Sbinburgb /Ifoatbematical tTracts No. S
RELATIVITY by
A.
W. CONWAY,
D.Sc., F.R.S.
Xon&on: CVJ
G.
BELL & SONS,
LTD.,
YORK HOUSE, PORTUGAL 1915.
Price 2$, net.
ST,
Edinburgh Mathematical Tracts
RELATIVITY
LINDW*-CO
PRINTERS 'BLACKFRJARS:
EDINBURGH
RELATIVITY
by
ARTHUR
W.
CONWAY
M.A. (Oxon.), Hon. D.Sc. fR.U.l.J, F.R.S. Professor of Matliemalical Physics in University College, Dublin
lon&on: G.
BELL & SONS,
LTD.,
YORK HOUSE, PORTUGAL STREET 1915
PREFACE ""THE
four chapters
before
the
Edinburgh Mathematical Colloquium on the
subject of Relativity. in
interests
which follow are four lectures delivered
other
As many
of the audience
branches of
mathematical
to treat the subject in the historical order to the stage in
If I
which
it
was
have stimulated any of
further,
I shall be
taken.
I
my
left
my
amply repaid
wish to express
science,
;
I
was
it
The best method appeared
necessary to start ab initio.
down
had their chief
to be
have brought
by Minkowski.
audience to pursue the matter for
any trouble that
I
have
thanks to the Edinburgh Mathe-
matical Society for the honour that they have conferred on in inviting
me
it
to give these lectures.
ARTHUR
325366
W.
CONWAY.
me
CONTENTS PAGE
CHAPTER EINSTEIN'S DEDUCTION OF
I.
FUNDAMENTAL RELATIONS
CHAPTER
MINKOWSKI'S TRANSFORMATION
-
15
III.
APPLICATIONS TO RADIATION AND ELECTRON THEORY
CHAPTER
i
II.
TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS
CHAPTER
-
25
IV.
35
CHAPTER
I
DEDUCTION OF FUNDAMENTAL RELATIONS
EINSTEIN'S
The subject of Relative Motion is one which must have presented itself at a very early stage to investigators in Applied Mathematics, but we may regard it as being placed on a definite dynamical basis by Newton. In fact, his first Law of Motion defines the absence of force in a manner which is independent of
any uniform velocity
When
frame of reference.
of the
the wave
theory of light became firmly established, and when a luminiferous aether was postulated as the medium, universal in extent, which carried such waves, attention was again directed to relative motion.
The questions at once arose Are we to regard the aether as being fixed Does matter, e.g. the Earth, disturb the aether in its passage through it ] The attempts to answer such questions gave rise to numerous investigations, experimental and theoretical, on the one hand to determine the relative motion of matter and aether, and on the other to give an adequate explanation in :
1
?
mathematical terms of the results of such experiments. As an two such experimental I shall briefly describe
introduction
researches which will bring us at once to the root of the difficulty. For those who wish to pursue the historical introduction to this
can refer to Professor Whittaker's
subject, I
"History of the
Aether," a book the value of which can be appreciated by anyone who has ever wished to trace the somewhat tangled line of thought in these matters during the last century. I shall first refer to Bradley's observation of the aberration of light. Let
S
(Fig.
be a source of light and
1)
observer moving with a 0'
by from
S
v.
A spherical
in
all
each point one equal c.
P
velocity represented
wave
of light diverges
Relative to
directions.
of the sphere has
to
an
the velocity
of
two
velocities,
light
c
and
in
the
direction i
,2
RELATIVITY
.
0'. The wave surface SP, and the other equal and opposite to is thus a sphere, the radius of which
relative to the observer is
expanding at the rate at the rate
moving
c
Or
v.
and the centre is again, the wave
surfaces are spheres, the point S being a centre of similitude. The ray direction, or direction in which the radiant energy travels, is along S to (Fig. 2), whilst the
P
the line joining
wave velocity is along S'P where SS'/SP v/c. The position of a fixed star as seen by an observer in a telescope moving with the earth
Fi$ 2
be the centre of the sphere of the system which passes through 0, and the real position of the star can be calculated from will
the triangle SS'O (Fig. 3) where SS'/OS = v/c and the angle OS' S is known. thus get
We
on the wave theory the same construction Underfor S as on the corpuscular theory. have we this however, explanation, lying
assumed that the aether of the earth
in the
at rest and
is
the motion of
the
earth,
is
neighbourhood undisturbed by
Fig 3
and we are thus
driven to the hypothesis of a stagnant aether, matter passing through without changing its properties. We now come to the classical researches of Michelson and Morley, which
were designed to test this hypothesis. A ray of light coming from a source
A
on plate of glass B (Fig. 4) at an angle of 45, and is partly transmitted to a mirror J/2 and whence it is reflected back along Z B,
falls
,
M
observing telescope at C by the other part goes over l twice in opposite directions and traverses the to
finally
the
reflexion at
B
BM
;
Fly*
B
and reaches C. Thus the two parts glass at of the beam are united and are in a position to
BM
is not equal to BM^ 2 This, produce interference bands if is on the supposition that the apparatus is at rest. Suppose, however, that it has a velocity v along AB the time taken for a wave to travel from B to J/ is then I
however,
;
light
where
I
is
the length of
2
BM^
and
it
takes
a time
/ (c
l/(c
+ v)
v),
to
EINSTEIN'S DEDUCTION OF travel back along
BM
{
FUNDAMENTAL RELATIONS
Thus the whole time taken
.
21 C
3
is
v
( \
*\ C'J
if v is small compared with c. For the reflexion from J/ M will have moved to Thus light travelled from B to M,' (Fig. 5). if t is the time taken from B to J//, B J// = ct, = vt, and if M^B = I, we have I* -f vH* = c 2 t 2 J/! ./ and therefore t = l/ J(c 2 - v 2 ), and the time
approximately
1}
1
MJ
whilst the
,
taken to come back to
Comparing have
B
is
this with the expression above,
time equal to the time over J5M being less than a
difference
of
we s
lv~/c
,
M
by that amount. By rotating the whole apparatus through 90, the conditions are reversed, and the time difference of path will be of the opposite In the actual experiment was about 11 metres, and sense. l l
2
BM
on taking v to be the velocity of the earth in its orbit, a displacement of 0-4 of an interference fringe width might be expected. The unexpected, however, happened, and the observed displacement was certainly less than one-twentieth of this amount,
and probably less than the one-fortieth. Here we have at once a result at variance with our hypothesis An explanation was put forward that the aether is at rest. and Lorentz, and developed in detail independently by Fitzgerald If force of the attraction the latter. between two molecules in by motion is less when they are in motion at right angles to the line joining them than when they are in motion along that line, then a line such as BM^ will measure less when it takes up the position and we can easily see that this contraction (called the Z
BM
,
Fitzgerald- Lorentz contraction) could compensate for the difference of path in such a way that the null effect of the Michelson-Morley
The mathematical experiment would be adequately explained. treatment of this theory in the hands of Lorentz and Larmor shows that all experiments on the relative motion of matter and aether must give approximately a null result. The interpretation of these experiments received a different
treatment from Einstein,
whose
paper in
the
"
Annalen
der
RELATIVITY
4
Physik," 4te Folge. 17, 1905, gave rise to most of the recent theoretical investigations which are included under the head of I shall now proceed to the deduction of the "Relativity."
fundamental equations of
this theory, following the
above-named
paper.
