Relativity

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 ST,

1915.

Price 2$, net.

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 which follow are four lectures delivered
before the Edinburgh Mathematical Colloquium on the
subject of Relativity. As many of the audience had their chief
interests in other branches of mathematical science, it was
necessary to start ab initio. The best method appeared to be
to treat the subject in the historical order ; I have brought it
down to the stage in which it was left by Minkowski.

If I have stimulated any of my audience to pursue the matter
further, I shall be amply repaid for any trouble that I have
taken. I wish to express my thanks to the Edinburgh Mathe-
matical Society for the honour that they have conferred on me
in inviting me to give these lectures.

ARTHUR W. CONWAY.

325366

CONTENTS

PAGE

CHAPTER I.
EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS - i

CHAPTER II.

TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS - 15

CHAPTER III.
APPLICATIONS TO RADIATION AND ELECTRON THEORY 25

CHAPTER IV.
MINKOWSKI'S TRANSFORMATION 35

CHAPTER I

EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS

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 of the frame of reference. When 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
fixed1? 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
mathematical terms of the results of such experiments. As an
introduction I shall briefly describe two such experimental
researches which will bring us at once to the root of the difficulty.
For those who wish to pursue the historical introduction to this
subject, I can refer to Professor Whittaker's "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. 1) be a source of light and 0 an
observer moving with a velocity represented
by 0 0' — v. A spherical wave of light diverges
from S in all directions. Relative to 0
each point P of the sphere has two velocities,
one equal to the velocity of light c and in the direction
c. i

,2 . RELATIVITY

SP, and the other equal and opposite to 0 0'. The wave surface
relative to the observer is thus a sphere, the radius of which
is expanding at the rate c and the centre is
moving at the rate v. Or 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
the line joining S to P (Fig. 2), whilst the
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

will be the centre of the sphere of the system which passes
through 0, and the real position of the star can be calculated from
the triangle SS'O (Fig. 3) where SS'/OS = v/c
and the angle OS' S is known. We thus get
on the wave theory the same construction
for S as on the corpuscular theory. Under-
lying this explanation, however, we have
assumed that the aether in the neighbourhood
of the earth is at rest and is undisturbed by
the motion of the earth, 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
falls on plate of glass B (Fig. 4) at an angle
of 45°, and is partly transmitted to a mirror J/2,
whence it is reflected back along MZB, and
finally to the observing telescope at C by
reflexion at B ; the other part goes over BMl
twice in opposite directions and traverses the
glass at B and reaches C. Thus the two parts
of the beam are united and are in a position to
produce interference bands if BM2 is not equal to BM^ This,
however, 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 light wave to travel from B to J/2 is then I / (c - v),
where I is the length of BM^ and it takes a time l/(c + v) to

Fig 3

Fly*

EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS 3

travel back along BM{. Thus the whole time taken is

21 ( v*\

C \ C'J

approximately if v is small compared with c.

For the reflexion from J/1} M1 will have moved to MJ whilst the
light travelled from B to M,' (Fig. 5). Thus
if t is the time taken from B to J//, B J// = ct,
J/! .¥/ = vt, and if M^B = I, we have I* -f vH* = c2t2,
and therefore t = l/ J(c2 - v2), and the time
taken to come back to B is

Comparing this with the expression above, we
have a difference of time equal to lv~/cs,
the time over J5Ml being less than £M2 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
sense. In the actual experiment BMl was about 11 metres, and
on taking v to be the velocity of the earth in its orbit, a displace-
ment of 0-4 of an interference fringe width might be expected.

The unexpected, however, happened, and the observed dis-
placement 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
that the aether is at rest. An explanation was put forward
independently by Fitzgerald and Lorentz, and developed in detail
by the latter. If the force of attraction between two molecules in
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
BMZ, and we can easily see that this contraction (called the
Fitzgerald- Lorentz contraction) could compensate for the difference
of path in such a way that the null effect of the Michelson-Morley
experiment would be adequately explained. The mathematical
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

4 RELATIVITY

Physik," 4te Folge. 17, 1905, gave rise to most of the recent
theoretical investigations which are included under the head of
"Relativity." I shall now proceed to the deduction of the
fundamental equations of this theory, following the above-named
paper.

Let us consider a set of axes or frame of reference, which we
shall term (merely for the sake of distinguishing) the fixed axes,
and let there be another system of axes moving with reference to
the former, which we shall call the moving axes. We shall
suppose that the directions of these axes are parallel, and that the
origin of the latter system moves along the #-axis of the former
with a uniform velocity v. Fixing our attention for a while on
the fixed system, let us suppose that at various points of space we
are provided with clocks, the position of each clock being sup-
posed 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 synchronization. Suppose that two
points A and B are provided with clocks, and that a beam of light
is flashed from A to B and reflected immediately back from B to A.
The time of departure of the beam from A being <]5 and of arrival
at B being tz, whilst the final arrival at A is at the time £3, the
times £j and ts are indicated on the clock at A, .and t2 is indicated
on the clock at B. Then the condition of synchronization is
tz-tl = ts- 1.2 or tl + t3 = 2t.2. This of course agrees with our
ordinary definition of synchronization ; in fact, it states merely
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 this method of deducing the
relations of Relativity.

We now lay down two fundamental assumptions.

