670 116 14MB
English Pages 241 Year 1952
Table of contents :
The Principle Of Relativity......Page 1
TitlePage......Page 3
Copyright......Page 4
Translator's Preface......Page 6
Contents......Page 8
1 MICHELSON'S INTERFERENCE EXPERIMENT Lorentz (1895)......Page 10
2 ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT Lorentz (1904)......Page 18
3 ON THE ELECTRODYNAMICS OF MOVING BODIES Einsten (1905)......Page 44
4 DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGYCONTENT? Einsten (1905)......Page 76
5 SPACE AND TIME Minkowski (1908)......Page 82
Notes By A. Sommerfeld......Page 101
6 ON THE INFLUENCE OF GRAVITATION ON THE PROPAGATION OF LIGHT Einstein (1911)......Page 106
7 THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY Einstein (1916)......Page 118
8 HAMILTON'S PRINCIPLE AND THE GENERAL THEORY OF RELATIVITY Einstein (1916)......Page 174
9 COSMOLOGICAL CONSIDERATIONS ON THE GENERAL THEORY OF RELATIVITY Einstein (1917)......Page 184
10 DO GRAVITATIONAL FIELDS PLAY AN ESSENTIAL ROLE IN THE STRUCTURE OF THE ELEMENTARY PARTICLES OF MATTER ? Einstein (1919)......Page 198
11 GRAVITATION AND ELECTRICITY Weyl (1918)......Page 208
CATALOUGE OF SELECTED DOVER BOOKS IN ALL FIELDS OF INTEREST......Page 227
Back Cover......Page 241
THE PRINCIPLE OF RELATIVITY A COLLECTION OF ORIGINAL MEMOIRS ON THE SPECIAL AND GENERAL THEORY OF RELATIVITY
BY
H. A. LORENTZ, A. EINSTEIN H. MINKOWSKI AND H. WEYL WITH NOTES BY
A. SOMMERFELD
TRANSLATED BY
W. PERRETT AND G. B. JEFFERY
WITH SEVEN DIAGRAMS
DOVER PUBLICATIONS, INC.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2.
This Dover edition, first published in 1952, is an unabridged and unaltered republication of the 1923 translation first published by Methuen and Company, Ltd. in 1923. This edition is reprinted through special arrangement with Methuen and Company and Albert Einstein.
Standard Book Number: 486600815 Library of Congress Catalog Card Number: A529845
Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014
TRANSLATORS' PREFACE
T
HE Theory of Relativity is at the moment the subject of two main lines of inquiry : there is an endeavour to express its principles in logical and concise form, and there is the struggle with analytical difficulties which stand in the way of further progress. In the midst of such problems it is easy to forget the way in which the theory gradually grew under the stimulus of physical experiment, and thus to miss much of its meaning. It is this growth which the present collection of papers is designed chiefly to exhibit. In the earlier papers there are some things which the authors would no doubt now express differently ; the later papers deal with problems which are not by any means yet fully solved. At the end we must confess that Relativity is still very much of a problemand therefore worthy of our study. The authors of the papers are still actively at work on the subjectall save Minkowski. His paper on " Space and Time" is a measure of the loss which mathematical physics suffered by his untimely death. The translations have been made from the text, as published in a German collection, under the title "Des Relativitatsprinzip" (Teubneri 4th ed., 1922). v
vi
THE PRINCIPLE OF RELATIVITY
The second paper by Lorentz is an exc~ption to this. It is reprinted from the original English version in the Proceedings of the Amsterdam Academy. Some minor changes have been made, and the notation has been brought more nearly into conformity with that employed in the other papers.