Let us consider a set of axes or frame of reference, which we term (merely for the sake of distinguishing) the fixed axes, and let there be another system of axes moving with reference to We shall the former, which we shall call the moving axes. suppose that the directions of these axes are parallel, and that the origin of the latter system moves along the #-axis of the former shall
with a uniform velocity
Fixing our attention for a while on
v.
the fixed system, let us suppose that at various points of space we are provided with clocks, the position of each clock being supposed fixed with reference to the axes, i.e. they are not in motion.
We
will further suppose that these clocks are synchronized, and that the synchronization is affected by the following process, which is in effect our definition of Suppose that two synchronization.
points A and B are provided with clocks, and that a beam of light flashed from A to B and reflected immediately back from B to A.
is
The time at
B
of departure of the
being
tz,
and
beam from A being
whilst the final arrival at
A
is
< ]5
and
of arrival
at the time
3,
the
are indicated on the clock at A, .and t 2 is indicated on the clock at B. Then the condition of synchronization is t z -t l = t s - 1.2 or t + t 3 = 2t.2 This of course agrees with our ordinary definition of synchronization ; in fact, it states merely
times
j
ts
.
l
that the time taken for light to travel from A to B is the same as that taken to go from B to A. This definition, obvious as it appears, is the very foundation of relations of Relativity.
this
method
of
deducing the
We
now lay down two fundamental assumptions. The equations by which we express the sequence of natural phenomena remain unchanged when we refer them to a set of axes moving without rotation with a uniform velocity. 1.
We
the " Principle of Relativity." see at once that it forbids us to hope ever to be able to determine the absolute motion
This
of
is
our reference system.
scheme
of dynamics,
It
is
although
it
obviously true for the ordinary should be borne in mind that a
physical quantity measured in one system
may
not be equal to the
EINSTEIN
S
DEDUCTION OP FUNDAMENTAL RELATIONS
5
corresponding quantity measured in the other. Thus while it is true that the change of Kinetic Energy in each system is equal to the work done by the forces, yet the number expressing the in fact, the Kinetic Kinetic Energy itself is different in each of a particle ra moving with velocity w along the #-axis is when referred to the fixed system, and is equal to ^m(w - 1?) 2 \rrawr ;
Energy 2
referred to the other system. This, as I stated, refers only to or Newtonian dynamics, which as we shall see, constitute ordinary
a particular case of the more general theory which we shall develop later. This principle of Relativity is in accordance with the null effect of the Michelson-Morley and other experiments on
the motion of the aether, and being thus in accordance with all known physical facts, is a valid basis for a scientific hypothesis. It will be noticed that the null effect is approximate after the
theory of Lorentz and Larmor, and that conceivably by a very great increase in the accuracy of our instruments an effect other
than null might be observed, but that this principle makes the null and incapable of ever being observed. The former
effect absolute
theory starts from certain principles and deduces the Relativity the latter starts at Principle as a final (and approximate result) Both theories traverse the this null effect and works backwards. ;
same ground, but in opposite directions, and experimental science The newer at present incapable of deciding between them. admirers more its to be to elegant (or theory, however, appears according to others, more artificial) in its formal presentation.
is
2.
The second
principle
we make
use of
is
that the velocity of
which is reflected from a mirror is the same as that of light coming from a fixed source. This principle, which follows from the
of light
ordinary equations of electrodynamics, has received definite experi-
mental proof recently in an ingenious experiment by Michelson (Astrophysical Journal, July 1913.)
We now proceed, having thus laid down our two principles. Supposing that in the moving system we have a system of clocks at rest relative to their axes, and that these clocks have been synchronized by observers in this system unconscious of their own motion. Suppose that there are two clocks in the system A and B at rest relative to the axes, and that a ray of light goes from A to B
and
is
by
the
reflected
back to A, the times being
Principle
of
r + rz =2r l must hold.
Relativity the
T,
r lt r2 as above.
relation
rl
-
T
Then
= r2 - ^
or
RELATIVITY
6
there will be a of this system (coordinates ?/, ) clock situated there. a indicated time the r, by synchronized This point has for coordinates x, y, z, referred to the fixed system,
To each point
,
time
and the clock
of this
system denotes a time
discover relations between
,
rj,
,
and
r
We have thus We can see
t.
x, y, z,
t.
to
at
once that the relations must be linear, otherwise the connections between infinitesimal increments of space and time would depend
For instance, the relation between a small and as observed in the other
on the coordinates.
triangle as observed in one system
must be the same wherever its
orientation
this triangle
In other
unaltered.
is
is
situated, provided that we ascribe no
words,
particular properties to the origin, which may be any point of This property is referred to as the homogeneity of space. space.
We
have then as the most general form of the relations the
four equations
,y
We
+C
z
z
could add an arbitrary constant to each equation, but without we can suppose the values of x, y, z, t, r to ?;, f,
loss of generality
,
more precisely, suppose that all the clocks in the fixed system are synchronized from a clock at the origin, and the same is done for clocks in the moving system, then at the instant that the "moving" origin 0' and the "fixed" origin coincide, the two clocks show the time zero. Suppose now be simultaneously
zero, or
in the moving system a ray of light starts from the origin 0' at the observed time r, and proceeds towards a mirror It is there reflected at placed at a fixed distance along the -axis. the time T I} and arrives back at the origin 0' at the time r.2 Let
that
.
us see
how
appears to observers in the fixed system. They will see an origin 0' moving along the axis of x, and at an invariable distance (which they find 011 measurement to be x') in front is all this
situated a mirror.
and the coordinates light is mirror.
The coordinates
of 0' to
them are
(v
t,
0),
A
of the mirror are (x +vt, 0, 0). ray of observed to start from 0' at a time t, and to overtake the
It
is
thence reflected back to the origin which is moving it. The reflexion thus takes place at the time
forward to meet
FUNDAMENTAL RELATIONS
EINSTEIN'S DEDUCTION OF
7
+ x'j ( V - v), and the reappearance of the light at 0' will be at the We have thus three events the t + x'j (V-v) + x'l (V+ v). We know arrival of the light. and the the reflexion, departure, t
time
the coordinates and the times of each of them in both systems, and on inserting in the last of the above linear equations, we get
The
relation of
= r-hT2 gives then synchronism 2r 1
Let the mirror now be placed on the observations be made.
We
rj-axis,
taneous values for the coordinates and the times
Making
and
let similar
have, as before, the following simul:
use, as before, of the equation
B
we get In a similar
0.
manner we get C =
X
and we thus have the
relation T
= D{t-VX/C Z }.
Referring again to the equations from which T and r were determined in the moving system, the beam of light takes a time - T to go from the origin 0' to the mirror. Thus the coordinate TJ l
must be equal to substituting the values of
of the mirror
On
Comparing
this with our
see that
TJ
assumed
= ^ A x+ = ^ ^ = C^ = l
we
c (TJ
l
-
and
T), r,
we
get
linear relation
y+C
l
and
V^- o
^-**v~
D
t
RELATIVITY
8
Let us next consider the relation between y and 77. Let A be a point on the Tj-axis, and let the distance AO' be found on measurement according to the fixed axis system to be y, and according to
The time t taken for light to go from 0' to ??. time T - T in the moving system is The yj J(c~- ). 2 - vx c' and on substituting x lit and t = y / v2 ), we ^/(c ), I
the moving axis to be
A
D
is
v
2
I
J
(t
get
In a similar manner we find
We
=Z>/\/l
.
z.
have thus got so far with our linear relations
77
r
= = D(t-vx/c-).