1. 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.

This is the " Principle of Relativity." We see at once that it
forbids us to hope ever to be able to determine the absolute motion
of our reference system. It is obviously true for the ordinary
scheme of dynamics, although it 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
Kinetic Energy itself is different in each ; in fact, the Kinetic
Energy of a particle ra moving with velocity w along the #-axis is
\ rrawr2 when referred to the fixed system, and is equal to ^m(w - 1?)2
referred to the other system. This, as I stated, refers only to
ordinary or Newtonian dynamics, which as we shall see, constitute
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
effect absolute and incapable of ever being observed. The former
theory starts from certain principles and deduces the Relativity
Principle as a final (and approximate result) ; the latter starts at
this null effect and works backwards. Both theories traverse the
same ground, but in opposite directions, and experimental science
is at present incapable of deciding between them. The newer
theory, however, appears to its admirers to be more elegant (or
according to others, more artificial) in its formal presentation.

2. The second principle we make use of is that the velocity of
of light which is reflected from a mirror is the same as that of light
coming from a fixed source. This principle, which follows from the
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 reflected back to A, the times being T, rlt r2 as above. Then
by the Principle of Relativity the relation rl - T = r2 - ^ or
r + rz=2rl must hold.

6 RELATIVITY

To each point of this system (coordinates £, ?/, £) there will be a
time r, the time indicated by a synchronized clock situated there.
This point has for coordinates x, y, z, referred to the fixed system,
and the clock of this system denotes a time t. We have thus to
discover relations between £, rj, £, r and x, y, z, t. We can see at
once that the relations must be linear, otherwise the connections
between infinitesimal increments of space and time would depend
on the coordinates. For instance, the relation between a small
triangle as observed in one system and as observed in the other
must be the same wherever this triangle is situated, provided that
its orientation is unaltered. In other words, we ascribe no
particular properties to the origin, which may be any point of
space. This property is referred to as the homogeneity of space.

We have then as the most general form of the relations the
four equations

,y + Czz

We could add an arbitrary constant to each equation, but without
loss of generality we can suppose the values of x, y, z, t, £, ?;, f, r to
be simultaneously zero, or 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 0
coincide, the two clocks show the time zero. Suppose now
that in the moving system a ray of light starts from the
origin 0' at the observed time r, and proceeds towards a mirror
placed at a fixed distance along the £-axis. It is there reflected at
the time TI} and arrives back at the origin 0' at the time r.2. Let
us see how all this 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
situated a mirror. The coordinates of 0' to them are (v t, 0 0),
and the coordinates of the mirror are (x +vt, 0, 0). A ray of
light is observed to start from 0' at a time t, and to overtake the
mirror. It is thence reflected back to the origin which is moving
forward to meet it. The reflexion thus takes place at the time

EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS 7

t + x'j ( V - v), and the reappearance of the light at 0' will be at the
time t + x'j (V-v) + x'l (V+ v). We have thus three events — the
departure, the reflexion, and the arrival of the light. We know
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 synchronism 2r1 = r-hT2 gives then

Let the mirror now be placed on the rj-axis, and let similar
observations be made. We have, as before, the following simul-
taneous values for the coordinates and the times : —

Fixed System.

Moving
System.

Departure

of Light

vt,

o,

o,

t

0 0 0 T

Reflexion

vt+vy'/

J(c2-v~),

2/'5

o,

t + y'l J (<?-*)

0, i/,0 r,

Arrival

vt + '2vy'/

V(<2-2),

o,

o,

t+tyl Jc*-v*

0 0 0 r2

Making use, as before, of the equation

we get B — 0.

In a similar manner we get C = 0X and we thus have the
relation

T = D{t-VX/CZ}.

Referring again to the equations from which T and rl were
determined in the moving system, the beam of light takes a time
TJ - T to go from the origin 0' to the mirror. Thus the coordinate
£ of the mirror must be equal to c (TJ - T),

On substituting the values of TJ and r, we get

Comparing this with our assumed linear relation V ^ - o ^-**v~ Dt

^ = Alx + £ly + Cl
we see that ^ = ^ = C^ = 0 and

8 RELATIVITY

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 measure-
ment according to the fixed axis system to be y, and according to
the moving axis to be ??. The time t taken for light to go from 0' to
A is yj J(c~- v2). The time TI - T in the moving system is
D (t - vx I c'J), and on substituting x — lit and t = y / ^/(c2 - v2), we
get

In a similar manner we find £=Z>/\/l — — . z.
We have thus got so far with our linear relations

77 =

r = D(t-vx/c-).

