The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 35 of 39

652. A case of great interest is that in which the velocity of a moving electron undergoes a very sudden change, such as would occur during a collision with matter of any kind. Let us represent such a sudden change by supposing that eu, ev, ew vanish except through a very small interval surrounding the time t = 0, during which they are very great. At a point at distance r, [eu], [er] and [ew~\ will vanish except through a small interval of time surrounding the instant t = r/a. During this short interval, the electric and magnetic forces will be very great; before and after this interval they will have the smaller values arising from the steady motion of the electron. Thus the sudden check on the motion of the electron results in the outward spread of a thin sheet of electric and magnetic force, the forces being very intense but only of brief duration.

The radiation which is emitted when rapidly moving electrons impinge on matter is generally called X-radiation or Rontgen-radiation. It was suggested by Stokes that this consists of thin sheets or "pulses" of electric and magnetic force of the type we have just investigated. Although there is no doubt that this is true in a general way, yet the growth of the new d}rnamics already referred to has made it clear that there is far more in the problem of X-radiation than can be explained by the theories of Maxwell and Stokes.

650-653] Forces on Moving Charges 579

Mechanical Forces on Moving Charges.

653. Whether we assume Maxwell's localisation of energy in the medium or not, the total energy of an electromagnetic field, as we noticed in § 649, will be T + W, where

W= [[^{X* + Y* + Z*)dxdydz (664),

T=[[^(^+^ + ^)dxdydz (665),

and the integrals extend through the whole of space.

Let us suppose that, on account of the electromagnetic forces at work, each element of charge experiences a mechanical force of components E, H, Z per unit charge. We can find the forces H, H, Z by the methods of § 196 and the general principle of least action.

Let us imagine a small displaced motion in which the coordinates of any point x, y, z are displaced to x + 8x, y + 8y, z + 8z, while the components of electric polarisation are changed from f, g, h to f + 8f, g + Sg, h + 8Ii, these new components of polarisation as well as the old satisfying relation (615). Thus if p is the density of electricity at any point in the original motion and p + 8p the corresponding density in the displaced motion, we must have

dx dy dz r'

o8f d8g d8h_ g dx dy dz "'

Let us denote the total work performed -by the mechanical forces in this small displacement by — {811} (cf. §551), so that

{8U\ = f f J p(B8x+R8y + ZSz) dxdydz (666).

Then the equations of motion are contained in (cf. equation (507))

l\8T-8W-{8U})dt^O (667).

Jo

We have 8T = ~ ffj{a8a + b8/3 + c8y)

on applying Green's Theorem ; and on further using equation (635), this becomes

£r=c///ii's('>t'+l) + es('3,'+l)+/f8('J"'+S)}"^

37—2

580

The Motion of Electrons

[ch. XTX

Let 8, -j- refer to a point fixed in space, and let A, y- refer to a point moving with the moving material. Then we have the two formulae for Au,

Au = -^ 8x = -j- 8x + u «— 8x + v ^- 8x + w— 8x, JJt dt ox oy oz

~ du -. du * du «s Au= 8u + ^ 8x + ~ 8y + -7r 8*, 3# 3y * 3.2

so that on comparison

8u = -r: 8x + u^ 8x + v 7- 8x + w ^- 8x at ox oy oz

du s du ^ , 3*7. dx-Bx + dy^+dz-Sz

)•

We now have

d

8(pu + f)=u8P + p8u + Jt8f

d

= u8p + ^(p8x + 8f)-8xd£

( 3 , 3 3\ r. /3c/ * 3^7. 3f/' \

On substituting for dp/dt and 8p their values (cf. § 622) and simplifying, we obtain

• (pu+f) = ^ 0>&s + 8/) + 1 (prS* - Pf%) - 1 (ptffc -pir&O,

dt

whence

8T= ~ fffFJt (P %x + ¥) dxdydz + terms in G, H

• C III F ^ (Pv%x — puty) — ^ (Pu^z ~ Pw%x)\ dxdydz + .... Transforming by Green's Theorem, the second line in 8T becomes

clll{{dy~ ~ Tz) (Pw8V~Pv82) + •••} dxdydz — -^\{p8x (cv—bw) + p8y (aw — cu) + p8z (bu—av)} dxdydz.

On integrating with respect to the time, and transforming the first term on integration by parts, we have

7 8Tdt=f dt -^!jjd~(p8x + 8f) + p8x(cv-bw)+... dxdydz.