The constant D, which can be a function only In the
undetermined. then
<j>(-v)
=
<j>
(v),
first place,
the
for
we may
relation
of
notice that
between
v,
is
if
D
so far
9
(v),
and y from we might have
77
symmetry must be the same for v and - v. Now considered the moving axes to be fixed and the fixed axes to be moving, and have repeated all the above transformations which would have the same form, with, however, - v in place of r, y=
On
D
X/I--IT/C-
?/
substituting these values in the equations above,
so that
if
we denote
-
v~/c~
get
=
D = l//3,
we have and we arrive
1
we
at, finally,
the fundamental equations of Relativity
=
y
FUNDAMENTAL RELATIONS
EINSTEIN'S DEDUCTION OF
9
Before discussing these equations, it ought to be observed that they have been obtained by the consideration of light rays which passed along one of the coordinate axes, and as a verification we ought to consider the case of a ray which passes obliquely. Let a source of light be at the moving origin 0', and suppose that a beam of light starts from 0' at the time t and goes to a mirror " fixed " rigidly attached to the moving axes, and whose and that the beam arrives coordinates at the time t are x, y and
which
is
,
back again at 0' at the time
moving system since 00' vt,
the times observed in the
if
for these events are respectively TO
T
We
Now
tr
=$
(
-
t
,
T,
ru
we
have,
2
VX/C
)
have then to verify the equation
or, to
put
it
in a convenient form,
c(ti-t)-c(t-
t
Q)
= c
(v
{\
-
x)
t,
-(x-v to)}, J
In the diagram (Fig. 6) let 0' and A be the position of the origin and mirror at the time 0" and A' their positions at the time t, and ,
the origin at the time by an observer in the fixed O'A' = c(t-t Q )- 0"A' c (t, - t) 0'"
--=
f
0"0'"
= v(t
l
:
0"0'"
O'A' - 0"'A'
and
O'O"
= v(t-
= O'A
:
O'A'
) ;
0"'A, c
is
easily seen to be identical
with that given above. Returning to our fundamental ordinates of any
number
of
y z^, (x l
2 2/0 z.2 ) t
/*2 ,
suppose the coetc., in the moving
r/j
),
equations,
points system, and rigidly attached to it are that the corresponding coordinates (x
t
W-o-'&'-OW^
a relation which
l
Then
system. :
seen
as
all
lt
So that
-t).
O'O"
t
P
15
(
in
we have then equations
1}
etc.,
the
(,
>; 2
fixed
of the type
etc.,
and
system
are
Q,
RELATIVITY
10 SO that
fa-fi
=
*2-i,
etc.
Thus any geometrical figure in the moving system appears deformed to the observers in the fixed system. The lengths parallel to the #-axis appear to be decreased in the ratio of
1
:
whilst transverse lengths are unaltered. Thus the ellipse 2 + fP -rf = fP a?, the eccentricity of which = o?" v/c becomes the circle (x -vt)- + y~
/3,
is
t
A
line
Ag + J}rj+C = Q
transforms into
Ax +
/3rj
+ C/3 = Avt,
so that parallel lines transform into parallel lines, but the angle between two lines is usually changed. If, situated in the fixed
system,
we caught
a passing glimpse of a teacher in the moving
system proving to a class that the sum of the three angles of a triangle was equal to two right angles, we should get the impression that he was demonstrating a rather involved question in non-
Euclidean geometry, and that both teacher and class were either
all
stouter or all thinner than similar individuals in these countries, according as the axis of x or of y is the vertical.
As regards the
time,
it is
T
clear
from the equation
= (3(t-VX/C*)
will be behind the local time at the correspondTo take a numerical example, ing point of the moving system. suppose that v/c = 4/5, so that ft = 5/3, and that at noon the clocks and 0' mark the same time, at at that time coinciding with 0'.
that the clock at
If a person in the
and works
moving system
until his clock
shows
1
starts a certain task at
p.m.,
and
if
noon
at that instant he
catches sight of the clock at 0, which is passing by, he will find that it registers only 12.36 p.m. So that if he regulated his work to the he latter, according might easily achieve the result of getting more than twenty-four hours into the day. Again, if we suppose an observer situated at 0' looks at various clocks in the fixed system as they come opposite to him, we have, on putting in 0' = x = v t, t = fir, or in our example t = 5r/3. shall now consider some kinematical results. The com-
We
ponent velocities of a point in the moving system equal to
dr
and dr
,
and thus
w^ w^
are
11
%
dg
dx-vdt
dr
dt-vdx/c'
l-vW /c
2
x
where in
W
and
x
the fixed
W
are the components of the velocity as observed From the above we get the reciprocal system. y
relation w*
wt + _ __1__
v
l+vw^/c*' another form of which
is
Any one of these formulae gives us a solution of the problem of the composition of velocities having the same direction, or of finding the relative velocity of one moving point with respect to Some remarks may be made with
another.
In the
equations. if
the velocities
first place, if c
W
w
and
are
velocity of light, these formulae Wj.
W
x
=
= GO
,
respect to these the same thing, compared with the
what
or,
very small
is
become
W
v
x
= v + W$
which reproduce
the ordinary equations of relative motion. a second moving system, moving with respect to the first moving system with a velocity v in the same direction, and if wJ w are the components of velocity, we have
Again,
if
there
is
'
f But
c-
wJ + (v + v') I (1 + v v'/c"
2
I
+ w'
(v
+
2
v')
/
c (1
)
+ vv' /c 2
'
)
1
RELATIVITY
2
More
we
if
generally,
consider any
number
the relative velocities of the origins are
of
moving
v, v,' v", ...,
we
axes,
and
if
get
+u *
where sn
u=
{
sl
meaning the
1
+W
2
U/C
+ s3/c2 + s 5/c 4 +...}/{!+ sum
2 s.2 /c
+s
4 4 /c
+...},
of the products of the
quantities v, v', etc., to be noticed that the expression for u is symmetrical in the velocities v, v', v".... The various operations are in fact commutative, that is, we might have taken these
taken n at a time.
velocities in
It
is
any other order
For the velocity
Wrj
v", v, v',
= d^/dr we
....
get the expression
w.
w
W=
or
W is given by W*-W*+W?
The resultant
velocity
iv
2
+ T~ +
2
w v cos 6 -
v-
w~ sin 2
~~~F^l+wvcosB/c
We
}~
the angle between the velocity w( = Jw? + w -) and v. can draw conclusions similar to those above for different axes
where 9 all
1 c~
2
is
moving with the same velocity
We find,
however, that
if
v
and
in the
same
the velocities are not
direction. all in
the same
direction, the final result is not independent of the order in
the velocities
v,
v\
...
are taken.
To
which
illustrate, let us consider a point
P, which moves with a velocity w parallel to the ^-axis of a moving system, the origin moving with a velocity v along the -axis, and let us compare the resultant with the velocity of a point Q, which
moves with a velocity v along the f-axis of a moving system which moves parallel to the ^-axis with a velocity w. We have then for P
W
v
=
EINSTEIN'S DEDUCTION OF
FUNDAMENTAL RELATIONS
13
and for Q
W
'
x
W
'
y
= >/l - ^2 /c 2 = w.
