The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 37 of 39

and these will be the equations of motion as observed by S.

If the particle is an electron of charge e, the values of P', Q', R' will be eX', eY', eZ', where X\ Y', Z' are given by equations (723). Substituting these, we find for the equations of motion of the electron as observed by 3,

m/c0 — = Jrm3

= Q';

m/c2 -T7 = li

dt

.(737),

rriK6

1)1 K'

mic*

du dt

dv

dt

dw ~di

= eX

= e/c

= etc

Z + ?/3

.(738).

The observer S will suppose the electron moving with velocity u to have longitudinal and transverse masses vii and mt; and his equations of motion for the electron will be

du v dv

dv ( u \ dw („ u „\ By comparison with equations (738)

.(739).

mi = mic*; mt = mK

.(740).

692. These are precisely Lorentz's expressions for the longitudinal and transverse mass of a moving electron (§ 664), which we have seen to be fully confirmed by experiment (§ 665). In deducing these expressions in § 664 we supposed the inertia to be produced by a magnetic field in the ether, so that u denoted the velocity relative to the ether, but the theory of relativity shews that U may legitimately be supposed to mean merely the velocity relative to the observer by whom the accelerations are measured. A further difference between the two calculations may also be noticed. When the mass

39—2

612

The Theory of Relativity

[CH. XX

was regarded as arising from an ethereal magnetic field, it was possible to estimate the radius of the electron from a knowledge of the value of m; the relativity calculation does not make any such estimate possible.

We may notice that

m/r

du d

dv d

= T,(mfcu); mic-T.=-;,(m>cv), etc.,

dt dt

dt dt

whence it follows that the equation of motion of an electron moving with any velocity U, V, W relative to any observer must be

It(mKu) = e(x + ^y-^^

d_ dt

A

dt where k is now given by

(mKV) = e^Y+-a-^ryJ

(ni/cw) = e(z + ^j3-~a\

T2 j_ tt-2 J. t^2\ — f

.(741),

1 Oi )

.(742).

To shew that these are the true equations, it is sufficient to notice that they are invariant as regards transformations of the x, y, z axes, and reduce to equations (738) for one special direction of these axes.

Momentum and Eneegy.

693. The expressions on the right of equations (741) are the components of force on the moving electron as they would be measured by S (cf. §§ 629, 653). If we denote these by P, Q, R, and if we regard

vikV, vikV, micw

as the components of momentum of the moving electron, then the equations of motion (741) assume the form

{Force) = (rede of change of momentum) (743).

The rate at which work is done on the electron will be

Pu+Qv+Rw = mUj-(KU) + mv-j~(/cv) + mw -j- (kw),

and after simple algebraic transformation this becomes

d

Pu+ Qv + Rw= j (micC2).

.(744).

Thus if we suppose the energy of the electron to be w/cC2 + a constant we have the equation

(Rate of doing work) = (rate of increase of energy).

692-695] Mass, 31 omentum and Energy 613

In the equation,

Energy = m/cC2 + cons (745),

the additive constant is entirely at our disposal. If we take it equal to — mC2, the energy is given by

Energy = wC2(«-l) (746),

and this reduces to the Newtonian kinetic energy Jra (u2 + V2+ w2) when the velocity is small compared with that of light. But it is generally more con- venient to put the additive constant equal to zero, so that the energy is simply ttikG2. This may be regarded as representing kinetic energy hiC2{k — 1) and intrinsic electronic energy mC2.

694. Let us put

r = mC2(l--)=mC2

7)T' then -=— = ttikU, etc. and equations (741) become

C2

£(£H £©-**

These are analogous to the classical Lagrangian equations of motion of a particle. We must note, however, that T' is not the kinetic energy, but is equal to the kinetic energy divided by k.

Conservation of Mass, Momentum and Energy.

695. In denning the energy of an electron to be micC'2, we have already arranged, by definition, that conservation of energy shall hold. The total energy of a system of electrons is Sm/cC2, and from equation (744) it is at once apparent that if no work is done from outside the total energy remains constant.

As a system of electrons change their velocities under their mutual inter- actions, the values of k for the different electrons will be continually changing. If the total energy remains constant, it is clear from equation (745) that 2m# must remain constant. Thus if in future we agree to define the mass of a moving electron as ra/e — the "transverse" mass of § 664 — then it appears that conservation of energy will imply conservation of mass.

