Einstein's comprehensive 1907 essay on relativity, part III

H. M. Schwartz University of Arkansas Fayefteville, Arkunsas 72701 (Received 30 November 1976; accepted 4 April 1977)

This is the concluding part of the English rendition of Einstein's 1907 essay on relativity, of which part I appeared in the June 1977 issue of this Journal and part I1 in the September 1977 issue. It consists of a direct translation of the last part of the essay, part V, entitled "Principle of Relativity and Gravitation," and of a few added footnotes.

A. INTRODUCTION

Einstein's 1907 essay on relativity' does not appear to be widely known. Yet, as noted in the Introduction to the first part of the present English rendition of this essay,2 it is of substantial interest both on didactic and historic grounds. Its didactic value, relating to the treatment of a number of basic topics in Special Relativity, is particularly in evidence in the portion of Einstein's essay that is dealt with in the second part of the present rendition.' Its his- torical importance is associated mainly with the genesis of special relativity, and also with the genesis of general rel- ativity. Part V, the last part of Einstein's 1907 essay, con- tains Einstein's first published expression of his initial highly important seminal ideas on the latter subject. It is translated here, as far as seemed feasible, verbatim, with a few added mainly explanatory notes.

It may perhaps not be amiss to point out here that the latter notes, as well as those presented in the other two parts of the present rendition, and in the partial translation of Einstein's first paper on relativity,4 have for their principal aim only the facilitating of a close reading of the respective fundamental papers of Einstein, whether historically or pedagogically motivated.5

As in the previous parts of this rendition all the original footnotes are labeled by lower-case roman letters, and the added footnotes by arabic numerals.

B. TRANSLATION OF THE GRAVITATIONAL PART OF EINSTEIN'S 1907 MEMOIR

V. Principle of Relativity and Gravitation

17. Accelerated Reference System and Gravitational Field

Until now we have applied the principle of relativity-i.e., the assumption that the laws of nature are independent of the state of motion of the reference system-only to nonaccelerated reference systems. Is it conceivable that the principle of relativity holds also for systems which are ac- celerated with respect to each other?

This is not really the place for the exhaustive treatment of this subject. Since it forces itself, however, on the mind of anyone who has followed the previous applications of the principle of relativity, I shall not refrain here from taking a position on the question.

We consider two systems of motion, and Z2. Suppose is accelerated in the direction of its X axis, and y is the

magnitude (constant in time) of this acceleration. Suppose 2 2 is at rest? but situated in a homogeneous gravitational field, which imparts to all objects an acceleration -y in the direction of the X axis.

As far as we know, the physical laws with respect to 21 do not differ from those with respect to X2; this derives from the fact that all bodies are accelerated alike in the gravi- tational field. We have therefore no reason to suppose in the present state of our experience that the systems Z1 and 2 2 differ in any way, and will therefore assume in what follows the complete physical equivalence of the gravitational field and the corresponding acceleration of the reference sys- tem7

This assumption extends the principle of relativity to the case of uniformly accelerated translational motion of the coordinate system. The heuristic value of the assumption lies therein that it makes possible the replacement of a ho- mogeneous gravitational field by a uniformly accelerated reference system, the latter case being amenable to theo- retical treatment to a certain degree.

18. Space and Time in a Uniformly Accelerated Reference System

We consider first a body whose individual material points possess relative to the nonaccelerated reference system S , at a fixed time t of S, a certain acceleration but no velocity. What influence does this acceleration y have on the shape of the body with respect to S?

If such an influence exists, it will consist in a dilatation of constant ratio in the direction of the acceleration, and possibly in the two directions perpendicular to this direc- tions; since an influence of another kind is precluded by considerations of symmetry. Those dilatations arising from the acceleration (if they exist at all) must be even functions of y; and they can be thus disregarded when one restricts oneself to the case when y is so small that terms of the sec- ond and higher powers in y may be neglectedg Since we wish to confine ourselves in the sequel to this case, we do not have therefore to assume any influence of the acceleration on the shape of a body.

We consider now a reference system 2 which is uniformly accelerated relative to the nonaccelerated reference system S in the direction of the X axis of the latter. Let the clocks and the measuring rod of Z, when examined at rest, be the same as the clocks and the measuring rod of S . Let the or- igin of coordinates of 2 move along the X axis of S , and let the axes of 2 remain parallel to those of S . There exists at every instant a nonaccelerated reference system S', whose coordinate axes coincide with the coordinate axes of Z at that instant (for a fixed time t' of S'). If a point-event oc- curring at this time t' has the coordinates ,$q,fwith respect to 2, then

since by the foregoing discussion we must not assume any

899 American Jouml of Physics, Vol. 45, No. 10, October 1917 Copyright 8 1977 American Association of Physics Teachers 899

Einstein's comprehensive 1907 essay on relativity, part II

H. M. Schwartz Department of Physics University of Arkansas, Fayetteville, A r k o ~ 72701 (Received 30 November 1976; acccptad 4 April 1977)

This continuation of the English rendition of Einstein's 1907 essay on relativity, of which the first part appeared in the June 1977 issue of this Journal, is devoted. to Parts II-IV of the essay, dealing with the relativistic treatments of electrodynamics, optics, mechanics, and thermodynamics. The original text of these parts covers 27 printed pages. However, owing to the nature of the subject matter and the character of the original exposition, it was possible to reproduce here the essential content of the original text intact in tenns of a free rendition, using modem notation, and retaining all the original formulas with their numbering, and also including direct translations of all passages of possible historical interest. Mathematical amplifications of a few of the key derivations in the original text are presented in added footnotes.

A. INTRODUCTION

This is the second part of a three-part English rendition of Einstein's 1907 memoir on relativity.' As noted in the introductory remarks to the first part,2 in addition to pos- sessing obvious historical interest;Einstein's memoir is also of considerable didactic interest. This is particularly true of the portion of the memoir presented here, consisting of Parts Il-IV, which contain concise and instructive discus- sion of the relativistic treatment of a number of basic topics in electrodynamics, optics, mechanics, and thermody- namics.

Because Parts II-IV of Einstein's paper are, with only a few possible exception^,^ mainly important for the phys- ical results and the methods of their derivation, their es- sential content can be represented accurately, as is done here, without resorting to a complete direct tran~lation.~ But in those instances where any doubt existed at all whether an original idea would be strictly conveyed by a free rendition, or where possible historical interest might attach to the original phraseology, the corresponding passages or phrases have been translated fully, and these are enclosed in quotation marks. On the other hand, the free rendition of the rest of the material, including the employment of direct vector notation, has.resulted in increased compactness in the presentatiod, without any essential loss in clarity.

Except for the notation, all the formulas in the original text are reproduced together with their original numbering; and Eqs. (1)-(4) and Secs. 1-6 to which reference is made in this paper, are contained in Ref. 2. In a few instances, for convenience of reference, originally unnumbered formulas have been indicated by asterisks. All original footnotes are labeled by lower-case Roman letters, and the added foot- notes by Arabic numerals. The latter are largely concerned with expanding or clarifying mathematical steps in deri- vations presented in the original text.

B. SUMMARY OF PARTS II-IV OF EINSTEIN'S 1907 MEMOIR

11. Electrodynamic Part

7. Tramjbrmation of the Maxwell-Lorenfz Equarioms

The equations

where6 E,H are, respectively, the vectors of electric and magnetic "field-strength," and

is the "4~-fold density of electricity,'" together with the "assumption that the electric charges are immutably bound to small rigid bodies (ions, electrons), form the foundation of Lorentz's electrodynamics and optics of moving bod- ies."

The reference systems S and S' being defined as in Sec. 3, and taking Eqs. (5) and (6) [as well as Eqs. (*) and (**)I to hold with respect of S , an application of the transfor- mation Eqs. (1) connecting S and S', yields equations rel- ative to S'of the same form as Eqs. (5) and (6), i.e.,

(p' ur + bEr/bf ')/c = Vr X H', ( 5 ' )

provided [introducing now the current notation: P = v/c, -, " (I - B2)-'/2]8

p' = V' E'. = y (1 - U U J C ~ ) ~ . (8)

In arriving at this conclusion, one must show that a resulting arbitrary factor depending on u which can be applied to E' and H', is necessarily unity, and this "can be easily shown in a way similar to that used in Sec. 3 in connection with the function 4(u)."