.
v
Thus, though the resultant velocity has the same magnitude in each case, the directions as referred to the fixed axes are different.
As
a final example, let us consider a problem of a very familiar Two points A and B are moving along two rectangular lines
type.
AO
and 0, with velocities v and iv, the distance AO being equal and BO equal to b. What will be their shortest distance We find by ordinary methods apart, and when will this occur? to a
that the shortest distance
is
(bv~aw)
2
/ ^/(v
2
+
and that the
w; ),
2 reaching the shortest distance is (av + bw) / (v 4- w). These are the distance and time as observed by a fixed observer,
time
but
of
we
if
seek for the measurement that would be
A0 =
w//3 and to be replaced
by
aw)
(bv
/
av
2 J(v +
w
2
+ bw /3~ + w- /3~ 2
fi~~) l
The expressions for the transformation more complicated, thus d
d-
2
/3(dt-v dx/c
dw
i)
)
vy x
y 2
dr
dr"
of the acceleration are
dw*
w^
dr
dr-
and
/3
v2
d~
made by an
A we
Thus w becomes get different results. a becomes /3a, so that the quantities given above are
observer moving with
/3
{1
- v x/z 2 }'2
2
j3
{
1
-v x/c
3
2
}
Supposing a wave of light diverges from a point ocQ y zQ at a time the equation of the spherical wave is at the time t ,
(x If
we transform
-
2
x,)
this
+ (y-
y,Y
+ (*- *tf = c 8 (t
-
8
*
)
.
by our fundamental equations, we get - t/0 ) 2 + -T 2 - v (r - T 2 +
&
)]
(n
(r
-r
)
2 )
,
14
RELATIVITY
so that a spherical wave of light transforms into a spherical wave as it ought to do, after the Principle of Relativity. This identity (x
- x,Y +
(y
-
2
2/ )
=
(
+ (* - zo) 2 - c 2 (t - ) a 2 + (n - *ioY + (t - Q* - c >)
2
(T
-r
2 )
has been employed conversely to deduce the equations of Relativity.
We can make some interesting deductions from this identity. we suppose that the quantities x, z, t differ infinitesimally from xQ */o, z t m and if we put x - XQ = dx, etc., then we have the If
?/,
,
,
following relation between infinitesimals
If
we
:
+ dy- + dz~ - c dt 2 ds 2 3 d^ + dvf + dt?
dx~
call
,
2
and
,
then we have at once the relations
d
/dx
dt\
n = P(^~~ v ~r } ~r dads/ \ds
d^
=
dy ds
da-
dj; _
dz
da-
ds
dr
n (dt
dx
so that the differential coefficients
dx
dy
dz
dt
ds
ds
ds
ds
are transformed by the same transformation as x, y, z, t, or, to use the algebraic term, they are cogredient. Obviously this is true, if in place of x, y, z, t we had any set of four cogredient quantities.
Hence we have
as examples of cogredient quantities
CHAPTER
II
TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS
WE
pass on
now
to the
most remarkable application
of
these
equations.
In free aether the electric force (X, Y, Z] and the magnetic are related to one another by the Hertz-Heaviside (a., ft 7)
force
form
of
Maxwell's equations as follows _
i
_
]
^
a^__8^_8^dt
dz
dy
L _8^ |ar_^z ~
8T
dz
8
:
dL
'
dt
JbZ 87 dy
_^M _dx '
ct
1
a.
dz
dz
cz ox'
These equations involve two others. If
we
differentiate the first of each of those triads
by
x,
the
the third by z, and add each triad together, the righthand sides vanish and the left-hand sides become respectively
second by
y,
8
1
8
d
t
x
\
8
(
cL
\.
c
x
c
y
cM d
y
from which we deduce SJT
ex
cY_
cy
dM oF = 0. + -^ ox ^4--^ c y d z cL
Here (X, Y, Z) denote the force in dynes on an electrostic and (L, J/, J\T) the force in dynes on unit-magnet pole in the
unit,
electromagnetic system. Let us transform the second of these equations
16
RELATIVITY
-- vfti
=8 ^8r
we have dt
8
v
9
=
9
(3
^8~~~7~
8^ so that the equation
^8
8
becomes
9F
where
In the same way we can deduce the complete set of transformed equations in the form
_ __
cX' rt
cN'
8r
8
r/
_j
9J/'
_
.
9}
=
"
_ 1 8^'
o
cL' r;
8F 8
8 t
9 A"
8
8
8r
8
M'
_?2T
8r
A
9Z/'
8
^8^
a-\T
"" 8
dr
rj
T77
91
87;
8 o
77
f7t
d^
8 ^
TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS where
X' =
X
17
=L
L'
c
We
have thus the remarkable fact that the equations which express the interconnection of electric and magnetic forces remain of the same form when transferred to moving axes. This, granting the Principle of Relativity, may be looked on as a proof of these equations. If we in the fixed system observe any point charge to have unit strength, then in the moving system the charge must be observed to
have the same strength, and so we have the following stateLet a unit charge be situated at 0' and move with it. It
ment
:
will experience a
fixed system
we
mechanical force of (X', Y', Z') dynes. will observe a mechanical force
In the
we have an electric force X, /3 Y, /3 Z, and an electromotive which is /2 times the vector product of velocity and the magnetic force. When the ratio v/c is small, /3 is nearly equal to 1, and these expressions agree with those of Maxwell for the above so that
force
forces.
We now
pass on to the case in which, instead of free aether, we electric currents of strengths in electrostatic units given by the components U, V, and W. The electrodynamical equations are then modified by the introduction of certain terms.
have present
Thus,
^ A'
\_cA
T
cM
,dt
Joy
dz
+ 47rH/ ^~87 C.
_dM )~^~x
18
RELATIVITY az;
.
_
_
1
t
Cy
cz
cM
cX
Ct
d z
cZ ex
l
dtf_dY
cX
dx
dy
'
c
If p is the
t
volume density, we have also the equation of con-
tinuity
cV_
cV^
ex and
cy
cW
cp
cz
ct
finally
cX
cY
+ ^ox Tcy
cZ
=
+ ^r
47rp
cz
dM cX = - 3L
On
+
+
ex
transforming as before we get equations of the type
cX' C-
, 1
T /
etc.,
where X',
F', Z', L',
notice that
U
Jf
cN
\
1
cM'
47T
9
We
0.
cz
cy
V'=
V
W'=
W
Wp
V
?;
etc.,
have the same meaning as before, and
JT"
,
a
is
cogredient with
xy zt.
also notice that the velocity of a convection current at
We may any point
given by the vector U/p, V/p, W/p, and from the above expressions these satisfy as they ought the laws for the composition of is
velocities.
If there is a distribution of electricity p' at relative rest in the = and and moving system, we have U' =
W
V
= fip(l -v z /c-) = p/3~ p' forms into p dx dy dz
l ;
;
an element
into an equal element of charge, for p'
d
U=vp
volume d^ drj d transso that the element of charge transforms of
d-r]
d = p dx dy
dz,
TRANSFORMATION OP ELECTROMAGNETIC EQUATIONS
19
were spread on a surface, the element of which is d*2 and if the measurements in the fixed and direction cosines (A, v), d and are S m, (I, n) respectively, we have then from the system If the charge
yu,,
equations
d=/3dx',
d-r\
= dy\
d=dz
z= IdS = /3mdS
and so the surface density
a-'
and
a-'d2
cr,
on account of the equation
=
which expresses the invariance of a charge, give
We
have now materials for a complete transformation of any problem from one set of axes to another. We proceed
electrical
to
some examples.