The full principle of conservation of energy states that as energy is inter- changed between different modes of energy the sum total of energy always remains constant. Hence in order that the sum total of masses shall remain constant — i.e. to secure complete conservation of mass — it is necessary that all forms of energy should possess mass, and energy E of any kind whatever must possess mass of amount E/C2.

For instance suppose that a system of electrons of energy E, and therefore of total mass 2m« equal to EjC2, radiates away energy of amount R. The

614 The Theory of Relativity [ch. xx

final energy of the electrons is E- R, so that their final mass is E/C2-R/C2. But the radiation possessing energy R must also possess mass R/C2, so that the total mass of electrons and radiation remains equal to EIC2, and the total mass is entirely conserved.

696. Our typical electron has been supposed to move with a velocity of components U, V, w relative to an observer S. Let us suppose its velocity relative to a second observer S' to be u', v' w' , when 8' moves relative to £ with a velocity u, 0, 0. Then the two sets of velocities are related by the kinematical equations (702) and (703) of § 679.

Let us write

= 1-

u2 + v2 + w^

C2

, A, U'2+v'2 + w'2\

•-T c2 J

Kq

<*rZ>

-k

•(747),

so that tc0 is identical with the k of § 679. On using the values of u', v', w' given by equations (703), we find

f ll2\ f _u*+ v2+ w^

so that, on raising each side to the powTer — ,

K' = KKe(l-^j (748),

or, again using the first of equations (703),

k'u' — kk0 (u — ll).

Multiplying both sides by m, and summing over all the particles in the

field,

l.'niKU' = k0 Hm/cU— uk0 XniK.

Let M, fix denote the total mass and total ^-momentum as observed by S,

and let M' , /xx' denote the same quantities as observed by S'. Then our

equation may be written

fix =k0ixx—uk0M (749).

Similarly, since S is moving relative to S' with a velocity — u,

fjix = K0fix' + uk0M (750).

695-698] Mass, Momentum and Energy 615

If the total energy of the system remains constant throughout any motion, M and M' must remain constant, so that /xx and /xx necessarily remain constant. Thus conservation of energy implies conservation of momentum.

697. As a particular case, let us suppose the velocity of the axes of S' chosen so that fix' = 0. Thus S' moves with the centre of gravity of the system, and this centre of gravity moves relative to S with a velocity u along Ox. Putting fxx = 0 in equations (749) and (750), we find

fj,x = uM,

M = k0M\

Thus the ^-momentum observed by S is u times the total mass observed by S. The total mass observed by S is k0 times the total mass observed by S'. And, if we take the energy equal to MC2, the total energy observed by 5 is k times the total energy observed by S'.

698. Returning to the analysis of § 696, let us suppose that the system emits a beam of radiation along Ox. Let the energy of this beam as estimated by S be E, so that its mass is E/O2, and let its momentum as estimated by S be Rx along Ox. Let accented letters denote the same quantities as estimated by S'. Then in order that equation (749) may be true both before and after the emission of the radiation, both mass and momentum being assumed to be conserved, we must have

Rx' = k0Rx-uKoE/C2.

This equation is true for all values of u. Take u equal to C, so that S' moves with the beam of light. Then Rx = 0, and the equation becomes

i4 = f (751).

Thus the momentum of a beam of light, as measured by any observer whatever, is equal to (I/O) times its energy. Or, again, the momentum is equal to C times the mass. If W is the energy of the beam per unit volume, the momentum per unit volume will be W/C, and the flow of momentum per unit area of cross-section will be C times this, and so equal to W. Thus the pressure of radiation is equal to the energy per unit volume.

This result was obtained in §592c as a consequence of the hypothesis that electric action was transmitted by an ether. It now appears that the result is in accordance with the hypothesis of relativity, and can be deduced as a direct consequence of this hypothesis.

616

The Theory of Relativity

The Energy and Momentum of Radiation.

[ch. XX

699. In free space the fundamental equations (613) and (614) of p. 559

assume the form

4,-rrpU a ldX_dy_dJ3 ^ ^^

.(753).