W. P. G. B. J.
TABLE OF CONTENTS PAGBB I. MICHELSON'S INTERFERENCE EXPERIMENT. By H. A. Lorentz • 17 § 1. The experiment. § 2. The contra.ction hypothesis. §§ 34. The contraction in relation to molecula.r forces. II. ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT. By H. A. Lorentz • 934 § 1. Experimenta.I evidence. § 2. Poinca.re's criticism of the contra.otion hypothesis. § 3. Ma.xwell's equations for moving a.xes. § 4. The modified vectors. § 5. Retarded potentia.Is. § 6. Electrostatic fields. § 7. A pola.rized particle. § 8. Corresponding sta.tes. § 9. Momentum of a.n electron. § 10. The influence of the ea.rth's motion on optica.l phenomena.. § 11. Applica.tions. § 12. Molecula.r motions. § 13. Ka.ufma.nn's experiments. 3565 III. ON THE ELECTRODYNAMICS OF MOVING Booms. By A. Einstein. KINEMATICAL PART. § 1. Definition of simultaneity. § 2. On the relativity of lengths a.nd times. § 3. The tm.nsforma.tion of coordina.tes a.nd times. § 4. Physica.l mea.ning of the equations. § 5. The composition of velocities. ELECTRODYNAIIIICAL PART. § 6. Tra.nsforma.tion of the MaxwellHertz equations. § 7. Doppler's principle a.nd a.berra.tion. § 8. The energy of light ra.ys a.nd the pressure of m.dia.tion. § 9. Tra.nsforma.tion of the equations with convection currents. § 10. Dyna.mies of the slowly a.ccelera.ted electron. IV. DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGYCONTENT? By A. Einstein . 6771 V. SPACE AND TIME. By H. Minkowski • 7391 I. The inva.ria.nce of the Newtonian equations a.nd its representation in four dimensions.I spa.ca. II. The worldpostula.te. III. The representation of motion in the continuum. IV. The new mecha.nics. V. The motion of one a.nd two electrons. Nof,es on this paper. By A. Sommerfeld • 9296 VI. 0N THJII INFLUENCE OF GRAVITATION ON THE l'RoPAGATION OF 97108 LIGHT. By A. Einstein § 1. The physics.I na.ture of gravitation. § 2. The gravitation of energy. § 3. The velocity of light. § 4. Bending of lightmys. vii
viii THE PRINCIPLE OF RELATIVITY PAGB8
VII. THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY. By A. Einstein •
109164
A. FUNDAMJIINTAL CONSIDERATIONS ON THE POSTULATE OF RELA· TIVITY, § 1. Observations on the specie.I theory. § 2. The need for a.n ext.ension of the postulate of relativity. § S. The spa.catime continuum; genera.I cova.ria.nce. § 4. Measurement in Space a.nd Time. B. MATHEMATICAL Arns TO THE FORMULATION OF GENERALLY CovARIANT EQUATIONS, § 5. Contra.va.ria.nt a.nd cova.ria.nt fourvectors. § 6. Tensors of the second a.nd higher ranks. § 7. Multiplication of tensors. § 8. The funda.menta.l tensor § 9. The equation of the geodetic line. § 10. The formation of tensors by differentiation. § 11. Some cases of specie.I importance. § 12. The RiemannChristoffel tensor. THEORY OF THE GRAVITATIONAL FIELD. § 18. Equations of motion of a. ma.teria.l point. § 14. The field equations of gra.vita.tion in the absence of matter. § 15. The Ha.miltonia.n function for the gra.vita.tiona.l field. La.ws of momentum a.nd energy. § 16. The genera.I form of the field equations. § 17. The la.ws of conservation. § 18. The la.ws of momentum a.nd energ;i,. D. MATERIAL PHENOMENA, § 19. Euler's equations for a. fluid. § 20. Maxwell's equations for free spa.ce. E. APPLICATIONS OF THE THEORY. § 21. Newton's theory a.s a. first a.pproxima.tion. § 22. Behaviour of rods a.nd clocks in a. static gra.vita.tiona.l field. Bending of light ra.ys. Motion of the perihelion of a. pla.neta.ry orbit. HAMILTON'S I'mNCIPLE AND THE GENERAL THEORY OF RELA· TIVITY, By A. Einstein § 1. The principle of va.ria.tion a.nd the fieldequations. § 2. Sepe.rate existence of the gra.vita.tiona.l field. § 8. Properties of the field equations conditioned by the theory of inva.ria.nts. COSMOLOGICAL CONSIDERATIONS ON THE GENERAL THEORY OF RELATIVITY, By A. Einstein § 1. The Newtonian theory. § 2. The boundary conditions a.ccording to the genera.I theory of relativity. § 8. The spa.tia.lly finite universe. § 4. On a.n a.dditiona.l term for the field equations of gravitation. § 5. Ca.lcula.tion and result. Do GRAVITATIONAL FIELDS PLAY AN ESSENTIAL PART IN T!IE STRUCTURE OF THE ELEMENTARY PARTICLES OF MATTER? By A. Einstein § 1. Defects of the present view. § 2. The field equations freed of sea.la.rs. § 8. On the cosmologica.l question. § 4. Concluding remarks. GRAVITATION AND ELECTRICITY. By H. Weyl .