The above result combined with the principle of relativity implies that E',H' represent, respectively, the vectors of "electric and magnetic field-strength referred to St." Moreover, comparison of Eqs. (3) and (9) shows that u' is the velocity of the electric particles relative to S', and hence p' is the density of electricity with respect to S'. "The electrodynamic foundation of the Maxwell-Lorentz theory agrees, thus, with the principle of relativity."

In interpreting Eqs. (la), it should be observed that in

811 American Journal of Physics, Vol. 45. No. 9, September 1977 Copyright Q 1977 American Association of Physics Teachem 811

Einstein's comprehensive 1907 essay on relativity, part I

H. M. Schwartz Deparlment of Physics, Universi~y of Arkansas, Fayet~eville, Arkansas 72701 (Received 18 June 1976; revised 30 November 1976)

As an important sequel to his first paper on relativity, Einstein's extensive discussion of the subject in the 1907 issue of Jahrbuch der Radioaktivitat und Elektronik, including a first exposition of his embryonic ideas on gravitation, is of intrinsic as well as considerable historical interest. In this, the first of three parts dealing with Einstein's essay, a translation is presented of the Introduction and of Part I, which is concerned with relativistic kinematics.

Two years after the appearance of his first paper on the theory of relativity,' Einstein published in the then prestigious Jahrbuch der Radioaktivitat und Elektronik an extensive survey article on the s u b j e ~ t , ~ entitled "IJber das Relativitatsprinzip und die aus demselben gezogene F~lgerungen."~ In addition to a renewed painstaking analysis of the basic kinematic principles of the theory of relativity, and a treatment of relativistic mechanics and thermodynamics inspired by Planck's work on the subject of the same year, the memoir includes a note on the clash with conventional causality of assuming the existence of signals faster than light, an extension of Einstein's earlier discussions concerning the inertia of energy, and, most importantly, a first step towards the eventual creation of the general theory of relativity in the brilliant idea of an intimate link between "acceleration" and gravitation, as suggested by the proportionality between inertial and gra- vitational mass.

Some of the topics treated in the memoir and the methods employed there are even today not devoid of at least edu- cational interest. But its greatest interest by far is certainly historical. The Introduction, in particular, contains valuable information bearing on the history of special relativity, when considered alongside the relevant material in Ref. 1 . As for the history of general relativity, the significance of Ein- stein's first inspirational flashes, which in the course of a personal scientific odyssey culminated in his remarkable gravitational equations, is of course abundantly clear.

It is because of this considerable historical interest pre- sented by the memoir that a translation of the introduction and of the first and last parts together with a concise mod- ernized rendition of its other parts appeared worth under- taking. Nevertheless, a few words of explanation are in order why there is, in fact, any need of an English rendition when the chief significance of this essay today is, after all, on1 y to historians of science. What must be strased here is that in addition to the professional historians of science, who are naturally in a position to avail themselves of the original material at first hand, there are many scientists who for one reason or another have serious but subsidiary historical interests and who may not find it feasible to go directly to the original work. Moreover, the comments presented in the footnotes that relate to the ideas or mathematical steps in the memoir, as well as the modernized rendition, may be helpful not only to the latter group but to some members of the former group as well.

The present paper consists of a translation of the Jntro-

duction and of Part I of Einstein's essay. As in the earlier work,4 of which the present is a sequel, it is endeavored to strike some reasonable balance between a measure of ad- herence to prevailing English idiom and as close a repro- duction of the original in content, style, and even punc- tuation, as seemed possible. Unlike Ref. 4, the present translation reproduces the numbering of formulas as found in the original. Thus the only changes that are introduced in the translation are in the mathematical notation of a few formulas (for typographical convenience) and in the addi- tion of a few footnotes. The latter are numbered by Arabic rtumerals, while the footnotes of the original paper are in- dicated by lowercase Roman letters.

B. TRANSLATION OF THE INTRODUCTION AND THE KINEMATIC PART OF EINSTEIN'S PAPER

The Newtonian equations of motion retain their form upon a transformation to a new coordinate system which is in a state of uniform translational motion relative to the original system, according to the equations

As long as one held to the opinion that all of physics could be built upon the Newtonian equations of motion, one could entertain no doubt that the laws of nature come out the same when referred to any one of a family of coordinate systems which move uniformly (free of acceleration) with respect to each other.5 That independence from the state of motion of the employed coordinate ~ y s t e m , ~ which will henceforth be called "the principle of relativity," appeared however all at once open to question through the brilliant confirmations which the electrodynamic theory of H. A. Lorentz has e~perienced.~ That theory is founded, namely, on the assumption of a stationary immobile ether: its fun- damental equations are not so constructed that they go over into equations of the same form upon application of the above transformation equations.

Since the emergence of that theory one had to expect that one would succeed in demonstrating an influence of the motion of the Earth relative to the luminiferous ether upon optical phenomena. As is known, Lorentz proved indeed in that work that according to his basic assumptions no in- fluence of the relative motion upon the path of a [lightI6 ray was to be expected, as long as one restricts oneself in the calculation to terms in which the ratio v/c of that relative velocity to the velocity of light in vacuum enters only in the

512 American Jouml of Physics. Vol. 45. No. 6, June 1977 Copyright O 1977 American Association of Physics Teachers 512

Einstein's first paper on relativity

H. M. Schwartz Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701 (Received 4 April 1975; revised 24 May 1976)

Because of its exceptional significance in the history of great ideas in science, Einstein's first paper on relativity, especially its first part, deseroes a more careful translation into English than presently exists. A new and annotated translation of this first part is presented here, together with a brief discussion of certain aspects of Einstein's paper.

A. INTRODUCTION

With the recent perceptible awakening of an interest among physicists in the history of their subject, the older of the two pillars of current terrestrial physics, the special theory of relativity, has become the subject of intensive historical studies.l Yet rather remarkably, seven decades after its publication, Einstein's trailblazing paper,2 "Zur Elektrodynamik bewegter KBrper" ("On the Electrody- namics of Moving Bodies"), does not yet have a fully sat- isfactory translation into English. The only English trans- lation available until recently, contained in the widely known collection of original papers on relativity? falls short as a completely reliable historical document, not so much because it is marred by a few outright mistranslations- which are in any case easily spotted-but because of its failure in important instances to convey properly the intent of a passage or of the nuances in its original pre~entation.~ A new translation has indeed appeared recently,5 but un- fortunately its newness is limited essentially only to cor- recting the few flagrant mistranslations.6

For these reasons, and since Einstein's first paper on relativity represents one of the most remarkable intellectual achievements? a thoroughgoing English translation of this work would be of obvious'interest. In the meantime, an at- tempt in this direction is presented in Sec. B of the present paper. It covers only the first of the two parts of Einstein's paper. It is this part which is of greatest interest historically, and the one where changes in the older translation are especially in order. A few observations concerning the second part of the paper are contained in Sec. C.

The references to Secs. B and C include comments on the text of the paper, and remarks concerning Refs. 3 and 5. The translations of the original footnotes to the first part of the paper are labeled with lower case roman letters (a, b, and c) and are located at the end of the present paper as the first three references. Except for one formula which is labeled as (A), none of the formulas are numbered in the original. Thus all the numbering of formulas by Arabic numerals is supplied in the translation. This is, however, the only essential deviation from the original. Even the punctuation of the original paper has been followed as closely as feasible, and the lengthy sentences of the original text have been retained almost throughout without dissec- tion, so as to preserve as far as possible a sense of the flow of thought in the original ~ r i t i n g . ~

In addition to the comments on the original text con- tained in the footnotes, a few observations of a more sys- tematic character are presented in the concluding section. A more complete discussion relating to this subject is de- ferred for presentation in a future paper dealing with certain

18 American Journal of Physics, Vol. 45, No. 1, January 1977

questions in the history of special relativity (referred to in Ref. 1 ).

