two coordinate
frames have p a r a l l e l p o s i t i o n spaces. By the corollary to the previous theorem, isometric t r a n s f o r m a t i o n s between coordinate systems
the set of all
(orthogonal cartesian)
in p o s i t i o n spaces, having these two
particles as origins,
is the set of all space coordinate
t r a n s f o r m a t i o n s of the form
(2) where
&
: za
÷ b a + aa6
[a ~] is any o r t h o g o n a l
The only space-time
zB
(~=1,2,3)
3×3-matrix.
coordinate t r a n s f o r m a t i o n s which
are consistent with the t r a n s f o r m a t i o n s
(2) and with equations
(2) of the above theorem have
(3)
~
: z0
÷ c + z0
where c is a constant.
,
Clearly c = b 0 .
The proof of the converse p r o p o s i t i o n is similar.
284
§9.s]
COROLLARY 4. number
Given
a coordinate
frame
~, there is a coordinate
the given
coordinate
frame
and a positive which
real
is related
to
frame by the transformations
(I)
x.
=
~
w.
This eoroiiary is used in the proof of T h e o r e m 63 (§9.7).
PROOF.
We shall caii the given coordinate frame the
w - c o o r d i n a t e frame.
There is some particle Q, p e r m a n e n t l y
coincident with the origin in space of the w - c o o r d i n a t e which has a natural time-scale an event
[Qx] are
frame,
such that the coordinates of
[ph,0,0,0]
By the above t h e o r e m there is
a coordinate frame, w h i c h we shall caii the y - c o o r d i n a t e frame, w h i c h has Q as an origin in space.
Equations
(2) of the above
t h e o r e m apply to the y - c o o r d i n a t e frame as welt as to the x - c o o r d i n a t e frame,
(2)
2
so 2
2
Yl + Y2 + Y3 =
U2
2
Therefore there is some o r t h o g o n a l
(3)
2
2
(Wl+W~+W3) 3x3 matrix
[a~6] such that
y~ = ~ a~6 w6
In accordance w i t h the previous
corollary, we define an
x - c o o r d i n a t e frame such that
(4)
Xo = YO
Combining equations
and
x
= a6a Y6
(3) and (4) ~hows that the x - c o o r d i n a t e
285
frame
§9.6] is related to the y-coordinate
§9.6
Homogeneous
frame by the transformations
Transformations
of Space-Time
(i). []
Coordinates
Having established the relationships between space-time coordinates and homogeneous
coordinates of particles
(three-dimensional hyperbolic) transformation
in each
SPRAY, the homogeneous
Lorentz
formulae can be derived by considering trans-
formations of homogeneous
coordinate
systems in three-
dimensional hyperbolic space. THEOREM 62
(Homogeneous
Lorentz Transformations)
Let Q and S be two d i s t i n c t instants
QO E Q and S O E S such
synchronous that QO = So
[Wo,Wl,W2, W 3] be the c o o r d i n a t e s coordinate an o r i g i n
frame
of any e v e n t
Q as an o r i g i n
is a n o n - s i n g u l a r
(I)
z. = a.. w. ~j J [zO, zl, z2, z3] are
relative
"
with
Let relative
in space
and
to a
[Qo ] as
in s p a c e - t i m e .
There
where
having
particles
to a c o o r d i n a t e
and [S O ] as an o r i g i n [aij] s a t i s f y
4x4 m a t r i x
and
w. = a*.. z . ~ ~J J
the c o o r d i n a t e s frame
[aij]
having
in s p a c e - t i m e ,
the c o n d i t i o n s :
286
such
that
,
of the same
S as an o r i g i n
event in space
and the c o e f f i c i e n t s
of
§9.6]
3
(2)
a2
2
~0
-
ao0
_
a208
=
-1
=
1
3 a 2
(%=1
~
(8=1, 2, 3)
,
8
(i,k=0, I,2,3
aoi aOk = 0
~=I aai a~k
and i ~ k)
3 a 2
~=1
0o~
2 ao0
-
=
-1
3
(8=I, 2, 8)
,
3
[ 0;=1
det[aij]
(3) theorem
Corollaries
and
PROOF. see f r o m
is a c o n s e q u e n c e
1 and
Corollary
the p r o o f
2 of T h e o r e m
3 of T h e o r e m
of T h e o r e m
If we d e f i n e Corollary
consistently are
= ~1
ao0 > I
This
and
(i,k=0,1, 2,3 and i / k)
ai~ ak~ - aio akO = 0
coordinate
frame
58
(§9.3),
(§9.5).
a mapping
(§9.3),
Theorem
61
It is u s e d
in
~S as in T h e o r e m
same
an a n a l o g o u s
theorem, mapping
~ o w by T h e o r e m which
58
(§9.5)
(§9.8).
i of the
define
synchronous.