The constant D, which can be a function only of v, is so far
undetermined. In the first place, we may notice that if D — 9 (v),
then <j>(-v) = <j> (v), for the relation between 77 and y from
symmetry must be the same for v and - v. Now we might have
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 = D X/I--IT/C- ?/

On substituting these values in the equations above, we get

so that if we denote 1 - v~/c~ =

we have D = l//3,

and we arrive at, finally, the fundamental equations of Relativity

= y

EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS 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 t0 and goes to a mirror
which is rigidly attached to the moving axes, and whose " fixed "
coordinates at the time t are x, y and «, and that the beam arrives
back again at 0' at the time tr Now if the times observed in the
moving system for these events are respectively TO, T, ru we have,
since 00' — vt,

T =£$ ( t - VX/C2)

We have then to verify the equation

or, to put it in a convenient form,

c(ti-t)-c(t- tQ) = — { (v t, - x) -(x-v to)},
c \ J

In the diagram (Fig. 6) let 0' and A be the position of the origin
and mirror at the time £0, 0" and A' their positions at the time t, and
0'" the origin at the time tlt all as seen
by an observer in the fixed system. Then
O'A' = c(t-tQ)-f 0"A' --= c (t, - t) : O'O" = v(t- t0) ;
0"0'" = v(tl-t). So that

O'O" : 0"0'" = O'A : 0"'A,
O'A' - 0"'A' O'A' c
and W-o-'&'-OW^

a relation which is easily seen to be identical
with that given above.

Returning to our fundamental equations, suppose the co-
ordinates of any number of points P15 /*2, etc., in the moving
system, and rigidly attached to it are (£ r/j £), (£, >;2 Q, etc., and
that the corresponding coordinates in the fixed system are
(xl yl z^, (x2 2/0 z.2)t we have then equations of the type

1} etc.,

10 RELATIVITY

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 : /3,
whilst transverse lengths are unaltered.

Thus the ellipse £2 + fP -rf = fP a?, the eccentricity of which is
v/ct becomes the circle (x -vt)- + y~ = o?"

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 clear from the equation

T = (3(t-VX/C*)

that the clock at 0 will be behind the local time at the correspond-
ing point of the moving system. To take a numerical example,
suppose that v/c = 4/5, so that ft = 5/3, and that at noon the clocks
at 0 and 0' mark the same time, 0 at that time coinciding with 0'.
If a person in the moving system starts a certain task at noon
and works until his clock shows 1 p.m., and if 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
according to the latter, he 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 0' = x = v t, t = fir, or in our example t = 5r/3.

We shall now consider some kinematical results. The com-
ponent velocities of a point in the moving system w^ w^ are

equal to — and — , and thus
dr dr

11

dg dx-vdt

% dr dt-vdx/c'

l-vWx/c2

where Wx and Wy are the components of the velocity as observed
in the fixed system. From the above we get the reciprocal
relation

wt + v
w- _ __ 1__

* 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
another. Some remarks may be made with respect to these
equations. In the first place, if c = GO , or, what is the same thing,
if the velocities W and w are very small compared with the
velocity of light, these formulae become

Wj. = Wx — v
Wx = v + W$

which reproduce the ordinary equations of relative motion.
Again, if there is 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

f
But

c-

wJ + (v + v') I (1 + v v'/c"2)
I + w' (v + v') / c2 (1 + vv' /c2) '

1 2 RELATIVITY

More generally, if we consider any number of moving axes, and if
the relative velocities of the origins are v, v,' v", ..., we get

+ u

* 1 + W £ U/C2

where u = { sl + s3/c2 + s5/c4 +...}/{!+ s.2/c2 + s4/c4 +...},

sn meaning the sum of the products of the quantities v, v', etc.,
taken n at a time. It is 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
velocities in any other order v", v, v', ....

For the velocity Wrj = d^/dr we get the expression

w.

w

or W=

The resultant velocity W is given by
W*-W*+W?