653] Forces on Moving Charges 581

We have from variation of equation (664),

8W = jfj(X8f+ Y8g + Z8h)dxdydz.

Hence, freed from the integration with respect to the time, equation (667) becomes

clil —-rff(p&x + 8f) + P&x(cv-bw) + ••• dxdydz -fjf(X8f+ Y8g + Z8h) dxdydz

-fjfp(B8x + H8y + Z8z)dxdydz=0 (668).

We may not equate coefficients of the differentials, for 8f, 8g, 8h are not independent, being connected by

d8f , B8g d8h _ 3 , . x d , *, . d , -> .

&+^+"^ = sp=~35(/,&)"^(p8y)"ai(^z)-

We multiply this by an undetermined multiplier ty, a function of x, y, z, and integrate through all space. We obtain

or, after integration by parts,

Adding this integral to the left hand of equation (668), we may equate coefficients, and obtain

X = -C-dt-dx->etC (669)'

ldF dV 1, . N

• = -Cdi-dx-+c(CV-bw)

= X + hcv-bw),etc (670).

The first equation is simply equation (639), of which we have now obtained a proof direct from the principle of least action (cf. § 575) ; the second gives us the mechanical forces acting on moving charges. It will be seen that the forces given by formula (670) are identical with those obtained in § 629, but they have now been obtained without any limitation as to the smallness or steadiness of the velocities.

582

The Motion of Electrons

[ch. XIX

Stresses in the Medium.

654. We can next evaluate the stresses in the medium, following the method of § 193 and assuming the medium to be free ether.

Let X be the total ^-component of force acting on any finite region of the medium, so that

X = japdxdydz = I \pXdocdydz + p\ livP7 ' ~ ftpw)dxdydz. On substituting for pV, p\v from equations (635), the last term becomes

On substituting for p from equation (615), and for d/3/dt, dy/dt from equations (636), and collecting terms, this becomes

1

X =

b7T

79X BY d_Z\ x _ \dx dy dz)^

-<-

-£V<-

dx) ^

dxdydz

-*(1-

dyj * 7 \dz

' dx)_

dxdydz

• [[[u,diL

....(671)

4tt JJJ

in which IT,,, as in § 576 denotes the as-component of the Poynting Flux.

On transforming the volume integrals in the first two lines into surface integrals, this becomes

X = -~-fj{il (X2 - F2 - Z*) + mXY + nXZ) dS

• j=- \ [\l (a8- /Sa-7*) + ma/3 + nay} dS

-^jtjfjnxdxdydz (672).

Since the last volume integral cannot be transformed into a surface integral, it is clear that the mechanical action is not such as can be transmitted by a system of stresses in a medium at rest.

655. On the other hand it is clear that if we suppose the medium to possess momentum of components

n* n„ n,

C2

G'2 ' C'2

.(673)

654-656] Motion with Uniform Velocity 583

per unit volume, then equation (672) would become exactly the equation of motion of this medium, if it is supposed to be acted on by a system of stresses defined by

PXX=~(X2 - P - Z2 + a2 - /3* - 72) etc.l

8?r I (674).

P^=-ii-(ZF+ay8)etc.

Thus the mechanical action is such as can be transmitted by a medium in motion, the momentum per unit volume being given by formula (673). The vector whose components are given by formula (673) is commonly called the "electromagnetic momentum." We see that it is of amount equal to I/O2 times the Poynting Flux, and in the same direction.

For an electrostatic or magnetostatic field existing alone, the electro- magnetic momentum vanishes, and the stresses reduce to those previously found in §§ 193 and 471.

Motion with Uniform Velocity.

656. Let us again return to the general equations, and examine the special form they assume for a system moving with uniform velocity. This may for convenience be supposed to be a velocity u parallel to the axis of x.

As in § 624 we may replace -j- by — u -j- and the general equation (648) becomes

a?) ox2 oy oz2

Let us now write k for f 1 j , and the equation becomes

7>dx2+dy2 + dz> 4?r<7' or, if we write x for kx,

S+S+S--4- <6Y5>-

We may conveniently speak of x', y, z as the " contracted " coordinates corresponding to the original coordinates x, y, z, since if two surfaces have the same equation, one in x ', y, z and the other in x, y, z coordinates, the former will be identical with the latter contracted in the ratio 1/k parallel to the axis of x.