C + G dt'' By dz,etc

\da_dZ__dY

Cdidy dz,QtQ

Multiply these six equations by X, Y, Z, — a, — /3, — <y respectively and add corresponding sides. We obtain

~p(Xu+Tv + Zyr) + ~^t(X'+Y' + Z' + ci' + l3'+7')

Multiply both sides by C/4-7T and integrate throughout any closed space. Assuming that the distribution of electric density p arises entirely from electrons, we obtain

d ^///(X2 + Yl + Z2 + a2 + @2 + 7") dxdydz

Xe(Xu+Yv + Zw) +

dt

&

G_

4>7T

In § 693, we put

l(Yy- Zj3) + m (Za - Xy) +n(X/3- Ya)

v w

e (X+^7 — -^/3) = P, etc.

dS

.(754).

Multiplying these relations by u, V, w and adding we find, after a further use of equation (744),

e(Xu+ Yv + Zw) = Pu + Qv + Rw = j (vikC2). Thus equation (754) becomes

d dt

Xm/cC2+ ^.jffiX2 + Y* + Z* + a2 + /32 + r) dxdy dz~\

= ^(( I (Yy-Z/3)+..^,dS... (755).

Now let the closed space be allowed to extend to infinity, so that the integration is through the whole of space. The surface integral on the right of equation (755) now vanishes, so that the left-hand member must also vanish. In other words, throughout the motion of the system of electrons,

2m«(72 +

7T

(Z2 -1- Y- + Z* + a2 + yS2 + r) dxdydz = cons. . . .(756).

699, 700] The Energy and Momentum of Radiation

(517

If R is the energy radiated away from a system of electrons, we have already had the relation

XmrcC2 + R = cons.,

whence it appears that the volume-integral in (756) must represent radiated energy.

If we assume the radiated energy to be localised in space according to the distribution of the integral, then the flow of energy into any closed surface must be represented by the surface-integral on the right hand of equation (755), and this flow is precisely that given by the Poynting Flux of § 576.

700. Again, let us multiply the six equations (752) and (753) by 0, 7, -/3,0, Z,-Y and add. We obtain

— a

dec 3/3 dy

• dJ3 + dy_x(dX + dY + d_Z\ dy dz ) \dx dy dzj'

\dx dy dz J \dx dy Dividing throughout by 47r, and using equations (615) and (616), this

becomes

P(x+l

" g^ + aIc^-2®

_ d

dx

1- {X2 - Fs - Z2 + a? - /S2 - 72)

07T

a

dy

J(xfW

d

• —

cz

~£(X*+.i)_

,(757).

Integrating through a closed surface as before, this becomes Xe{x+^y-^)+^fJf(Yy-Z^dxdydz

= -jj i-(^-F2-Z2 + a2-/32-72) + ^:(ZF + a/3)

n

b7T

(XZ + 07)] dS.

Using the first of equations (741), the left-hand member becomes

dt

5?

mtcU+

^-cjjJ(Yy-Z/3)dxdydz),

.(758).

If the integral is taken through the whole of space, this expression must vanish, so that the quantity inside square brackets must remain constant. The first term 2m«i7 is the ^-momentum of the electrons, so that the second term must represent the ^-momentum of the emitted radiation. This is identical with the formula found for the momentum in the ether in § 655 ; it

618 The Theory of Relativity [ch. xx

has now been obtained, independently of the assumption of an ether, as a general expression for the momentum of radiation.

If we assume the momentum to be localised in space according to the distribution of the integral, then equation (757) can be interpreted as shewing that there is a flow of #- momentum per unit area at any point, whose com- ponents will be Pxx, Pxy> Pxz, where

PXX=~(X>- Fa-^ + aa-/S3-73),

Pxy = ~(XY+a/3),etc.

These are precisely the quantities we obtained in § 655 to represent the components of stress in an assumed ether. On the relativity-theory, they appear in a much more general way as representing the flow of momentum in space.

The Existence of an Ether.

701. Throughout the earlier chapters of this book, we treated the existence of an ether as a working hypothesis. Maxwell and Faraday appear to have had no doubt that the ether had a real objective existence, but no proof that it exists outside our own minds has ever been obtained, and it seems best to regard it merely as a working hypothesis, to be discarded if it is found to lead to contradictory or impossible results, and to be retained if it proves to be useful as well as self-consistent.