iv2 + T~ + 2 w v cos 6 - v- w~ sin2 0 1 c~
~~~F^l+wvcosB/c2}~

where 9 is the angle between the velocity w( = Jw? + w -) and v.
We can draw conclusions similar to those above for different axes
all moving with the same velocity v and in the same direction.

We find, however, that if the velocities are not all in the same
direction, the final result is not independent of the order in which
the velocities v, v\ ... are taken. To 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

Wv=

EINSTEIN'S DEDUCTION OF FUNDAMENTAL RELATIONS 13

and for Q

Wx' = >/l - ^2/c2 . v
Wy' = w.

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
type. Two points A and B are moving along two rectangular lines
AO and £0, with velocities v and iv, the distance AO being equal
to a and BO equal to b. What will be their shortest distance
apart, and when will this occur? We find by ordinary methods
that the shortest distance is (bv~aw) / ^/(v2 + w;2), and that the
time of reaching the shortest distance is (av + bw) / (v2 4- w°).
These are the distance and time as observed by a fixed observer,
but if we seek for the measurement that would be made by an
observer moving with A we get different results. Thus w becomes
w//3 and A0 = a becomes /3a, so that the quantities given above are
to be replaced by (bv — aw) / J(v2 + w2 fi~~) and

av /3 + bw /3~l
v2 + w- /3~2

The expressions for the transformation of the acceleration are
more complicated, thus

d- £ d w^ dw*

dr- dr /3(dt-v dx/c2)

d~ i) dw y vy x

dr" dr /32 {1 - v x/z2}'2 j32 { 1 - v x/c2}3

Supposing a wave of light diverges from a point ocQy0zQ at a
time £0, the equation of the spherical wave is at the time t

(x - x,)2 + (y- y,Y + (*- *tf = c8 (t - *0)8.

If we transform this by our fundamental equations, we get
& - v (r - T0)]2 + (n - t/0)2 + (r - T0)2

- r0)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 - 2/0)2 + (* - zo)2 - c2 (t - «0)a

= (£ - £>)2 + (n - *ioY + (t - Q* - c2 (T - r0)2

has been employed conversely to deduce the equations of Rela-
tivity.

We can make some interesting deductions from this identity.
If we suppose that the quantities x, ?/, z, t differ infinitesimally
from xQ, */o, z0, tm and if we put x - XQ = dx, etc., then we have the
following relation between infinitesimals : —

If we call dx~ + dy- + dz~ - c2 dt2 — ds2,

and d^2 + dvf + dt? - <r dr- = da3,

then we have at once the relations

d£ n/dx dt\

~r = P(^~~ v~r }

da- \ds ds/

d^ = dy
da- ds

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

X

y

z

t

dx

dy

dz

dt

ds

ds

ds

ds

d-x

tfy

d-z

dH

w

~d/

ds*

ds2 '

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
force (a., ft 7) are related to one another by the Hertz-Heaviside
form of Maxwell's equations as follows : —

_ia^__8^_8^- ^ dL JbZ 87

dt dy dz ' dt dy dz

_]|ar_^zL_8^ _^M _dx cz

8T ~ dz 8 a.1' ct dz ox'

These equations involve two others.

If we differentiate the first of each of those triads by x, the
second by y, the third by z, and add each triad together, the right-
hand sides vanish and the left-hand sides become respectively

8 1 \ 8 x c y
8 ( cL cM
d t \. c x d y

from which we deduce

SJT cY_

ex cy

cL dM oF
-^— + ^—4--^— = 0.

ox c y d z

Here (X, Y, Z) denote the force in dynes on an electrostic
unit, and (L, J/, J\T) the force in dynes on unit-magnet pole in the
electromagnetic system.

Let us transform the second of these equations

16 RELATIVITY

we have — = 8 -- vft—i

dt ^8r ^8

8 9 v (3 9

8^ = ^8~£~~7~ 8

so that the equation becomes

0

9F

where

In the same way we can deduce the complete set of transformed
equations in the form

cX' cN' 9J/'

rt — . _ _ _ _

8r 8 rj

_j 9 }r/ 9Z/' 9 A"
" = ""

_1 8^' 8 M' cL'

dr 8 £ 8 r;

^8^ _?2T 8F

8r 8 8

8r 8 £ 8 77

a-\T t o T77 o f7t

A 91 d^

8 £ 87; 8 ^

TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS 17

where X' = X 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 state-
ment : — Let a unit charge be situated at 0' and move with it. It
will experience a mechanical force of (X', Y', Z') dynes. In the
fixed system we will observe a mechanical force

so that we have an electric force X, /3 Y, /3 Z, and an electromotive
force 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
forces.

We now pass on to the case in which, instead of free aether, we
have present 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.
Thus,

^ A' \_cAT cM

,dt Joy dz

_dM
° ^~87 + 47rH/ )~^~x

C.

18 RELATIVITY

. az; t

Cy cz

_1 cM cX cZ
Ct d z ex

_ldtf_dY cX
c t dx dy '

If p is the volume density, we have also the equation of con-
tinuity

cV_ cV^ cW cp

ex cy cz ct

and finally

cX cY cZ

^- + T— + ^r— = 47rp
ox cy cz

3L dM cX
- + - + - = 0.
ex cy cz

On transforming as before we get equations of the type

cX' \ cN1 cM'

,

C-1 47T

9 T / a ?;
etc., etc.,
where X', F', Z', L', Jf , JT" have the same meaning as before, and

V'= V
W'= W

We notice that U V W p is cogredient with xy zt. We may
also notice that the velocity of a convection current at any point
is given by the vector U/p, V/p, W/p, and from the above expres-
sions these satisfy as they ought the laws for the composition of
velocities.

If there is a distribution of electricity p' at relative rest in the
moving system, we have U' = V = W — 0 and U=vp and
p' = fip(l -vz/c-) = p/3~l; an element of volume d^ drj d£ trans-
forms into p dx dy dz ; so that the element of charge transforms
into an equal element of charge, for

p' d£ d-r] d£ = p dx dy dz,

TRANSFORMATION OP ELECTROMAGNETIC EQUATIONS 19

If the charge were spread on a surface, the element of which is d*2