63
61
of T h e o r e m
has
61
TQ,
(§9.5),
S as an o r i g i n
287
that
58 we
since there
in space
(§9.3),
we
can Q and is a and t h e r e
§9.6] is a coordinate frame w h i c h has Q as an origin in space, both frames having the event
For events within
IS 0]
= [Qo ] as the origin in space-time.
the light cone whose vertex is
[QO ]
(= [S0]) , equations
(7),
(2),
(i) and (2) are equivalent to equations
(3) and (4) of §9.2.
In order to e s t a b l i s h equation
(3), we observe that the first of equations
Ia00 I ~ I ;
also
ao0
must be positive,
(2) requires that
since otherwise events
w i t h i n the upper half light cone w o u l d t r a n s f o r m onto events within the lower half light cone. that the t r a n s f o r m a t i o n s
We have now e s t a b l i s h e d
(i) apply to the coordinates of
events w i t h i n the light cone whose vertex is
Q
S
Fig.
58
288
[Qo ]
§9.6]
In order to show that the t r a n s f o r m a t i o n s
(i) apply to
events w h i c h are not w i t h i n the light cone whose vertex is
[Qo ] , we take an a r b i t r a r y instant Qb e Q w i t h Qb < QO and consider the t r a n s f o r m a t i o n s which apply to events within the upper half light cone whose vertex is [Qb ] (see Fig.
58).
By
T h e o r e m 61 (§9.5) and Corollary 2 of T h e o r e m 58 (§9.3), there is a particle
U such that:
(i)
U coincides with Q at the event
(ii)
u II S ,
(iii)
U is congruent to S (see Fig.
[Qb ] (= [U0]) ,
and 58).
Then coordinate frames, having S and U as origins in space, have p a r a l l e l p o s i t i o n spaces so, by Corollary
3 of
Theorem 61 (§9.5), there is a linear t r a n s f o r m a t i o n of the form
61 : zi
~ xi = ci + zi
"
between coordinate frames having S and U as origins in space
and [QO ] and [Qb ] , respectively, as origins in space-time. Again, by the same corollary,
there is a linear t r a n s f o r m a t i o n
of the form
62 : Yi
-~ wi = di + Yi
"
between coordinate frames having origins in space which are p e r m a n e n t l y coincident with Q and having origins in space-time.
[Qb ] and [Q0] as
Now, as in the previous paragraph,
there is a linear t r a n s f o r m a t i o n of the form
289
§9.7]
D : xi between and
coordinate
Yi = A~~j xj
~
frames
having
U and Q as origins
[Qb] as an origin in space-time.
coordinate
transformations,
,
Combining
we obtain
in space
these
three
the t r a n s f o r m a t i o n
62 o p o 61 : zi-~ w i = d i + A~j cj + A*ij zj between
the w - c o o r d i n a t e
applies
to all events
frame
within
and the z - c o o r d i n a t e
the
light
cone whose
frame,
vertex
is
[Qb ] , and w h i c h must be i d e n t i c a l w i t h the t r a n s f o r m a t i o n within
the upper half
light
cone whose
vertex
which
(i)
[Qo ]
is
[Qb ] is a r b i t r a r y and since each event is contained
Since within
some
instant
light
cone h a v i n g
a vertex
[Qc ] , for some
Qc e Q , we see that the t r a n s f o r m a t i o n s
(i) apply to
all events.
§9.7
Minkowski
In this all
Space-Time
concluding
coordinate
frames
mations
between
optical
lines
them.
section,
we c h a r a c t e r i s e
by d e s c r i b i n g The
the coordinate
trajectories
are then d e s c r i b e d
frame.
290
the
transfor-
of p a r t i c l e s
relative
set of
and
to any coordinate
§9.7]
THEOREM
, such (i)
by
There
63.
the
coordinate
and
set
of all
class
z
~j
x.
is
of
- d.
j
[do, dl, d2, d 3] is
[aij]
coordinate
transformations
z. = a..