B. TRANSLATION OF THE INTRODUCTION AND THE FIRST PART OF EINSTEIN'S PAPER

That Maxwell's electrodynamics-as presently con- ceived-when applied to moving bodies, leads to asymme- tries which do not appear to be inherent in the phenomena, is well known. Consider, for instance, the electrodynamic interaction of a magnet and a conductor. The observed phenomenon depends here only on the relative motion of the conductor and the magnet, whereas according to the usual conception, the two cases in which either one or the other of the two bodies is in motion must be strictly differ- entiated. For, if the magnet is moving and the conductor is stationary, there arises in the vicinity of the magnet an electric field of a certain energy, which generates a current in those places where parts of the conductor are situated. However, if the magnet is stationary and the conductor is moving, then no electric field arises in the vicinity of the magnet, whereas in the conductor there appears an elec- tromotive force to which in itself there corresponds no en- ergy, but which-assuming equality of the relative motion ' in the two considered cases-gives rise to electric currents of the same magnitude and the same course as those pro- duced by the electric forces in the first case.

Examples of a similar kind, as well as the unsuccessful attempts to verify that the earth moves relative to the "light medium," lead to the conjecture that, not only in me- chanics but in electrodynamics as well, no properties of phenomenal0 attach to the idea of absolute rest, but rather that the same electrodynamic and optical laws hold in all coordinate systems in which the equations of mechanics are valid, as has already been proved for first-order quantities.' 1

We shall raise this conjecture (whose content will be called in the sequel the "Principle of Relativity") to the status of a postulate, and in addition we shall introduce another postulate that is only seemingly inconsistent with the for- mer, namely, that light in empty space always propagates with a definite velocity V which is independent of the state of motion of the emitting body. These two postulates suffice for arriving at a simple and consistent electrodynamics of moving bodies on the basis of Maxwell's theory for sta- tionary bodies. The introduction of a "luminiferous ether" will prove to be superfluous, inasmuch as according to the conception that will be developed, there will be no need for the introduction of an "absolutely stationary space" en- dowed with special properties, or for the assignment of a

Copyright@ 1977 American A d a t i o a of Physics Teachers 18

Bulletin of the American Physical Society, v 20, Issue 1, p 99-99, 1975

s l i c e s . P a r t i c u l a r a t t e n t i o n has been devoted t o Ni t s New Theory of GravityI1 a s p e c i a l case of which has the same post-Newtonian l i m i t a s General Re la t iv i ty . For t h i s s p e c i a l case, the re i e only one homogeneous, i so - t r o p i c cosmological model; and i t has a bounce (po in t of maximum d e n s i t y ) a t a r e d s h i f t zb s 1/m0 5 300. Because of t h i s bounce the model cannot explain, i n any reasonable way, t h e ex i s tence and isotropy of the cosmic microwave r a d i a t i o n . *

Supported i n p a r t by the National Aeronautics and Space Adminstrat ion [NGR 05-002-2561 and the National Science

tFoundation [GP-36687x1. Submitted by WARD WHALING.

Iw. T. N i , Phys. Rev. D 3 2880 (1973).

d , Cal i forn ia I n s t i t u t e of Technology.-- We presen t a new formalism f o r c a l c u l a t i n g t h e gravi ta- t i o n a l waves emi t ted by any aystem with i n t e r n a i gravi- t a t i o n a l f i e l d s t h a t a r e weak ( Ig - 11 ( << I) , but n e v e r t h e l e s s a r e s t rong enough tonYnflu!;ce t h e system's motion. This formalism~.expresses the r a d i a t i o n i n terms o f a r e t a r d e d Green's funct ion f o r s l i g h t l y curved apace- time, and then breaks the Green's-funct ion i n t e g r a l i n t o f i v e p ieces : d i r e c t rad ia t ion , produced d i r e c t l y by the motions of t h e source; whump rad ia t ion , produced by the " g r a v i t a t i o n a l s tresses" of the source; t r a n s i t i o n radia- t i o n , produced by a time-changing time delay ("Shapiro met") i n t h e propagation of the nonradiative, l / r f i e l d of the source; focussed rad ia t ion , produced when one p o r t i o n o f the source focusses, i n a time-dependent way, t h e nonrad ia t ive f i e l d of another port ion of the source; and t a i l r a d i a t i o n produced by "backscatter" of t h e nonradia-regions of focussing.

i Supported i n p a r t by NASA [NCR 05-002-2561 and by NSF [GP-36687x1.

t ~ u b m i t t e d by KIP 3 . THORNE.

This paper a= c l a s s i c a l electrodynamics f o r charges of f i n i t e exten- s i o n . Charges a r e required t o maintain a , prescribed " r i g i d " shape throughout the course of t h e i r motion. An a c t i o n p r i n c i p l e i s formulated f o r the coupled system c o n s i s t i n g of a charge plus e lec t rnnsgne t ic and gravi ta- t i o n a l f i e l d s . The a c t i o n pr inc ip le y ie lds a complicated s e t o f c o u ~ l e d i n t e u r o - d i f f e r e n t i a l equation8 for the

On Hoylcls l ien Theor o f t i rav i ta t ion. T.L, CHOil, c a l i f . S ta te Gullet c Sta:islaus, T u r I ~ c l i , CA.--1Ioyle Z T a i o v rcccnil; f o ~ ~ i u l a t e b a confon~ia l l y i n v a r i a n t t l~eory o f g rav i ta t ion by extehding ilach's p r i n c i 1c t o tile masses o f atomic par t i c les -and found t h a t t(o p a r t i c l e r.lasscs are quadrat ic func t ion o f cosmic t i ~ ~ l e i n the Hinko\rskl representation, This leads t o a redsh i f t c f f cc t . The existence o f sonle natu ra l rad ioac t i ve elements docs n o t seem t o support t ime var ia t ion o f the atomic masses. About G6;: of the eleaent n i s the P-active isotope m le7 1111ich decays t o o s l e 7 w i t h a I1alf;life o f G , ~ x 1 0 ' ~ y r and a decay energy o f A = 2.6 kev. I f t h e a t a ~ i c [lass increases quadrat< c a l l y i r i t h t iclo, Ulan t l lc r e s t mass o f the e lect ron during the e a r l y h i s t o r y o f the ear th, say 3x103yr ago, was about 286 Lev, i f l i ich i s smaller t h a ~ ~ e lec t ron 's p resc~ l t r e s t mass by 222 kev. Then the decay energy o f J ( E ~ ~ ~ 3x103yr ago was 222 + 2.6 * 224.6 kev, and the h a l f - l l f c of Rels7 3x109yr ago was could not have been qreater than 1 ,4x l0~yr . This means t h a t the ~ e " ' would no t have survived. Sne f a c t o f t i le cx is tcnce o f n e l E 7 today thus docs not support time v a r i a t i o n o f atomic Izsses.

1Hoy7e, F., : lar l ikov, J,V., t la turc 233 (1971) 01

'Dyson, F.J.- Pily. k?v, Let t . 18 (1967) 1291

KF 11 A New Tes t of the Equivalence Pr inc ip le Based on c r y o g e n i c equivalence pr inc ip le experiment has been b u i l t which invest igates the use of cryogenic techniques f o r g rea te r po ten t ia l accuracy. Superconducting shielding, magnetic bearings, and SQUID position monitors a r e combined with modern control techniques t o achieve high s t a b i l i t y and accuracy. The apparatus and preliminary r e s u l t s a re described.

*Work supported i n part by Cal Tech PF-066. tsubmitted by J. T. Anderson.