and direction cosines (A, yu,, v), and if the measurements in the fixed
system are d S and (I, m, n) respectively, we have then from the
equations

d£=/3dx', d-r\ = dy\ d£=dz

z= IdS
= /3mdS

and so the surface density a-' and cr, on account of the equation

a-'d2 = <rdS,
which expresses the invariance of a charge, give

We have now materials for a complete transformation of any
electrical problem from one set of axes to another. We proceed
to some examples.

A unit charge fixed at 0' produces in the moving system
components of electric force as follows : —

whilst L' = M' = N' = 0.

In the transformation we have

? + rf + ? = P*(x-vt)* + y' + zi,

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,
p is independent of r, and we get U=vp, F=0, W=0 and p = /3 p;
and if the equation of the conductor is /(£, 77, f ) = 0, the equation
of the transformed conductor is f(fi(x - vt), y, z) = 0.

Thus the surface density a-' on a conductor which has the form
of an ellipsoid of revolution is given by

where the equation of the ellipsoid is

a2 62
From previous equations we find that

where 6 is the angle which the normal to the ellipsoid in the fixed
system makes with the #-axis, is the surface density on the
conductor

which is moving with the velocity v along the x-axis. We find,
on reduction, if we put a' = a/ ft,

e
' 62 J (x2 / a'4 + (y- + z2) / 64) '

TRANSFORMATION OF ELECTROMAGNETIC EQUATIONS 21

The forces can easily in any case be obtained by making use of
the potential. Thus, in the electrostatic system,

X'- dV

x- -fl

r-'f

Si)

Z> *V
"8?

where F, the potential, is a function only of f , 77, £ ; we get then
easily

X--/. L=0.

dx

-r

c 3 z

So that in the case of electricity at rest on a moving conductor the
various forces may be obtained by differentiating a certain
potential function.

As an example, the electrostatic potential of a line of length
'21 and of charge e per unit length is given by

r + r' + l
V= e log

&

+ r'-l
where the origin 0' is the middle point of the line, and

By means of the equations above we get the forces due to a moving
charged line where V has now the transformed value in terms of
xt y, z, and t, i.e., if we put I - (31' and e = /3e (for the total charge
2el must be equal to 2eT), we have

and

22 RELATIVITY

This then gives us the potential of a moving charged line, but
it also gives the potential of a charged moving conductor, the
electricity being in relative equilibrium, and the form of the
conductor being a surface of the family

V— const.
This family is

= constant = 2/3 a (say).
On rationalisation we get

a- +p*(a*-f*) = l'

This gives a family of spheroids, and by properly choosing I' we
get a spheroid of any required eccentricity moving with a velocity
v. An important case is when we take fi2 (a - I"-) = a2, 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
spheroid which gives the same forces as a moving point charge.

We shall now consider a case of the propagation of plane light
waves. If we take

X = XQ sin ^— (Ix + my + nz + ct)
A

27T

Y = Y0 sill — (Ix + my + nz + ct)

27T

Z = Z sin — (Ix + my + nz + ct)
A

L = L0 sin ^— (Ix + my + nz + ct)
A

M = MQ sin — (Ix + my + nz + ct)
A

27T

J\ = J\\ sin — (lx + my + nz + ct),
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

23

vibrations being X0 Y0 Z0t LQ J/0 JV0, etc. On insertion of these
values in the electromagnetic equations we find that

X0 = m JV0 - n MQ
Y»=nL,- IN,
ZQ = I M0 - mL0

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 this train of waves would be measured by the
moving system. We find that the argument of the circular
function

2-jr

— (lx + my + nz + ct)
A.

becomes

or, writing this in the form

—

we get

l + v I c
1+lv/c

This last equation gives the equation for the Doppler effect, and
the first three give the effect of aberration. These latter show
that since m / m = n / n, the aberration displacement is in a plane
through the o>axis.

If I' = cos (6 + e) and I = cos 0, we have

COS0 + V/C

cos (6 + e) = £7- .

1 + v cos 6 / c

24 RELATIVITY

If v I c is small we get

cos (6 + e) = cos 0 + (v I c) sin2 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,

Y0-mp(v/c)X0

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 of the fourth equation and of the

do. 8/2 8y
equation - — I --- 1 — - = 0, we find

dx dy dz

In the same way we find that every one of the six quantities
JT, T, Z, L, J/, ^V is annihilated by the operator

This operator plays a part in the theory of radiation similar

to that of Laplace's operator ^— — + ^— TT + ^~ir in the theory of

8 #2 8 y dz2

attractions and electrostatics. In passing, we may notice that

8880 8888

since — , — , — , — are cogredient with — , — , — , —

dx oy oz ot 8 1 drj 8f 8r

we have

82 82 82 0 82 82 82 8'2 , 82

8 x2 8 y- 8 z2 8 tr 8 £? 8 rj'2 c £~ or2 '

I may also remark here that we owe to Prof. Whittaker a
general solution of the equation 82 F/8or + 82 Vfiy1 + 82 Vjcz- = c~282
in the form

V = I /(<& sin ^ cos $ ~^~ y sin ^ sin (fa -{• z cos $ — c£, $, ^!>) d 0 d <f>

Jo Jo
where / is an arbitrary function.

26 RELATIVITY

If x and ^ 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 dy ct

o2 $ 32 \j/

cy ct

If we make use of our Relativity transformation, we find the
very interesting fact that these remarkable functions \ and \//
transform into themselves, and are in fact " absolute invariants "
for our transformation. It will be noticed that these functions \
and \f/ are symmetrical about the a;-axis. We can get a more
symmetrical but not more general form by introducing six
