origin and concept of relativity iii[1]

The British Society for the Philosophy of Science

Origin and Concept of Relativity (III)Author(s): G. H. KeswaniSource: The British Journal for the Philosophy of Science, Vol. 16, No. 64 (Feb., 1966), pp. 273-294Published by: Oxford University Press on behalf of The British Society for the Philosophy ofScienceStable URL: http://www.jstor.org/stable/686829Accessed: 19/05/2009 08:46

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ORIGIN AND CONCEPT OF RBLATIVITY (III)*

G. H. KESWANI

I Relativity of all Motion

IN parts I and II of is paper 1 we mentioned Poincare's doubts about ie validity of the principle of relativity generally. He found the pnnciple valid for displacements in space and for uniform motion of trans- lation but saw incontrovertible evidence for absolute e?ects in ie case of accelerated and rotational mojaon. For example, even uniform rotational motion, say, of the earth, could be detected th Foucault's pendulum, by observations made on the eardl itself, i.e. not relanve to any other body. Poincare reflected: Experiments have reference not to space but to bodies. Therefore, which are those bodies with respect to which bodies that tum, turn? Is it, he conjectured, that to apply the law of reiativity in all its rigour, it must be applied to the entire universe? If we did so, how cotlld the principle of relativity remain valid any more? If motion, in general, must be defined with reference to the universe of' fixed ' stars, then it is surely absolute, and cannot be relauve. Such were the questions that troubled Poincare 2 1n I902.

Although Poincare returned to the subject of relativity of motion again and again, ie first serious attempt to extend the pnnciple of relatinty to all types of motion was made by Einstein,8 culminating in is paper of I9I6. Einstein thought that he had succeeded in this attempt through the ' principle of equivalence ', postulating equivalence of gravitational fields and accelerated cbordinate systems, and the ' principle of covariance ' showing that it is possible to generalise the metric for alZ coKrdinate systems (CS) nto a single invariant quadratic form, and to these ideas we now tuin.

*Roceived I8.i.65

ThisJounsal, I965, I5, 286 ad I6, I9.

2 Science and HnoNsis, Dover, pp. 77, 84, II3, II4, 169I72.

8 lXse Foundations of the Gral Theory of Relativity, Sc Principle of Relativity, Dover, p. I I I.

273

G. H. KESWANI

2 The Principle ofEquivalence

The following CS are considered in various formulations of the principle of equivalence.

SO, an inertial CS (we shall comment on inertial CS later). S1, a CS having acceleration (a) relative to SO.

S2, a CS falling freely in a gravitational field of any intensity. S3, a CS held at rest in a gravitational field (which can produce acceleration a).

The systems S0 and S2 are considered eqliivalent in regard to ' the motion of particles or any otherphysical process s 1 and likewise S1 and S3 are considered physically equivalent between themselves.2

In one of Einstein's formulations,3 S0 is considered equivalent to ' 1 with a gravitational field (a) superimposed on it. The system S1 is then regarded by Einstein as ' at rest ', and equivalent to the inertial system S0. We consider this formulation. 'three questions now arise.

(i) How is a gravitational field to be produced to change S1 from a non-illertial to an inertial system so that S1 may then be regarded as ' at rest ', as Einstein proposed?

Obviously, a gravitational field can be brought into play only by introducing suitable masses. Can this be done without contradicting the relativity-thesis? Let us go back to the vexing rotating sphere of Poincare. What is the system of bodies which will produce a gravitational held such that the rotating co-ordinate system can be regarded as ' at rest ' ? According to a calculation made by Thirring,4 a mass in the shape of a concentric shell, rotating with the same angular velocity but in the direction opposite to that of the inner sphere under consideration, will induce the same centrifugal and Coriolos field in the inner sphere as the inner rotating sphere alone does, provided it has such enormous proportions that

MKIc2RI or M/R I *3 5 X I028 gm. cm.

W. Pauli, Theory of Relativity, Pergamon, I958, p. I45.

2 R. C. Tolman, Relativity Thermodynamics and Cosmology, Oxford, I949, pp. I74-

I 75. Einstein, op. cit. p- I I 4

3 A. Einstein, The Meanirlg of Relativity, Methuen, I950, p. 56.

4 H. Thirring, ' Uber die Wirkung rotierender ferner Massen in der Einsteinschen, Gravitaiionstheorie', Physik. Z., I9I8, 19, 33; 192I, 22, 29. For a succinct remark on these papers, see: W. Pauli, op. cit. pp. I74-I75.

274

()RIGIN AND CONCEPT OF RELATIVITY (III)

where M is the mass of the outer concentrlc shell, R is the radius of the shell, K is the gravitational constant,

and c is the velocity of light in vacuo.

As we shall see, for a mass to have M/R of tin order c2/K, it has to

be the universe itsel? We might now be tempted to say that this only

shows rather (lefinitely the relaiivity of rotational motion; we may regard doe sphere as rotaiing relative to the universe or the universe rotating relative to the sphere with the same angular velocity in the opposite direction. Eiier of dlem, when rotaiing, subject each particle of the inner sphere tO the same field of accekraton. But this is not

correct. The universe is one, while the rotating bodies are many and their axes of rotation and velocities varied. Are we to regard the u verse as urldergoing rotations of all klnds simultaneously, each school- boy spinning his top, producing a fresh rotation? Also, how are we to rotate the masses around to convert a non-ineriial rotating CS into an inertial one? The task appears to be operationally impossible. More- over, rotation of the outer material of ie universe vXll induce enormous centrifugal and Coriolos forces in it, wlich have not been observed. The correct way is to regard rotation as moton relaiive to the masses of the uIiiverse, producing as a consequence the observed contrl-

fugal and Coriolos field. And motion relative tc) the universe and physical effects of rotation are both absolute.

(ii) Is the equivalence between S1 widl a gravitational field superimposed, and the ineriial Cs, SO, complete, as Einstein

asserted? TSs is not so even according to what Einstein himself taught us.

If a gravitational field-exists, the Riemann-Christoffel (R-C) tensor

cannot vanish at all points of the space-time, but fUr accelerated systeins

not subjected to a gravitational field, the R-C tensor vanishes at all

points of the space-time, which is then flat. Evvalence cannot therefore be complete. EddLington 2 had slearly pointed out that the

1 Einstein always rqarded the equ*aknceto be complete, ie. for all events. See, for example, Id and Opiniogs by AlbertEinstein, Croom Pub., I95X, p. 287. In his original paper he said that a gravitational fie}d could be ' produced ' merely by cOgmg the system of coordinates, i.e. by having sutably ccelerated co{>rdite systerns. The Principle o+Relativity, Dover, p. I14.