K F 1 2 Dynamics of a Two Dimensional General Covari- ant Field Model. JOSEPH KLARFELD and ALEX HARVEY, De- partment of physies. Queens College of The City Univer- s i t y of New York, N.Y. 11367.--The c laes ica l and quantum canonical f o b l i s m of a general covsriant f i e l d model i s analyzed on a two-dimensional i n d e f i n i t e metr ic . The dynamical consistency of the quantizat ion procedure i s discussed and the c lass ica l equat ion f o r t h e non- polynomial f i e l d (metric) which is of hyperbolic type i s solved i n closed fonn.

motion a n d ' f i e l d s . -A perturbat ion expaneion of the t r a n s l a t i o n a l and r o t a t i o n a l equat ions of motion i n pow- KF 13 Uniqueness o f E ins te in ' s Theory o f Gravi tat ion. e r a of the s i z e of the charge d i s t r i b u t i o n i s obtained. H. M. SCHWARTZ, U. o f Arkansas.--It w i l l be shown In t h e l i m i t t h a t the s i z e of the charge tends to zero t h a t Einste in 's theory o f g r a v i t a t i o n fo l lows i n an o n l y a tew k inemat ica l fea tures survive i n the equations essen t ia l l y unique manner from t h e requirements t h a t o f motion; t h e s e a r e charac te r ized by i n t r i n s i c parame- the Newtonian g rav i ta t iona l theory and the specia l t e r s such a s t h e t o t a l charge, moments of i n e r t i a , e tc . theory of relativity are contained i n f t as the The r e s u l t i n g equations of motion have the DeWitt-~rehme appropriate l i m i t i n g cases, and t h a t the i n e r t i a l - [Ann. Phys. 3 220 $1869)] form, but with an addit ional reference-systems puzzle i s resolved w i t h a minirnun c u r v a t u r e coup1 ing term. nmber o f natural p r im i t i ve assumptions. A b r i e f 4+ i nd ica t ion w i l l also pe made o f t h e connection w i t h +Supported i n p a r t by NSF [GP-36687x1.

Permanent address : Ca l i fo rn ia S t a t e , Fuller ton. re la ted current work, and w i t h cu r ren t investigations on Mach's p r inc ip le .

"Submitted by R.W. KAVANAGH 'F n . K - S. Thorne. D. S. Lee, and A . P. Lightman,

KF 9 P o s s i b l e Detect ion of t h e Lense - F h i r r i n g E t f e c t . JACK JAFFE, Boston College & Weston Observa tor , and IRWIN I. SHAPIRO, MIT. - The weLl knownLLense - Thi r r i ng ~ f f e c t , - which has n o t y e t been observed, i s considered w i t h i n t h e framework of r ad io t racking measure- ments o f s p a c e probes. Calcula t ions show t h a t such probes i n h e l i o c e n t r i c o r b i t s permit a s e p a r a t i o n of t h e Lense - Thi r r i ng term from a solar quadrupole nwment e f f e c t . The value of t h e Lense - T h i r r i n g term i s a few orders of magnitude t o o smal l f o r p r e sen t ly achievable ~easurement accu rac i e s .

- . , . , . . - - . . - . . . - , Phys. Rev. D7. 3563 (1973). -

KF 14 The Vacuum Expectation Value o f t h e S t r e s s Ten- s o r Near t e Horizon of a Black Hole.* S.M. C W U S ~ , U, of Texash a t Austin. - - Recent worE o f S. W. ~ a w k i n g l ,

llnruhi. and others has s h m t h a t t h e surface a rea of a b l a c k hole can decrease through the spontaneous emission of p a r t i c l e s . They ca.lculate.the fly of angular momentum and mass a t infinity l n d i c a t l n g t h a t in the case of very small b l a c k ho les the l o s s of energy i s explosive. The Hawking Area Theorem is transcended, thus shedding some doubt on the nature of t h e horizon.

An Axiomatic Deduction of the Pauli Spinor Ihcory

ll;~uli'i ttrc~lry 01 llic sp111 111 IIIL* c l c c t r ~ ~ r t t f la~t l r , 1027) Ior11is a11 t ~ r r p r t a r t t part o l '~ io r i rc la l~v~s l~c rlii;rtiltrtrr I I I ~ C ~ I ~ ~ I I J , t ~ r l 11% r~rr@n;ll lirrri1111a11oti US

is ritrl irlircqucrllly ( rue rtf plorlccrlng cor~lrrhnlic~rrs lcuvcs ruolli ILr I'~rrllrer cb~il'icaliue. l l ~ i s Is tlonc cc~~rcr r t lp hp Wlgtlcr ( IOSV) ;uitl 11j V ~ I I dcl Wslurdcr~ (19741 ill tcrriis uf' tlrc l11cc)ry IIS firtrup r ~ p r c s c r i t a ~ r ~ t ~ ~ . ~ . 111 I t~i$r iote arb allcrllativc L I I ) ~ ~ L I O L ' I I t o l)a1111's tlrc~rry IS ~ C Y C ~ I I [ ) O ~ 11rii1 t ' l t ~ p l o y ~ o ~ i l y clcri1cril;1ry ~ ~ r ; l ~ i t u ~ r l - r ~ r ~ r . I ~ ; ~ ~ s lwbra Irr addrtrurt to a rnirr~liinl postuli~lc ~ ~ [ I c c ~ I I I ~ LIic les\rlls 11l'thc SIC~II-( icr lucl~ c x l w r r ~ i ~ r ~ r t . As i l l t l ~ c appruaclics 01 Wiglisr nird trl'valr tlcr Wilcrtlcn. llrr ~rrgular ~lri,~ilcrilusi clraraclcr o f Ilic h u l l spit1 opcrdlors is r u t assunled hul iti dctluccd; and i t is u l ~ u slri~wti, irlcidcntelly, Illict sunslstcnry w r l l ~ I ~ I C rcr t~lrs trbrrilicd rcquircs csstlr~lially 1l1;lt plrysical tqt;lct' h ~ I I ~ C C - ~ I I I I C I I Y I O I I ~ ~ ,

We curisidcr uri ~ - c I ~ ~ i i u r t s i a a ~ I 01 3 3) Euchdeari Kclor ~ p a c c and 8 I lerrnilian opcrator S(rc), I cli1anlul11-rtrucllalical "nbxrvablc." cleiirlcd fur every ~ttric vcrtor r r rrl' this space, arrd sllh~ecr 10 I ~ I E I 'OIIBWIII~ C O I I ~ ~ I ~ O ~ I :

(1) 'f'lw Lrlxriirirr S ( u ) lrrs fur cvcry u ~ h c sanic cigcrivcctor spucc alld the U I I I C IWO i i l~ i l e ~~nndcgencri i te c i g c n v a l u ~ u s ~ , s t , witl~oul its d c p n d e n c c on II bcilry vucuous.

.cllls luurnrl is copyr&hrctl by Plorturn, Gel1 nrrlclr is itvuilrrblc rut $7.50 from Pknum I'ubll.rltl~tg C ; ) r ~ ~ ~ r ~ r ~ t r n . 227 Wcat 17111 Street, NEW York, N.Y. 10011.

Satisfaction of the boundary condition ~ + , ( x ) = 0 is, in these terms, equivalent to finding the zeros of U,; these are l2

which is the same as the result obtained by the usual method in Eqs. (5) and (6).

~urthermok, the general expression for these polyno- mials is1=

clearly equivalent to Eq. (4). Thus all important details of the standard solution are obtained by this method.

Some author^'.^.^.^-^ regard Eqs. (2) and (3) as n homogeneous, linear equations for the unknowns a,, a,, . . . , a,. Consistency of these equations requires that the n X n determinant of coefficients be zero:

For a given value of n, this leads to an nth-order polynomial in x, whose zeros must be found: This is just what Symon's suggestion accomplishes. Marion5 actu- ally carries out this procedure for a few small values of n,

but we have not seen the solution carried forward from this point, for the general case, except by the introduction of Eqs. (4) or (3, which our method avoids.

We feel that the theory of difference equations provides the best basis for treating problems of this kind, and we do not claim any generality for our method. But there is an appealing directness in Symon's suggested procedure of building the polynomials (1, using Eq. (2), and we therefore feel that our generalization of his approach is of potential interest to the student.