functions, <£1} <£2, <£3, i^1? ^2> ^v where <£j and ^ are symmetrical
about the .x--axis, <j>z and ^.2 symmetrical about the y-axis, <f>3 and
\[ts about the 0-axis. These get transformed into functions
<£/, ^o', ^>3', ^/, $.'5 V^s according to the following scheme : —

so that we have the curious result that these functions
</>!> ^-25 ^3? ^Ai> V'a? '/'s are cogredient with JT, lr, ^, Z/, Jl/, ^V.

When we wish to treat generally the solutions of the electro-
magnetic equations we begin by introducing, after the manner of
Maxwell, a vector F, G, II called the vector potential. It is

APPLICATIONS TO RADIATION AND ELECTRON THEORY 27

defined as follows— if (L M N) denote the magnetic force, we
have

M=c(-

This representation is possible on account of the fact that

If we introduce these values in the equations

1 dX dN dM
c -^ — = -^ -- ~ — i etc.,
dt dy dz'

we find first that

are differential coefficients of a function - <£ (say), so that we have

8 <£ 8 F

~v j - '. /"¥

o <p o^7"

8 t/ 8 £

^_ ecft 8//

8 z d t

3 "F" 3 V 3 ^

C/^rl. 0 JL CiZj

From the equation — h — h — — = 0

8 x dy dz

and the equation — — + 5—5- + -r-^- = c~2^-^-

oas- oy* 8s- or

87^ dG 8/7 8 c/>

we find -r— + - — + -5— + c— -r— = 0.

8ic 8^/ 9^ o«

The quantity <£ is usually called the scalar potential, and we
have, in terms of <£ and the vector (F, G, 77), a means of represent-
ing the field, convenient for many purposes. If we seek now for
the transformed quantities <£', F', G', 77', we have

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
system of 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
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
longer restricted to such small velocities 1 We have such velocities
in the famous experiments of Kaufmarm. In these experiments the
/^-particles of radium moving with velocities almost as great as
three-fourths the speed of light were subjected to transverse
electric and magnetic forces, the direction of these forces being
the same. 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
plane arid was proportional to the velocity. From the observations
recorded, it was found that the " mass " increased as the speed
increased. Various theoretical formulae were deduced by Abraham
and others which agreed well with these results. The formula of
Lorent/, m / J(l -?r/c2), or m ft, where m is the mass for slow
speeds, i.e., the Newtonian mass, gives, however, probably the best
agreements with the observed numbers.

We shall now see what account the Relativity Principle gives
of this. To begin with, the velocity changed very little in actual
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 29

Suppose, then, that e is the charge and that the electric force is F,
and the magnetic force is M, we have then, in the moving system,

We have thus, in the moving system, the equations of motion

where m is the Newtonian mass. If we recall the formulae
which we deduced earlier for c2 77 / c -r and c2 £ / ? r2, on putting
x = 0 and x = v, we 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
theory of inertia as applied to an electron. Suppose that we have
a distribution of electricity throughout a certain volume and
contained inside a certain surface, and let us term this system an
electron. The volume density is p, and the current vector is
(U, F, JF). This current may be simply the convection of the
electrical volume density, or it may consist partly of this and
partly of a relative motion of portions of the electron. The forces
acting will be assumed to be of two kinds — (1) non-electrical,
(2) electrical. Of the first kind we will assume that they form a
system in equilibrium amongst themselves, or, in other words, that
they obey Newton's Law of equality of action and reaction or its
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

* 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- VL).

We may also add the expression for 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

P dx dy dz = 0

Q dx dydz = 0

R dx dy dz = 0.
If we also assume that the total electrical activity is zero, we have

I

A dx dy dz = 0.
We have also the equations of the couples

(yft-zQ)dxdydz =

1

ii!
I

(z P-xR)dxdydz = 0
(xQ-yP)dxdydz = 0.

These equations form the basis of the electrical theory of inertia
and of the motion of electrons in the same way that D'Alembert's
Principle enters into Dynamics.

To enter into this more fully would lead us too far into the
Dynamics of Electrons, but a simple example may make the matter
clearer. Suppose that an electron of charge e of any symmetrical

APPLICATIONS TO RADIATION AND ELECTRON THEORY 31

shape is moving with a slow motion along the axis of x under the
influence of an electric force Xm then the total electric force is
XQ + Xi, where Xt arises from the motion of the electron, and so
the equation

becomes M^° P^x dy dz + I I Xi pdxdydz = 0

or XQ e + Xtp dx dy dz = 0.

Now, on calculating Xt and finding the value of the integral, we
find that the equation becomes

XQ e - mf= 0,

where m is a constant — 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

d mx

d my j-,

dt x/ { 1 - (x2 + y~ + z2)c~2lf

d mz

where F& Fy, Fz is the mechanical force. We may notice that if
this mechanical force is of electrical origin we have from the
assumption of Lorentz and Larmor

where e is the charge.

32 RELATIVITY

If we introduce a vector Px, Py, Pz, such that

P,

p -

* y~

p.

the equations can be put into a more symmetrical shape.

Writing

d cr2 = dt~ - c~- (dx2 + dy~ + dz")
we get

m ^ = Pz.