(i)
where
a maximal
of c o o r d i n a t e
frames,
that: is
~
is
an
an a r b i t r a r y
the
and
~
frames
~
= a..
z. + a..
zj
j
quadruple
4×4-matrix
are
related
form
x.
arbitrary
which
which
~j
of r e a l
satisfies
d.
j
,
numbers the
conditions: 3 ~ s=l
(2)
a2 sO
2 - ao0
= -1
3 a 2
s=l
s6
2_
- Co6
=
1
(6=1, 2, 3)
,
3 ~. s=l
asi
ask
- aoi
aOk
(i,k=0,1,2, 3 and
= 0
i ~ k)
3 s= 1
a2 OS
2
- Co0
= -1
3 a2 Bs
_
2 aBo
=
I
(6=1, 2, 3)
s=1
aks
- aio
ako
=
0
(i,k=0,1,2,3
3
s=l
ais
det[aij]
ao0
= ±I
,
and
~ 1
;
and
291
and
i ~ k)
,
§9.7]
given
(ii)
any frame
and some p o s i t i v e are r e l a t e d
not in
real
7,
number
by a c o o r d i n a t e
(3)
there
is some frame
~ # I , such
transformation
that
in
the two frames
of the form
x. = ~ w.
REMARK.
There are many classes of frames having the same
properties as
~
.
This is easily d e m o n s t r a t e d by choosing
any positive real number ~ $ 1 and applying the t r a n s f o r m a t i o n (3) to all frames in ~ .
This t h e o r e m is a consequence of Theorems 57 (§9.1), 59 (§9.4), 61 (§9.5) and Corollary T h e o r e m 62 (§9.6).
PROOF.
3 of 61 (§9.5) and
It is used in the proof of T h e o r e m 64 (§9.7).
It should be noted that the set of t r a n s f o r m a t i o n s
(i) with d. = 0 form a group
(as e x p l a i n e d in §9.2
which has a subgroup r e p r e s e n t e d by m a t r i c e s
Co0 = I ,
aso = 0 ,
and therefore the submatrix
[aij]
above) for which
Cos = 0
[a 8] is an o r t h o g o n a l 3x3 matrix.
We shall first prove a special case of (i) with
[do, dl,d2, d 3] = [0,0,0,0] Take any p a r t i c l e Q w i t h a natural time scale; then for each p a r t i c l e T e SPR[Q 0] there is some p a r t i c l e S e T such that S and Q are synchronous.
By T h e o r e m 59
292
(§9.4), this
§9.7]
sub-SPRAY consists of an equivalence particles.
class of synchronous
By C o r o l l a r y 3 of T h e o r e m 61 (§9.5) we see that,
to any given p a r t i c l e in this class, coordinate frames,
there is a set of
each having the given particle as an origin
in space and the event
[QO ] as an origin in space-time, whose
coordinates are related by t r a n s f o r m a t i o n s of the form:
(4)
YO = Xo "
where
y~ = b 6 x 6
[b 6] is any o r t h o g o n a l
symbol
~
to denote
in this way.
(~=1,2,3)
We shall
3x3 matrix.
the set of all
,
coordinate
Then, by the previous theorem,
frames
the
defined
for any two
coordinate frames in / , there is some 4x4 m a t r i x satisfying equations
use
[aij]
(2) such that the two coordinate frames
are related by the t r a n s f o r m a t i o n s
(5)
Yi Conversely,
frame in ~ there is a
(6)
=
aij xj
by T h e o r e m 57 (§9.1), given any coordinate
and any qx4 m a t r i x
[a..] satisfying equations ~J
(2),
particle T which coincides with the set of events
{[haoo,-Xao1,-Xao2,-ha03]
: -~ <
which are s p e c i f i e d in the given coordinate shall call the
x-coordinate
frame.
such that Q and S are synchronous.
X
< ~} frame, which we
Take a particle S e Then T h e o r e m 61 (§9.5)
implies that there is a w - c o o r d i n a t e frame which has S as an origin in
293
§9.7]
space and
[S O] (= [Q0 ]) as an origin in space-time, and
T h e o r e m 62 (§9.6) implies that the two coordinate frames are related by t r a n s f o r m a t i o n s of the form
w k = bkj xj
(7)
and
x k = bkj wj
where b.. is the inverse of b.. and ~J
~J
boj
(8)
in accordance w i t h
(6).
