The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 36 of 39

These experiments do not prove either that there is no motion through the ether, or that the Fitzgerald-Lorentz contraction does not occur. They prove that if there is motion through an ether, and if the contraction does occur, then the effect of this contraction is somehow veiled or compensated by some other effect. Thus Lorentz shewed§ that the null result of the experiments of Rayleigh and Brace would be exactly accounted for on his own theory of the constitution of the electron, on which the electrons would be contracted in just the same ratio as the transparent matter. And Trouton and Rankine shewed, in their original paper, that the null result of their experiment is an inevitable consequence of the electron theory of conduction through matter

• Phil. Mag. 4 (1902), p. 678. t Ibid. 7 (1904), p. 317.

J Proc. B. S. 80 (1908), p. 420. § Theory of Electrons, p. 217.

38—2

596 The Theory of Relativity [ch. xx

(cf. § 345 a), provided the electron has the transverse and longitudinal masses assigned to it by Lorentz (§ 664). Thus these experiments, undertaken originally in order to find velocity through the ether, resulted finally in pro- viding confirmation of Lorentz's theory of the constitution of the electron.

In other experiments, the compensatory effect is still more easily dis- covered. A charged body moving through the ether ought to set up a magnetic field, so that every charged body in a laboratory ought to be sur- rounded by a magnetic field proportional to u/C. Every other charged body in the laboratory is moving across the lines of force of this magnetic field with velocity ic and so ought to be acted on by a mechanical force pro- portional to u?/C2. Trouton and Noble * suspended a parallel plate condenser by a torsion thread and looked for a couple, proportional to u2/C2, tending to turn the plates parallel to the direction of motion through the ether. No such couple was observed.

The null result of this experiment is readily explained as a consequence of the Fitzgerald-Lorentz contraction. A shrinkage of the distance between the plates decreases the energy of the condenser. There is therefore a mechanical couple tending to turn the system into its position of minimum potential energy — i.e. into a position in which the plates are at right angles to the direction of motion through the ether. It is readily verified that this couple exactly neutralises the couple of magnetic origin, which the original experiment tried to detect. Indeed, granted the Fitzgerald-Lorentz shrinkage, the theorem proved in § 659 shews at once that the system would be in equi- librium in all orientations.

The Relativity-Condition.

674. These and similar experiments have one and all failed to detect motion through an ether. They have not proved that there is no motion through an ether, but shew that if this motion exists, its effects are in every case veiled by some other effect, and, in every case, it has proved possible to discover this veiling effect as an effect predicted by general electromagnetic theory.

The question arises whether there must always and of necessity be a veiling effect in every experiment. In other words, are the electromagnetic equations of such a nature that it is inherently impossible to detect motion through an ether by electromagnetic means ?

It is well known that the ordinary Newtonian equations of dynamics are of this nature. For the equations

,t

./•

• Phil. Trans. A, 202 (1903), p. 165 and Proc. R. S. 72 (1903), p. 132.

673-675] The Relativity -Condition 597

do not change their form when referred to axes moving with a uniform velocity u — i.e. when x is replaced by x — ut. Thus all phenomena governed by these equations are the same on an earth moving with a uniform velocity u as they would be on an earth at rest, so that it is necessarily futile to attempt to determine the earth's velocity in space by means of such phenomena.

Systems of equations or natural laws which are such as to make it im- possible to determine absolute motion may be said to satisfy the "Relativity- condition." The characteristic of such equations will be that they do not change their form when referred to axes moving with a uniform velocit}^ relative to the axes to which they were originally referred. We have seen that the Newtonian equations satisfy the relativity-condition, and the continual failure of experiment to determine the earth's velocity through the ether leads us to consider whether the electromagnetic laws may not also satisfy the relativity-condition.

If we simply change the electromagnetic laws by replacing x by x — ut, it is at once seen that a change of form results. But the hypothesis of the Fitzgerald-Lorentz contraction has already given grounds for suspecting that the required change may not be so simple as this. For instance, it may be that on changing to moving axes, all lengths parallel to the #-axis ought to

be contracted in the ratio (1 — u2/C2)~. In this case the transformation would be from x to a new coordinate x defined by x = k (x — ut), where re denotes

(1 — ii2/C2)~ • The analysis of § 659 has already shewn that all electrostatic phenomena conform to the relativity-condition when this transformation is made.