A
unit charge fixed at
components
whilst L'
0'
produces in the moving system
of electric force as follows
:
= M' = N' = 0.
In the transformation we have
? + rf + ? = P*(x-vt)* + y' + z
i
,
and we have for the electric force due to a unit charge moving with a uniform velocity along the #-axis with a velocity v
and therefore
M=
Z
c
20
RELATIVITY
so that finally
P(x-vt)
Y=
y
Knowing the distribution of electricity on a conductor at rest, we can by the above methods obtain the distribution of electricity on a certain moving conductor. In fact, if V = V = W = 0, is of and we r, p independent get U=vp, F=0, W=0 and p = /3 p; '
and
if
the equation of the conductor
of the transformed conductor
Thus the surface density of
an
is
a-'
f(fi(x
we
previous equations
where 6
is
/(,
77,
vt), y,
= 0, = 0. z) f
)
the equation
by
of the ellipsoid is
a2
From
-
on a conductor which has the form
ellipsoid of revolution is given
where the equation
is
6
2
find that
the angle which the normal to the ellipsoid in the fixed #-axis, is the surface density on the
system makes with the conductor
which is moving with the velocity v along the on reduction, if we put a' = a/ ft,
x-axis.
e ' '
6
2
J
2
(x / a'
4
+
(y-
+ z2 )
/ 6
4
)
We
find,
TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS
The
forces can easily in any case be obtained Thus, in the electrostatic system,
21
by making use
of
the potential.
X'x
dV
-
-fl
r-'f Si)
V "8? *
Z>
where
the potential,
F,
a function only of f
is
,
;
77,
we
get then
easily
L=0.
X--/. dx
-r c
So that various
in the case of electricity at rest
forces
may
obtained
be
by
3 z
on a moving conductor the differentiating
a certain
potential function.
As an example, '21
and
of charge e
the electrostatic potential of a line of length per unit length is given by
V= e log& where the origin
0' is the
r
+ r' + l + r'-l
middle point of the
line,
and
above we get the forces due has now the transformed value V where charged x y, z, and t, i.e., if we put I - (31' and e = /3e (for the 2el must be equal to 2eT), we have
By means
of the equations
line
t
and
to a
moving
in terms of total charge
22
RELATIVITY This then gives us the potential of a moving charged
also gives the potential of a charged electricity being in relative equilibrium, it
line,
but
moving conductor, the and the form of the
conductor being a surface of the family
V This family
const.
is
= constant = 2/3 a
On
we
rationalisation
(say).
get
=
+ a-
l
'
p*(a*-f*)
This gives a family of spheroids, and by properly choosing I' we get a spheroid of any required eccentricity moving with a velocity 2 2 important case is when we take fi (a I"-) = a or I' = va/c. We have then the case of a moving sphere. Or, again, by taking I' = 0, we reproduce a case given above in which we get a prolate
v.
An
,
spheroid which gives the same forces as a moving point charge.
We waves.
now consider a we take
shall
If
X = XQ
case of the propagation of plane light
sin ^
A
(Ix
+ my + nz +
(Ix
+ my + nz + ct)
(Ix
+ my + nz + ct)
ct)
27T
Y= Y
sill
Z= Z
sin
L=L
sin ^
A
(Ix
+ my + nz + ct)
M= M
sin
A
(Ix
+ my + nz + ct)
= J\\
sin
(lx
+ my + nz + ct),
27T
Q
J\
A
27T
A
we have the electric and magnetic vectors for a plane-polarised beam of light coming from the direction (/, ra, n\ the wave length being
A,
and the amplitude
of the
component
electric
and magnetic
TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS
X
Z
23
vibrations being Y 0t LQ J/ JV etc. On insertion of these values in the electromagnetic equations we find that ,
X = m JV
-n
Y=nL,Z = Q
I
M
Q
IN,
M - mL
equations which express the facts that the electric and magnetic vectors are in the wave front and are at right angles to one another.
Let us see how
moving system.
this train of
We
find
waves would be measured by the
that the argument of the circular
function 2-jr
(lx A.
+ my + nz + ct)
becomes
or,
writing this in the form
we get l
+vIc
1+lv/c
This last equation gives the equation for the Doppler effect, and These latter show the first three give the effect of aberration. that since
m /m=n
through the If
I'
/ n,
the aberration displacement
o>axis.
= cos (6 + e) and I = cos cos (6
0,
+ e) =
we have COS0 + V/C 7v cos 6 c
1
+
/
.
is
in a plane
24
RELATIVITY
If v I c is small
cos (6
we get
+ e) = cos + (v I c)
sin
2
0,
or
e
= -v/c
sin
0,
which agrees with the observations on aberration. The relations between the values of the electric vector are
X' = X,
Y -mp(v/c)X
from which we could determine the change in the position of the plane of polarisation with reference to the axes.
CHAPTER
III
APPLICATIONS TO RADIATION AND ELECTRON THEORY In dealing with radiation generally we start with the electro-
dynamic equations. If we differentiate the third of these equations with respect to y, and the second with respect to z, and subtract the latter result from the former, we obtain
from which, on making use do. -
equation
--8/2
I
1
dx
8y- = dz
dy
0,
the fourth equation and of the
of
we
find
In the same way we find that every one of the six quantities Z, L, J/, ^V is annihilated by the operator
JT, T,
This operator plays a part in the theory of radiation similar to that of Laplace's operator ^ 8
8880
#2
,
dx
,
,
ot
oz
oy
8
TT
y
+ ^~ir 2
the
in
dz
theory of
8888
In passing, we may notice that
attractions and electrostatics. since
+^
are cogredient with
,
8
,
,
1
8f
drj
8r
we have 8
2
8
2
8
2
8
2
8
2
8
2
2
8
8'
2
,
'
8
I
x
2
may
8 y-
also
8 z
2
8
tr
/(<& sin ^ cos
I
$
~^~
8
2
F/8or
y sin ^ sin
Jo Jo
where
/
is
8
?
2
c
rj'
~
or
2
remark here that we owe to Prof. Whittaker a
general solution of the equation in the form
V=
8
an arbitrary function.
+8
(fa -{
2
1
Vfiy
z cos $
+ 82
c,
= c~ 8 2 2
Vjcz-
$,
^!>)
d
d
26
RELATIVITY
If x an d ^ are two general solutions of the above form, Prof. Whittaker has also shown that any solution of the electromagnetic equations can be put in the form
dx dy
dz dt
dx dz
~
o
2
we make use
3
$
cy If
dy
ct
2 \j/
ct
our Relativity transformation, we find the
of
very interesting fact that these remarkable functions \ and \// transform into themselves, and are in fact " absolute invariants " for our transformation.
and
\f/
but
symmetrical functions,
<
about the \[t s
<
1}
2,
.x--axis,
about the
^o', ^> 3 ',
so
that
!>
^-25
It will be noticed that these functions \
are symmetrical about the
^3?