g,,...
c.
VIII.
IX.
X.
XI.
165173
175188
189198
200216
MICHELSON'S INTERFERENCE EXPERIMENT BY
H.A.LORENTZ Translated from " Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Korpern," Leiden, 1895, §§ 8992.
MICHELSON'S INTERFERENCE EXPERIMENT BY H. A. LORENTZ
1. A S Maxwell first remarked and as follows from a very simple calculation, the time required by a ray of light to travel from a point A to a point B and back to A must vary when the two points together undergo a displacement without carrying the ether with them. The difference is, certainly, a magnitude of second order ; but it is sufficiently great to be detected by a sensitive interference method. The experiment was carried out by Michelson in 1881. * His apparatus, a kind of interferometer, had two horizontal arms, P and Q, of equal length and at right angles one to the other. Of the two mutually interfering rays of light the one passed along the arm P and back, the other along the arm Q and back. The whole instrument, including the source of light and the arrangement for taking observations, couln be revolved about a vertical axis; and those two positions come especially under consideration in which the arm P or the arm Q lay as nearly as possible in the direction of the Earth's motion. On the basis of Fresnel's theory it was anticipated that when the apparatus was revolved from one of these principal positions into the other there would be a displacement of the interference fringes. But of such a displacementfor the sake of brevity we will call it the Maxwell displacementconditioned by the change in the times of propagation, no trace was discovered, and accordingly Michelson thought himself justified in concluding that while the Earth is moving, the ether does not remain at rest. The correctness of this inference was soon brought into question, for by an oversight Michelson had * Michelson, American Journal of Science, 22, 1881, p. 120. 3
4
MICHELSON'S EXPERIMENT
ta.ken the cha.nge in the pha.se difference, which wa.s to be expected in accordance with the theory, a.t twice its proper va.lue. If we ma.ke the necessary correction, we a.rrive a.t displacements no greater than might be ma.sked by errors of observation. Subsequently Michelson* took up the investigation a.new in collaboration with Morley, enhancing the delicacy of the experiment by ca.using ea.ch pencil to be reflected to and fro between a. number of mirrors, thereby obtaining the same advantage a.s if the arms of the earlier apparatus had been considerably lengthened. The mirrors were mounted on a massive stone disc, floating on mercury, a.nd therefore easily revolved. Each pencil now had to travel a total distance of 22 meters, and on Fresnel's theory the displacement to be expected in passing from the one principal position to the other would be 0·4 of the distance between the interference fringes. Nevertheless the rotation produced displacements not exceeding O 02 of this distance, and these might well be a.scribed to errors of observation. Now, does this result entitle us to a.ssume that the ether takes pa.rt in the motion of the Earth, a.nd therefore that the theory of aberration given by Stokes is the correct one? The difficulties which this theory encounters in explaining aberration seem too great for me to share this opinion, and I would ra.ther try to remove the contra.diction between Fresnel's theory and Michelson's result. An hypothesis which I brought forward some time a.go,t and which, a.s I subsequently learned, has also occurred to Fitzgerald,! enables us to do this. The next pa.re.graph will set out this hypothesis. 2. To simplify matters we will assume that we a.re working with a.ppa.ra.tus as employed in the first experiments, and that in•the one principal position the a.rm P lies exactly in • Michelson and Morley, American Journal of Science, 84, 1887, p. SSS; Phil. Mag., 24, 1887, p. 449. t Lorentz, Zittingsversl.a.gen der Akad. v. Wet. te Amsterdam, 189298, p. 74.. t AB Fitzgerald kindly tells me, he has for a long time dealt with hie hypothesis in his lectures. The only published reference which I ca.n find to the hypothesis is by Ledge, " Aberration Problems," Phil. Trans. R.S,, 184 A, 1898.