IT. C. Bradbury, Theorericnl Mechanics (Wiley, New York. 1968), Sec. 12.5.

'G. R. Fowles, Analytical Mechanics (Holt. Rinehart and Winston, New York, 1970). 2nd ed., Sec. 1 1.7.

=A. P. French, Vibrations and Waves (Norton, New York. 1971). pp. 136144.

'E. 1. Konopinski, Classical Descriptions of Motion (Freeman, San Francisco, 1969). Sec. 11.3.

SJ. B. Marion, Classical Dynamics (Academic. New York, 1965), Sec. 14.8.

'K. R. Symon, Mechanics (Addison-Wesley. Reading. MA. 1971), 3rd ed., Sec. 8 .4 .

'J. L. Synge and B . A. Griffith, Principles of Mechanics (McGraw- Hill, New York, 1959), 3rd ed., pp. 498-503.

%L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw.Hil1, New York, 1958), 2nd ed., Chap. 10, Sec. I I .

OF. B . Hildebrand, Methods of Applied Marhemarics (Prentice-Hall, New York, 1952). Sec. 3.6.

'°Ffandbook of Mafhemarical Functions, edited by M . Abramowitz and I. Stegun (National Bureau of Standards, Washington. DC. 1964), Sec. 22.

llG. Attken, Marhematical Merhods for Physicisrs (Academic, New York, 1970). 2nd ed.. Sec. 13.3.

lPR "See Ref. 10, Table 22.16, p. 787. lasee Ref. 10, Eq. 22.3.16, p. 776.

On the logical foundations of special relativity*

H. M . Schwartz Deyrtment of Physics Unfwrsity of Arkansas Fayettcville. Arkansas 72701 (Received 15 February 1974; revised 2 May 1974)

The logical foundations of special relativity insofar as they relate to non-Newtonian ideas, concern mainly kinematics, and this is already in evidence in Einstein's original paper .on relativity. '

In fact, all the essentially new ideas introduced by Einstein in that paper belong to the domain of kinematics. If nevertheless, in his derivation of the associated Lorentz group, Einstein took as his fust postulate a principle of relativity that was intended to encompass all laws of physics, he had of course compelling reasons. The prin- ciple had to insure the complete equivalence of all inertial frames, and to serve at the same time as a heuristic guide in the development of all branches of relativistic physics

(with the exception of gravitational p h e n ~ m e n a ~ ) . ~ How brilliantly the latter objective has been accomplished with the aid of this broad relativity principle together with the principle of the constancy of light velocity and Einstein's revolutionary idea of clock synchronization, has been common knowledge for over half a century.

Understandably, the exposition of the logical structure of the theory of relativity to be found in the textbook lit- erature has in general followed closely the original dis- cussion. From a pragmatic point of view. Einstein's ap- proach was of course a stroke of supreme genius, re- sembling in significant respects Newton's logical ap- proach in the Principin. Logicians and philosophers can come and make sundry formal illuminating refinements, and many have, but the essence of the original framework remains. Still, from a purely cultural or didactic point of view it may be of interest, perhaps ultimately even to physical theory itself, to re-examine the conventional ap- proach to the logical foundations of relativistic kinematics.

To begin with, it is desirable to separate as far as pos- sible the region of kinematics from that of the rest of

362 /Am. J . Phys. Val. 43, No. 4 , April 1975 Notes and Discussions

Bulletin of the American Physical Society, v 18, Issue 4, p 645-645, 1973

GF 10 Charge Q u a n t i z a t i o n i n CUrved Space. M. J . TRINKALA and A. INOMATA, SUNY a t Albany.

attempt i s made t o quant ize the charge i n a curved space by employing the Ka uza Klein- "offman s i x dimensional formalism.i m e underlying t ransformations a r e four-dimensional general coordinate t r a n s f onnation8 , gauge transformations, and two-dimensional ro ta t ions . we next cons t ruc t t h e equations of motion f o r a dual ly charged p a r t i c l e i n the electromagnetic f i e l d . The equat ion of motion corresponds to the w u a l Lorent2 equatioll of motion i n f l a t space except f o r t h e e x t r a coupling t e r n . From the equat ion of motion we i n t e r p r e t the two-dimensional r o t a t i o n as the dual ro ta t ion associated with electrodynamics. From the compactness c h a r a c t e r a ssoc ia ted with the dual r o t a t i o n , we seek ou t terms t h a t lead to a minimal quan t iza t ion condit ion.

1. B. Hoffman, Quart . J . Math. 1, 19 (1936); 32 (1936).

GF 11 Conventional Relat l v i t y and Tachyons. H.M. S C W T Z , U of Arkansas.--The incompatibi l i ty with conventional rela- t iv i tv of the a s s u m ~ t i o n of the exis tence of tachvons i s - , shown as a coro l l a r ; o f a new derivat ion of the ~ b r e n t z qroup. This de r iva t ion r e s t s on the pos tu la te of kinemati r e l a t i v i t y , without invokinq E ins te in ' s postulate of the constancy of l iqh t ve loc i ty . Such an approach has been considered f requen t ly s i n c e 1910; the method to be present i s new in i t s l ead ing In d i r e c t and simple fashion to the necessi ty of e x i s t e n c e of an invariant maximum veloci ty. This i s accomplished by makinq e x p l i c i t u s e the fol lowi special instance o f the kinematic r e l a t i v i t y p r inc ip le : the con t inu i ty of t h e ve loc i ty of a p a r t l c l e i n a well- s p e c i f i e d o d in i t s h l s to ry i s an invariant property. We require , in f a c t , only the weaker Invariance condition t h a t i t i s imposslble t o have such continuous behavior in one permissible frame of reference and a jump i n the p a r t i c l e ' s ve loc i ty from plus t o minus in f in i ty I n another permissible frame. Since the f in i t eness of the invariant veloci ty charac te r izes the Lorentz qroup, I t s property of beinq t h e r e a t e s t p o s s l b l e p a r t i c l e veloci ty shows tha t the e x i s t e h a c h y o n s i s not consis tent with the con- ventional assumptions of kinematic r e l a t i v i t y .

GF 12 Tachyons v s . Anti-Mass P a r t i c l e s on Re la t iv i ty . R B.P. SINHA, Univ. of Guelph, Canada.--Recent speculations' about Tachyons' have gained much bu t confused s t a t u s . It i s shown2 wi th in s p e c i a l - r e l a t i v i t y theory t h a t t o a Brad- yon (our world) r e s t mass, m o , t he re corresponds an equal and opposi te mass p a r t i c l e , iio--mo, pas t the l i g h t b a r r i e r . Such anti-mass p a r t i c l e s cal led t h e JUGNONS w i l l have unique phys ica l property t o ann ih i l a te t h e i r pair-Bradyons on c o l l i s i o n . Thus macro-annihilation i s possible . The energy-release i s 3moc2 i n con t ras t t o 2rnoc2 of micro-pair bradyons. The Jugmons could be the cons t i tuen t s of much speculated anti-galaxy of which ~ 0 1 - l i s i o n with our galaxy is then normally forbidden because of mutual G r a v i t a t i o n a l Repulsion. Are Tachayons and Jugmons t h e same? I f so , these auperluminal p a r t i c l e s w i l l be a n n i h i l a t e d t h e moment they appear and c o l l i d e with our l a b o r a t o r y pair-bradyons. A recent research' ru les i n t h i s p o s s i b i l i t y .

posi t ion i s shown t o be t h e n a t u r a l one f o r solving t h e momentum constraints in general r e l a t i v i t y and f o r cas t - ing the action in to canonical form without cons t ra in t s . The transverse-traceless part of the momentum i s con- formally invariant and t h i s makes i t a l so i d e a l l y s u i t e d f o r solving the remaining (Hamiltonian) cons t ra in t l .2 (following paper).

* WOKE supported i n pa r t by N.S.F. Grant GP30799X.

l~ermnnent address beginning September. 1973: Depart- ment of Physics, The University of North Carolina, Chapel H i l l 27514.