2 A. S. Eddington, The Mathematica1 Theory of Relativity, Cambridge, Ig3os pp.

4>4I. There is, however, a serious diSlculty here. The trouble is that one does not

always knowwhetheror noFtheR-Ctensorentersexplicitly. Weowethis remark

t 9* 275

G. H. KESWANI

pnnciple does not always hold good, only sometimes. A high degrec of physical eqaiivalence is, of course, to be expected sance dse equivalent grantational field is so chosen as to produce ie same linemaiical effects as the acceleraiion field it replaces or nullifies.

(iii) Even if iere were complete equivalence between the two, would it establish relatinty of all motion?

It is said Yes, because it is argued that by internal observations we cannot disjunguish between a state of non-uniform motion and a gravitationsl field.l Quite apart from the consideraiion wheier the curvature tensor does or does not figure in a given physical situation,

further reflection will show that iis assumed ambiguity in deciding whether it is mechanical acceleration or gravlty, canrlot lead to relativity of all motion. It appears to us that one can always distguish between the two possibiliiies. A cosmonaut can know exactly whether it iS an increase in the grantational pull from some body or acceleraiion, and iis by observaiions made in his capsule. He can measure t-he momentum of the escapq gases relaiive to the capsule and arnlre at fdle acceleration of his capsule.

We may fitangly round offthis discussion math a comment by one Ofthe foremost living ' relaiivists ' za

It appears as if general theory contained within itself the seeds of its own con- ceptual destructon because we can construct ' preferred ' coordinate systems. These preferred coordinate systems are not ' dat ' bllt they are determined by the intnnsic conditions of the physical situation....

Our own point of new, as we explained earlier, lies farther sn the same direcion. Even ' flat' space-ome of accelerated CS, points towards an absolute frame of reference, and more about this in the

sequel.

3 Einstein's Theory of Gvitation

Einstein's general theory of relatsity is really only a dleory of grantaiion and of modificaiions in the laws of physics in grantaiional

to one of the referees. A coected que-stion is whether strictly homogeneous gTavi- taiional Eelds can erst. Also see: F. A. E. Pirani, Lecheres on General Relativity (Brandeis Summer Estitute in Theoreiical Physic:s), Prentice Hall, I96S, p. 260.

I R. C. TolmaIl, op. at. pp. I76I79. A. Einstein, Se Principi of Relativity, Dover, p. II4.

2 p. G. Bergmann, ' Observables in General Relaiinty ', Rev. Mod. Phys., I96I,

33, SI4-

276

ORIGIN AND (CONCEPT OP RSLATIVITY (III)

fields. Even the principle of covaliance, as Fock 1 has pointed out so emphaiically, does not make it in any sense a general ieory of relaiivity. Here we give an outline of tEhe theory partly to see its conneciion widl dle nojaon of general relativity of moaon and partly to get insight into the nojaon of an itlerdal CS. In this process, we slull develop a static cosmological model consistent wii principal astrono- niical observations, purely from the field equaaons widlout using the cosmological term, and verifying Mach's pnnciple exactly in a some- what novel mantler.

Elnstein postulated an invariant 44imensiona1 quadratic metric of spacFte, a generalisation of Minkowski's metric ofthe special theory, as dP gFv(x) dxz dx,. ,^ I, 2, 3, 4) (I)

where x's are ie coKrdinates of the 4 dimensional manifold. It is assumed that form (I) has the signature 3 +I- 2. As Forsyth 2

has pointed out vigorously, this iS a stupendous assumption, without mathematcal justification. But it seems to ' work ' physically. The 4 dimensions of 4-dimensional geometry are co ordinate among them- sel_s. Not so are the 3 dimensions of space and the 4th dimension of \/-I X ame.

The coefficientsg,sv(x), all functions of x's, depend on the gravitaung masses present and their motions. Einstein regarded the space-time manifold as non-Euclidean,3gE,v then defg the metrtcal properties of iis manifold. How do the components of g.^ vary with the distn- bution of matter? From g..v, which constitute a covariant tensor, by differentiation of gv only, we may build up a mixed tensor of the 4th rank (RZ or curvature tensor):

BVa-{Ha*a}{aloe}-{#Vva}{afre}+dx {#J1##e} ax {/1IJe}>

where {yc,a} i8 - (t-+ ax^, dx2, )

1 V. Fock, lXe 171eory of Space Time and Gravitation, Pergamon, IgS8, pp. xviii, 3S?, 37>374; ' The Prulclple of Relanvity and of Eqliivalence inEinsteinian Gravita- tion Theory ', K. Norske ViSsk. Selsk. Forhandl (Norway), x963, 369 I6.

9 A. R. Forsyth, Geometry of Four Dimensions, Cambridge, I930, Preface. Edding- ton, who showed great concem for ' first principles ' has left the follog coy remark: ' The 3 minus signs with I plus sign pariicularise the world in a way which we could scarcely have predicted from first prinapIes.' op. at. p. 25-

s N. Rosen suggested that spacFtime be regarded as Euclideatl, withgv defining the grantaiioml field but having no conneciion with the geometry of space-time, Phys. Rev., I940, 57. I47. It appears to us that Rosen's point of new is the more satiF factory. See for exampIe Appendix.

277

G. H. KESWANI

If ie R-C tezor sshes, ien co ordinate systems can be found such iatgHV are constant throughout.

This part of ie analysis ts purely maiemaiical. Einstein then postulated iat thelawB^,^?-o describes space-timewithouta gravita-

tonal field. The R-C- tensor may be contraaed (e - ), giving the

Ricci tensor:

Gu,v = {,ucr,a} {av,(r}-Wu,a} {aantr} f 3x @(r,(r}-dx {yv,(r}.