ff.A.LORENTZ
5
the direction of the motion of the Earth. Let v be the velocity of this motion, L the length of either arm, and hence 2L the path traversed by the rays of light. According to the theory,• the turning of the apparatus through 90° causes the time in which the one pencil travels along P and back to be longer than the time which the other pencil takes to complete its journey by
There would be this same difference if the transla. tion had no influence and the arm P were longer than the arm Q by tLv 2/c2. Similarly with the second principal position. Thus we see that the phase differences expected by the theory might a.lso arise if, when the apparatus is revolved, first the one arm and then the other arm were the longer. It follows that the phase differences ca.n be compensated by contrary changes of the dimensions. If we assume the arm which lies in the direction of the Earth's motion to be shorter than the other by tLv 2/c2, and, at the sa.me time, that the translation has the influence which Fresnel's theory allows it, then the result of the Michelson experiment is explained completely. Thus one would have to imagine that the motion of a solid body (such as a brass rod or the stone di!!C employed in the later experiments) through the resting ether exerts upon the dimensions of that body an influence which varies according to the orientation of the body with respect to the direction of motion. If, for example, the dimensions parallel to this direction were changed in the proportion of I to I + o, and those p.erpendicular in the proportion of I to 1 + e, then we should have the equation .
(I)
in which the value of one of the quantities o and e would remain undetermined. It might be that e = 0, o =  tv 2/c2, but also e = tv2/c2 , o = 0, ore= t;v2/c2, and o=  t;v2/c 2• 3. Surprising as this hypothesis may appear at first sight, * Cf. Lorentz, Arch. N,er1,, 2, 1887, pp. 168176.
6
MICHELSON'S EXPERIMENT
yet we shall have to admit that it is by no means farfetched, as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we a.re able at the present time to make this assertion definitely. If they are so transmitted, the translation will very probably affect the action between two molecules or atoms in a manner resembling the attraction or repulsion between charged particles. Now, since the form and dimensions of a solid body are ultimately conditioned by the intensity of molecular actions, there cannot fail to be a. change of dimensions as well From the theoretical side, therefore, there would be no objection to the hypothesis. As regards its experimental proof, we must first of all note that the lengthenings and shortenings in question are extraordinarily small. We have v2/c 2 = I0 8, and thus, if e = 0, the shortening of the one diameter of the Earth would amount to about 6·5 cm. The length of a meter rod would change, when moved from one principal position into the other, by about rilf micron. One could hardly hope for success in trying to perceive such small quantities except by means of an interference method. We should have to operate with two perpendicular rods, and with two mutually interfering pencils of light, allowing the one to travel to and fro along the first rod, and the other along the second rod. But in this way we should come back once more to the Michelson experiment, and revolving the apparatus we should perceive no displacement of the fringes. Reversing a. previous remark, we might now say that the displacement produced by the a.Iterations of length is compensated by the Maxwell displacement. 4. It is worth noticing that we a.re led to just the same changes of dimensions as have been presumed above if we, firstly, without taking molecular movement into consideration, assume that in a solid body left to itself the forces, attractions or repulsions, acting upon any molecule maintain one another in equilibrium, and, secondlythough to be sure, there is no reason for doing soif we apply to these molecular forces the law which in another place* we deduced for • Viz., § 23 of the book, "Versuch einer Theorie der elektrischen und opti· schen Erscheinungen in bewegtQn Korpern."