^/,
more
not <
3,
<j> z
i^ 1?
and
0-axis. $.'5 V^s
^Ai>
V'a?
'/'s
general
^v where
form <j
and
can get a more
by
introducing six
^
are symmetrical
and 3 symmetrical about the y-axis, These get transformed into functions ^.2
according to the following scheme
we have the
When we
^
2>
We
a;-axis.
curious
result
are cogredient with
that
:
these
functions
r
JT, l
,
^,
Z/, Jl/, ^V.
wish to treat generally the solutions of the electro-
magnetic equations we begin by introducing, after the manner of It is Maxwell, a vector F, G, II called the vector potential.
APPLICATIONS TO RADIATION AND ELECTRON THEORY defined
as
follows
if
M N)
(L
27
denote the magnetic force,
we
have
M=c(-
This representation
If
is
we introduce
possible
on account of the fact that
these values in the equations 1
c
dX = dN -- dM ~ -^
-^
dt
we
i
dz'
dy
etc.,
find first that
are differential coefficients of a function 8
-
j
~v
'.
8
^_ From
the equation
8
and the equation 87^
we
find
-r
x
oas-
dG
-
+
8ic
8^/
The quantity have, in terms of
<
<
is
3
t/
JL
h
dy
d
=
8s-
oy*
8/7
8
+
t
dz
+ 55- + -r-^- =
9^
"
^
3 CiZj
h
+ -5
we have
8//
ecft
V
(say), so that
8
8z 3 "F" C/^rl.
/"
o^7
o
<
F
8
<
c
-r
c/>
=
c~ 2 ^-^-
or
0.
o
usually called the scalar potential, and we (F, G, 77), a means of represent-
and the vector
If ing the field, convenient for many purposes. have we the transformed quantities <', F', G', 77',
we
seek
now
for
28
RELATIVITY
and similar equations, so that
so that the four quantities cF, cG, c//, ^c" 1 , are cogredient with x, y, z,
L
We
have next to consider the question of dynamics. The Newton is known, by the accuracy of astronomical predictions and otherwise, to be true, at any rate, to a very high degree of approximation but, on the other hand, the velocities system of
;
relative to our axes of reference
which we consider in ordinary
dynamics, are very small compared with the speed of light. The inquiry then arises, What are the laws of dynamics when we are no
We
have such velocities longer restricted to such small velocities 1 in the famous experiments of Kaufmarm. In these experiments the /^-particles of
radium moving with
three-fourths
the
speed of
light
velocities almost as great as
were subjected to transverse
and magnetic forces, the direction of these forces being The displacement due to the electric force was in the plane containing this force and the direction of the velocity the displacement due to the magnetic force was at right angles to this From the observations plane arid was proportional to the velocity. " mass " increased as the was found it that the recorded, speed
electric
the same.
;
Various theoretical formulae were deduced by Abraham and others which agreed well with these results. The formula of 2 Lorent/, m / J(l -?r/c ), or m ft, where m is the mass for slow speeds, i.e., the Newtonian mass, gives, however, probably the best
increased.
agreements with the observed numbers. We shall now see what account the Relativity Principle gives To begin with, the velocity changed very little in actual of this. magnitude during the experiment, so that the motion is what is
termed "quasi-stationary." In other words, suppose the particle starts with a velocity v, then if we take axes moving with velocity v, the motion of the particle relative to those axes will be slow, and therefore the laws of Newton can be applied to such a motion.
APPLICATIONS TO RADIATION AND ELECTRON THEORY Suppose, then, that
e is
and the magnetic force
We have thus,
where
m
in the
the charge and that the electric force is F, M, we have then, in the moving system,
is
moving system, the equations
the Newtonian mass.
is
which we deduced earlier for x
=
and x = v, we
29
c
2 77
we
If
/ c -r
and
of motion
the
recall c
2
/ ?
r2
,
formulae
on putting
find
i.e.
m /3 y = Y
so that the mass is m /3, which agrees with Lorentz's formula and with experiment. To carry this theory further, we must consider the electrical Suppose that we have theory of inertia as applied to an electron.
a distribution of
electricity
throughout a
contained inside a certain surface, and
The volume density
electron.
(U, F,
JF).
electrical
This current
may
volume density, or
it
is
p,
certain
volume and
us term this system an and the current vector is let
be simply the convection of the
may
consist
partly of this and
The forces partly of a relative motion of portions of the electron. of will be assumed to be two kinds non-electrical, (1) acting first kind we will assume that they form a in themselves, other or, words, that amongst system Law of of and or its Newton's action reaction equality obey they more general expression, D'Alerubert's Principle, as used in deducing the equations of motion of a rigid body.* As to (2) we assume that the expressions for them are those given by the
(2) electrical.
Of the
in equilibrium
*
An
example of such a force would be a uniform hydrostatic pressure
over the boundary.
30
RELATIVITY
theory of Maxwell, i.e. the mechanical forces of electrical origin per unit volume are given by
(VN- WM)
c
c
(UM-
We
also
may
add the expression
for
VL). the
rate
of
working or
activity of these forces
A=XU+YV+ZW. If
we
express the fact that these forces are also in equilibrium
amongst themselves, we get
iff
I HI If
we
also
Q dx dydz =
R dx dy dz = 0.
assume that the total
I We
P dx dy dz =
electrical activity is zero,
we have
A dx dy dz = 0.
have also the equations of the couples
1 ii!
I These equations
and of
the
(yft-zQ)dxdydz = (z
P-xR)dxdydz =
(xQ-yP)dxdydz = 0.
form
the basis
motion of electrons in
of the electrical theory of inertia the same way that D'Alembert's
Principle enters into Dynamics. into this more fully would lead us too far into the but a simple example may make the matter of Electrons, Dynamics clearer. Suppose that an electron of charge e of any
To enter
symmetrical
APPLICATIONS TO RADIATION AND ELECTRON THEORY
shape
is
moving with a slow motion along the axis
Xm
influence of an electric force
X
X
Q + Xi, where the equation
t
becomes
P^x dy dz +
M^
X
e
Q
calculating
X
find that the equation
t
X
+
is
t
I
X pdxdydz = i
p dx dy dz
= 0.
and finding the value
Q
m
I
we
of the integral,
becomes
X where
of x under the then the total electric force is
from the motion of the electron, and so
arises
or
Now, on
31
a constant
-
e
mf= 0,
the electromagnetic mass
and
f
is
the
acceleration.
The particular shape of the formula will depend on our assumptions as to the structure of the electron but for motion, where the loss from radiation can be neglected, a convenient form which has many arguments in favour of it is the Lorentz mass;
formula.
This gives as the equations of motion of a particle
mx
d
my
d dt
x/
{
1
-
Fy F ,
z
is
(x
the mechanical force.
this mechanical force
is
of
electrical
assumption of Lorentz and Larmor
where
e is
j-,
+ y~ + z 2 )c~ 2lf
mz
d
where F&
2
the charge.
We may notice
origin
that
if
we have from the
32
RELATIVITY If
we introduce a
vector
Px P y P ,
,
such that
z,
P,
py~ *
p.
the equations can be put into a more symmetrical shape.