Einstein assumed that G{AV = ? is the law of gravitaiton in empty space surrounding matter. gyv are then given by dle solutions of partial diflerential equations G2.V-o. Since gv and G#V are symmetrical tensors of 2nd rank, each of iem has io independent components and itwould appear that iere are ten equations from G80ZV-o, todetere the ten unknowngHv, but there are four identical relatons between Gl,^, leaving only 6 independent equaiions. In general, four additonal ' coKrdinate conditons ' are reqtiired. It appears that there is no way of telling which condijaons to choose. This sltuaton is regarded as unsatisfactory by some.l

If the components of a tensor vanish in one system of co ordinates, they vantsh in all systems, i.e. they are covanant with respect to arbi- trary transformations of coKrdinates. Einstein 2 believed thst covariance iml)lies a ' general postlllate of relativity , but as pointed out by Fock3, dlis is by no means the case. The basic point is that covariance only ensures that & form of a law is the same for all CS. II1 the case of invanant form (X), for example, physical laws dependent ongLXV vary from CS tO CS, the different CS can therefore be disiingtiished from one another by observaiions made Withill each CS and a doctrine that relauve- motion alone is significant cannot be maintained for all types of mojaon.

In order to fmd gv in nonwmpty space, Einstein introduced the energy-momentum tensor, T,SV or TFv or TS,. For a perfect fluid the mised form, which is the simplest, is

Z bo+pls2) fiV _ aV p/cs {8t I r -8} (2)

V. Fock, op. at. p. tp4. N. Rosen, op. at. p. 149. 2 The Principle of Relativity, Dover, p. II7. 8 V, Fock, op. at, p. 3S?? Rica and LenXivitahad shown earlier in I90I tllat

fiom any assumed law one can always denve another law in covariant form which does not diffier fFom the assumed law in any observable consequence: E. Whittaker, A Hisforoy of ffie Tiories of Atther and Elaritity, Thc Modern 17xeories (rpooIg26), I'homas Nel-son, I953, p. I 59*

278

ORIGIN AND CONCEPT OF RELATIVITY (III)

where p0 is the scalar proper density, p is the scaLar proper pressure or sttess,

and ddXz z the the 4-velocity of the fld.

Under static conditions, only the ddX4 component (equal to I) of the 4-

velocity survives and then pl 2 _ _ _

r_ - p/C2 p/Cs -

-- Po

For our present case we may regard the pressure as arising Som the radial grantational lnteractions ofthe ' parjacles ' constituting the fluid. The divergence of TH always vanishes and it therefore satisfies the principle of conservation. Other energy tensors could be constructed but if the law of gravitation is tO be expressed by second order differential equations, there is no alternative.

The divergence of G,, does not vanish identically but that of another tensor, G,] i8,rG, doe?s. Einstein then gave the following ' field equations > for finding gfi, in matter-occupied space:

87rKTE,-G, i8E,G (3) where G-8 G;, is a scalar called the Gaussian cutrvatureand K is the

. . gravltatlonaw . constant. Here we are interested in a particular solution of eq. (3). We con-

sider a sphere of perfect fluid (this means that only radial interactions can be transmitted and that the fluid has constant proper density) and follow Synge.l We impose no limit on the size or mass ofthis sphere. Firsdy, we set up Gaussian polar coKrdinates (p, @, +, r), where p is ehe geodesic distance from the centre of the spherical mass,

@, + are polar angles, and T iS the proper iime measured at the centre. The metric form is then:

dss-772dr2 2 [dp2+r2(dd92+sin28j</>3)] {X4)

where 7 and r are funciions of p only. 1J. L. Syage, Relativity: lSe General IfSeory, North-Holland Pub. Co., I960, pp.

22&229, 276, 287-289.- For a sidar approa&, se T. >w-vita, o Absolute Differential Calcglus, Blackie, I947, pp. 408-4I4.

279

G.RIZESWANI

Coder dle ospace for which p and r hasre fised values,and t1 and + are current coordinates. On account of sphencal symmetry, all poinjcs of this 2-space are equivalent and it has a constant intinsic Gallssian curvature I/r2. The merc on it is tien

dss- rs (dW2+sAl+8). 11Le ospace has, therefore, an invariant area, 47n2,

Obuously p and 7 could be regarded as funcaons of r, when the metric form (3) may be expressed as

dss = ewdr2 s[eodr2+r2(dS2+sin249d+2)],

ea and eY being funcitons of r. We ius have what Syage calls ' curvature cbordinates ' (r, 9, +, T), ie last one, as before, being the proper iime measured at the centre ofthe sphere.

The field equaiions (3) contain p and pO on -L.H.S. and ea and eY on RH.S. as unknowns. OthergE,v are ty or zero. It tums out iat to determine the four quantities p, p0, ea and eY, (3) pelds ree equations only. Itis is to be expected because p and p0 could be related. We dlen get

t-a-I qrs

eY {(3 vI qR2 vI-qP)/(3 vI qR2-I)}2

and p/c2 - po{(A/I qr2 vI qR2)/(3vI qR2 vI-qr2)} (5a) wsth q- {37rKpO/c2 2MK/c2R3 (5)

R-radius ofthe sphere and M massofthesphere.

111e metric jnside the sphere is then

ds2-eYdr2 --a 0 2 +r2(dW2+26Jd?>2,]. (6)

Reverung to e radial geodesics, we find

dp2- or ( ) (I-2)- (7)

The soluton of this diSerental equaton is r _ f isMqip which is plot- ted in Fig. I . As p increases, r increases to the maum value R,,-+ and t;hen diminishes. Beyond this value, one-toone correspond- ence does not exist between the events @, 8, i, ) and the co-ordinate tetrad (r, 0, +, z). Therefore only the portion OP of the graph is phwcally admissible.l

For points of similarity: see L. O'Raifeartaigh Proc. Roy. Soc. (A), I 958, 24f, 2?2.

280

ORIGIN AND CONCBPT OF RELATIVITY (III)

FIG. I. Cunrature coordinate/geodeac Fictawe relaiionship: Same value of r (r) corresponds to more d one value of p tpl. ps, . )

4 World ModeI

We now ideni*R,, 6 iecurvatureofdsewhole 'universe'{>f tche biggest sphere (mass M") accesuble to dle observer at the cente. The maximum geodesic distance observable is dsen

pmax-2 * = 2 Ru 2 +/87rKpo

The coeficientI/(I-qr)t of dr becomes itnsfyinaryfor r>f}. NerF fore if r>f*, only dle poriion R">rzo is physically accble.

: z * - * * * * - * * w b * . * *

* ^ * ̂ * * * *#/-- * * *

- <w.*@w*@@

* o * X* i* * * * ? z * . * * * * -

* e # 0 ? * * * * * * v * * *

* * * * #

* * * * * *

OSservers ot centtes of zapcdist Spheres ba;ng

ZM,, cX

Ru K

PIG. 2. Umverse open to observajaon.