H.A.LORENTZ
7
electrostatic actions. For if we now understand by 8 1 and 8 2 not, as formerly, two systems of charged particles, but two systems of moleculesthe second at rest and the first moving with a velocity v in the direction of the axis of zbetween the dimensions of which the relationship subsists as previously stated ; and if we assume that in both systems the z components of the forces are the ea.me, while the y and z components differ from one another by the factor .,/1  v~/c\ then it is clear that the forces in 8 1 will be in equilibrium whenever they a.re so in 8 2• If therefore 8 2 is the state of equilibrium of a solid body at rest, then the molecules in 8 1 have precisely those positions in which they can persist under the influence of translation. The displacement would na.tura.lly bring about this disposition of the molecules of its own accord, a.nd thus effect a shortening in the direction of motion in the proportion of 1 to .,/1  v2 /c2 , in accordance with the formulm given in the abovementioned pa.re.graph. This leads to the values
.,,z
S=.1.~c2•
e0
in agreement with (1). In reality the molecules of a body a.re not a.t rest, but in every " state of equilibrium" there is a. stationary movement. What influence this circumstance may have in the phenomenon which we have been considering is a question which we do not here touch upon ; in any case the experiments of Michelson and Morley, in consequence of unavoidable errors of observation, afford considerable latitude for the values of Sande.
ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT BY
H.A.LORENTZ Reprinted from the English version in Proceedings o/ the Academy of Sciences of Amsterdam, 6, 1904.
ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT BY H. A. LORENTZ
§ 1. T H E problem of determining the influence exerted on electric a.nd optica.l phenomena. by a. tra.nslation, such as a.11 systems have in virtue of the Earth's annual motion, a.dmits of a. comparatively simple solution, so long as only those terms need be taken into account, which a.re proportional to the first power of the ra.tio between the velocity of translation v and the velocity of light c. Ca.sea in which qua.ntities of the second order, i.e. of the order v2/c2 , ma.y be perceptible, present more difficulties. The first exa.µiple of this kind is Michelson's wellknown interferenceexperiment, the negative result of which ha.a led Fitzgerald and myself to the conclusion tha.t the dimensions of solid bodies are slightly altered by their motion through the ether. Some new experiments, in which a. second order effect was sought for, have recently been published. Rayleigh• and Brace t have examined the question whether the Earth's motion may cause a body to become doubly refracting. At first sight this might be expected, if the just mentioned change of dimensions is admitted. Both physicists, however, have obtained a negative result. In the second place Trouton and Noblet have endeavoured to detect a. turning couple a.cting on a charged condenser, the plates of which make a certain angle with the direction of translation. The theory of electrons, unless it be modified by some new hypothesis, would undoubtedly require the * Rayleigh, Phil. Ma.g. (6), 4, 1902, p. 678. t Brace, Phil. Mag. (6), 7, 1904, p. 817. :;: Trouton a.nd Noble, Phil. Trans. Roy, Soc. Lond., A 202, 1908, p. 165. 11
12
ELECTROMAGNETIC PHENOMENA
existence of such a couple. In order to see this, it will suffice to consider a. condenser with ether as dielectric. It may be shown that in every electrostatic system, moving with a. velocity v,• there is a certain amount of "electromagnetic momentum." If we represent this, in direction and magnitude, by a vector 0, the couple in question will be determined by the vector product t (0. v] . (1) Now, if the a.xis of z is chosen perpendicular to the condenser plates, the velocity v having any direction we like; a.nd if U is the energy of the condenser, calculated in the ordinary way, the components of O a.re given t by the following formulre, which a.re exact up to the first order, 2U 2U G,z: = ~Vz, Gy = """:! Vy, G. = 0. c c Substituting these values in (1), we get for the components of the couple, up to terms of the second order, 2U
c2 VyVz,

2U
' + vgra.d' A'z c ?,t c
(13)
H' = curl' A'
(14)
• bb . t' f The symbol 'v'll 1s an a rev:1a. ion or
?,2 c>x's
+
?,2
?Jy's
+
?,2 ?,z' 2,
and grad' cf>' denotes a. vector whose components a.re ?,cf>' ?,:i;"
?,cf>'
?,cf>'
;,y' ,
'lJz'.