=
aoj
We now define a m a t r i x
[Cik] such
that
Cik
(9) and note that equations
def * - aij bjk
(6) of §9.2 and the fourth of equations
(2) above imply that 3
(i0)
CO0
=
aoj bjo
Now by definition,
both
=
Co0
-
a=l
[a..] and [b..] satisfy equations ~J
having the form of equations
~J
(2) above.
these equations form a group, matrix
Matrices
satisfying
as explained in §9.2, so the
[cij] satisfies equations (2).
above and equations
Oa
Therefore equation
(i0)
(15) of §9.2 imply that the submatrix
[caB] is an orthogonal matrix.
Corollary
implies that there is a y - c o o r d i n a t e
3 of T h e o r e m 61 (§9.5)
frame w h i c h is related to
the w - c o o r d i n a t e frame by the t r a n s f o r m a t i o n s
294
§9.7]
Yi = Cik Wk
(ii) Combining equations which completes
(7) and
(ii), we obtain equations
(5),
the proof of the converse proposition.
have now proved a special
We
case of (i) with
[do, dl,d2, d 3] = [0,0,0,0] Now given any event and any coordinate given event has coordinates, to the coordinate frame.
~
, the
[do,dl, d2, d 3] with respect
frame, which we shall call the y-coordinate
By Corollary
z-coordinate
say
frame in
3 to Theorem 61 (§9.5),
there is a
frame, whose origin in space-time
is the given
event and which is related to the y-coordinate
frame by
the transformations
(12)
zi
=
We shall use the symbol ~ frames defined in this way.
Yi
-
di
to denote the set of all coordinate Thus,
combining
equations
the form of (5) and (12), we find that any frame in ~ related to any frame in (i).
This
two frames
~
set of equations in
~
by equations
having is
having the form of
forms a group and therefore
are related by a coordinate
transformation
having the form of (i). Conversely,
given an x - c o o r d i n a t e
arbitrary quadruple
of real numbers
295
any
frame in
~
, an
[do, dl,d2, d 3] and an
§9.7]
arbitrary
4x4 matrix
a y-coordinate frame
in
~
[a..] satisfying
frame in
~
related
equations
(2), there is
to the x-coordinate
by
(13)
Yi
and, by definition
of
=
~
aij xj
ei
-
, there is a z-coordinate
which is related to the y-coordinate
(14)
~i = Yi + (ci-di)
Combining equations
(13) and
frame in
frame in ~
by
(14), we obtain equations
(i)
which completes the proof of (i). (ii)
Given any coordinate
frame, there is some
y-coordinate
origin in space-time. synchronous that the
origins
x-coordinate
say the x-coordinate
frame in
If these coordinate
in space,
not have synchronous
frame,
Theorem
frame is in origins
~
~
with the same
frames have
62 (§9.6) .
in space,
implies
If the two frames do Corollary
4 to
Theorem 61 (§9.5) implies
that there is some constant
and a w-coordinate
related to the given
frame,
frame by the transformations of the
w-coordinate
space of the frame is in that
~
~
.
frame.
This completes
is maximal.
x-coordinate
(3), such that the origin in space
frame is synchronous
y-coordinate
~ $ 1
with the origin in
Therefore
the
w-coordinate
the proof of (ii) and shows
D
296
§9.7]
THEOREM
64
(Particle
Given time
a coordinate
scale,
given
Trajectories)
the
frame
coordinates
and
a particle
of the
events
Q with
coincident
a natural with
Q are
real
,
by: z.(~)
(i)
= z.CO)
+ kv.
(i=O,1,2, SJ
,
h
where
v 02 - v I2 - v 22 - v 32 > 0
(2) Conversely,
given
any
the
above
specified
by
a natural
time
are
[zi(~)
This
scale
coordinate
such is
that
the
[Qk]
v0 > 0
frame
conditions,
: i=0, I,2,3]
theorem
and
of e v e n t s
there
is a p a r t i c l e
event
whose
where
is a c o n s e q u e n c e
a n d a set
Q~
Q with
coordinates
e Q .
of T h e o r e m s
61
(§9.5)
and
63 ( § 9 . 7 ) . PROOF.