15. W. York, to appear i n J. Math. Phys., 1973.

L ~ . O'Murchadha, Princeton thes i s , January, 1973; N. O'Murchadha and J. W. York, t o be published.

GF 14 Hamiltonian Cons t r a i n t: Existence and Unique- ness of Solutions,* N. O'MURCWHA and JAMES W. YORK,JR., Princeton Univ.--The solut ion of the momentum cons t ra in t as obtained by the method of tensor decompositions of the preceding report may be eubst i tuted i n t o t h e Hamil- tonian constraint , which becomes a quasi-linear e l l i p t i c equation f o r the conformal or scal ing function of t h e Riemannian metric on the i n i t i a l three-space. This re- s u l t s from the f a c t tha t the tensor decomposition i t s e l f depends essen t ia l ly only on the scale- invariant p a r t of the three-geometry. Thus, the scal ing funct ion has a value fixed by the Hamiltonian constraint alone. We show tha t the resu l t ing "scale equation" has a s o l u t i o n f o r a l l assignments of the independent Cauchy da ta ex- cept on a "set of measure zero" in the configurat ion space. The solut ion obtained is always unique, on open or closed manifolds. The independent ("minimal") Cauchy data i n t h i s procedure a re the conformal +geometry, the Hubble parameter of the 3-space, and an a r b i t r a r y t race- f r e e syametric tensor used i n constructing eolut ions t o the momentum constraints . External sources can be read- i l y included i n the present framework.

*Work supported in p a r t by N.S.F. Grant GP30799X.

GF 15 A Derivation of Symmetries f o r Type (2,2} Spacetimes.* P. SOMMERS**, U. of Texas, and L. P. HUGHSTON, Oxford U.--Two independent Kil l ing vector f i e l d s ar-~cted out of the complex amplitude of the self-dual Weyl tensor and i t s two pr inc ipa l n u l l direct ions. The construction i s val id f o r a l l epacetimes of the algebraic c lass {2,2} which s a t i s f y the Einstein vacuum equation o r the E i n s t e i n - W e l l equations with the two principal n u l l direct ions of the Maxwell f i e l d coaligned with those of the Weyl tensor. For a ce r ta in subclass of these spacetimes, four Ki l l ing vector f i e l d s a r e derived.

*Supported by National Science Foundation and Rhodes Trust. **Submitted by A. Schild.

1

G. Feinberg, Phys. Rev. 159, 1089 (1967); A. Anitapa, GF 16 Classif icat ion of the Time-l ike Geodes ics of

Nuo. Cim. a, 389 (1972). the R e i s s n e r - N o r d s t r o m Metric*. Angelo Arment i , J r . ,

B. P. Sinha, ( t o be publiehed). I anova . , a n e t e r Havas, T e m le U --The e x a c t

'From a Sanskr i t word (meaning ant i -character p a i r s ) . d e C f t h e h w a r z schid- have b e e n d e -

'9. B a r t l e t t e t a l . , Phys. Rev. D6, 1817 (1972). t e r m i n e d and c lass i f i ed i n varying degree by m a n y ail- thnra We have in tegra ted the equations of mot ion f o r -..-- ". .. - - un-harged t e s t p a r t i c l e s moving i n t h e c o m b i n e d e l e c t r o - gravi tat ional f ie ld of the Re i ssner -Nords t rBm m e t r i c of

GI? 1 3 Decompositions of Symmetric Tensors* JAMES W. g e n e r a l relat ivi ty . T h e r e s u l t s ( a s i n the c a s e of t h e S- VoRK, J R . ~ and N . O~HURCHADHA, Princeton Univ. - Arbi- m e t r i c ) c a n b e e x p r e s s e d i n t e r m s of ~ a c o b i a n e l l ip t i c t r a r ~ symmetric t ensors can be decomposed on Riemannian functions. A complete classif icat ion of t h e g e o d e s i c s

manifolds i n t o t r ansverae - t race less , t r a c e ( sca la r ) , was obtained; the o r b i t s a r e a l l quas i -Keple r ian e x c e p t and long i tud ina l (vector) parts1. The decomposition f o r s o m e asympto t ica l ly c i r c u l a r motions which h a v e n o i s orthogonal and involves no coordinate condit ions on Newtonian counte rpar t . T h e m e t h o d employed 1s a g e n - the manifold, which may be closed or open. The decom- e ra l i za t ion of one u s e d original ly b y ~ e a v i t t ' f o r a P a r -

AJP Volume 40

Poincark's Rendiconti Paper on Relativity. Part 111

H. M . SCHWARTZ Department of Physics University of Arkansas Fayetteville, Arkamas 72701 (Received 27 March 1972)

This is the concluding part of a modernized rendition of Poincarb's Rendiconti paper on relativity, of which the first two parts appeared i n th.e November 1971 and June 1978 issues of this Journal. It covers the last section of that paper, in urhieh Poinear& develops in masterful, even i f incomplete, fashion, a gewalization of Newtonian gravi- taiional theory, involving retarded action-at-a-distance interaction that i s covariant u n d ~ the Lwentz group. Aa 1h.e first such attempt i t is of obvious historical significance. I n addition, just as the first two parts, ao this part, too, contains material of independent interest to the historian of the gemsis of special relativity.

For the purpose and scope of the present modernized rendition of Poincm6's Rendiconti paper on relativity, and for the notation that is being employed, the reader is referred to the introductory remarks to Pt. 1 of thie study [Amer. J. Phys. 39, 1287 (1971)l. The additional remarks concerning notation made in the introduc- tion to Pt. I1 [Amer. J. Phys. 40, 862 (1972)l are also applicable here. Because this part of Poincar6's paper is of particular interest in con- nection with its methodological aspects, including an anticipation of the four-vector calculus, certain relevant portions of the original text are repro- duced here more closely than would have been otherwise indicated. h for the notes or comments-which, as in the

earlier parts, are either given in footnotes or enclosed in braces in the t ex t these are intended in general, as previously, to serve only as explana-

tion of the original text. In addition, there are included here a few footnotes that point out nontrivial misprints in exieting French and English literal reproductions of PoincarB's paper and a footnote containing references to later work on the subject of Poincar6's pioneering investigation in relativistic gravitational theory.

9. HYPOTHESES CONCERNING GRAVITATION

1271 "Thus the impossibility of making evident the existence of absolute motion would be fully explained by Lorentz's theory, if all forces were of electromagnetic origin."

But there exist forces, such as gravitation, which are not of electromagnetic origin.

[28] "Lorentz was therefore obliged to com- plete his hypothesis by supposing that forces of any origin, and in particular, gravitation, are affected by a translation (or, if one prefers, by a Lorentz transformation) in the same way as are the electromagnetic forces."

It follows from this assumption, as applied to gravitation, that we can no longer retain the Newtonian theory involving an attraction between two bodies that depends only on their relative position at each imtant under consideration. The gravitational attraction must also depend on ('the velocities of the two bodies." In addition, it is to be expected that "the force which acts on the attracted body at an instant t depends on the position and velocity of that body at the instant t ; but also on the position and velocity of the at- tracting body, not at the instant t , but at an earlier instant, as if it took gravitation a certain time to propagate itself."

The equation for this propagation must there- fore be of the form

where, x=xl- xo, xo is the position vector of attracted body at time f , XI is the position vector of attracting body at time tl=k+t, and u, ul are

1382 / September 1872

Physical Rev. D, v 8, Issue 6, p 1915-1915, 1973

The Comments and Addenda section is for short communications which are not of such urgency as to justify publication in Physical Review Letters and are not appropriate for regular Articles. It includes only the followlng types o f communications: ( I ) comments on papers previously published in The Physical Review or Physical Review Letters; (2) addenda to papers previously published in The Physical Review or Physical Review Letters. in which the additional information can be presented without the need for writing a complete article. Manuscripts intended for this section should be accompanied by a brief abstract for information-retrieval purposes. Accepted manuscripts will follow the same publication schedule as articles in this journal, and galleys wiN be sent to authors.