We also have

m — - = A

d<r-

, dx _ dv ^ dz . dt

where Px -- + P,, -f- + P.- - c- A — = 0

- - - -

on account of the fact that

M

and that therefore

dx d? x dy d2 y dz dzz 2 dt dz t

APPLICATIONS TO RADIATION AND ELECTRON THEORY 33

The quantity A is c~2 -—

and the quantities Px, Py, Pz and A are cogredient with a?, y, z
and t.

We shall only give one example of these equations of motion.*

Suppose that a particle of mass m describes an orbit in the plane

of (x, y), about a centre of force which varies inversely as the

square of the distance, say m p. / r2, where /x is a constant ;

we have then

d y

where v2 = x* + y2,

or on putting *J 1 - v2/c" = /3 and performing the differentiations
P'x 4- /39xvv/c2= -

Py + /33yw/CZ= -

On multiplying by 02, y and adding we get

VV UL',

or = — —r

Hence we have the Energy Integral

1 _2 f

where A is a constant. From which

v2 r .A uu-2

* For many other problems see Schott, Electromagnetic Radiation.
C.

34 RELATIVITY

In the same way, by multiplying by - y and a?, we get the angular
momentum integral

/3 r1 6 = h (a constant)

which leads to

- [l+ c-2(A + -£)]~2

If we put & = 0 (1 - /r2 / c2 A2), we get an integral of the type
L u = 1 + e cos 0', where L is a constant.

The general effect is 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,
the form in which the preceding results have been stated by
Minkowski. In the fundamental Relativity transformation let us
put »„ xz, a?s, £, £>, £, instead of x, y, z, £, r/, f respectively, and
let us further put x4 and £4 instead of i ct and i cr respectively,
i being the "imaginary" of algebra. We then get

Introducing the imaginary angle 6 given by the equations
cos 6 = /3 } - sin O^iv /3 / c, the transformation becomes

f ! = xl cos ^ - x4 sin ^

S2 = X2

4 = £Cj 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 rectilinear motion of a point along the re-axis is defined by
two variables, x and t. It can be geometrically represented by a
curve, the familiar space-time graph. In attempting to represent
in the same way the relation between xl and x4 considered as
rectangular coordinates, we are met by the difficulty that the
corresponding graph may be wholly or partially imaginary, e.g. as

36 RELATIVITY

in the cases x = ut • x = ut + J gtz. 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
to conies, etc.

Let the curve PQ (Fig. 7) be then sup-
posed to represent the motion of the particle
as denned by a^ and x4. If the motion be
referred to the origin moving with velocity
v, the equations

£ = xl cos 0 + #4 cos 0 ;

£4 == xl sin 6 - x4 sin 6

show that this transformation is geometrically equivalent to
referring the system to new axes 0^, 0£4, making an angle 0 with
the former axes respectively. This throws also a new light on the
transformation of velocity

dx_
d£ dt V

dr v dx

c2 dt

In fact, noticing that dxl/dx4 is the tangent of the angle a.,
which AP (the tangent at P) makes with Ox^ and that c?£i/c?£4 is
the tangent of the angle /3 which AP makes with 0£15 the above
relation is easily seen to be equivalent to

„ tan a. - tan 6

tan p — — — z or p = a. - 6.

1 + tan «. tan v

The relation between the accelerations is more complicated, but it
is easily put in the form

#6 r, , (*& Yl"1 - **
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
three coordinates xly x2, #4, we see that the motion can be repre-
sented by a curve in the three dimensional space of xlt xz, x4, the
projection of this curve on the plane of (a,, x9) being the actual path

37

of the particle. 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 xlt x.2, x4 about the axis of a5a, and
if the motion is referred to system moving with velocity v along
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,
if performed in the reverse order they would give a different result.
If, however, the velocities v and v' are small compared with c, these
operations are commutative. This fact throws a light on the
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) space, which remains invariable, whilst we can choose
any axes to describe its properties.

Before dealing with the general motion of a particle, we shall
recall some results in the transformation of rectangular axes ; if we
consider the scheme

£= llxl + I.2x2 + I3xs

£2 = ™>\ x1 4- m.2 x2 + m3 x3

Then the quantities 119 lzt etc., satisfy various relations, such as
l\ + 1? + If = 1 ; l-> m-i + k ™-2. + k ms = 0 ', li = m2 n3 - m3 n2, etc.
A vector, that is, a directed quantity obeying the parallelogram

law, can have its three components represented by distances taken

along the three axes, and will thus obey the same transformation

as above. We may notice that in all such cases

£ i2 + & + ( s2 = *i + x* + x3\
Certain operators obey the same laws, and may thus be called

vector operators. Thus we easily see that

3 3,8 3

rr- = li r- + k — + 13 ^~
3f ! dx1 3#2 3#3

3333
— - = ml — - + ra2 -— +m3 —
3£.2 oxl cx.2 ox3

3333
^r = nl;— + n^ — - + ns —
8^3 dxl dx2 dxs

32 32 32 32 32 32

38 RELATIVITY

This latter result expresses the "in variance" of Laplace's
operator. Conversely, the above transformations may be regarded
as tests by means of which we can recognise whether three quan-
tities 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 (x1 x.z xs) and (a;/ x» x3) are two
vectors, which become (^ £2 £3) and (£/ £>' £/) when referred to new
axes, we have at once

Each side is called the inner product of the corresponding vectors,
and is equal to the product of the magnitudes of the vectors
J(x* + x.? + x.A') and ,J(x^- + x»2 + #3'2) into the cosine of the angle