The considerations which have seemed to favour the hypothesis of an objective ether are mainly the following:

(i) That light and other forms of electromagnetic action are propagated with a uniform velocity C, which is most naturally interpreted as a velocity of propagation in a medium of some sort.

(ii) That the hypothesis of an objective ether explains electrical forces with comparative simplicity as arising from systems of stresses transmitted by the ether.

(iii) That the hypothesis of an objective ether gives a simple account of electromagnetic energy as being the energy of a medium in a state of strain and stress.

The force of the first consideration is very much weakened by the dis- covery, resulting from the Michelson-Morley experiment, that the velocity of propagation is the same for an observer moving through the supposed ether as for one at rest. We have seen in the present chapter that this fact, whether we assume an ether to exist or not, requires us to adopt systems of kinematics and dynamics which are different from the old classical systems. The con- sequences of these new systems of kinematical and dynamical laws are found

700, 701] The Existence of an Ether 619

to be confirmed by experiment. If they had not been confirmed, we should have reached an impasse ; the circumstance that they are confirmed provides no definite information on the question of the existence of an ether. But the hypothesis of an ether is weakened to this extent, that the theory of relativity has shewn that the results in question, although possibly admitting of an ex- planation in terms of an ether, are necessary consequences of the simpler supposition that phenomena are the same for all observers, no matter with what velocity they are moving. And the simplest way of all of arranging that phenomena shall be the same for all moving observers, is to suppose that there is no ether at all ; all moving observers then necessarily stand on the same footing, for there is no fixed framework by which their motion can be estimated. The hypothesis that there is an ether may give a possible explanation of the phenomena, but the hypothesis that there is no ether provides an equally possible and very much simpler explanation.

If we still wish to retain the hypothesis of an ether through which light and electromagnetic phenomena are propagated, we must adjust the properties of this ether to agree with experiment. Now we have seen (§ 688) that, no matter how an observer and a source of light move, the wave-surface formed by the light emitted at any instant will be a sphere having the observer as its centre. If the observed constant velocity of light is simply the constant velocity of propagation through an ethereal medium, it would seem to follow that each observer must carry a complete ether about with him. This at least robs the ether of the greater part of its reality. We cannot quite go so far as to assert that the ether is reduced to a subjective imagination, as a simple analogy will shew. A number of travellers may all see what they would describe in ordinary language as being the same rainbow. The angle of the rainbow would be the same for each traveller, and no amount of travelling towards the rainbow would cause it to subtend a greater angle. If the travellers compared observations they would have to conclude that each traveller carried his own rainbow about with him. This would not, however, prove the rainbow to be merely a sub- jective illusion ; when the rainbow disappeared for one traveller it would disappear for all. Considerations such as we have mentioned do not prove in strictness that light cannot be propagated through an ether; what they prove is that if an ether exists, it must be something very different from the absolutely objective ether imagined by Maxwell and Faraday.

The hypothesis of an ether shewed great aptitude for explaining either electric or magnetic forces in systems at rest ; a simple system of pressures and tensions was found to account perfectly for the observed forces. On the other hand the same explanation cannot account for both electric and magnetic forces simultaneously. If, as is usually assumed, the electric forces are accounted for by simple pressures and tensions, then some other ex- planation must be found for magnetic forces, and the hypothesis of ether-

620 The Theory of Relativity [ch. xx

stresses loses its principal advantage. Further it has been found that the hypothesis of an ether at rest fails entirely to account for the forces in systems in motion (§ 655). To account for these forces, it appears that the ether must be supposed endowed with momentum. On the relativity-theory also we have seen (§ 700) that the forces can be explained in terms of a flow of momentum. The relativity-theory has thus shewn that what is essential to the ethereal explanation is not the ether but the momentum with which it was supposed to be endowed. It is quite easy to imagine a flow of momentum without there being an ether to carry it, and the conception of forces and pressures arising from a flow of momentum is one with which we have become familiar in other branches of physics, as for example the Kinetic Theory of Gases.