The expression grad' A'x has a similar meaning. In order to obtain the solution of (11) and (12) in a simple form, we may take x', y', z' a.a the coordinates of a. point P' in a space S', and a.scribe to this point, for ea.ch value of t', the values of p', u', cf>', A', belonging to the corresponding point P (x, y, z) of the electromagnetic system. For a. definite value t' of the fourth independent variable, the potentials cf>' and A' a.t the point P of thersystem or at the corresponding point P' of the space S', are given by t cf>' = 4~ I[~)dS' A'=
.
(15)
_!_J[p'~1as·
(16)
4'71'C
' M.E.," S§ , and 10.
r
t lbia., §§ 5 and 10.
H. A. LORENTZ
17
Here dS' is an element of the space S', r' its distance from P', and the brackets serve to denote the quantity p' and the vector p'u' such as they are in the element dS', for the value t'  r'/c of the fourth independent variable. Instead of (15) and (16) we may also write, ta.king into account (4) and (7),
q,'
=
A'
= _!_J[pu]dS
~J[p ]dS 4'7T r 4'7TC
r
(17)
(18)
'
the integrations now extending over the electromagnetic system itself. It should be kept in mind that in these formul!B r' does not denote the distance between the element dS and the point (x, y, z) for which the calculation is to be performed. If the element lies at the point (x1, y1, z1), we must take r' == l,.j /3 2(:i:  :i:1)2 + (y  Y1)2 + (z  z1)2 • It is also to be remembered that, if we wish to determine A' for the instant at which the local time in P is t', we must take p and pu', such as they are in the element dS at the instant at which the local time of that element is t'  r'/c. § 6. It will suffice for our purpose to consider two special cases. The first is that of an electrostatic system, i.e. a system having no other motion but the translation with the velocity v. In this case u' = 0, and therefore, by (12), A' = 0. Also, ' is independent of t', so that the equations (11), (13), and (14) reduce to r1•2,1..• ' v 't' =  P• . (19) D' =  grad' cp', H' = 0
q,' and
l
After having determined the vector D' by means of these equations, we know also the ponderomotive force acting on electrons that belong to the system. For these the formul!B (10) become, since u' = 0, .
...
(20)
18
ELECTROMAGNETIC PHENOMENA
The result may be put in a simple form if we compare the moving system $, with which we are concerned, to another electrostatic system $' which remains at rest, and into which I is changed if the dimensions parallel to the axis of x are multiplied by f)l, and the dimensions which have the direction of y or that of z, by la. deformation for which (fJl, l, l) is an appropriate symbol. In this new system, which we may suppose to be placed in the abovementioned space S', we shall give to the density the value p', determined by (7), so that the charges of corresponding elements of volume and of corresponding electrons are the same in I a.nd I'. Then we shall obtain the forces acting on the electrons of the moving system I, if we first determine the corresponding forces in I', and next multiply their components in the direction of the a.xis of x by l 2 , and their components perpendicular to
z2 ff
that axis by formula.
This is conveniently expressed by the F(I) =
z2 z2) (l 2, ~· fJ F(I')
.
. (21)
It is further to be remarked that, after having found D' by (19), we can easily calculate the electromagnetic momentum in the moving system, or rather its component in the direction of the motion. Indeed, the formula 0 =
Uco.
H]dS
shows that
Gx
=
~J(DyH,;  DzHy)dS.
Therefore, by (6), since H' = 0 Gx =
.e:~
4
vJ(Dy12
+ Dz'2)dS = ~~vJ(Dy' 2 + D,?)dS'. (22)
§ 7. Our second special case is that of a particle having an electric moment, i.e. a small space S, with a total charge
J
pdS
= 0, but with such a. distribution of density that the
H.A. LORENTZ
J
J
19
J
integrals pxdS, pydS, pzdS have values differing from 0. Let E, µ,, t be the coordinates, taken relatively to a fixed point A of the particle, which may be called its centre, and let the electric moment be defined as a vector P whose components are Pw
=
JpfdS, P:v
=
Jp11dS, Pz
=
JptdS .
.