However, for each observer, the '-verse' available for observazaon is different. The cosmology is dlen ofthe type indicated in Fig. 2. As we shall see presody, ese ideas cohere phwcally.

l1teaveragedensityofmatterin the universel probablylies between . A*Wider, heares in 11xeoretzal Physics, edv by W.E. Bn et al., htersacnct

Pub.J I963, pe 5?7*

28I

G. H. KESWANI

(I|3) X IO-30 and 3 x Io-30 gm. cm.-3. The energy density of neutrinos is not known and we take the upper value, 3 x IO-3O, and get

pmax-3 *8 x Iol? and Ru - 2-4 x Iol? light-years.

Puting- R3, eq. (6) becomes q

d52 = (I--)dFS-- (I--) dr2+r2(dS5+sin2tild+2) . (8)

For a photon travelling radially from r Ru to ie centre (observer), ds-o and

(I r2/R2) dT2 = dr2/c2 (I r2/R2) ro

and r-limg dr/c2(I r2/R2")- . sR r

No photons from r R,, can therefore reach the observer. For r<R", the emitted and received wavFlengths are related as follows.

Aemitted _ eY/2 evaluated at emitter Areceived eY/2 evaluated at receiver

* * Areceived = Aemitted (I-r /R") } = Semitted (I + R2+ 8 R4 + * )

andshift 8S +SemittedX R2 for R I.

This red-shift is obtained purely on account of the fact dlat light has to travel through a verse having the metric of (8) relative to the observer. The law-ofshifts is quadratic in the first approximation and not linear. Hawkins's 1 analysis of the data published by Humason, Mayall and Sandage shows that for field gala2ues, the law 8A/A < r2-2 best fits the data for galaxies brighter than I4 magnitudes and that in general the result-s are more consistent with the quadratlc law than with the linear.

We have been quietlyusing a metric ofthe satne form asde Sitterfs, -but this metric has now been obed sn an entirely natural way from the held equations, as a result simply of recognition of a singularity2 of

G. S. HawkiB, ' Expansion of the Universe ', Nattlre, I96e, I94, 563-S64. 2 This singdanty is, of course, the same as the famous Schwarzschild singularity.

For tlle exterior metric also, the singularity arises at 2mlR = I or 2MK/c2R = I, or M/R Io28 gm/cm. For no known object (fundamental pariicle, quasar, galaxy) is 2nt,/R>I . 'rhe object would be inYisible-but detectable in certain cases if zjR ZI. It is an interestiw corg3ecture whether, for certain partides (for example, neutrinos) zmlR>I. Rosen ad Fock get an lsotropic and unique metric-form in which the singdar surEace lies at m/R-I.

282


these equaiions. We must now discuss a difficulty of the de Sitter wii- verse which arises in the present case as well. 5 (a) for the ' pressure ' or radial interaction reduces to

p/c2 pO- (9)

The pressure is constant but negative and equal jr magnitude to c2 x density. It had long been believed that the notion of such a negative pressure was inadmissible. The way has now been cleared for the admissibility of (g) by tlle work of Jordan,l McCrea,2 McAlittie,3 Sciama4 and Davidson.5 But the development here has, of course, nothing to do with the creation thories of cosmolcBgy. What (g) says is that ' the gravitational cnergy of any particle arising from its interaction with the rest of the universe is exactly equal and opposite to the energy due to the mass itself'. On this account, to use McVittie's6 words, we have a ' gravitationally steady state '. As A. Hass and P. Jordan suggest, we may put the equation M"Klc2Ru-I, where Mu is the mass of the universe, in the form KM2/RU-M c2, which inJordan'ss words means, ' that the negative potential energy of gravitation for the whole universe is equal to the sum of the rest energies of the stars '. It may appear that a negative pressure may, somehow, offend Newton- ian nlechanics. McCrea8 has pointed out that if the pressure is uniform,

its gradient is zero, and it will, therefore, neither appear in the equation of motion nor in the equation of continuity. This stress cannot therefore be observed.

The relationship p poc2, or ' gravitational energy of a mass equals minus its inertial energy', may then be looked upon as a neoteric verification of Mach's principle.

S- Inertial CS

An inereal CS is usually defined as a system in which ' force-free ' bodies are not accelerated with reference to the CS. However,

t P. Jordan, ' Formation of the Stars and Development of the Universe ', Nature, I 949, I64, 63 7640.

2 W. H. McCrea, ' Relativithr Theory and Creation of Matter ', Proc. Roy. Soc. (A), I95I, 206, S62-575

3 G. C. McVittie, ' A Mode} Universe Admitting Interchangeability of Stress and Mass, Proc. Roy. Soc. (A), I952, 2II, 295-30I.

4 D. W. Sciama, ' On the Ongin of Inertia ', M.N.R.A.S., I953 II3, 34.

6 W. Davidson, ' General Relativity and Mach's principle ', M.N.R.A.S., I957,

II7, 2I2-224. ff G. C. McVittie, op. cit., Summary, p. 295.

7 P. Jordan, op. cit. We do not discuss here a discrepancy of a factor equal to 2. 8 W H. MCcreas op. cit. p S7I

283

G.H. KESWANI

Enstein 1 himself pointed out rEhat this slefiniiion involves a circtllar argument: ' A mass moves widlout acceleration if it is sufficiently far removed fiom other bodies; we know that it is sufficiens;ly far from ther bodies only by the fact it it moves wiiout acceleratom'

Obnously, with Einstein's theory of gravitajaon-before us we csnot ignore the masses of ie 1lniverse. However (8) shows dlat at the observer (r o) the metrsc is

@-dT2-IS [dr9+r2(dd+sin3Sd+2)],

which is locally flat with g4^-diag [I, I, I, I]. Denation from flatness due to sun's granty at Pluto is found to be Ioll times greater than iat due to the masses of ie universe. Therefore CS at rest with reference to the universe are forcFfree and inertial.2 Do other mertial systems in a state of motion relative to ie system of the stars exist? It is an observed fact that systems in uniform motion of translajaon relative to the universe are also inertial, if the definition is ie one given at the beginning of this section, but iis does not neceF sarily imply that CS in uniform motion relative to the universe are all equivalent for ie descripnon of the laws of physics. Aldlough force- free material bodies may not be accelerated relative to such CS, there may be other e?ects of uniform translaiion and this we shall examine in the sequel. That bnngs us to the special theory of relativity again.