Are the Einstein and Brans-Dicke Theories of Gravitation Equivalent?

H. M. Schwartz Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701

(Received 16 November 1972)

The identification o f Einstein's general theory o f relativity with the Brans-Dicke gravitational theory implicit in a recent paper by Harrison in this journal is shown to be based on a misunderstanding o f Einstein's paper. "Hamilton's Principle and the General Theory o f Relativity."

Since the Brans-Dicke gravitational theory1 rep- resents unquestionably, both by design and by formulation, an essential extension of Einstein's gravitational theory, i.e., of his general theory of relativity, an affirmative answer to the question posed in the title of this note, a s i s implicit in a paper that appeared in a recent issue of this jour- nal: can only mean that the te rm "general theory of relativity" i s employed in that paper in an es- sentially extended sense quite remote from that contemplated in Einstein's original papers on gen- e ra l relativity. This i s indeed the case, even though the paper (Ref. 2) re fe rs to Einstein's arti- cles in support of the contention. The quantities

from the field quantities represented by the metric tensor g,, ,: They all refer to what Einstein and most of the subsequent wri ters on the subject, including R. H. Dicke, denote by the t e rm "mat- ter." It i s therefore inconsistent with Einstein's theory to identify one of the q(,) with a scalar quantity that does not characterize the State of matter but that of the gravitational field. In fact, if one supposed that the Einstein quantities q(,,) of Ref. 3 need not be restricted (in Einstein's words) to "matter including the electromagnetic field," but could be identified with all possible speculative quantities, the resulting theory presented in Ref. 3 would become so general as to be of questionable

AJP Volume 40

4 See, for example, Ref. 1 or W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964)) pp. 191 ff and 2_55 ff.

The behavior of b ( t ) or b ( ~ ) , ~ i s of course much more complicated than this. That b(s) has, among other singularities, a simple pole at a = - y [see Eq. (3.13)] can be seen from the exponentially decaying behavior of b ( i ) shown by Eqs. (4.1) and (6.1).

OR. J. Glauber, Phys. Rev. 131,2766 (1963).

?Thii3 enhancement is not shown by the solutions of linearized equations of motion, which show the same behavior as the harmonic oscillator. See, for example, B. R. Mollow and M. M. Miller [AM. Phys. 62, 464 (1969)l who explicitly drop the higher order t e r n in the equations of motion, or H. Sauermann [Z. Physik 188,480 (1965)l who neglects the dependence of the Langevin form on the state of the atomic system, and thus effec- tiveIy linearizes the equations of motion.

Poincare's Rendiconti Paper On Relativity. Part I1

H. M. SCHWARTZ Department of Physics Universily of Arkansaa Fayettdle, .Arkanam 78701 (Received 14 January 1972)

This is a continuuticm of the modaized presentation of Poinearct's Rendiconli paper (begun in the Novemler 1971 issue of this Journal). I t mers Secs. 6-8 of that paper, dealing with its central theme, as indicated by its ti&, "On the dynamics of the electron," the mbject being of interest to both the historian of the classical theory of the &ron and the historian of relatzirity.

The purpose and scope of the present modernized rendition of Poincarb's Rendiconti paper is described in the introductory remarks to Pt. 1.1 A few additional words are needed to introduce Pt. 11.

Although the contents of Pt. I1 (Sec 5-8 of the original work) deal with the structure and dynamics of "electrons," and are, therefore, in the first place, of interest to the historian of the development of the theory of charged particles a t the turn of the century, there is also a great deal here which is relevant to the early history of relativity theory.

Because of the latter fact, the listing of some of the translated portions with the aid of bracketed numbers, begun in Pt. I, is continued. In general, all numberings in Pt . I1 (except for the references) are a continuation of those in Pt. I, so that in referring to Pt. I, no explicit mention of "Pt. I" is needed.

Although a number of new symbols arise in Pt. 11, it seemed unnecessary to extend Table I of Pt. I. Wherever a t all feasible, the original notation is retained, and the few nontrivial changes that are introduced are explained in the references. However, no reference is made (nor was it made in Pt. I ) to the replacement of Poincarb's antiquated notation for different types of differentiation.

Because Pt. I1 contains material which is somewhat intricate and not now too familiar, and explanations and references in the o r i g i d text are

86B / June 197B

Poincark's Rendiconti Paper on Relativity. Part I

H. M. SCHWARTZ paper. In addition, the last section of the paper Department of Physics contains a discussion of a relativistically covariant University of Arkansas theory of gravitation-the first such published FayeUeville, Arkansas 7B7Ol theory. (Received 5 January 1971; revised 25 June 1971) The contributions of the paper fall thus into

three groups: ( I ) foundations of special relativity,

'tSUr la dynamique de 1lBlectron,~1 in 1906 (11) theory of the electron, and (111) relativistic in Rendiconti del cirwb j fdemat im di Palerrno, is theory of gravitation. The present rendition of Poincart's principal publication dealing with relatiuity. the Introduction and Pt. I is intended to be an It i s , among other things, of vety c o ~ ~ r a b l e histotical accurate representation of both the text and the inkrest. To make it more readily accessible, the contents mathematical analysis. H ~ ~ ~ ~ ~ ~ , direct transla- of this comprehensive paper will be presented, i n a mod- ernized ver&n, in throe parts that r n e s p a d to the three tions are given of selected portions of the major topics treated in the paper. The first part, presented text. Nor is the mathematical part reproduced here, deals with the principle of relativity and the Lorentz verbatim whenever changes in arrangement of group us well as with the action prineipb of Mazwell- the argument facilitate its understanding. To Lorentz systems. the same end the notation is modernized through-

out, including the employment of the vector formalism.

These minor changes from the original, and also t,he different numbering of formulas which they have necessitated, need not introduce undue difficulties when comparison with the original is

Henri Poincar6's paper, "Sur desired; because all the results in the orignal and la d~namique de 1'61ectr0n" LRendiconti del the essential ideas in the derivations are left Circolo Matematico di Palermo 21, 129 (1906) ; intact, and all comments that are not in the dated: Paris, July 19051 contains the illustrious original are enclosed in braces or are given in mathematician's principal contributions to the footnotes. To facilitate further such comparisons, theory of relativity. It is of considerable historic each formula that corresponds exactly to one in as well as intrinsic interest and deserves to be the original is numbered also as in the original, more fully ,and more widely known. It has not this numbering being enclosed in square brackets, been easily accessible to physicists both because with the first digit indicating the section (not of the journal in which it is published and the shonfn in the original). In addition, a dictionary style in which it is written. A presentation of the

of symbols is presented in Table I. Last, for ease paper in the pages of this Journal, true to its of future reference, some translated portions are content but modifying its form whenever indi- listed by bracketed numbers. cated, is therefore in order.1o2

As its title indicates, the principal subject of the paper is the mechanics of the electron, the "elementary particle physics" topic around the turn of this century, which engaged the attention of leading theorists of that period such as J. J. Thornson, H. A. Lorentz, and M. Abraham. The underlying considerations relating to the principles of relativity and of the Lorentz group, as con- ceived by Poincar6, are presented in the Intro- duction and in introductory sections of the

ON THE DYNAMICS OF THE ELECTRON

Introduction

[I] "It seems at first sight that the aberration of light and the related optical and electrical phenomena would provide us with a means of determining the absolute motion of the earth, or rather its motion not with respect to the other stars, but with respect t o the ether."

AJP Volume 39 / 1.987

WTTERB AL NUOVO CIXENTO I'OL. 2, H. 24

On Wyler's Derivation of the Fine-Structure Constant.