(23)
Then
Of course, if E, 11, t are treated as infinitely small, Ux, u,, u~ must be so likewise. We shall neglect squares and products of these six quantities. We shall now apply the equation (17) to the determination of the scalar potential cf,' for an exterior point P (x, y, z), at a finite distance from the polarized particle, and for the instant at which the local time of this point has some definite value t'. In doing so, we shall give the symbol [p], which, in (17), relates to the instant at which the local time in dS is t'  r'/e, a slightly different meaning. Distinguishing by r' 0 the value of r' for the centre A, we shall understand by [p] the value of the density existing in the element dS at the point (E, 11, ~. at the instant t0 at which the local time of A is t' 
r0/e.
It may be seen from (5) that this instant precedes that for which we have to take the numerator in (17) by
f1 2vf + fJ(r' 0 e2

le
r') = f1 2":!f + fi(E?ir' + ?Jr' + t"r') 11 ay e2 le ?ix ?iz
units of time. In this last expression we may put for the differential coefficients their values at the point A. In (17) we have now to replace [p] by
vf["P]
[p] + fl2c2 °'i:Jt +
fJ ( "'i:Jr' 11 0. The former, the front cone of 0, consists, let us say, of all the worldpoints which " send light to O," the latter, the back cone of 0, of all the worldpoints which " receive light from O." The territory bounded by the front cone alone, we may call "before" 0, that which is bounded by the back cone alone, " after" 0. The hyperboloidal sheet already discussed F
lies after 0.
= c2t2

x2

y2

z2 = 1, t > O
The territory between the cones is filled by the
84
SPACE AND TIME
onesheeted hyperboloidal figures _ F = x2 + y2 + z2
_ 02t2 =
ks
for all constant positive values of k. We are specially interested in the hyperbolas with O as centre, lying on the latter figures. The single branches of these hyperbolas may be called briefly the internal hyperbolas with centre 0. One of these branches, regarded as a worldline, would represent a motion which, for t =  oo and t = + oo , rises asymptotically to the velocity of light, c. If we now, on the analogy of vectors in space, call a directed length in the manifold of x, y, z, ta vector,. we have to distinguish between the timelike vectors with directions from O to the sheet + F = 1, t > 0, and the spacelike vectors
Fm. 2.
with directions from O to  F = 1. The time axis may run parallel to any vector of the former kind. Any worldpoint between the front and back cones of O can be arranged by means of the system of reference so as to be simultaneous with 0, but also just as well so as to be earlier than O or later than 0. Any worldpoint within the front cone of O is necessarily always before O ; any worldpoint within the back cone of O necessarily always after 0. Corresponding to passing to the limit, c = oo , there would be a complete flattening out of the wedgeshaped segment between the cones into the plane manifold t = 0. In the figures this segment is intentionally drawn with different widths. We divide up any vector we choose, e.g. that from O to :i:, y, z, t, into the four components x, y, z, t. If the directions
H. MINKOWSKI
85
of two vectors are, respectively, that of a radius vector OR from O to one of the surfaces + F = 1, and that of a, tangent RS at the point R of the same surface, the vectors are sa.id to be normal to one another. Thus the condition that the vectors with components x, y, z, t a.nd x1, y1 , z1 , ti may be normal to each other is c2 tt1  xx1  yy1  zz 1 = 0. For the measurement of vectors in different directions the units of measure are to be fixed by assigning to .a. spacelike vector from O to  F = 1 always the magnitude 1, and to a timelike vector from O to + F = 1, t > 0 always the magnitude 1/c. If we imagine at a. worldpoint P (x, y, z, t) the worldline of a. substantial point running through that point, the magnitude corresponding to the timelike vector dx, dy, dz, dt laid off along the line is therefore dT =
!c../ctdt
2 
dx 2
 dy 2

dz 2.