6 Special Theory and Lorentz Transiormation Equations As we saw in Part II of this paper, Einstein used: (I) the principle

of relativity, and (ii) the principle of constancy of velocity of light, tO

establish Lorentz transformation equations (L.T.E.). We now propose to show (not for tie first time !) 3 that L.T.E. can be

derived from the second postulate alone, thout recourse to the pnnciple of relativity. The novelties and sometimes weird results predicted by L.T.E. and verified expenmentally-really arise from the second

Se Meaning of Retativity, MeEhuens x9S?. p 57- 2 The following dichotomy mentioned by R. H. Dicke is thus resolved for CS at

rest w.r.t. the universe. '. . . on the one hand wc consader inertial effects to be related Ln an iniimate way to the rest of the universeX and on t}le other hand we assume that the universe has no observable effiects on the laboratory and on the numencal content of physical hws.' ' Eotvos Expeent and the Gravitaiional Red Shift ', Amer. J. Phys. I960, 28, 345.

8 For a similar demonstration, see: G. Stephenson and C. W. Ki]ister, Special Relativityfor Physicists, Lontmans, Ig58, pp. 9Io.

284

ORIGIN AND CONCEPT OF RELATIVITY (tII)

postulate and are indepadent of the noton of relaunty. L.T.E., no doubt, leave certain quanjaiies invariant, for example Fe wave equa- iion of electromagnetic propagation. Experiments dependent on electromagneac propagaiion will, therefore, naturally conform to the relat;inty-principle, but khere ts no ground, really, for asseriing that L.T.E. will conform to dle relaiinq hypodsesis always.

Let S (x, y, z, t) and S'(x', y', z', t') be two inertial fFames as defined earlier, with S' moving wsth velociq v relanve to S in khe directon of X-X' axes which comclde. Let Y' and Z' axes be respeciively parallel to Y and Z axes. The problem is to find dle wansformaiion equaiions, i.e. x', y', z' and t' as funciions of x, y, z, t and v. We expect these equations to be linear, such as x' - glo+gllx+glsy+glaz+gluts where gik are some coefflcients mvolg v. Nis simply means that dle physical content of transformaions is independent of dle origin of tdle coKrdinates. The total differential lx' - glldx+ . . . is ie same iroughout, which would not always be dle case wids non-linear trans formatons. By a suitable choice of the owins, glO may be made zero. Y' and Z' axes are similarly located w.r.t. the XO ans and the direction of v. Obviously x' and t' are independent of y and z, and y' and z' independent of x and t. By shifting the orip of S', Y and Y' axes may be made to coinclde. Therefore y' is independent of z. LikF wise z' is independent of y. These judgements are based on dle cy- lindrical symmetry of the problem and Euclidean geometry. Finally, therefore, these coderaiions lead tO the following transformatons

X =gllX+gl t. y g22yt Z g22Zs 4 (IO)

gss z a sche fictor mdependent of v. We now e we of ie second postulate. Since the velocity of light is independent of dle velocity of an obsenrer relative to dle source of light, the spherical wave iont of a light pulse leag the coincident origins of S and S' at time t o t' will bave the equatons in the two systems as given in (II) and (I2) below.

Xs +y2 +Z2 _C2t2 = O (t I) X 2 +y/2 +Z'2 _c2t'2 = o (I2)

Since y' = gs2Y and z'-gs2z, the conditons dlat (I2) should trans form lNtO (II) are ie same as ie ones for tX'2 dt'2)/g22 to tform lNtO (X51 C2t2). We comider it, transfomlaton of (X 2-C2t 2) StO (x2-c2t2). Subsiituting for x' and t' Som (IO), we have

(gllX+glt) c2(gulX+gst)2 = Xs csts 285

G. H. KESWANI

This is an ideniity in the variables x2, xt and t2, and it funiishes three equations for the four unknowns gll, gl4, g4t and g44. Leavmg aside some tedious algebra, the solutions are found to be:

X - g22(X--- At)/(I +2/C2)+

t - g22(t--- AX/C2)1(I /C2)i.

The coefficientg22 has been restored. + is some function of v, which is the physical parametercharacterising the problem. We may putAas a

power series, thus: ir aO+alv+a2v2 + . . .

Now for v-o, x'-x. .* x' g22(x aOt)l(I-a2/c2)*-x,forallvaluesoft.

Obviously ien, g22 = I and aO =?.

The transformaiton equations thus, simplify into the form: X/ (X-+t)|(I-+21C2)+, y = y, Z = Z, t i (t-AX|c2)/(I- +2/s2)+ + alv+a2v2 + . . .

These equations will transform a spherical pulse of velocity c in S' into a spherical pulse of the same velocity in S. Further simplifications in the above equations do not appear possible without further assump- tions. We now suppose that these equations are valid for all events. We can hardly expect different equations for electromagnetic and other phenomena. This is not tantamoullt to the assumption of relativity or equivalence of all inertial CS. We might recall that in pre-relativity days we expected evidence counter to relativity, if GaJilean transformations held good for electromagnetic phenomena. In fact, at the present stage, we may nurse the expectaiion that application of the new transformations to mechanics may now show that the principle of relativity is not valid for mechanicsv

Consider now the events (o, o, o, tr) i.e. the time recorded by a clock (mechanical or electromagnetic) at the origin of S'. For this clock,

X O (X t)|(I 0 /C ) or + x/t. Now x/t for this clock is itS relative velocity v as measured in S. The transformation equations are then found tO be the usual ones:

x' = ,l3(x-vt), y'-y, z'-z, B(t Vx/c ) with ,8= (I_v2/s2)-i (I3)

286

ORIGIN AND CONCEPT O-F RELAT-IVITY {III3

7 Second Postulate

lEe second postulate was questioned many years ago by Ritz.l He suggested what has recently been revived by Dingle that light is emitted with constant Yelocity relative to dle soulrce or emitter ' just as though it were a matenal patcle fired by an unvarying mechanism ', as Dingle 2 puts it.

Firstly, we must clearly recognise that if the stecond postulate goes, I.T.E. go th it too. Lorentz-transformazon 1s now so iniimately woven into the warp and woofofmodem physics theit if it was invalid, this would surely have been detected somewhere.

And, what do we exactly meantya constant velocity relaiiere to dle emitter ' ? It is radwer difficmlt to say what emits light. Nudeons may havevelocitieso IC aIld orbital electrom O-OIC. h thecaseoffight, it is not at alleasy to defineY vclocity relanve to dle etter '.