This change really amounts to a change in the measurement of the unit of length, as regards lengths parallel to the axis of x, when we change the velocity of motion parallel to the axis of x. Following a method originated by Einstein* we proceed to examine whether similar changes in all the units can result in the electromagnetic laws conforming to the relativity-condition.

675. Consider first the condition that the simple phenomenon of the transmission of a light-signal shall satisfy the relativity-condition. Imagine an experimenter & moving with an unknown but uniform velocity, and using coordinates x, y, z, t to record the result of his observations. If the pheno- menon of light-transmission satisfies the relativity-condition, a signal started from the origin at any instant t = 0 will after time t have reached points lying on a sphere

x" + y2 + z2 - CH- = 0 (691),

where C is the velocity of light determined by the observer S.

Let a second observer S' move with a different velocity, and let him use coordinates x, y', z', t' to record the result of his observations. The sphere

• Ann. d. Phyx. 17 (1905), p. 891.

598 The Theory of Relativity [ch. xx

whose equation is (691) for 8 will have an equation expressed in terms of x , y ', z, t' for S', and if the relativity-condition is satisfied, this equation must be

x2 + y'2 + z'2-C'n'2 = 0 (092),

where C" is the velocity of light determined by 8'.

If S' changes his units of length or time he will change his value of C, which is the distance light appears to him to travel in unit time. We may without any loss of generality suppose 8' to use units which make C equal

to a.

We may also suppose that light will appear, both to 8 and to 8', to travel in straight lines with uniform velocity*. Thus for 8 the equation connecting the position x, y, z of a light-signal with the time t must be linear in x, y, z and t. The similar equation for 8' will be linear in x, y ', z and t '. Thus x, y', z and t' will necessarily be linear functions of x, y, z and t. And we have already supposed that the equations of transformation from x, y, z, t' to x, y, z, t must be such that equation (691) transforms into equation (692), C" being equal to C.

Let us introduce new variables r, r' in place of t, t', these being given by t = iCt, t' = iCt' where i = a/(- 1). Then equations (691) and (692) become

x2 + y2 + z2 + t2 = 0,

x'2 + y'2 + z'2 + t'2 = 0.

The relativity-condition is satisfied if a linear transformation transforms the one equation into the other. Since the equations of transformation are linear this requires that

x2 + y2 + z2 + t2 = k (x2 + y'2 + z'2 + t'2) (693),

where k is a constant.

Imagine a four-dimensional space constructed in which x, y, z, r are orthogonal rectilinear coordinates. On account of the linearity of the equations of transformation, x, y', z', r may also be regarded as rectilinear coordinates in this same space, but these have not yet been required to be orthogonal. Now x2 + y2 + z2 + r2 is the square of the distance of the point x, y, z, t from the origin when expressed in x, y, z, t coordinates, so that, by equation (693), k (x2 + y'2 + z'2 + t'2) must be the square of the distance of x', y' , z , r from the origin. It follows at once that x, y ', z, r must be orthogonal coordinates; if

• According to Einstein's theory of generalised relativity, to which we shall return below (§ 702), light does not travel in straight lines in the presence of a gravitational field. The assumption we have just made marks the parting of the ways between the old physical theory of relativity and the new generalised theory. On the new theory the assumption just made is strictly true only at an infinite distance from all matter; it may nevertheless be regarded as a very accurate first approximation to the truth except in gravitational fields enormously more intense than any of which we have experience.

675-677] The Relativity -Condition 599

they were not orthogonal, cross products x'y\ x'r etc. would enter into the expression for the square of the distance from x, y ', z , r to the origin. Thus the axes of x, y\ z, t can be obtained from those of x, y, z, r by a pure rotation in the four-dimensional space.

We have already fixed the ratio of S"s units of length and time by making C = 0. If we further change the absolute values of these units, we can alter the value of k, and we may agree to fix these absolute values so that k—1. The change from coordinates x, y, z, t to x, y', z', r', or conversely, is now effected by a pure rigid body rotation of the axes.

We may notice in passing that if the relativity-condition is satisfied as regards the transmission of light-signals, no set of axes x, y, z, r in the four- dimensional space is geometrically more fundamental than any other. A change of velocity of translation is merely effected by turning the axes about, and no observer can claim on purely geometrical grounds that his system of axes provides a standard set from which all other positions of the axes ought to be measured.