Almost similar remarks apply to the interpretation of electromagnetic energy. Stress in the ether would account quite simply for either electric or magnetic energy, but not for both. Usually the energy of ethereal stress is regarded as electrostatic energy, so that kinetic energy of the ether must be invoked to account for magnetic energy. Again the ether has to be supposed endowed with motion, but the motion requisite to account for the magnetic energy is something quite different from that corresponding to the momentum required to account for electromagnetic forces. So far from the ether providing a simple explanation of all phenomena, it is found that highly complex pro- perties must be ascribed to it in order to account for electrical and magnetic properties simultaneously.

If an ether existed, it would provide a fixed set of axes relative to which all positions and velocities could be measured. To account for the result of the Michelson-Morley experiment, it would be necessary to postulate a real shrinkage of all bodies moving through the ether. This shrinkage could not be detected by mechanical means, for a measuring rod would shrink in precisely the same ratio as the body to be measured, but it could be detected by gravi- tational means unless every gravitational field of force shrunk in just such a way as to conceal the shrinkage of matter. For instance, if the gravitational field did not shrink, the geoid, or surface of mean sea-level on the earth, might be a gravitational equipotential for some one velocity through the ether, but could not remain an equipotential as the earth's velocity through the ether changed from point to point of its orbit. Thus we might anticipate seasonal and daily tidal surgings as a result of the earth's motion through the ether. No such events are observed. It is true that even if these occurred the earth's motion through the ether might not be sufficiently rapid for them to be capable of observation, but the generalised theory of relativity, explained in the next section, makes it clear that such events could not be observed whatever the earth's motion might be. There is no longer any room for reasonable doubt that gravitational phenomena conform to the relativity condition.

701, 702] Generalised Relativity 021

If, then, we continue to believe in the existence of an ether we are compelled to believe not only that all electromagnetic phenomena are in a conspiracy to conceal from us the speed of our motion through the ether, but also that gravitational phenomena, which so far as is known have nothing to do with the ether, are parties to the same conspiracy. The simpler view seems to be that there is no ether. If we accept this view, there is no conspiracy of concealment for the simple reason that there is no longer anything to conceal.

Generalised Relativity.

702. In discussing the transmission of light-signals in § 675, we made the assumption that light travelled in straight lines with uniform velocity. If space and time were known to be uniform throughout their whole extent, no such assumption would be needed; the uniformity of velocity and the straightness of path would be direct consequences of the uniformity of time and space.

The theory of relativity has however developed in such a direction that space and time can no longer be supposed to be everywhere uniform. In the early days of the theory it was noticed that Newton's inverse-square law of gravitation did not conform to the relativity-condition, and in 1915 Einstein put forward a theory of generalised relativity according to which all gravita- tional phenomena are the consequences merely of departures from uniformity of time and space.

According to Einstein's theory, the properties of both time and space in the neighbourhood of a gravitating mass differ from those in regions far removed from all matter. The properties of space in the latter regions can be adequately described by the geometry of Euclid, but those in the neighbour- hood of gravitating matter need a new geometry for their description.

In ordinary space, as described by Euclid's geometry, parallel lines never meet. Other geometries and other spaces can, however, be imagined. For instance, two lines drawn upon the earth's surface, through two points on the earth's equator, both running due north, and so running exactly parallel to one another, will ultimately meet in a point — the North Pole. We see that the geometry of a spherical surface is different from that of a plane, so much so that almost all the ordinary theorems of Euclidean geometry fail when applied to a spherical surface.

The geometry required by Einstein's gravitational theory is much less simple than the spherical geometry we have just used as an illustration. It is not concerned with two-dimensional surfaces, or even with a three-dimensional space. It is concerned with a four-dimensional continuum, of the type con- sidered in § 675, in which three space-coordinates and one time-coordinate are plotted parallel to four rectangular axes. In the neighbourhood of gravi-

622 The Theory of Relativity [ch. xx

tating matter this four-dimensional continuum is supposed to be curved somewhat in the way in which the earth's two-dimensional surface is curved. A circle of diameter 1000 miles drawn round a point on the earth's surface will not have a circumference of 10007T miles, as it would be if the circle were drawn on a plane, but of only about 997-77- miles. In the same way, according to Einstein's geometry, the circumference of a circle of radius r drawn about a gravitating mass is not precisely 2irr, but is less by a fraction which is pro- portional to the mass and falls off as we recede from it.