Equation (675) is Poisson's equation in contracted coordinates. Its solution is

f ffadx'dydz _ fffadxdydz v t

X~]JJ F KJj! S=K~r" where r denotes distance measured in the contracted space.

584 The Motion of Electrons [ch. xix

Hence (cf. equations (644), (645)) the values of M* and F, G, H are given by

F=1^Xp, G = H=0

•(676),

so that the potentials are the same in contracted coordinates as they would be in ordinary coordinates if the system were at rest, multiplied by the factor k.

Motion of a uniformly electrified sphere.

657. To illustrate the method just explained, we shall examine the field produced by a uniformly electrified sphere of radius a, moving with velocity U.

The surface in the contracted space is a sphere of radius a, so that that in the uncontracted space is a prolate spheroid of semi-axes ica, a, a, and there- fore of eccentricity u/C. To find the distribution of electricity, we imagine the charge on the sphere to be uniformly spread between the spheres r = a and r = a + e, where e is infinitesimal. The charge on the spheroid is now seen to be uniformly spread between the spheroid itself and another similar spheroid of semi-axes k (a + e), a + e, a + e. Thus the distribution of electricity in the spheroid in the uncontracted space is just what it would be if the spheroid were a freely charged conductor, and is given by the analysis of §§ 283, 284.

658. The field has been discussed in detail by Searle* and Abrahamf. The electric and magnetic energies W and T are found to be given by

w e2 [SC*-U\ G+u

T-f- f °2 + °* 1 G+U 8a 1 Cu gC-u

2a [u ° C — u j

while the total electromagnetic momentum G in the whole of space is given by

e2 (Q*+u2 C+u 2

Va\Cu^ gC^u u this direction of G being of course that of U.

Motion of any system in equilibrium. 659. When a material system moves with any velocity u, the electric field produced by its charges is different from the field when at rest. The difference between these fields must shew itself in a system of forces which must act on the moving system and in some way modify its configuration.

• Phil. Trans. A, 187 (1896), p. 165.

t Phys. Zeitschrift, 5 (1901), p. 576, or Theorie der Elektrizitat (2nd ed.), p. 165.

656-660] Electromagnetic Mass 585

Let us consider first a simple system which we shall call S in which all the forces are electrostatic, and all the charges are supposed concentrated in points (e.g. electrons). Let us suppose that when the system is at rest there is equilibrium when a charge ex is at x = xx, y = yx, z = zx\ e2 at x = x2, y = y2, z = z2, and so on.

Let us compare this with a second system S' consisting of the same electrons but moving with a uniform velocity u, and having the charges ex at x = x1} y = ylt z = zx\ e2 at x = x.2, y = yi,z — zi, etc., so that each electron has the position in the contracted space which corresponds to its original position in the original space. Then if V denotes the electrostatic potential in the original system, the potentials in the moving system are (cf. equations (676))

W = KV, F = ^% G = 0, H = 0,

and the forces in the moving system are

dx C dt

_?W vdF

dx C dx

_a^/ K/iv*\ _id_v

dx\ C2 )~ KOX'

T= — ^-= — k^- , etc. dy dy

We notice that the electrostatic forces in S' are l//c times those in S as regards their ^-components, but k times those in S as regards their y-com- ponents. As a special case we notice that if the system S was in electrical equilibrium, then S' will also be in electrical equilibrium, so that a system which is in equilibrium when at rest can regain equilibrium after being set in motion with velocity u by contracting in a ratio 1/k'.

Electromagnetic Mass.

660. Consider a charged body, which will ultimately be identified with an electron, moving with a uniform velocity U parallel to the axis of x. Let us first consider the simple case in which u is so small that u2/C2 may be neglected.

The moving charge creates a magnetic field. If the charged body is supposed to be a sphere of radius a, whose surface is uniformly electrified to a total charge e, then there is no field inside the sphere, and the components of magnetic force outside the sphere are given by

a o uez UeV

586 The Motion of Electrons [ch. xix

If we assume localisation of energy in the medium, then at a distance r greater than a from the centre of the sphere there will be magnetic energy per unit volume of amount

1 / , no ,n e* U2 sin2 e

where 6 denotes the angle between the radius r and the axis of oc. On inte- gration, the total energy of this magnetic field is found to be

S//^^^*=§^ (6").

This result is of course only true provided we suppose the energy to reside in the medium as imagined by Maxwell. In this case the energy, being magnetic, must be supposed to be kinetic energy.