The
theorem
and T h e o r e m
frame
which
first
has
proposition 6 1 (§9.5):
Q as an o r i g i n
in s p a c e - t i m e ,
so for
the e v e n t
are
[Q~]
is a c o n s e q u e n c e
any
we can in space
instant
wi(~)
and
a w-coordinate [Qo ] as an o r i g i n
QI e Q , the
= 6iOk
297
find
of the p r e v i o u s
coordinates
of
§9.7]
By combining the coordinate t r a n s f o r m a t i o n s
(i) and
(3) of the
previous theorem, we obtain the equations
zi(~)
= ~aio ~ - d i
which are equivalent to equations
v i = ~aio The inequalities
and
•
(i), with
z i(O)
= -d i
(2) are consequences of (the first equation
and the last inequality of (2) of) the previous theorem.
Conversely, matrix
[aij]
given equations
and define
previous theorem,
(3)
[d i : i = 0 , 1 , 2 , 3 ]
2
(2), we can find a and ~, as in the
such that
aio = ~-lu i 2
(i) and
2
and
d i = -zi(O)
2
where p = ( V o - V l - V 2 - V 3) By the previous theorem,
there is an x - c o o r d i n a t e
which is related to the z - c o o r d i n a t e frame by equations of the previous theorem,
and there is a w - c o o r d i n a t e
which is related to the x - c o o r d i n a t e of the previous theorem.
frame (i)
frame
frame by equations
(3)
Combining these coordinate trans-
formations, we obtain.
z i = ~aij
wj
298
- di
§9.7]
Inverting these equations and substituting from (i) and (3), we find that
Wk(~) = 6kO~ Now let Q be an origin in space of the w - c o o r d i n a t e
THEOREM 65
[]
frame.
(Optical Lines)
Given a coordinate frame and an optical quadruples of real numbers such that the coordinates
[x~,x~,x~,x~] of all events,
line, there are
and [c0, Cl,C2, C3] corresponding
to
instants of the optical line, are given by the equations (l)
(xi(~)-x~)/c
i
=
2
=
where ~ is a real variable and (2)
2
c0 Conversely,
2 -
c I
2 -
c 2
-
given quadruples
[c0, c1,c2, c 3] satisfying
0
c 3
[x~,x~,x~,x~]
and
(2), there is an optical line whose
instants are elements of the events specified by equations PROOF.
Both p r o p o s i t i o n s
of T h e o r e m 61 (§9.5).
are consequences of equations
D
299
(2)
(I).
§i0.0]
CHAPTER i0
CONCLUDING REMARKS
Our task is now complete in that we have described Minkowski
space-time in terms of u n d e f i n e d elements
"particles" and a single u n d e f i n e d
called
"signal relation".
We have
d e m o n s t r a t e d that our axiom system is categoric for M i n k o w s k i space-time.
However,
as m e n t i o n e d in the introduction, we
have not d i s c u s s e d the q u e s t i o n of independence of the axioms. It is quite likely that there is some i n t e r d e p e n d e n c e between the axioms and that the axiom system could be improved by the substitution of w e a k e r axioms.
H o w e v e r the author is aware
of c o u n t e r e x a m p l e s w h i c h can be used to demonstrate the independence of some of the axioms; namely, Axioms VII VIII
(§2.10), X (§2.12) and XI (§2.13):
(§2.9),
also certain subsets
of the other axioms can be shown to be i n d e p e n d e n t from those remaining.
Consequently,
the p o s s i b i l i t i e s
for m o d i f i c a t i o n
of the axioms are subject to a number of constraints.
Minkowski space-time signature
(+, +, +, -).
is a p s e u d o - e u c l i d e a n In many waFs,
space of
the de Sitter universe
is the c o r r e s p o n d i n g analogue of the n o n - e u c l i d e a n h y p e r b o l i c space and, in the present context, the most r e l e v a n t analogy is that p a r a l l e l i s m is not unique in these spaces.
The present
system of axioms can be m o d i f i e d so as to be valid p r o p o s i t i o n s
300
§lO.O]
in a de Sitter universe: Signal A x i o m
the p r i n c i p a l a l t e r a t i o n is to the
(Axiom I, §2.2) which must be m o d i f i e d to take
the de Sitter "event horizon"
into account.
All but one of
the remaining axioms can be altered slightly so as to be in accordance with the new Signal Axiom; the A x i o m of Connectedness
the only e x c e p t i o n being
(Axiom X, §2.12) which can be
r e - e x p r e s s e d in two different forms to c o r r e s p o n d to the two n o n - i s o m o r p h i c models of the de Sitter universe which are d i s c u s s e d by S c h r o d i n g e r positions
[1956].
W h e t h e r or not these pro-
form categorical axiom systems is a question w h i c h
remains to be investigated.