It appears to a that Dungle is driven- to reJect the second postlilate because he feels that althcsughwhere is ourerwhelming evidence for relativity, L.T.E. contradict it. The fact of the matter is that we have evidence for L.T.i. and only to that extent for relatierity. No more. Lndeed Einstein 3 himself later took the wew that ' *le whoSe content of the special theory of relativity is inckded in the postXate that the laws of nature are invariant with respect to a Lorentz transformation'.

It is then obviousiat if a ' law of physics' could be discovered which was not invariant under a Lorentz transformation, the principle of relativity would be violated. Here we must note a distiIictton. If the space and jume {neasures -of inertial CS are generally different but constant within various systems (as they are with L.T.E.), by experiments carried out within the respective systems, differences m the laws

1 W. Ritz, Collected Works, pp 3 I 7, 427, 447. Also see W. Pauli, op. cit. pp. 5<. 2 t1. Dingle, ' On Inertial Reference Frames ', Ssience Prugress, I962, 50, 5. Dingle

has taken the viewflthat there is no reliable direct experimental proof of the second postulate. In support, J. G. Fox has argued that in the older observations purporting to support this postulate, [ight from moving sources passed through intervening media (air,glass, envelopeofgas)and notvacuum,and thataccording totheextinction theorem of light, these media became the secondarysourcesofemission,Setc. W. R.Haseltine has pointed outthatextinctiozof light in the inter-vening mediadoes notoccur if thesignals from the primary source have a veloaty different from c. More recent experimcnts avoiding Fox's objeciion, have confirmed the second postulate (D. Sadeh, T. Alva- ger et al., J. F. James and R. S. Sternberg3 excepting an experiment by W. Kantor. P. Btlrcer has shown that the ' relativistic ' t}leory of Kantor's experiment is in con- formity wii bis result.

3 Ideas andOpi1zions by Albert Einstein, I954, Crown I'ub., p. 370. Also see p. 283.

a87 2 o

G. H. KESWANI

of physics are not to be expected. Basically th?, two types of relajuv- ity may be mentoned. The ' pure ' vanety to usc Sherwin's 1 word for it in which all inereal CS are idenecal, not only in regard to ie

laws of phyncs as observed thin any CS but also when a comparison

Of the laws in different CS is made. And ' apparent ' relatinty, in wbich although tbe laws of phpcs are seemingly the same onthi vanous CS, a suitable comparison of two or more systems may reveal di?erences in ese laws. The quesiion then is, how dlis comparison may be made. A comparison of len;ths cannot, perhps, be made meaningfully. Could the iime-measures be compared? Let us see what is to be expected theoreiically from L.T.E. and what the experental results are.

8 L. T.E. and Tinz of Different CS

Consader two contiguous events assoaated wii a point moving widl velocity v relaiive to S. Let dseir measures in S be [x, t] and [x+8x, t+8t]. Transformed by L.T.E. ie corresponding event in ,

z are

[x', t'l-[,l(x-vt), ,8(t vxlc2)],

and [x' +8x', t'+8t']-[08(x+8x vt- vat), ,8(t+8t- vxls2_vax/c9)].

We note dlat (x, t) and (x', t') are the measures sviiin the respeciive systems.

.-. at /8t-(I V2/C2) 0f zt 18t _ t1, (8x/8t = v).

We consider the linlit as at zd At'o,i.e.,thecase when the twoevents coincide in dz limit. Then:

dt'/dt = ,&1 This is a fanliliar result, but what does it mean? Why is the result dt'/dt _ t1 and not vice-versa? te explanation is that in deriving L.T.E. we have assumed one system (S) to be at rest relative to which the oier system moves with velocity v. Odlerwise we shollld not

have got dlis asymmetncal relationship. We may remark that the above result follows m a straightforward manner from L.T.E. but how can it be regarded as in confoty with relaiinty, or ie notion of equivalence of the two CS ? t and t' are the iime measures within two

1 C. W. Sherwin, ' Some Recent Experimental Tests of the " Clock Paradox " ', Phys. Rev., I960, I20, I8. He remarks, ' Thus whatever its theoretical difElcultes may be, pure relativism is untenable earperimentaBy'.

288

ORIGIN AND CONCEPT OF RELATIVITY (III),

CS. h dle law of passage of iime not a law of physics? It is, but this hw is not invanant under a Lorentz transformation. It is to be noted, however, that the time measure of all devices is equally a?ected m S', so that the form of laws wiffiin dsis system will re the same. Einstein himselfasserted later that tche time-measures of di?erent inertisl systems are different and tit movlng clocks change their rhythm. Wniing joindy witEh Infeld 1 he said, ' In clazicAl mechanics it was tacidy assumed that a monng clock does not rhange its rhythm.... We can well imatine that a moving dock changes its rhythm, so long as the law of dlis change is dle same for a11 m rtial CS.... We must accept dle concept of relaiive iime in every CS.... Every CS must be equipped 6 ies own clocles at rest, since motons change the rhythm....' We should like to emphasise iat it could not be the ' rhythm' or ' time ' as it ' appears ' or is measured from the ' other ' system. A change in rhythm of Fis kind (first order Doppler ffiift in frequency) had long been known and caued no trouble in understand- mg. Also, no meaning can be attached to the descripiion ' iime of S' as measured in S'. All difficulites have arisen because relativity has wrongly been identified wii L.T.E.

Let us dear up some irrelevant subtleties. There is nothing in our story about clocks as such. We talked only ofthe time-measures, t and t', of two CS. No observationsfrom one system on another are involved. One need not fuss about the ' simultaneity of dBtant events '. We coder the Simiting case when two events coincide and thus get a relationship at the point of intersection of two world lines. AgatnX one might protest: S' could as well have been chosen as at ' rest ', leading to the perfeely reciprocal relaiionship dt/dt' tt. One codd have, but obnously it is nonsense to say dt'/dt t1 and dt/dt'-t1 @ * I).

One makes a last ditch stand and says that only ie relauve velocity enters into L.T.E. This is true because no other velocity was considered, but we can proceed consistently only if it is conceded that v is measured relative to an absolute system.

The quesiion then is: which is the physical Jystem that is always at rest? The systemof iefixed starsqunlifies for this. The time measures (t') of all oier itlel systems are related to the time measure (t) of the system of dle fixed stars (S) by the equation dt'-dt/,l3.

If we have tvo systems S' and S", so dlat S" has velocity v relative to S' and S' has velocity w relative to the systcin of the universe S, the

1 A. Fktes and L. Jmfeld, rhe EvoZution o+Physics, Cambridge (Paperback), I96I,

pp. I 84, I 86, I 89.