676. The simplest case of rotation of the axes occurs when every point moves parallel to one of the coordinate planes, say x, t. The formulae of trans- formation then assume the simple forms

x = x cos 0 + t sin 0\

T' = TCos6-xsind- (694).

y'=y; z' = z J

To determine what physical meaning is to be assigned to 0, we notice that

x = 0 when

x = - r tan 6 = - iCt tan d (695).

Thus a point which the experimenter S regards as moving along the axis of x with a velocity — iC tan 0 will appear to 8' to be at rest. In other words the axes of S' move relative to those of S with a velocity — iG tan 9 along the axis of x. Let us put

u = -iCtan6 (696),

then the transformation (694) is that appropriate to the case in which the axes of S' have a velocity (u, 0, 0) relative to those of S.

then k = cos 0, and the formulae of transformation (694) become

x=K(x-ut), y' = y, z' = z, {/ = «U__J (697).

677. Following Einstein we have found that the transformation relations (697) express the necessary and sufficient condition that the propagation of light-signals shall conform to the relativity-condition. We have already

600 The Theory of Relativity [ch. xx

noticed (§ 674) that the first relation x = k (x — ut) is simply an expression of the Fitzgerald-Lorentz contraction which is necessary if the Michelson-Morley experiment is to conform to the relativity-condition. We now have the further information that if all experiments of light transmission are to satisfy the relativity-condition, we must have the further relation

, I xu

t = K [t ~ JT2

The transformation (697), although we have obtained it by a method due mainly to Einstein, is commonly known as Lorentz's transformation. For Lorentz had shewn*, before the appearance of Einstein's paper, that precisely the same transformation expresses the condition that the ordinary electro- dynamical equations shall conform to the relativity-condition.

678. Before proving this, let us examine some of the purely kinematical properties of the Lorentz transformation expressed by equations (697).

Transforming to axes moving with a relative velocity u is equivalent, as

we have seen, to turning the axes through an angle 6 in the x, r plane, where

6 is given by equation (696). Transforming to axes moving with a velocity u'

relative to these new axes is equivalent to turning through a further angle 6'

given by

u' = - iC tan 6'.

But these last axes can be obtained from the original axes on turning through an angle 6 + 6', and we have

• iC (tan 6 + tan 6') u + u'

-iC tan (6 + 6')

1 — tan 6 tan 6' ., uu'

1+Ci

Thus the velocity of the last set of moving axes relative to the first is not u + u; it is u, given by

u=^±±, (698),

UU

1+CJ

and we notice that u is necessarily less than u + u' when both u and u' are positive. We should only have a right to expect that u would be equal to u + u if both 8 and S' measured their lengths, times and velocities in similar ways, and this, under the Lorentz transformation, they do not do.

As a direct consequence of equation (698),

(C-u)(C-u')

G-u=

l+%

so that if u and v! are each less than C, then u is necessarily less than G.

• Amsterdam Proc. (1901), p. 809.

677-679]

The Relativity- Condition

601

Thus no possible compounding of velocities less than G can ever give a resultant velocity u greater than G. As a special case if u' = G then u= G, regardless of the value of u ; the resultant of the velocity of light and any other velocity is the velocity of light.

Similar analysis will give the result of superposing two velocities not in the same direction, but the required formulae can be obtained rather more directly from the formulae of transformation (697), as we shall now see.

679. Let a point move with velocity u, v, w relative to the axes used by

S, so that

x = x0 + ut, y = y0+vt, z = za + wt (699),

and let the velocity of the same point relative to the axes used by S' be u', v', w\ so that

x' = xj + u't', y = y0' + v't', z = z0' + w't' (700).

In these last equations, let us substitute Lorentz's values for x', y', z , t', as given by equations (697). We obtain

K(x — ut) — x0 + U Kit — j^J,

y=yd+ v ' K\t -

XXI

&■■

and a similar equation for z. Differentiate these three equations with respect

ctx to t, putting -j-—u, etc., in accordance with equations (699), and we find

dt

u-u=u (l--^fj

from which follows directly

V = v'k (1 - W=W/k ( 1 -

u =

uu\

uu ~G'<

)

u' + u

1 +

u u

~&

v'

H

w = .

' u u W

K 1 +

.(701),

.(702).

u u

7F

In these equations U, v, w may be regarded as the resultant velocity obtained by compounding velocities u, 0, 0 and u' , v', iv*.

602

The Theory of Relativity

[CH. XX

From equations (701) we obtain directly

u— u

u'=-

1-

Ull

IT*

r' =

*K")

TT' = .