Finally, we remark that other directions of more general space-time Busemann
for i n v e s t i g a t i o n
structures have been d e s c r i b e d by
[1967] and Pimenov
axioms to ordered structures
[1970], who have applied topological called "space-times" whose
undefined elements are called "events".
Both of these authors
aim at extending our knowledge of p o s s i b l e space-time structures. They express space-time theory in terms of a single r e l a t i o n (before-after) and so their approaches are more akin to that of Robb
[1921, 1936] rather than to that of Walker
[1948, 1959].
Their methods have much in common with those of geometry and topology.
301
A p p e n d i x i]
APPENDIX
I
C H A R A C T E R I S A T I O N OF THE ELEMENTARY SPACES
In the present treatment we are interested in showing that each SPRAY is a h y p e r b o l i c space of three dimensions,
for
this p r o p e r t y of each SPRAY is intimately related to the Lorentz t r a n s f o r m a t i o n formulae
(see
39.6).
The p r o b l e m of
c h a r a c t e r i s i n g h y p e r b o l i c spaces is a special case of the famous " R i e m a n n - H e l m h o l t z "
or "Helmholtz-Lie" p r o b l e m which
is reviewed by F r e u d e n t h a l
[1965].
by Tits
[1953,
reasons:
A recent c h a r a c t e r i s a t i o n
1955] is used in the present treatment for two
firstly the dimension of repidity space need not be
assumed, and secondly the
"double
transitivity"
of the
motions of rapidity space is a consequence of the A x i o m of Isotropy of SPRAYs
(Axiom VII,
32.9).
The c h a r a c t e r i s a t i o n
by Tits and its proof are discussed by Busemann
[1955, 1970].
Given a n o n - e m p t y set X, a collection of subsets a topology
(i)
~
is
on X if:
%, X ¢ 7
(# is the empty set),
(ii) the union of every class of sets in ~
is a set in
and (iii) the intersection of every finite class of sets in is a set in 7 .
302
~ ,
Appendix
The
l]
sets
in the class
ological
space
(X, ~
points.
A closed
A neighbourhood
ical
is an open
section
set w h i c h
of a t o p o l o g i c a l of all closed
A topological two subsets
which
A class
the open
set in a t o p o l o g i c a l
is open.
a subset
are called
) and the elements
complement space
~
space
in a topolog-
the point.
If A is
of A is the inter-
c o n t a i n A.
is said to be c o n n e c t e d
are both open and closed
{0 i} of open subsets
if the only
are X and ~.
of X is said to be an open to at least
one Oi;
if
U O. = X. A subclass of an open cover which i an open cover is called a subcover. A compact space ological
space
A topological
in w h i c h space
has a n e i g h b o u r h o o d
with
A set X is called elements
locally
the d i s t a n c e
lowing
three
compact
a metric
x, y s X there
called
is a top-
between
subcover.
if each of its points
closure.
space
if to each pair of
is a real number
d(x,
x and y, w h i c h
y) ~ O,
satisfies
the fol-
conditions:
(i)
d(x,
y) = 0 if and only
(ii)
d(x,
y) = d(y,
(iii) for
compact
any x, y,
x),
if x = y,
and
z s X, d(x,
that
is itself
every open cover has a finite
is
its
is a set whose
of a point
cover of X if each point in X belongs is,
space
the closure
sets w h i c h
of the top-
of X are called
contains
space,
sets
y) + d(y,
303
z) >. d(x,
z).
Appendix
i]
The diameter of X is
sup d(x, y). x, yeX
bounded if it has a f i n i t e
is d e f i n e d 4: a = t
X(x,
into
in the < t
0
X.
~) =
usual
< ...
1
~
d x(
i=i Then
The
x and
for
t
), x ( t )
interval
(~ ~ t ~ 8)
any p a r t i t i o n
and ~ ( x ) d-e-fsup X ( x ,
~).
i
x(8))
~ ~(x,
X is arewise
y~ there
A metric
is a c u r v e
x and y, the
of the
lengths
that
whose
if,
for any two p o i n t s
end-points
if for
f r o m x to y.
X is a r c w i s e
distances:
are x and y.
d(x, y) is e q u a l
curves
A motion of the space preserves
connected
distance
of all
4) ~ ~(x).
d on X is intrinsic
function
points
which
of a c l o s e d
4,
space
presupposes
X is
< t k = 8 we p u t
d(x(~), The
mapping
length l(x) of a curve x(t)
way:
i-i
for e a c h
space
diameter.