(ricevuto il 25 Ottobre 1071)

Tho rccont purported derivation of thu ul~ivorsal coi~stn~lt a = eg/fio is sltetchod out in ttw onotes ( l e a ) , invulvilig ilinthun~t~ticnl idcna with vb ic l~ ECJV pl~yeichts ar0 suiflcie~ltly fnmilinr, Moroovc!r, thc phgsicnl nrgurneuts wl~icll two thcru adduccd for tho idot~tifioatiol~ of a with n ccrtitiii iirntl~c~nnticnl qunntity tllnt misea ill n o~i~eiclo~~lt- ti011 of colm~plex reproso~~tntiol~ spiwcfi of confoi*ind groulps cnl hardly be tnltoil ns very cogeilt. If, I I O V O ~ ~ ~ G ~ ~ B B , Wylcr's llotes llavo curre~~tly nrvusod coneiderablo iu- teyoet amollg pl~ysicists (as4), it is bccnuee, regnrdlees oi t l ~ c morits of its derivatiou, tlio oxprcseion proposed by WYLEW to rupmsci~t 01-1, namcly

I agrees completely with the present representative o oxporimontd o vt~lue of a-'

This is nn ngreement to better thnn 1 p.p.m., nnd the odds ngainst its being purely for- tuitous are obviously enormous. n u t nrc thoy in faot entiroly coinpcUii~gP I t k tllo pur- pose of t l ~ i ~ lntter to indicate that this ilued not noce~sarily be the cwo; bu t that if, 011 tho other hnnd, the result reprcseiitod by the exprossioil in (1) is iudced well gro~mdod phmiody, then whntever the eve~~tnnl geaerd theory in whioli it is emnbeddod, tho ilnplicatione nro most fnr-renching, poiutilig to an intinintc oolinectio~~ botwcen doctro- dynamics and qunntum moohauics.

AIL indication that it is possible to hnvc n eiinplo oxprefision affording romarltnblo ngreomont with f i e mo~sured wlno of a univo~.sal climo~lsionless conatnut, althoI~gll derivation i~ nlmoet cortniuly devoid of phyeical eignificnnco, ie provided by ~ V Y L ~ R himself in his second noto, ref. (e). Tho exprmaiol~ 6n6= 1830.118 ... wee% within experimel~tnl error with the measured value of tho ratio of the massee of tho proton nlld

Arotes and Discussions

this experiment. The Gaussian distribution gives a chi square of 17.6. This has a probability of being exceeded in a similar experiment of less than 1%. On the basis of these chi-square probabilities, the plus-minus 10go resistance label fits a uniform dis- tribution much better than a Gaussian distribution with a 10% width?

I n conclusion, this experiment gives an introduction to the comparison of two different distributions to the same sample of data. The data can be collected very rapidly with a minimum of equipment. The analysis is compressed into calculations on five intervals and comparison to chi-square tables. An additional benefit in an electricity and magnetism laboratory is that this experiment gives a better feeling for the possible range of component values that may occur in later work.

1 An intuitive discussion of the chi-square test is given in many places such as: R. D. Evans, The Atomic Nuclews (McGraw-Hill, New York, 1955), or H. D. Young, Statistical T ~ e a t m a t of Expe~imenlal Data (McGraw-Hill, New York, 1962).

8 See H. D. Young, Ref. 1. a The mean of the sample of resistors fell a t 1.09 K and

not the nominal value of 1.0 K. The shapes of the uniform and Gaussian ditribntions are not affected by the shift of center point. However, the distributions are no longer completely a p ~ i 0 ~ i distribution of fifty events centered at 1 K with a 10q0 width. Centering the distributions, reduces the number of degrees of freedom from five to four in using probability table. Using four degrees of freedom, the probabilities become 0.05 and 0.001 for the uniform and Gaussian, respectively.

A New Method of Clock Synchronization without Light Signals

H. M. SCHWARTZ Department of Physics University of Afkanaas FayetteuzZle, A~kansas 72Y01 (Received 25 Febtuary 1971)

In a recent note in this Journal1 attention was called to an ingenious idea due to Einstein for defining simultaneity in a given inertial frame without the use of synchronized c10cks.~ The question suggests itself whether this idea could not be applied in principle for the synchronization of clocks in a given inertial frame without the employment of light signals.

Let us first describe Einstein's idea, which is, as are many ingenious ideas, very simple. One imagines two rods of equal rest length that are moving towards each other with constant velocities relative to an inertial frame S, so as to pass each other in perfect juxtaposi- tion. I t is further assumed that the speeds of the rods are equal. By symmetry considerations (i.e., by the principle of sufficient reason) it can then be concluded that the front end of each rod and the rear end of the other will touch a t the same instant of time in S. In other words, the two "coincidence events" will be simultaneous in S.

It is quite obvious how we can turn this idea around for the purpose of setting up in principle a method of clock synchronization. We need only imagine a master clock a t rest in S a t the position of one of the coincidence events: call it El; then an identically const.ructed clock a t rest in S a t the position of the other coincidence event, Ez, can be synchronized with the master clock by being set to read as the time of E2 the time shown on

the master clock for the event El.' This scheme of clock synchronization would thus represent an alternative to another and well-known method of clock synchroniza- tion dispensing with light signals, namely, the method of infinitely slow clock transport: an alternative which seems to possess the merit of involving only fixed velocities, rather than a limiting process.

Unfortunately, this alternative, as i t stands, is open to a logical objection: How are we supposed to know the speeds of our rods before we have available at least two spatially separated synchronized clocks? We need to know, it is true, only that the speeds are equal, and we could imagine some perfectly reproducible mechanism for accelerating rods from rest to a given speed, which could be used to ascertain the desired state of motion of our rods. It is possible that some such idea was in Einstein's mind when he proposed his thought experi- ment. Nevertheless, for our present purpose we would want to employ only kinematic concepts or more precisely, concepts that relate only to rods and clocks? This can indeed be accomplished, but at the cost of introducing a limiting process-a self-consistent process of successive approximations, which can be made, how- ever, to converge very rapidly.

In formulating this process we can begin with estimating in some fashion either the equality of the speeds of our rods or the state of synchrony of our clocks. Suppose for definiteness that we begin with the latter. We can then measure the speeds of the rods, and with their aid, by our synchronization method reset the clocks, thus completing one step of a recursive process. In order to estimate the improvement in the state of synchrony of our clocks obtained in one step, we may, as in the case of the analysis of slow clock transport synchronization, employ the kinematic formulas of special relativity. We then find that the error in the setting of a clock is diminished by a factor of the order of magnitude of ( V / C ) ~ in each step, where v is the measured

AJP Volume 39 / 1889

American Journal of Physics, v 38, n 7, July 1970, p 927-929

N O T E S A N D D I S C U S S I O N S

small model 170 Optics Technology H & N ~ laser. Good transmission quality was obtained in the entire audiorange. Outside the clsssroom we transmitted over a 2-mile distance. Another interesting experiment is to use a slit and detect the modulation in the several diffraction orders. The student then "hears" the intensity distribution in the

Visualizing Dispersion of a Space Charge Due to Mutual Repulsion

Ho KIT-FUN

Blechicd Enpineering Department, Uaivsrm@ of Hang Kong [baeived 30 December 18.59)

Sand grains charged by friction in a perspex chute are collected in a pile on a horizontal sheet insulator on top of an earthed metal plate which forms an equipotential plane. The three-dimensional distribution of charged particles so

Wo. 2. Diswrsion of the same charge distribution released from itd image on dropping the plate.

collected hes its space charge imaged in the plate thus permitting a geometry otherwise unstable. On reducing the image attraction, accomplished by dropping the metal plate, the mutual repulsion among the charged particles is sufficient to overcome their weights and static friction, resulting in e rapid dispersion of the space charge, as illustrated in the photograph. The energy necessary for the dispersion is accounted for by the work done during the removal of the met,al plate.

Generalization of an Elementary Fonnula

FIG. 1. Distribution of 35 oc of sand graws (mostly 100-200p), initially The mere fact that T in (1) is a Lorentz matrix follows charged by triation in a 28 om perspez chub, with a thin esrthed at from the conservation equation P ~ + P , = ~ I + ~ z , plate beneath the supporting aheet insulator. since in elastic collisions maa=p,p. =pap0 and hence also