IF

UU

~C2

.(703).

These are also a necessary consequence of equations (702), for if, v', w' is the velocity obtained by compounding velocities U, v, and — u, 0, 0.

Electromagnetic Equations.

680. Following Lorentz* and Einstein f, let us now proceed to transform the general electrodynamical equations of Chap. XIX (§§ 621, 622), namely

4-7T /

pU +

df\ dy d/3

dt) dy

5- , etc. oz

l_da_dZ_d_Y Gdt~dy dz'

df dg dh __ dx dy dz ^

da db ?c_n doc dy dz

to the new variables x , y', z', t' connected with x, y, z, t by relations (697).

If v is any function whatever of x, y, z, t we have

dx = he te_ + he M_ = K fix _ }L bc\

.(704), •(705), •(706), .(707),

dx dx' dx dt' dx

= K

dx c*dt

dt dxdt^dt'dt \dt' dx'

d_X=dJC. hc^hc

dy dy' ' dz dz'

.(708).

The three equations (704) and equation (706) accordingly assume the form

(709),

47T ~C

47T

~C

• I.e. ante.

(df \dt

pU + K[

dx

dy_d/3 dy' dz

t Ann. d. Physik, 17 (1905), p. 916.

679-681]

Electromagnetic Equations

603

47T

~0

, dh dh VK^di'-UM

3/3 u 3/3

= * k-, -

,9a:' C'2 3*'

9a

.(711),

Kdl+?l + M_KUdf_ 7

and Kat7+^ + a?~^^"' = ^

If we introduce a, /3', 7',/', #', h', defined by

/' =/ 4^

2).

£' = *(/3+^)' ff' = K{^^)

r/ = K\y-^g), h' = K(h +

u

4>irC

0

•(713);

(715).

then equations (710) and (711) may be written in the form

4-7T / dg'\ _ da dy'

~C

4tt / cZA'N _ 3# _ da'

C \pW + dt')~dx dy

681. We still require to transform pU, pV, pw to the new coordinates. The density p in the new coordinates must be such that

p dx dy' dz = pdxdy dz.

Since the coordinates x, y', z', r are derived from x, y, z, r by a pure rotation in four-dimensional space, we have

nil* 7/ z t* )

— — ,-^-L — '—. -£ = 1. or dx'dy'dz'dT = dxdi/dzdr. d{x,y,z,r) u J

^ .(716),

Thus

dxdydz _ dr' dt' _ l dx'dy'dz ~fc = Tt = K{ "C2

so that

p' = pic\l-

.(717).

Combining this with equations (703), we find at once that p(u — u) = p'(f, pv = p'v' and pw = p'w'. Thus equations (714) and (715) become

4-7T

dg'\ _ da' dy'

dt'

dz' dx

•(718), .(719).

C \pW + dt')~dx' dy'

• On multiplying throughout by k and using relations (713), equation (709) becomes

4?r ~G

df ( df da dh~] dy d$

which, by the use of equation (712), reduces further to

x df] dy' 3/3'

4_7T

C

/cp

dt

dy' dz

604 Hie Theory of Relativity [ch. xx

Using the relation p(u— u) = p'u' just obtained, and also the relation / = /', this becomes

^(p>v> + gi) = *L_W (72o).

G [p U + dt'J dy' dz l ;

dy'

Finally, again using relations (713), equation (712) transforms into

df_ 1 idg_ dhf\ w (dy _ dj3\ _ ku df K dx7 + K \dy' + dz') + 4ttG \dy' dz') C2 df ~ p'

Using the relation (720), which has just been obtained, this becomes

K

1 ,'dq' dh'\ /n Uu

+«w+w=p{1-ir*j>

doD'V G'\ or, multiplying throughout by k and using equation (717),

df djf dhf_

dx' + dy' + dz'-p {'-i}-

682. We have now seen that if the new quantities a, /3', y'} f, g, h' are denned by equations (713), then the electric equations (704) and (706), when transformed to coordinates x, y', z, t', resume their original form exactly. By precisely similar analysis we find that if

a' = a, X ' = X

b'=K(b + %z), Y' = k(Y-^c

.(722),

G )' \ 0

then the magnetic equations (705) and (707), when transformed to the new coordinates x ', y , z', t', will also resume their original form exactly.

