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
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PRINTERS
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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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