289

G. H. KESWANI

timeZilation factors for dle two syste are as follows (for samplicity

we assume w to be in the same direcison as v; this is the worst case for the present argument, the result is easily generalised):

[I w2/C2]-i for S',

and -I_ V+W 8 IC2 i forS" VI +VW/C2} |

The relative timeRilation factor between S' and S" is then

I v2/c2 +

_(I +VW/C ) _ (I4)

the time-processes being slower m S". Now, in various experiments carried out on the earth, w, the velocity of the earth relative to the universe, is certainly less than IO-3C as indicated by the present degree of accuracy in the observation of red-shift of distant nebulae. If w was higher, a defuiite anisotropy would appear with preponderance of blue-shifted galaxies in one direciion.l

It appears tO US that no experent has yet been carried out to test (I4) for itS denominator. As we remarked earlier, experiments dependent on electromagnetic propagation, of necessity, fully comply with the relativitywonditlon. Also experiments like that of Tomlinson and Essen 2 cannot discriminate between the ' pure ̂ and ' apparent ' relativistic points of view. The vibrating rod in their experiment essen- tially functioned as a clock which was contained in one CS (that of the earth). Orientation of a clock in the same CS has, of course, no effect on its rhythm. In theexperiments on ,u mesons, v/cI,and the accuracy is still about 9 ?/O Ollly.3 It must be noted that slowing of time is observed to take place in ie fast-moving system of the mesons and not

in the slower systemofthe earthhaving much smaller velocity relative to theuniverse. It mightbesaid that incertain recentexperiments using the Mossbauer effect and invohring velociiies v of the order IO- 6 c, absolute velocity of the earth, if any, should have shown up. We shall pre- sently see that these particular experiments could not, in theory, have nzade this possible.

1 R. H. Dicke, op. cit. p. 344. 2 G. A. Tomlinson arZL. Essen, Proc. Roy. 50a (A), I937, I58j 606-633.

3 D. H. Frisch and J. H. Smith, tMeasurement of the Relaiivistic Time Dilation Using ,u Mesons ', Atater. J. Phys., I963, 3 I, 342.

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g Experiments Using Mosstauer EfJect

Two kinds of such experiments pertinent to the present inquiry have been performed. In one, performed by Hay et al.,l and repeated with greater accuracy by Kundig,2 a change in the characteristic frequency of absorption of gamma rays by an absorber situated near the outer periphery of a rotor was observed, the source of gamma rays being near the centre of the rotor. The gamma rays passing through the absorber reached a stationary counter outside the spinning rotor. The counting rate was observed to increase with the angular velocity of the rotor. In the other experiment, first performed by Pound and Rebka,3 the rate of absorption of gamma rays by an absorber was found to change with the temperature ofthe emitter, on account of changes in the characteristic frequency of emitting nuclei parficipating in the thermal velocities of the crystal latiice.

Synge 4 has given a beautiful demonstration to show that for a source at radius R and the absorber at radius R anywhere on the rotor, the ratio of the frequency of the source, v', and of the absorber, v, is

V {I- R2C,)2 +

V tI R'202J

exactly, Ct) being the angular velocity of the rotor. The shift to be expected is dependent on the magnitude of the radii R' and R and of the angular velocity but not on the relative velocities of the source and the absorber. Thus, -when the source and absorber lie at the two ends of a diameter, the expected shift is zero, although the relative velocity of the two is 2oR. Indeed in an experiment performed earlier, this theoretical result was verified ! 5

In the second experiment also, the changes in temperature result in changes in the accelerations of the emittirlg nuclei in harmonic motion. (In the theory of the experiment worked out by Josephson,6 it is

1 H.J. Hay et al., ' Measurement of the Red-Shift of an Accelerated System Using the MossbauerEiect in Fes7 ', Phys. Rev. Letters, I960, 4, I65.

2 W. Kundig, ' Measurement of Transverse Doppler ESect in an Accelerated System, Phys. Rev. I963, I29, 237I.

3 R. V. Pound and G. A. Rebka, ' Variation with Temperature of Energy of Recoil-free Gamma Rays from Solids ', Phys. Rev. Letters, I960, 4, 274.

4J. L. Synge, ' Group Motions in Space-time and Doppler ESects ', Nature, 1963, I98, 679.

6 D. C. Champeney and P. B. Moon, ' Absence of Doppler Shift for Gamma Ray Source and Detector on Same Circular Orbit', Proc. Phys. Soc., I961, 77, 350.

6 B. D. Josephson, Phys. Rev. Letters, I960, 49 34I.

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G. H. KESWANI

assumed that the forces coupling the atoms are harmonic.) In fact the situation is similar totherotorexperimentexceptthattheacceleraiion (-cl)2R'sin cl)t) of an emitting nucleus is variable here but is constant ( 02R') in the rotor experiment.

Concluding Remark

The principle of constancy of velocity of light leads to Lorentz transformation equations (L.T.E.), without the use of the principle of relativity, the two being independent schemes. An analysis of L.T.E. shows that a high degree of equivalence in the description of the laws of physics exists for different inertial systems. The principle of relativity, therefore, assumes verisimilitude by identification with L.T.E. Fur- ther analysis, however, shows that although the time measures of different inertial systems areconstant within, they varyErom system to system. This forces us to recognize the system assumed as at rest in deriving L.T.E. to be a primary preferential system. The system ofthe masses of the universe is the obvious choice for this. In fact, the aW solute effects observable in the case of accelerated motion demand this primary system.

Room No. 28A, 4th Floor, Baroda House, Annexe I

Curzon Road New Delhi, India.

- APPENDIX Motion of Mercury's PeriXelion

The present argument is based on ie consideration it the 3-space is non-Euclidean according to Einstein,gvbcdxvdeEming the metnc of space (The Principle of Relativity, Dover, p. I6I; The Meaning of Relativity, Methuen, p. 59), while with Rosen, gElv merely define the physical state, without any connection with the geometry of space which is regarded as flat. When the prevailing geometry is non- Euclidean, the ratio circumferencelradius is not 27r and in the present case it is in fact less than 27r.

The equation of the orbit is obtained from dle exterior Schwarz- schild metric:

ds2; (I- 2m/r)dwr2 i[(I-2mlr)ldr2+r2(d+sm2Sd?)2)]. (I)

292


Various symbols have the meaning given in section 3 of the present paper and m has the meaning given below. The exact orbit is given by an elliptic function but, as usual, we take the following trigonometric equaaon of the orbit which is suffsciently exact.