Thus it appears that the relativity-condition will be satisfied by all electromagnetic phenomena, if the relation between the forces as estimated by S and those estimated by S' moving with a velocity (u, 0, 0) relative to S can be supposed to be those given by relations (713) and (722). If these relations are found, in actual fact, to be satisfied, it will be impossible to determine absolute motion by any electromagnetic means whatever.

683. Consider first the form assumed by the relations in free space, for which we may take K = /jl = 1. Here a, b, c become identical with a, /3, 7, and a', b', c with a, ft, y'. Also /, g, h become the same as X/4nr, Y/4nr, Z/4>ir and similarly for/', g', h'. The two sets of equations (713) and (722) are now seen to become identical, each reducing to

«'=«, X' = X \

r-.(/»+5). r-„(r-5,)

y_«(7-£y), z- - * (2 + jj/s) I

681-684] Electromagnetic Equations 605

If iPjC- is neglected, k may be put equal to unity, and the forces X' , T', Z' are exactly those which we found in § 628 for the forces on a unit charge moving with velocity (u, 0, 0). Similarly the forces a, j3\ y are easily shewn by the method of §572 to be precisely those which would be acting on a unit magnetic pole moving with a velocity (u, 0, 0). Thus there is direct experi- mental verification of these equations when u2/C2 is neglected.

When u2/C2 is not neglected, it is naturally impossible to obtain direct experimental evidence of the accuracy of the equations. A complication arises from the fact that S and S' are using different units of length in the direction of Ox, and on allowing for this and treating the problem in the manner of §656, it is at once found that the presence of the factors k in equations (723) exactly represents the complication introduced by the finiteness of u2/G2.

Thus it appears, by what is not far short of absolute proof, that the relativity-condition is satisfied by all electromagnetic phenomena.

684. The problem presented by phenomena in dielectric and magnetic media is naturally more complex. Various hypotheses have been put forward as to the relation between a, b', c' and a', ft ', <y' in moving magnetic media, as also regarding the relation between f, g', h' and X', Y', Z' in moving dielectrics. Some of these hypotheses are in agreement with relations (713) and (722), while some are not. Experiments have been conducted by various physicists, and in particular by H. A. Wilson and A. Eichenwald, with a view to discriminating between these rival hypotheses. In each case the victorious hypothesis is found to be in conformity with equations (713) and (722) above.

Wilson* moved a dielectric body through a magnetic field and found that there was an electric polarisation (f, g , ti) set up of which the amount agreed very closely with that demanded by equations (713). And Eichenwaldf set a polarised dielectric in motion and found that it produced a magnetic field similar to that demanded by equations (713). Thus there seems to be experimental confirmation for every term in equations (713). Experiments on moving magnetic media have not been performed, but there seems to be little room for doubt that they would similarly confirm equations (722).

If, on the strength of this evidence, we assume equations (713) and (722) to be fully confirmed, then we have shewn that the electromagnetic equations conform to the relativity-condition. In other words, all experiments to determine velocity through the ether are necessarily futile. If for the moment we assume that we are moving through the ether with a velocity u in a direction which we call Ox, then we may consider that we are playing the role of our observer S', while an imaginary observer at rest in the ether may be supposed to be playing the role of our observer S. But we have seen that

• Phil. Trans. A, 201 (l'JOlj, p. 121. f Ann. d. Phys. 11 (1904), p. 121.

606 The Theory of Relativity [ch. xx

all electromagnetic phenomena would be exactly the same for us as for S. If we could deduce a velocity u through the ether for our motion, S would necessarily deduce a velocity u for his own motion, which would be contrary to the facts. By this argument, here put in the form of a reductio ad absurdum, we see the impossibility of determining our velocity through the ether.

If at any time equations (713) and (722) are proved to be untrue — and, as we have seen, the remaining opportunities for proving these equations untrue are very few — then it will become possible, in theory at least, to determine our velocity through the ether. But for the present we shall assume, as a working hypothesis, that it is in no way possible to determine velocity through the ether. This is commonly called the Hypothesis of Rela- tivity. We proceed to examine some of the consequences of this hypothesis.

The Relativity Hypothesis.

685. This hypothesis commits us, generally speaking, to all the equations of the present chapter. It does not commit us to any special physical interpreta- tions of them. For instance, the first equation of the Lorentz transformation, namely x = k (x — ut), may if we please be interpreted in terms of the Fitz- gerald-Lorentz contraction-hypothesis ; we may postulate a fixed ether and the equation is then taken to shew that any length moving with a velocity (w, 0, 0) through the ether will be contracted in the direction of the #-axis in the ratio 1/k. Alternatively we may interpret the same equation in such a way as not to assume a fixed ether at all. Any observer S measures out a sphere which remains at rest relative to him ; to a second observer S' moving relative to S with a velocity {u, 0, 0), this sphere will appear to be contracted in the ratio 1/k along Ox.