A curve in X is a c o n t i n u o u s of the reals
A metric
any p a i r
to the
This
of
infimum
concept
connected.
X is a m a p p i n g that
is,
of X onto
a motion
itself
is an i s o m e t r i c
mapping.
A space for any two a motion
X has
a doubly
ordered
point
which
maps
the
transitive pairs
first
with
pair
304
group of motions equal
into
the
distances, second
if, there
pair.
is
Appendix
I]
A space points which
y,
X is isotropic
z e X such that
sends y into
is isotropic
i80tropic,
at a point
d(x,
y) = d(x,
z and leaves
z), there is a m o t i o n
x invariant.
at all its points,
or that
x e X if, for any two
we will
X is an isotropic
If the
simply
space X
say that
X is
space.
LEMMA
An arcwise
connected
topological
space
is connected.
PROOF
This
is a w e l l - k n o w n
[1962,
result:
see,
for example,
Mendelson
§4.6].
LEMMA
An arcwise it has a
connected doubly
metric
transitive
space
is isotropic
group
if and only
if
of motions.
PROOF
Let x, y,
z be three points
d(x, Then the o r d e r e d distances a motion pair
and so which
point
sends
y) = d(x,
pairs
double
in X such that
z).
(x, y) and
transitivity
the o r d e r e d
pair
(x, z).
305
(x, z) have equal implies
that there
is
(x, y) onto the ordered
Appendix
I]
Let x, x',
y, y' be any four points
d(x, Since
X is arcwise
If we take
d(w,
connected,
an a r b i t r a r y
x) and d(w,
arc length
y) = d(x',
along
point
are continuous
the
curve,
of real v a r i a b l e
a point
z on the curve
Now there y".
x) = d(z,
X is isotropic,
the
functions
functions
[1961]),
of
Value there
is
x'). there
some point
= d(x',y")
composition
shown that there
from x to x'.
Intermediate
(see Fulks
is a m o t i o n
y" such that
about x' w h i c h
of two motions
is a m o t i o n w h i c h
(x, y) onto the ordered pair
which
= d{x',y')
is an i s o t r o p y m a p p i n g
Since the
curve,
real-valued
so by the
theory
sends x onto x' and y onto
d(x,y)
is a curve
such that
d(z, space
y').
w on this
x')
Theorem
Now since the
there
in X with
sends
(x', y').
306
sends y" onto
is a motion,
we have
the ordered p a i r
[]
Appendix
i]
THEOREM
(Tits
[1952,
If X is a locally transitive
1955])
compact
connected metric space with a doubly
group of motions,
then X i8 finite-dimensional
is either an elliptic, euclidean or hyperbolic elliptic
or hyperbolic
elliptic or hyperbolic
hermitian
or quaternion
Cayley plane;
distance
These Tits
[1952;
function
ones which are unbounded
].
that a maximal
hyperbolic
hermitian
members least
5 members
hence,
§53] and
it is known 4 members
spaces of 3 dimensions.
hermitian
2N and the hyper-
points has at most
space of dimension
in the hyperbolic h e r m i t i a n
by comparison with the hyperbolic
3
2, at
spaces of dimension
quaternion quaternion
Cayley plane.
307
The
4N, where N is any
A set of equidistant
in the hyperbolic
spaces and the
points has exactly
and in all of the hyperbolic
4 dimensions,
/1955,
Furthermore,
spaces have dimension
in the hyperbolic
2N(N ~ 2)
spaces.
spaces have dimension
integer.
that
Of the above spaces the only
and hyperbolic
bolic quaternion positive
by Busemann
set of equidistant
in the euclidean
or an
y)) where d(x, y) is an
are the euclidean
various types of hyperbolic
space,
on the space.
spaces are described 1955 §II.E.
or an
with the reservation
the distance may be of the form g(d(x, intrinsic
space,
and
spaces and space of
Appendix
i]
We conclude this section by summarising remarks,
the previous
the theorem of Tits, and the preceding
two lemmas in
the form: If X i8 an unbounded metric
locally
compact a r c w i s e - c o n n e c t e d
space such that any maximal
has 4 members,
set of equidistant
then X is either a euclidean
space of 3 dimensions.