Ilr [t +e cos(f @ 3m20/h2)]mlh2 (2) where m-M.G/c2 mass ofthe sun x G/c8,

h r2d0/ds (which turns up as a constant of integration), e eccentricity ofthe orbit,

and a)-longitude ofthe perihelion. The period of successive perihelia is then 27r/(I- 3m2/h2) 27r(I+

3m2/h2). If one revolution corresponded tO 27r this result would mean that the perihelion of Mercury advances by 6rm2/h2 per revolution of Mercury or by 43 sec. per terrestrial century.

However, the quantity 27r arisig above is a number and may not correspond to one revolution in a non-Euclidean space. Indeed R. C. Tolman (Relativity Thermodynamics and Cosmology, OxfordZ I949, p.

208, footnote) and G. C. McVittie (General Relativity and Cosmology, Chapman & Hall, I956, pp. 87-88), both consider the fact of non-Eucli- dean geometry but conclude that no significant error is introduced by

ignoringthisfact. Theyshowthatwhetheronetakes(i)rorJ( /)i.

(ii) s the proper time at Mercury, or T the co-ordinate time at the origin (or for that matter the proper time measured by an astronomer on the earth) in the calculation, the end result is not sensibly affected. But the fly in the ointment is +.

We now calculate the ratio circumference/radius. In fact we only carry McVittie's calculation forward. (A similar method is indicated by L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Pergamon, I959, p. 307). Using the interior and exterior Schwarzschild metrics, we have the radial geodetic distance

p-| (9(I 2mr2/R(i>3)idr+| (I 2m/r)idr (3) o R<Ez

where, R<i>- radius of curvature at the surface of ie sun (taken as bounded),

and R = radius of curvature at dle mean position of Mercury with reference to dle origin kentre of the sun).

Rpand R may, without sacrifice of the necessary accuracy, be ielciltified widl dle astronomerws measures ofthe radius ofthe sun and the average

293

Ge H. KESWANt

distance of Mercury fiom the centre of ie sun, respeciively. Assump ton of the average distance, again, does not a?ect the result matenally. Expanding (3) we get

P =| @(I+mf2/Ri3+...)dr+I( I +mlr+- )drR+M(ifl?geR/W) ? R(3

Then, the angle swept out per revoluion is

27T R|[R+m(++logeR|Rs)] 2T-7X65 X IO-7-

The second term represents departure from Euclidicity. And in one terrestrial century, the deficit of 7-65 x lo-7 per revoluiion, amounts to

6s sec. Taking care of the slgns, the centennial advance is found to be 43 +65-I08 sec.

According to Rosen, r is the radial distance itself arld (I-2m|r)-+,

etc., would be the components of the gravitational field which enter IntO the dpacs of motion but do not alter ie space-geometry. Thus the eq. (2) has the usual interpretaiion leading to the resdlt of 43 sec. Ln fact Rosen fmds the metric tO have dle unique form

I M/rdH I - +mlrdr2+(r+m)2(dS2+sm28d+2)]- I +m/r c X-mlr

One possible criiicism of this demonstration may be answered. Why take the radial geodetic distance p in the calculation? Because, the mcrement of radial distance is dp, ie radial merc being

dst (I 2m/r)dT2 P2 > etc.

Regarding appropriateness of Gaussian coordinates in ie present

case, also see: Introduction to General klotivity, R. Adler et al., McGraw Hill, I965, pp. S9 62.

ERRATA ' Origm and Concept of Relaiivity ', Ps I and II

Part I (I5, No. 60, I965)

p. 2-86,1ine 15: For' is ' read' are ' p 288, hne 5: For ' references 3 and 4 ' read ' references 2 and 3, p. 287 ' p. 304, line 2I: For 4 uzp' = 8(U5-v) ' read s utZp' = ,8(uX-v)p '

Part II (I6, No. 6I, I965)

p. 26, lme 2I: For ' word-9 ' read ' worId-Me ' Read k for K on p. 3I and 32, except in the figure on p. 3r, ad line 7 on p. 3z, where read K for K. Also, read k for k in ie figure on p. 3 I.

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Ge H. KESWANt

distance of Mercury fiom the centre of ie sun, respeciively. Assump ton of the average distance, again, does not a?ect the result matenally. Expanding (3) we get

P =| @(I+mf2/Ri3+...)dr+I( I +mlr+- )drR+M(ifl?geR/W) ? R(3

Then, the angle swept out per revoluion is

27T R|[R+m(++logeR|Rs)] 2T-7X65 X IO-7-

The second term represents departure from Euclidicity. And in one terrestrial century, the deficit of 7-65 x lo-7 per revoluiion, amounts to

6s sec. Taking care of the slgns, the centennial advance is found to be 43 +65-I08 sec.

According to Rosen, r is the radial distance itself arld (I-2m|r)-+,

etc., would be the components of the gravitational field which enter IntO the dpacs of motion but do not alter ie space-geometry. Thus the eq. (2) has the usual interpretaiion leading to the resdlt of 43 sec. Ln fact Rosen fmds the metric tO have dle unique form

I M/rdH I - +mlrdr2+(r+m)2(dS2+sm28d+2)]- I +m/r c X-mlr

One possible criiicism of this demonstration may be answered. Why take the radial geodetic distance p in the calculation? Because, the mcrement of radial distance is dp, ie radial merc being

dst (I 2m/r)dT2 P2 > etc.

Regarding appropriateness of Gaussian coordinates in ie present

case, also see: Introduction to General klotivity, R. Adler et al., McGraw Hill, I965, pp. S9 62.

ERRATA ' Origm and Concept of Relaiivity ', Ps I and II

Part I (I5, No. 60, I965)

p. 2-86,1ine 15: For' is ' read' are ' p 288, hne 5: For ' references 3 and 4 ' read ' references 2 and 3, p. 287 ' p. 304, line 2I: For 4 uzp' = 8(U5-v) ' read s utZp' = ,8(uX-v)p '

Part II (I6, No. 6I, I965)

p. 26, lme 2I: For ' word-9 ' read ' worId-Me ' Read k for K on p. 3I and 32, except in the figure on p. 3r, ad line 7 on p. 3z, where read K for K. Also, read k for k in ie figure on p. 3 I.

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origin and concept of relativity iii[1]

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