As a consequence of the special geometry of a spherical surface, it is not possible to draw a map on a plane surface so as to shew all the parts of the earth's surface simultaneously in their proper shapes and relative sizes. This results from its not being possible to select coordinates x, y on the earth's surface such that the element of length ds is given by

ds2 = dx2 + dtf (759)

at all points of the surface. The simplest coordinates it is possible to select are the ordinary 6, </> of spherical polar coordinates, in terms of which the element of length on the surface of a sphere of unit radius is given by

ds2 = dd2 + sin2 6 dcf>2 (7C0).

According to Einstein's generalised relativity, the element of length in the four-dimensional continuum can be expressed in the form*

ds2 = dx2 + dy- + dz2 + cZt2 (761)

only in regions which are far removed from all gravitating matter. This form for ds2 is of course analogous to expression (759) for the value of ds2 on a plane surface. As in § 675, r stands for iCt where i = «/(— 1).

If we replace t by its value iCt and transform from the space coordinates x. y, z to the usual spherical polar coordinates r, 6, (f), equation (761) assumes

the form

ds2 = dr2 + r2dd2 + r2 sin2 6d(p2 - C"-dt2 (762).

Einstein's theory requires that in the neighbourhood of a gravitating particle of mass m, equation (762) shall be replaced by

ds2 = drl + r2 dd2 + r2 sin2 ddd>2 - C2 ( 1 - ^) dt2 . . .(763). 27 no f \ rO- /

l7C2

A particle describing a "geodesic" or most direct path — defined by Sjds = 0 — in this space, can be shewn to change its coordinates x, y, z, t in very approximately the same way as a particle describing an ordinary ellipse or hyperbola in ordinary space about a mass m under a law of attractive force ym/r2. The agreement, however, is not quite exact, and Einstein's theory is found to require three phenomena which were not predicted by, and are in

• In technical investigations on Relativity - tin* is usually written for our ds3.

702, 703] Generalised Relativity 623

fact inconsistent with, the classical Newtonian theory; first, the perihelia of all the planets ought to advance at a rate which should be easily detected in the case of Mercury; second, light passing near to the sun ought to shew an appreciable deflection ; and third, the spectral lines emitted in a strong gravitational field such as that of the sun ought to be seen shifted slightly to the red when compared with the corresponding lines emitted in the weak gravitational field of the earth. Of these three phenomena, the first had been observed by Leverrier long before any explanation was forthcoming, the second was observed as soon as it was looked for, first in the solar eclipse of 1919 and subsequently in that of 1923, while the third is still in doubt, on account of the extreme difficulty of the observation and the smallness of the quantity to be measured. But the quantitative agreement in the case of the first two phenomena is so good that no doubt is felt as to the substantial truth and accuracy of Einstein's theory.

In brief Einstein supposes a particle in a gravitational field to describe a straight path through a curved space* whereas Newton had imagined it to describe a curved path through a straight space. Newton imagined the curvature of path to result from the action of "forces" which emanated from the gravitating mass, and tried, although without success, to interpret these forces as stresses transmitted by a gravitational ether. Einstein's theory abolishes the conception of gravitational "force" and so escapes altogether the dilemma of having to suppose these forces either to be transmitted through a medium or by direct action at a distance.

WeyVs Electromagnetic Theory.

703. This dilemma of action through a medium or action at a distance is precisely that which has led to most confusion in electromagnetic theory (cf. § 154). If it can be avoided in gravitational theory, it would seem reasonable to hope that an electromagnetic theory could be constructed which should also avoid it.

This has in actual fact been attempted. Weyl in 1918, followed by Ecldington in 1921, shewed that Einstein's geometry is far from being the most general geometry in which the relativity-condition is satisfied. In the expression (763) which specifies the element of length ds on Einstein's theory, the coefficients of the differentials dr, rd0, r sin 0d(f> and dt are functions solely of the position of ds in space. With certain conventions as to the meaning and

• It must always be remembered that the space in question is four-dimensional; otherwise the statement appears nonsensical. It would be absurd to say that the approximate semicircle described by the earth between perihelion and aphelion is the most direct path between these two points; it is only when the six months interval in time is taken into account that the statement begins to appear reasonable. We can get rid of the time-interval by supposing the particle to move with infinite velocity in which case the path becomes a straight line even in ordinary three- dimensional space.