308
isotropic points
or a hyperbolic
Appendix
2(i)]
APPENDIX
HOMOGENEOUS
COORDINATES EUCLIDEAN
(i) Projective
n-Space
2
IN HYPERBOLIC AND SPACES
(see Busemann and Kelly
We first discuss homogeneous space of n dimensions,
x = (zo, z,,
where n is a positive
..., x n) and y = (Yo" YI"
of real numbers
(not all zero).
in projective
(n+l)-tupled.
m-space
Let
..., yn) be (n+l)-tuples
If the
(n+l)-tuples
are pro-
~,
by eZa88e8
representing
them is linearly independent
..., ~n are given representations
arbitrary point,
inde-
classes
is (n+1).
independent points,
of n + 1
and ~ is a given representation
of an
then the equations
~k :
[
xi Pk
of
{x I, x 2, ..., x m} is i n d e p e n d e n t
The maximum number of linearly
of (n+l)-tuples If ~ 0
integer.
can be represented
A set of points
if the set of (n+l)-tuples pendent.
in a projective
z and y are said to be members of the same cZaee.
portional,
Points
coordinates
[1953])
(k:o,1 ..... n)
i=o determine
the {x~} uniquely,
since the matrix
singular,
due to the {pi } being independent.
-i
[pk ] is nonHowever to
specify the point ~ it is only necessary to specify the
309
Appendix
2(ii)]
{x~} to within an arbitrary n o n - z e r o m u l t i p l i c a t i v e the {x[} are called p r o j e c t i v e the basis
{>i}.
=
(6
~,
...
is the K r o n e c k e r delta, the c o r r e s p o n d i n g p r o j e c t i v e
8~
coordinates are called s p e c i a l change
of x r e l a t i v e to
coordinates
If we define a basis
p where
factor:
of basis
of p r o j e c t i v e
projective
coordinates.
A
results in a linear n o n - s i n g u l a r t r a n s f o r m a t i o n
coordinates.
Three points x, y, z are collinear if there are real numbers a and b such that for all i E {0,1,...,n}
z i = ax i + by i.
(ii)
n-Dimensional Hyperbolic Geometry
All
(real) h y p e r b o l i c geometries
and curvature are isometric.
of the same d i m e n s i o n
A model of n - d i m e n s i o n a l hyper-
bolic geometry which has a direct r e l e v a n c e to the r e l a t i o n s h i p s between rapidity,
velocity,
coordinate distance and coordinate
time is the Hilbert g e o m e t r y whose domain is the interior of the unit n-sphere
E:
where
x'
( x 1,)2
i = x~/xo"
coordinates.
...
+
(x~)2
"
x 'n
=
+
...
+
(x~) 2 ~ I,
X n / X ° are special p r o j e c t i v e
From the i n e q u a l i t y above xz 1
+
x ~ 2
+
...
+
x 2 n
310
-
x 2 o
~
O.
Appendix
2(iii)]
With
(1,
c =
O,
bolic distance between
(i)
..., h(x,
c)
...
(Xo,
is related
+
constant
More generally~
Xl,
...,
Xn),
the hyper-
to the e u c l i d e a n
distance
c by
(xi)2] ½ =
k is a p o s i t i v e
is -k.
and x =
x and the centre
[(x~) 2 +
where
O)
tanh
h(x,
e)/k,
and the curvature
if we define~
of the space
for any two points
x
and y, ~(x,
y)
def =
xoy °
-
+ x
+
IYI
x2Y ~ +
...
+ X n Y n,
then
(2) (iii)
h(x,y)
n-Dimensional
A model relevance time
= k Arcosh{l~(x,y)I[~(x,x)~(y,y)
of n - d i m e n s i o n a l
in N e w t o n i a n
E:
kinematics
euclidean
(x~) ~ +
x I' = X l / X o ,
e =
(x~) ~ +
the e u c l i d e a n
distance
0)
=
(I,
0,
e(x,
geometry, velocity,
w h i c h has distance
(but not in the k i n e m a t i c s
...
"" . " x'n = Xn / x
With
eCx,
between
is the g e o m e t r y
coordinates.
(3)
Geometry
to the r e l a t i o n s h i p s
special r e l a t i v i t y )
where
Euclidean
] "½}
whose
domain
special
projective
0)
and x =
(xo,
...
1
311
is
are
c) is g i ve n by
[(x') 2 +
of
(x~) 2 < ~,
+
...,
and
+
( x ~ ) 2 ] ½.
...,
Xn),
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