Thus if the charged body is supposed to be of mass m0, the total kinetic energy of its forward movement will be

4("'«+!^>' <678>'

in which the first term arises from the ordinary mass of the body and the second from the kinetic energy of the medium.

An analogy from hydrodynamics will illustrate the result at which we have arrived. Suppose we have a balloon of mass m moving in air with a velocity v and displacing a mass m' of air. If the velocity v is small compared with the velocity of propagation of waves in air, the motion of the balloon will set up currents in the air surrounding it, such that the velocity of these currents will be proportional to v at every point. The whole kinetic energy of the motion will accordingly be

£(m+J/)v2,

the term kmv2 being contributed by the motion of the matter of the balloon itself, and the term £ Mv" by the air currents outside the balloon. The value of M is comparable with m', the mass of air displaced — for instance if the balloon is spherical, and if the motion of the air is irrotational, the value of M is known to be \m! (cf. Lamb, Hydrodynamics, § 91).

661. Strictly speaking, formula (678) is true only when u remains steady through the motion. Any change in the value of u will be accompanied by magnetic disturbances in the ether which spread out with velocity G from the sphere. An examination of integral (677) will, however, shew that the energy is concentrated round the sphere — the energy outside a sphere of radius R is only a fraction a/R of the whole, and if R is taken to be a large multiple of a this may be disregarded. The time required for the energy to readjust itself after a change of velocity is now comparable with RJC.

Thus if we exclude sudden changes in u, and limit our attention to gradual changes extending over periods great compared with RjG, we may take expression (678) to represent the kinetic energy, both for steady and variable motion.

The problem gains all its importance from its application to the electron. For this a is of the order of 2 x 10 ~13 cms. (see below, § 6G6), so that all except one per cent, of the

660-662] Electromagnetic Mass 587

magnetic energy is contained within a sphere of radius R = 2 x 10ncms. Since C=3 x 1010, the time of readjustment of this energy is *66 x 1021 seconds, an interval small enough to be disregarded in almost all physical problems.

662. We shall now consider the same problem in a different manner, and shall remove the restriction that u/G is to be a small quantity. The electron will still be supposed to move with a uniform velocity u, v, w which may be of any amount. The field arising from its motion may be calculated as explained in § 647. So long as the electron has no acceleration, the forces X, Y, Z, a, fi, 7 fall off at infinity as 1/r2, so that the stresses defined by equations (67-i) fall off as 1/r4.

If we now apply equation (672) to the field of the single electron, allowing the closed surface S to recede to infinity, the equation becomes

X = -±jtffJnxda!dydz (679),

where the integral is taken through the whole of space. Here X will now represent the ^-component of the ponderomotive force on the electron from the field set up by its own motion through the ether.

When the electron moves with uniform velocity, the integral on the right retains a constant value. In this case X = Y = Z = 0 ; there is no resultant force acting on the electron from the ether.

Now suppose that the electron has not only a velocity u, v, w but also an acceleration U, v, w. The forces X, Y, Z, a, (3, y now contain terms in 1/r, but these depend only on the accelerations. When the surface S recedes to infinity in equation (672), the surface integrals will no longer vanish, but will contain terms dependent on the squares and products of the accelerations. If we suppose the accelerations to be so small that their squares and products may be neglected, then equation (679) remains true even for an accelerated electron.

We have seen that Ila. will depend on the values of u, v, w, u, etc., both at the instant t under consideration and also at preceding instants. Thus we may in general suppose that

jj-Al \Uxdxdydz=fx{u, v, w, u,... u, ... etc.).

Each side of this equation represents the ^-component of electromagnetic momentum, and equation (679) assumes the form

• dfx . dfx . dfx ••dfx du dv dw du

(680).

It is clear that the force X will depend on all the accelerations and their differential coefficients with respect to the time.

588 The Motion of Electrons [ch. xix

Consider first the case in which all the accelerations are steady and so small that their squares may be neglected. Then if, V etc. all vanish and equation (680) reduces to

X =

6u dv dw

.(681).

In general dfx/du etc. may depend on U, V, w, but if we agree that squares of u, v, w may be neglected in calculating X, then we may calculate dfx/du etc. on the supposition that U, v, w all vanish. In other words fx etc. may be calculated as if the motion were steady.