624 The Theory of Relativity [ch. xx

method of the measurement of length*, we may deduce from this expression that the length of a measuring rod would change as it was moved about from place to place in a gravitational field, but that its length at any instant would depend solely on its position in space. Indeed we may be even more precise, for the coefficients of the differentials in expression (763) all depend on the

single quantity 1 j^ which in turn depends only on the gravitational

potential ym/i~, so that the length in question depends only on the gravi- tational potential at the place.

In the most general geometry possible in a space of coordinates x, y, z, r, a measuring rod of length I moved parallel to itself through a displacement dx, dy, dz, dr may be expected to experience a change of length dl defined by

dl = l(Fdx + Gdy + IIdz + Kdr) (764),

where F, G, H, K may be the most general functions of the position of the point. If the rod is moved from one point P to any other point Q its whole change of length will be given by

\ogl-fi=[Q(Fdx + Gdy+Hdz + Kdr) (765).

lp J p

In Einstein's geometry, Iq and lP depend only on the positions of P and Q, so that the integrand on the right is necessarily a perfect differential. The condition that this integrand shall be a perfect differential is expressed by the six equations

dH_dG=Q

dy dz

dx dy ' dz dr

In Weyl's geometry, on the other hand, the integrand on the right hand is not in general a perfect differential, and the six quantities which constitute

• Unfortunately it is round these conventions that the difficulties of the subject mainly centre. It is meaningless to speak of a measuring rod changing its length unless there is something more absolute against which it can be measured, and the complexities of the theory, especially of Weyl's theory, turn on the properties of the imaginary gauges or meshes against which material objects may be measured. It is impossible to give a full discussion in the present book. The student who wishes to pursue the subject further may be referred to

Eddington, Space, Time and Gravitation.

Eddington, The Mathematical Theory of Relativity.

Weyl, Raum, Zeit, Materie (French Translation, Temps, Espace, Matiere).

It ought to be added that the brief sketch in the present book follows the exposition of Eddington rather than that of Weyl.

dK

dF

dx

' dr

dK

dG

*y~

' dr

dK

dH

dz "

"3r

.(766).

703, 704]

Generalised Relativity

G25

the left-hand members of equations (766), instead of vanishing, have values a, b, c, d, e, f so that

dG

dH

dy

dF_dH = b

dz dx

dG

dx

dF_

dy~C>

dK

dF

■d

dx

dr

dK

dG

dy

dr ~

■■ e

dK

dH

-f

dz '

dr '

.(767).

704. If we restore its value iCt to r, and replace K by ity and id, ie, if by X, Y, Z, the system of equations assumes the form

dH

dy

dG

zt: ~ ^z a> — ^Z~ n~^T — X

dz

dF_dH = b dz dx

dG_dF = dx dy

dx dy dz

ldF

c dt

ldG C dt

ldH

G dt

= Y

= Z

.(768).

On differentiating the three equations on the left with respect to x, y, z and adding, we obtain

da . db do

.(769).

dx dy dz

The seven equations (768) and (769) are formally identical with the seven equations from which we proceeded to develop the general equations of the electromagnetic field in § 639. Weyl's electromagnetic theory supposes that the identity is one not only of form but also of reality; he supposes an equation of the form

dl = Kl(Fdx+Gdy + Hdz-CVdt) (770),

to connect the change undergone by a rod on displacement with the three components of magnetic vector potential F, G, H and the electric po- tential ^. This equation is identical with our former equation (764) except for the occurrence of the constant k which is necessary to secure that F, G, H and M* shall be of the appropriate physical dimensions. The only result of introducing this factor k is that the left-hand member of equation (765) must be replaced by

-l°gf>

K Vp

after this k disappears and we reach equations (768) and (769) as before. We see that on Weyl's theory the values of F, G, H, ^ at any point are determined by the rate at which a unit measuring rod changes its length as it passes through the point.

On Weyl's theory a material body has no permanent size associated with it. By many this has been regarded as an objection to the theory. The j. 40

626 The Theory of Relativity [ch. xx

various attempts which have been made to avoid it, while retaining the obvious advantages of the theory, can hardly be discussed here.

705. From equations (768) and (769) it is easy to develop the whole of the classical electromagnetic theory.