In a similar way all the other equations can be interpreted so as to have no reference to a fixed ether : they may be taken merely as expressing relations between quantities as measured by one observer S and another observer S' moving with a velocity u relative to S.

The kinematical relations of Einstein, namely equations (702), may, on this interpretation, be regarded merely as laws for the composition of velocities. It appears that the simple laws of composition of velocities and of vector- addition — the so-called " parallelogram of velocities " — are no longer true if the hypothesis of relativity is true. The simple laws are true if u2/C2 and u'2/C2 are small, but not otherwise.

Startling though this result may appear, there is almost direct experimental confirmation of it, as we shall soon see.

684-687] The Relativity Hypothesis 607

Optical consequences of the Relativity Hypothesis.

686. The relativity hypothesis makes no claim to explain the nature of phenomena, it merely proposes, tentatively, a general law of a restrictive nature, which so far has appeared to dominate all known phenomena. All explanations of phenomena which conform to the limits of this restriction are equally permitted by this hypothesis, but the hypothesis serves to rule out, tentatively, all explanations which do not conform to the condition. Conse- quently it is only in rare cases that the principle of relativity by itself enables us to obtain a full solution of a problem. As an instance of such a case, we have seen that it enables us to determine the electric and magnetic forces in ponderable media. Other instances occur in optical phenomena, and to these we now turn.

Fizeaus Water Tube Experiment.

687. In Fizeau's water tube experiment, a stream of water was made to flow through a tube, its velocity of flow being u relative to the earth, and a ray of light was passed through the water in the direction of its motion. To an observer moving with the stream, the water would appear to be at rest, so that the light would be propagated relative to this observer with a velocity u' connected with the refractive-index v of the water by the relation u' = C/v.

According to the classical laws of kinematics, the light ought to travel, relative to an observer at rest on the earth, with a velocity u + u or

-+u (724).

v

Fizeau found it possible to measure the actual velocity by an interference method and formula (724) was not confirmed. The formula

£+«(1-*) (725)

was found to represent the velocity accurately both for water and other trans- parent media.

As we shall now see, formula (725) is not only consistent with the theory of relativity, but could also have been fully predicted by this theory. For the velocity in question is simply that which results from compounding the velocities u and C/v, and the resultant velocity obtained by the relativity formula (702) is

v G

u= ^- = -4

u (G\ v , (u G

If u2/C2 is neglected this reduces the formula (725). In this experiment we have very direct experimental confirmation of Einstein's formula for the composition of velocities.

608 The Theory of Relativity [ch. xx

Reflection of Light from a Moving Mirror and Emission from

a Moving Source.

688. According to the relativity hypothesis, the velocity of light in free space is always equal to G. If the light be observed by an observer moving with a velocity u relative to the source, the velocity is still equal to G, for we have seen in § 678 that the velocity obtained by compounding a velocity G with any other velocity u is itself equal to G.

This consequence of the relativity hypothesis has been tested by Majorana. He first examined the light reflected by a moving mirror and found its velocity to be exactly equal to G independently of the velocity of the mirror*. In a later investigation f he tested the velocity of light emitted by a rapidly- moving source and found this to be equal to G independently of the velocity of the source.

These experimental results are of very great importance, for it will be seen that the Michelson-Morley experiment and the experiments of Majorana taken in combination establish the Lorentz transformation equations (697) as a fact of observation. The Michelson-Morley experiment shewed that the average to-and-fro velocity of light reflected from a mirror back to the source was the same for all directions in space. The Majorana experiments now shew that the result is true for the separate paths before and after reflection, so that the velocity of light, as measured by any observer, is the same for all directions in space. We now have as an experimental fact that, independently of the velocities of the source and observer, the wave-surface is a sphere having the observer as centre. This is precisely the supposition from which we started in § 675 ; it was found to lead directly to the Lorentz transformation (697).

Aberration and the Doppler effect.