between them.

Also the three quantities

will be found to satisfy the test given above for vectors. They
constitute 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
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 us) is a
vector, the scalar quantity

8 MI 8 u<, 8 u.>

— + ^ + ^-?-

oxl ox* ox2

and the vector

8 us 8 u2 c ul 8 u3 8 u2 8 ul
8 x2 8 xs ' 8 x3 8 X-L ' 8 x1 8 x}

are related to the vector (u-^ u2 u3] in a manner independent of the
axes.

The above remarks will prepare us for a consideration of four-
dimensional space, in which we have no geometrical intuitions to

MINKOWSKl's TRANSFORMATION 39

guide us, but have to rely on analytical transformations. In a
general orthogonal transformation in four dimensions

£l = ^11 Xl + ^12 X1 + ^13 ^3 + 'l4 X4
b2 = 21 X1 ' 22 X2 T ^23 ^3 ' ^24 *^4>

etc.,
where £i2 = £>ji etc., we have

*11 *21 + ^12 ^22 + Aa £>3 + lu 124 = 0, etC.

and each quantity, such as Zu, is equal to the minor of that quantity
with proper sign in the determinant formed by the I's.

We may define a " four- vector," or set of four quantities
forming a vector, as a set satisfying the above transformation.
We may notice that the four quantities may be vector operators,

8888
such as — , — , — , — , and that

oxl cxz oxs dx4

J?!_ JL J!l _?_ J!_ _?!_ 82 8"

8|f 8f22 .8f32 8J42 ~ dx? dx2~ dx32 dx?
in the same way as

When we come to two vectors we meet as before the inner
product xl 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
try to form the vector product, we get not four but six quantities

3i>.->

which we may write for shortness

2fo ytl 'l/l'2 2/U 2/24 2/34 '

We can see at once that these quantities are transformed by the
transformation

*?23 = An ^o3 + A12 y31 + etc.,

where the A's are the second minors of the I determinant.

Conversely, any six quantities transformed by what we may
call the A, transformation 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 (zlt «2, z3, «4) is a four-vector, we have

*2 2/34 + % 2/42 + *4 2/23 5
Z3 2/41 + ^4 2/13 + «1 2/34 ,
*4 2/12 + 3j 2/24 + 3o 2/41 ,
»1 2/23 + «2 2/31 + »3 2/12 ,

and

*2 2/12 + ^3 2/13 + 242/14,
Z3 2/23 + *4 2/24 + *1 2/21,
Z4 2/34 + «1 2/31 + *2 2/32 5
*1 2/41 + *2 2/42 + *3 2/43 •

We 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 take, instead of (zlt zz, z3, «4), the vector operator

(8/a^, 8/e*2, 9/8*3, dpxj.

After this preliminary survey of four- and six-vectors, let us
return to the Relativity Transformation

£j = xl cos 0 - x4 sin 9

^4 = xl sin 6 + #4 cos 0.

It is seen at once that the substitution is an orthogonal one,
and that thus a point (x, yt z) and an associated time t correspond
to a point in space of four dimensions. A new meaning of certain
invariants as given above will now be at once evident. For
instance, the in variance of the expression or + y~ + £ - cH~ becomes
in our new variables £ a'J + £23 + £32 + £42 = X* + x.? + x.? + o;42, which
can be interpreted as meaning that the distance of the point
(ajj, #2j #35 a?4) from the origin remains unaltered by the orthogonal
transformation. As other invariants we might mention the
element of arc J(dx^ + dx£ + dxs2 + dx?) and the differential operator
for wave propagation ff/dx? + 92/8a:22 + d2/dx./ + o-/dx*. When we
apply these ideas to the electromagnetic relations, a surprising
symmetry becomes evident. If we denote the electric current
components U, V, and W by U^ U.^ and U3 respectively, and the

41

volume density of electricity p by (i c)~] U4, we have then, since it
was proved that £7, F, W 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

then (Flt F«, F^ F4) is a four-vector. We may notice in passing
that the equation of continuity

becomes

From the vector operator d/dxlt d/dxz, d/dx3, d/dx4 and the
generalized potential (Fl F2 F2 F4) we can form a six-vector

dF3 _8F2 dF, dF3 dF2 dF1 dF4 dF1
dx.2 8a:3 ' dxs dxl ' dxl dx2 ' dx} dx4 '

a^__8^_ dFj. dFs

dx.2 dx4 ' dx3 dx4 '
These become respectively

dy cz/ \dz dx / \dx dy /'

or A Jf, ^, -iX, -iY, -iZ,

which we may symmetrically write

^235 ^31) ^125 ^14) ^24) ^34'

Thus the magnetic and electric forces form a six-vector.

Using the operator (d/dxlt d/dx2, d/dx3, d/dx4) and this six-vector,
we can form two four- vectors. The first is

87/34 CL4Z 87/og

8Z12 37^24

•- • -p -p

87/23 8Z31

42 RELATIVITY

If we equate these four components to zero, we get in the
earlier notation

3 T "'7 3 V

0 Li (jZj 0 £

dt ~ cy dz

_ c— i

dt dz dy

_18^r_8T_8X
dt dx dy

a" + r" -^ = 0-
ex dy dz

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)
- VL)

The first three of these represent the mechanical force on an
electric charge, and the fourth is i c~l multiplied by the activity.

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 view of this theory. A point in
ordinary space and a definite time is represented as a point in
four-dimensional space. A moving particle is represented by a
fixed curve in this space. The question of absolute rest in ordinary

MINKOWSKI'S TRANSFORMATION 43

space ceases now to have any meaning. For, in the four-
dimensional isotropic space, one set of axes is as good as another
for describing its properties. The various electrodynamical rela-
tions take their position in a 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
Relativity. Both would fall together if any experimental fact
appeared which would upset one, and can it not be said that the
probability of both being true is increased by this elegant
symmetry *?

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