When the motion is steady the whole electromagnetic momentum G is clearly in the direction of the motion and its amount will depend only on C, where C* = u'1 + V2 + w2. Thus we may put

f=?G

where G is the whole electromagnetic momentum in the whole of space, a function of c only. On differentiation, we obtain

y.J.?!M. d£x=uvi(G\ etc

du o gBcKg)' dv c dc\o/'

Now suppose the whole motion to be in the direction of Ox, so that c = u, v= w = Q. The three equations such as (680) now assume the forms

„ . dG Tr . G „ ■ G

X = -vK-, Y=-v~, Z--w-.

dG G G

When u exists alone, v = w = 0, so that Y = Z = 0. Thus the electro- magnetic field exerts a force on the electron in the direction opposite to V. This force is the same as would be exerted if the electron possessed an ad- ditional mass equal to dG/dc. This is called the longitudinal electromagnetic mass of the electron. An electron of mass m„ will respond to a force in the direction of its motion in the same way as an electron, unencumbered by an electromagnetic field, of mass

dG

dG

m0 + u^ (682).

Similarly if V exists, along the opposing force of the electromagnetic field is —v(G/g). By a similar interpretation, G/c is called the transverse electro- magnetic mass. The electron will respond to a force transverse to its motion in the same way as an electron, unencumbered by a magnetic field, of mass

m^% (683).

662-665] Electromagnetic Mass 589

663. Abraham suggested in 1904 that the electron might be treated as

a rigid sphere of radius a, uniformly electrified over its surface. If so, the

longitudinal and transverse masses mj and mt would be given, from the

formulae of § 658, by

e2 C ( 2uG . C+u\

e2 C /C'+uK C+u n u\

e2 C fC'2+u\ U+u . u\ mt = ^¥A~C^l0gG^-u-2G)

664. Lorentz brought forward an alternative conception of the electron according to which it is spherical in shape only when at rest. The electricity is not supposed to be rigidly fixed in a spherical configuration, so that when the electron is set in motion with a velocity U, it contracts, in accordance with the theorem of § 659, in the ratio 1 : k in its direction of motion and so assumes the form of an oblate spheroid. Against this conception of the electron Abraham has brought the objection that the original electron cannot be simply a distribution of electric charges acted on by their own mutual repulsions; there must be other forces at work to keep the charges from flying apart. When these other forces are taken into account, there is no reason for supposing that the contracted electron would be in equilibrium, or if it were in equilibrium, that the equilibrium would be stable. We shall return to this point later.

The electromagnetic field of Lorentz's electron is readily calculated by the method of § 656, for the configuration, when expressed in terms of contracted coordinates, is spherically symmetrical.

If W is the electrostatic energy of the system of charges which constitute the electron when at rest, it is readily found that the electromagnetic momen- tum G of the contracted electron moving with velocity u is

• 3  (72      ' 
  

so that the longitudinal and transverse masses are

4 W A

(684).

m' = 3^*3

4 W

y

665. The formulae for the transverse mass can be tested experimentally. It was shewn in § 631 that an electron in a uniform magnetic field H would describe a path of constant curvature muC/eH, where U is the velocity perpendicular to the magnetic lines of force. When electromagnetic mass is taken into account, m in this formula must be replaced by m0 + mt, where m0 is the mass of the electron apart from its electromagnetic mass. Experi-

590 The Motion of Electrons [ch. xix

ments to determine the variation of m0 + mt with the velocity were first undertaken by Kaufmann in 1906. More recent experiments by Bucherer, Bestelmeyer and others shew that m0 + mt varies precisely as (1 — u2/C-) ~ ? or k. This is in exact agreement with the transverse mass of the Lorentz contractile electron if m0 is taken to be zero — i.e. if the mass of the electron is supposed to be wholly electromagnetic.

666. All experiments agree in giving a value for e/m at zero velocity very nearly equal to 1*767 x 10v in Electromagnetic Units (Bucherer's value). Com- bining this with Millikan's value for e, namely 4774 xlO-10 in Electrostatic Units, we find for the mass of the electron at rest

m = 9-00 x 10-28 grammes. The mass of the electron at rest is, from formulae (684),

m-3C,2.

If the charge e of the electron is supposed spread uniformly over the surface of a sphere of radius a, the value of W, the electrostatic energy, is e2/2a, so that

m=s^ <685>

in agreement with formula (678). In this equation we know the values of

m, e and C, so can deduce

a=l-S74xlO-13 cms.