Writing
d cr2 = dt~ -
we
c~- (dx
2
+ dy~ + dz")
get
m We
also
^
=
P
z
.
have
m
-
= A
d
,
Px
where
on account
dx -- + -
_
dv
P,, -f--
dz -
^ + P.-
.
c-
A
dt
-
=
-
of the fact that
M and that therefore dx
d?
x
2 dy d y
dz dz z
2
dt
dz t
APPLICATIONS TO RADIATION AND ELECTRON THEORY
The quantity A
is
c~ 2
and the quantities and t.
We
33
-
P P P x,
z
y,
and A are cogredient with
a?,
y, z
one example of these equations of motion.* a that Suppose particle of mass m describes an orbit in the plane of (x, y), about a centre of force which varies inversely as the 2 of r the where is a constant ; distance, say square p. / /x shall only give
m
,
we have then
d
y
where or on putting
*J 1
-v
P'x
On
multiplying by
/c"
9
2 xvv/c = -
3
yw/C =
4-
/3
Py +
/3
02,
= /3 and performing
2
v2
= x* + y 2
the differentiations
-
Z
y and adding we get
VV
UL',
=
or
r
Hence we have the Energy Integral 1
where A
is
C.
f
From which
a constant. v
*
_2
2
r
.A
,
uu-
2
For many other problems see Schott, Electromagnetic Radiation.
34
RELATIVITY
In the same way, by multiplying by - y and
momentum
a?,
we
get the angular
integral /3
r1 6
=h
(a constant)
2
-
[l+
If
we put & =
(1
-
2
/r / c
2
L u = 1 + e cos 0', The general
effect is
c- 2
(A
+
-)]~
which leads to
A 2 ), we get an integral of the type
where
L
is
a constant.
the same as that of a force varying inversely
as the cube of the distance.
CHAPTER
IV
MINKOWSKPS TRANSFORMATION
We now
come to the concluding portion of our subject, namely, in which the preceding results have been stated by form the Minkowski. In the fundamental Relativity transformation let us x z a?s instead of x, y, z, and r/, f respectively, put ,
,
let us further i
,
>,
,
,
put x4 and
instead of
4
i ct
We
being the "imaginary" of algebra.
and
i
cr respectively,
then get
Introducing the imaginary angle 6 given by the equations = /3 } - sin O^iv /3 / c, the transformation becomes
cos 6
f!
= x
S2
= X2
4
=
l
Cj
cos ^
- x 4 sin ^
sn
+ x
cos
.
In order to study and interpret these equations, we shall consider first the motion of a point along a straight line secondly, the motion of a point in a plane and lastly, the motion of a point in ;
;
space.
The two
rectilinear
variables,
motion of a point along the re-axis is defined by It can be geometrically represented by a t.
x and
In attempting to represent curve, the familiar space-time graph. in the same way the relation between x and x4 considered as l
rectangular coordinates,
corresponding
we
are
met by the
graph may be wholly
difficulty
that the
or partially imaginary,
e.g.
as
36
RELATIVITY
in the cases x = ut x = ut + J gt Yet the representation of such a curve helps us very much to visualize the different relations, and in fact we are accustomed to such a procedure in geometry, as, for instance, when we draw the circules and tangents through them z
.
to conies, etc.
Let the curve
PQ
(Fig. 7) be then supto the of the particle motion posed represent as denned by a^ and x4 If the motion be .
referred to the origin v,
moving with velocity
the equations
=x 4
==
cos + #4 cos ; l x l sin 6 - x 4 sin 6
show that
this transformation is geometrically equivalent to the 4 referring system to new axes 0^, making an angle with the former axes respectively. This throws also a new light on the ,
transformation of velocity
dx_ V
d dr
dt
In which
is
l
easily seen to be equivalent to
tan tan
The is
dt
(the tangent at
the tangent of relation
2
dx /dx 4 is the tangent of the angle a., P) makes with Ox^ and that c?i/c? 4 is the angle /3 which AP makes with above 15 the
fact, noticing that
AP
dx
v c
p
1
a.
+ tan
- tan 6 z
.
or
tan v
between the accelerations form
relation
p = a. -
is
6.
more complicated, but
it
easily put in the
#6
r,
,
1 (*& Yl" -
**
WA *WJ\ -d^
We
see at once that this merely asserts that the ordinary expression for curvature gives the same result no matter what rectangular axes are used.
Coming now
to the motion of a point in a plane denned by the we see that the motion can be reprea curve in the three dimensional space of x lt xz x4 the
three coordinates x ly x 2 # 4 ,
sented by
projection of this curve
,
,
on the plane of
(a,,
,
x 9 ) being the actual path
37 If, now, the motion is referred to an origin moving with velocity v along the ce-axis, we see that this is equivalent to turning the system x x.2 x 4 about the axis of a5a and if the motion is referred to system moving with velocity v along
of the particle.
lt
,
,
the y-axis of this latter system, this is equivalent to a subsequent rotation about the #-axis of this latter system. The two operations thus described are not commutative. In fact, being finite rotations,
performed in the reverse order they would give a different result. however, the velocities v and v' are small compared with c, these This fact throws a light on the operations are commutative. if
If,
conception of relative velocities in the Newtonian system. What is fixed then about the path of the particle is the curve in the (a?!,
#25 #4)
any axes
space, which remains invariable, whilst to describe its properties.
we can choose
Before dealing with the general motion of a particle, we shall some results in the transformation of rectangular axes if we
recall
;
consider the scheme
2
Then the l\
A
1 19
quantities
+ 1? + If =
1
vector, that
;
= =
m-i
l->
x1
>\
lzt
+
xl
ll
4-
I.
2
etc.,
+k
x2
m.2 x2
such as
satisfy various relations,
+km
-2.
+ I3 xs + m 3 x3
= s
= m 2 n3 - m 3 n2
li
',
,
etc.
a directed quantity obeying the parallelogram components represented by distances taken
is,
law, can have its three
along the three axes, and will thus obey the same transformation We may notice that in all such cases as above. 2 i
+
&+(
2 s
= *i + x* + x
3
\
Certain operators obey the same laws, and Thus we easily see that vector operators. 3 rr-
3,8
3
3333 3333 =
li
3f ! -
3.2
=m
r- + k dx 1 -
l
ox l
+ ra 2
^r = n dx + 8^3 l ;
3
3
2
3
2
3
2
13
+m
-
3
cx.2
n^
l
2
+
3# 2
3
2
-
dx2
+ ns
3
2
^~ 3# 3
ox3
dxs
may
thus be called
38
RELATIVITY
This latter result expresses the "in variance" of Laplace's Conversely, the above transformations may be regarded operator. as tests by means of which we can recognise whether three quantities define
a vector or
not.
In the case of two vectors there are two quantities, one scalar, and the other vector, which express invariant properties inde-
pendent of the axes. Thus, if (x x.z x s ) and (a;/ x x3 ) are two vectors, which become (^ 2 3) an d (/ /) when referred to new axes, we have at once 1
>'
Each side is and is equal
called the inner product of the corresponding vectors, to the product of the magnitudes of the vectors
J(x* + x.? + x.A ') and between them.