689. As in § 591, the equation of wave-propagation in free space, namely

dt2 x>

has a solution

X = A cos — =j (Ix + my + nz - Ct) (726),

where P -f m2 + n2 = 1, and this corresponds to the propagation of a plane wave of light of frequency v in a direction I, m, n. Suppose that the same ray of light appears to the observer S' to be of frequency v and to be propagated in a direction V, m, n', so that the solution of the wave-equation for S' will be

2tt

X = A' cos -77^(l'x'+m'y' + n'z'-Ct') (727).

• Phil. Mag. 35 (1918), p. 163. f Phil. Mag. 37 (1919), p. 145.

688, 689]

The Relativity Hypothesis

609

On substituting for x ', y , z , t' in terms of x, y, z, t from equations (697),

this becomes

xu>

x =A cos^

I'k (x — ut) + m'y + n'z — Gicyt —

C-

This expression must be identical with (726), so that by comparison we obtain

i

m' _n _k(C + Vu) _ v m n G v

.(728).

Aberration. Equating the first and fourth fractions in equations (728)

we find

u

l' +

I

G

l + l

, u G

.(729).

This must, according to the hypothesis of relativity, be the exact formula for astronomical aberration. Let the observer S be at rest relative to any system of axes in uniform motion, while $' moves relative to these axes with a velocity u along Ox. Then light which appears to S to arrive in a direction I, m, n will appear to 8' to arrive in a direction V, m , n, where I, I' are related by equation (729). Put I = cos $, I' = cos cf>' ; then

u

cos (f)' — cos <£ = I' — I = — sin2 <£'

C il + ^cosfi

If ujG is small, this reduces to the ordinary formula of practical astronomy,

(£'-<£ = sin 0'Qy.

Doppler Effect. Equating the last two fractions in equations (728), we find

U COS (£'N

• = (1

V

G

.(730).

This is the full expression for the Doppler effect. If u-JC2 is neglected, the right-hand member reduces to

u

1 + ^ cos (f)',

which is the Doppler factor usually assumed. If the observer is moving directly towards the source of light with velocity u, we have cos 0' = 1, and equation (730) becomes

v

V

u 1 + 0

1-

II

J.

.(731).

39

610 The Theory of Relativity [ch. xx

Acceleration, Mass and Force.

690. In formulae (702) we obtained equations for the velocity u, V, W obtained by compounding a velocity u, 0, 0 with a velocity u', v', w'. When the velocity u', v', w' is so small that its square may be neglected in com- parison with C2, these formulae reduce to

v = u+U-U=u+\

, % 1 **>

v=-; w= —

K K I

Suppose that u, 0, 0 is the velocity relative to S of a moving particle at an instant t = 0. Let it appear to an observer S', moving with a uniform velocity u, 0, 0, to have accelerations

du' dv' dw*

W' w w

these being measured in the coordinates used by S'. In formulae (732) let

us put

, dv' ,,. , dv' .. . dw/ -, zw^^x

u = Wdt' r = J7dt- ^=Wdt (733>'

then u, v, w will be the velocities, as measured by S at the end of a small interval dt' as measured by S'.

The times t, t' used by S and S' are connected by equation (697), namely,

so that on differentiation with respect to t following the particle in its motion,

dt' /., u dx\ /_ u2\ 1 ,w~,x

dt V Odt) V CV * Thus relations (733) may be replaced by

, ldu',

U=KWdt>etC»

and equations (732) become

1 du'

k* dt' a"' " «2 dt' ""' " ~ /C2 dt

1 du' 1± 1 dv' Jx 1 dw' 7 ,_ ,

U=u+Zi-Mdt> v=Z>17dt> w=Z,ZWdt (735).

If T7, -77 , -t- are the accelerations as measured by S, we must have

du _ = u> + -j- dt, etc.,

whence by comparison with equations (735),

du _ 1 du\ dv_1LdY, dw_ldw

dt'' k» dt' ' dtK*dt'' dtK*W (736)t

These formulae give the accelerations as measured by S in terms of the accelerations as measured by an observer S' moving with the particle.

690-692]

Acceleration, Mass and Force

611

691. Let the accelerations be supposed to originate from the action of a force. Since the particle is supposed to be at rest relative to 3' at the in- stant t' — 0, its equations of motion, in terms of the coordinates used by >S", will be

du' j), dv' n,

dw' ,

dt' ~ ' "" dif

where m is the mass, as estimated by 3', and P', Q', R' are the components of the force, also as estimated by S'.

From these equations and equations (736), we obtain

dv dt~~~ ' """ ~di