This must be the radius of the electron if its charge is spread uniformly over the surface of a sphere. If the charge is spread uniformly through the volume of a sphere, W = Se2/oa, giving

m = JL; a = 2-249xl0"13cms. 5 aU2

Other distributions of charge would give other values for a but always of the same order. We conclude that the value of a is of the order of 2xl0~13cms.

The Internal Mechanics of the Electron.

667. Let us regard the electron as a contractile sphere of radius a whose surface is uniformly charged with electricity. Then m is given by formula (685), and the electromagnetic energy of the electron, when moving with a velocity U, is found to be

T = iiiC2(k-~ j + a constant (686).

665-668] Electromagnetic Mass 591

Suppose an acceleration uto operate for an instant dt. Since the longitudinal mass is iuk3, the work done by the force producing the acceleration is

m/c3 v dt dt

which may be written as -j- (mC'2/c) dt. The increment in the electromagnetic

energy (686) is, however,

s^«-h(t)*

and this is not equal to the work done on the electron.

To satisfy the conservation of energy it appears that in addition to its

electromagnetic energy T the electron must have energy U of some type

unknown but of amount

1 mC2 U = -i 1- a constant (687).

TD fC

Then T+ U=m/c + a, constant, and the work done by external forces is equal to the increment of T + U.

668. If a charge e is spread over a conducting sphere of radius a, the force per unit area on its conducting surface is

R = 2ira*

e2

87ra4'

The electron is not a charged conductor, but the above formula makes it clear that the electron at rest could be held in equilibrium by the action of a normal tension R of amount e2/87ra4, per unit area. Poincare* has shewn that the electron in its contracted state would still be in equilibrium if tensions of this amount continued to act while the electron was in motion. Now if v is the volume of the electron at any instant, the work done on these tensions as the electron changes shape will be Rdv. When the electron is moving with velocity U, its volume is §7ra3//c, so that

D e2 1 mC*

Rv = £— = - .

ba/c 4 k

Thus if U is taken to be Rv in formula (687), the conservation of energy will be exactly satisfied.

There is no evidence as to whether these tensions do or do not exist; the possibility of their existence suggests a mechanism by which the electron can be held in equilibrium at all velocities, while its motion conforms to the con- servation of energy.

• Rendiconti del Circolo Matem. di Palermo, 21 (1906), p. 129.

592 The Motion of Electrons [ch. xix

The Reaction on an Accelerated Electron.

669. The whole force acting on a moving electron is given by equation (680), in which we have so far neglected all terms beyond those in (J, v, w. Lorentz * has calculated the effect of the terms in u, v, w etc., and finds that they give rise to a force acting on the electron of components Fx, Fy, Fz given by

2^ = -! ^s£ etc (688).

Lorentz also gives formulae from which the remaining terms in equation (680) can be calculated, but these terms are of little physical interest.

The force given by formula (688) may be regarded as a frictional resistance opposing the motion of the electron through the ether. The rate at which the electron does work to overcome this force is

uFx+ vFy+ wFz,

so that the work done by the electron in an interval from t = 0 to t = r will be

2 e2 fT

-oTTsj (UU+VV+WW)dt. On integrating by parts, this becomes

2e^

3 0s

UU+VV+ WW

o ^ ^ Jo

The last term represents the radiation emitted by the electron as calculated by Larmor's formula (662); the first term must represent changes in the energy stored in the ether.

• The Theory of Electrons, p. 251.

CHAPTEE XX

THE THEORY OF RELATIVITY

Motion through the Ether.

The Michelson-Morley Experiment.

670. When we have spoken of a system at rest we have so far meant, for all practical purposes, a system at rest in our laboratories. But if we have been right in conjecturing that all electromagnetic phenomena have their seat in the ether, then a system at rest would most naturally be taken to mean a system at rest in the ether. We have so far made no clear distinction between the conceptions of rest in the ether and rest relative to the walls of a laboratory.

The view was at one time held that a moving body drags the ether along with it. If this were a true view the distinction just referred to would not arise ; a body at rest relative to the walls of a laboratory would also be at rest in the ether. But in time it was found that this was not a true view ; it could not be reconciled simultaneously with results of laboratory experiments such as Fizeau's water-tube experiment (cf. § 687 below), and with the astro- nomical theory of the aberration of light* (cf. § 689 below). Finally it became established that the ether, if one existed at all, could not share in the motion of moving bodies ; it must be stagnant, and moving bodies must simply move through it without setting up mass-motions in it.