,J(x^-
+x
2
+ #3
'
2
into the cosine of the angle
)
Also the three quantities
found to satisfy the test given above for vectors. They what is termed the vector product of the two quantities. They represent a vector the magnitude of which is the product of the magnitudes of the two vectors into the sine of the angle will be
constitute
between their directions, and the direction is at right angles to the two vectors. It may be noticed that one of the vectors employed as above may be a vector operator, thus, if (u-^ u.2 u s ) is a vector, the scalar quantity 8 MI
+
ox l
^
8
u<,
8
u.>
+ ^-?ox2
ox*
and the vector
us 8 x2 8
8
u2
8
xs
are related to the vector
c
u
8
x3
'
(u-^
l
u 2 u3 ]
8
u3
8
u2
8
u
8
x
8
x
'
8
X-L
in a
1
l
}
manner independent
of the
axes.
The above remarks will prepare us for a consideration of fourdimensional space, in which we have no geometrical intuitions to
MINKOWSKl's TRANSFORMATION
39
but have to rely on analytical transformations. general orthogonal transformation in four dimensions guide
us,
l
b2
= ^11 X + ^12 X1 + ^13 ^3 + l
=
21
X
X2 T
22
'
1
^23
^3
'
'l4
In a
X4
^24 *^4>
etc.,
where
i2
=
we have
etc.,
>ji
+ ^12 ^22 + Aa
*11 *21
and each quantity, such as Z u
+ l u 124 = 0,
etC.
equal to the minor of that quantity
is
,
>3
with proper sign in the determinant formed by the
We may
define
" fourvector,"
a
or
set
I's.
four
of
quantities
forming a vector, as a set satisfying the above transformation. We may notice that the four quantities may be vector operators, such as
8888
oxl J?!_
,
cx z
,
ox s
JL
8|f in the
dx 4
_?_ ~ J!_ 2
J!l
2
2
8f2
and that
,
,
dx 2
dx?
8J4
.8f3
82
_?!_ ~
dx3
2
8
"
dx?
same way as
When we come
to
two vectors we meet
as
before the inner
'
product x l a;/ + x2 x2 + xs x.J + x+x^, which we can easily verify to be a quantity independent of the axes of reference. If, however, we form we to the vector not four but six quantities get product, try 3i>.->
which we may write
for shortness 2fo ytl
We
'l/l'2
2/U
2/24 2/34
'
can see at once that these quantities are transformed by the
transformation *? 23
where the
A's are
= An
^o 3
+ A 12 y + etc., 31
the second minors of the
I
determinant.
six quantities transformed
call
Conversely, any the A, transformation
may
by what we may
be called a six-vector.
40
RELATIVITY
A four-vector and a six-vector can be combined in two ways to form a four-vector. Thus, if (z lt 2 z 3 4 ) is a four-vector, we have ,
,
*2 2/34 + % 2/42 + *4 2/23 Z3 2/41 + ^4 2/13 + 2/34 1
*4 1
2/12 + 3j 2/ 24 + 3o 2/ 41
2/23
+
2 2/31
+
3 2/12
5
,
,
,
and
+ ^3 2/13 + 242/14, + *4 2/24 + *1 2/21, Z 4 2/34 + 2/31 + *2 2/32 *1 2/41 + *2 2/42 + *3 2/43 *2
2/12
Z3
2/23
1
We
5
can verify by actual substitution that these two sets of
four quantities each are actually four-vectors after the definition given above. Also it may be remarked that the above statement holds true
when we
(8/a^, 8/e*2 9/8*3, ,
take, instead of (z lt z z z 3 ,
,
4 ),
the vector operator
dpxj.
After this preliminary survey of four- and six-vectors, let us return to the Relativity Transformation j
^4 It
is
=x
l
cos
- x4
sin 9
=x
l
sin 6
+ #4
cos
0.
is an orthogonal one, and an associated time t correspond
seen at once that the substitution
and that thus a point
(x,
y
t
z)
to a point in space of four dimensions. invariants as given above will now
A new
meaning
of certain
be at once evident. For or + + - cH~ becomes
instance, the in variance of the expression J in our new variables + 23 + 32 + 42 = X* a '
y~
+ x.? + x.? + o; 4
2 ,
which
can be interpreted as meaning that the distance of the point # 2 #35 a?4 ) from the origin remains unaltered by the orthogonal (ajj, As other invariants we might mention the transformation. j
+ dx + dx 2 + dx?) and the differential operator When we wave propagation ff/dx? + 9 /8a: 22 + d 2/dx./ + o-/dx*.
element of arc J(dx^ for
s
2
apply these ideas to the electromagnetic relations, a surprising If we denote the electric current symmetry becomes evident.
components U,
V,
and
W
by U^
U.^
and
U
3
respectively,
and the
41
volume density of electricity p by (i c)~ U4 we have then, since it was proved that 7, F, and p are cogredient with x, y, z, and the fact that Ult U2 US1 U4 is a four-vector. In a similar manner, if we use the vector potential (F, G, H] and the scalar potential and if we take quantities (Fu F^ F3 F4 ) defined by the equations ]
,
W
,
,
,
,
then (Flt F, F^
F
We
a four-vector.
is
4)
may
notice in passing
that the equation of continuity
becomes
From
the
vector
generalized potential
dF3 _8F2 '
dx.2
8a: 3
operator
d/dx lt d/dxz
F F F we
(F
2
l
dF,
dF3
dxs
dx l
2
4)
dF
dF
dx
dx 2
2
'
1
l
d/dx3
,
d/dx 4
,
and
the
can form a six-vector
'
dF4
dF
dx
dx4
}
1 '
a^__8^_ dx.2
dx4
dFj.
dFs
dx3
dx4
'
'
These become respectively
dy
cz/
A
or
dx /
\dz
Jf,
\dx
dy
/'
^, -iX, -iY,
-iZ,
which we may symmetrically write ^235
Thus the magnetic and
^125
^31)
^14)
electric forces
^24)
^34'
form a
six-vector.
Using the operator (d/dxlt d/dx2 d/dx 3 d/dx4 ) and this six-vector, ,
,
we can form two
four- vectors.
8Z 12 87/ 23
first is
CL 4Z
87/ 34
-
The
87/og
37^24 -p
-p
8Z 31
42
RELATIVITY If
we equate these four components
to zero,
we get
in the
earlier notation 3 Li T
~
dt
_
V
"'7
3
cy
dz
(jZj
i
c
dz
dt
dy
r
_ 8^ 1
_8T_8X
dt
dx
a" ex
+ r"
dy
-^ =
-
dz
dy
In the same way, from the other four-vector which can be formed, we can derive the other four fundamental electrodynamical equations.
As
a last example, consider the four-vector
these become on substitution
(WL-UN) -
The
first
VL)
three of these represent the mechanical force on an and the fourth is i c~ multiplied by the activity. l
electric charge,
For further examples, reference must be made to the work of Minkowski above referred to. What we have said is, however, sufficient to indicate the point of
ordinary space and a
view of this theory.
definite time
four-dimensional space. fixed curve in this space.
A
is
A
point in
represented as a point in
moving particle is represented by a The question of absolute rest in ordinary
MINKOWSKI'S TRANSFORMATION
43
For, in the fourspace ceases now to have any meaning. dimensional isotropic space, one set of axes is as good as another The various electrodynamical relafor describing its properties.
manner which reveals a symmetry which was by no means apparent in the unsymmetrical equations founded on our experimental knowledge. The whole scheme, in one aspect, is merely an analytical development of the Einstein Both would fall together if any experimental fact Relativity. appeared which would upset one, and can it not be said that the
tions take their position in a
probability of
symmetry
both
being
true
is
increased
by
this
elegant
*?
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