The earth's velocity in its orbit is about 30 kms. a second, so that the velocity of the earth relative to the supposed ether must at some season of the year be at least 30 kms. a second. If an ether exists, there must be a stream of ether flowing through every laboratory which must attain velocities at least as great as 30 kms. a second.

Starting in 1887, Michelson and Morley conducted experiments with a view to measuring the actual velocity of this supposed stream of ether relative to their laboratory, or, what is the same thing, the velocity of the earth through the ether. The principle of the experiment is easily explained. Let the laboratory be moving with velocity u through the ether, then a ray of light travelling against the stream of ether will move with an actual velocity C in the ether, and so will have an apparent velocity C — u if measured relatively

• For a fuller account the reader is referred to special treatises — Larmor's Ether and Matter (Camb. Univ. Press, 1900) or Cunningham's Relativity (Camb. Univ. Press, 1914).

j. 38

594 The Theory of Relativity [ch. xx

to the moving laboratory. Similarly a ray of light made to travel in the reverse direction will have an apparent velocity C + u. If a ray travel over a path I and is then reflected back to its starting-point, the time £2 taken will be given by

t>"ffh+uk-"{1-$T (689)-

Suppose next that a ray is made to travel a distance L across the direction of motion and back to its starting-point, the system moving with velocity u as before. Let the whole time be t2, then the distance travelled by the system is uU. The actual path of the ray through the ether consists of two equal parts, one before reflection and one after ; each part is the hypotenuse of a right-angled triangle of sides L and \ut2, and the time of describing each part is ^t2. Hence

%t2C=(L* + %%%*)?,

2Z/, u"-~^

whence i2=^ 1-J (690).

From formulae (689) and (690) it appears that the times taken by a ray of light to travel a distance I and be reflected back, while the laboratory is in motion through the ether, will be different according as the path of the rays is along or across the direction of motion of the system. This time difference admits of measurement by optical means, and from such measurements it ought to be possible to determine u.

When the experiment was performed no time difference could be observed. The obvious explanation would be that, at the moment of performing the experiment, the laboratory was at rest in the ether, but this explanation was not found to be tenable, since no time difference could be discovered at any season of the year.

The Fitzgerald-Lorentz Contraction Hypothesis.

671. Fitzgerald in 1893 and Lorentz in 1895 suggested independently that the reason why no time difference was observed might be because the arm I of the apparatus which moved with velocity u longitudinally through the ether was contracted in a ratio (1 — u?/C2)- as a result of its motion. In such a case the arm I would have shrunk from an initial length l0 given by

measured in the system when at rest. Equation (689), expressed in terms of /0, now becomes

h~c[ o\

and so agrees with formula (690).

670-673] Motion through the Ether 595

Thus the Fitzgerald -Lorentz contraction hypothesis would account com- pletely for the null result of the Michelson-Morley experiment. The hypothesis in itself is not unreasonable, for we have already seen (§ 659) that an electrostatic system set in motion with a velocity u would only regain its equilibrium after contracting longitudinally in exactly the ratio (1 — u2/C2)- assumed by the hypothesis. It is true that the arms of sandstone and pine used by Michelson and Morley were not purely electrostatic systems. But neither is the electron (cf. § 664), and yet Lorentz's hypothesis that this contracts longitudinally in exactly the same ratio is found to lead to a value for the electromagnetic mass which is entirely confirmed by experiment (§ 665).

672. According to the contraction hypothesis, the Michelson-Morley experiment failed to detect the velocity of motion through the ether because this motion was exactly concealed by the shrinkage of the apparatus. If this were so, the velocity ought of course to become measurable if we could in any way measure the amount of this shrinkage.

It is at once obvious that the shrinkage could not be measured, or even detected, by any process of direct measurement, for any material measuring- rod would shrink in exactly the same ratio as the apparatus to be measured. Indirect means might, however, be expected to reveal the amount of shrinkage.

673. Lord Rayleigh* pointed out that an isotropic medium ought to become anisotropic when shrunk, so that ordinary transparent matter ought to be doubly refracting for a ray of light crossing it in a direction oblique to its motion through the ether. But no trace of double refraction was found either by Lord Rayleigh or by Bracej" who repeated the experiment with apparatus so sensitive that a fiftieth part of the expected effect would have been detected.

Following a similar train of thought, Trouton and Rankinej tried to detect changes in the resistance of a bar of metal as it was turned in various directions, but found no measurable change.