Ttle integral ~dT = T of this amount, ta.ken along the worldline from any fixed startingpoint P O to the variable endpoint P, we call the proper time of the substantial point a.t P. On the worldline we regard x, y, z, tthe components of the vector OPas functions of the proper time T; denote their first differential coefficients with respect to T by x, ff, z, i; their second differential coefficients with respect to T by ii, y, z, l; and give names to the appropriate vectors, calling the derivative of the vector OP with respect to T the velocity vector at P, and the derivative of this velocity vector with respect fo T the acceleration vector at P. Hence, since c2[2 _ :i;2 _
we have
c2 i't 
y2 _ z2
= c2,
xx  yy  zz =
0,
i.e. the velocity vector is the timelike vector of unit magnitude in the direction of the worldline at P, and the acceleration vector at P is normal to the velocity vector at P, and is therefore in any ca.se a. spacelike vector. Now, as is readily seen, there is a. definite hyperbola.
SPACE AND TIME
86
which has three infinitely proximate points in common with the worldline at P, and whose asymptotes are generators of a. " front cone " and a " back cone " (Fig. 3). ,, , , Let this hyperbola be ca.He'd the hyperbola. •• of curvature at P. If Mis the centre of this •\ hyperbola, wa here ha.ve to do with an interns.I hyperbola. with centre M. Let p be '\ \ :' p the magnitude of the vector MP ; then we I I I I recognize the acceleration vector a.t P a.s the \ I, \ l,'p: vector in the direction MP of magnitude M ,'t' : c2/p. f \ If y, t are a.II zero, the hyperbola of ,' \ 'I curvature reduces to the straight line touchI • I ' I ing the worldline in P, a.nd we must put I • p = 00. \'
.
I
I'
I'
..
I
I
'I '
\
I
I
...
x, z,
I
I
,
I
I
\
I
I
.,,,
\ I
,,
IV
To show that the assumption of group Ge for the laws of physics never leads to a contradiction, it is unavoidable to undertake a revision of the whole of physics on the basis of this assumption. This revision ha.s to some extent already been successfully carried out for questions of thermodynamics and hea.t radiation,* for electromagnetic processes, and finally, with the retention of the concept of mass, for mechanics. t For this last branch of physics it is of prime importance to ra.ise the questionWhen a force with the components X, Y, Z parallel to the axes of space a.eta at a ~orldpoint P (x, y, z, t), where the velocity vector is x, y, t, t, what must we take this force to be when the system of reference is in any way changed ? Now there exist certain approved sta.tements as to the ponderomotive force in the electroma.gnetic field in the cases where the group Ge is undoubtedly admissible. These statements lea.d up to the simple rule :When the system of reference is changed, the force in question transforms into a force in the new space coordinates in such a wa.y that the appropriate vector with the components iX, FIG. 8.
* M. Planck, " Zur Dyna.mil!: bewegter Systeme," Berliner Berichte, 1907, p. 542; a.lso in Ann. d. Phys., 26, 1908, p. 1. t H. Minkowski, " Die Grundgleichungen fiir die elektroma.gnetischen Vorgii.nge in bewegten Korpern," Giminger Na.chrichten, 1908, p. 58.
H. MINKOWSKI tY,
iz,
87
iT, where
1 (" . . T = 2 + ~Y + c t t t is the rate at which work is done by the force at the worldpoint divided by c, remains unchanged. This vector is always normal to the velocity vector at P. A force vector of this kind, corresponding to a force at P, is to be called a "motive force vector" at P. I shall now describe the worldline of a. substantial point with constant mechanical mass m, passing through P. Let the velocity vector at P, multiplied by m, be called the " momentum vector" at P, and the acceleration vector at P, multiplied by m, be called the " force vector" of the motion at P. With these definitions, the law of motion of a. point of mass with given motive force vector runs thus : * The Force Vector of Motion is Equal to the Motive Force Vector. This assertion comprises four equations for the components corresponding to the four axes, and since both vectors mentioned are a priori normal to the velocity vector, the fourth equation may be looked upon as a consequence of the other three. In accordance with the above signification of T, the fourth equation undoubtedly represents the law of energy. Therefore the component of the momentum vector along the axis of t, multiplied by c, is to be defined as the kinetic energy of the point mass. The expression for this is
~x
mc2 q,J = mc2 / dT
~z)
11
...,
 v2 /c2
'
i.e., after removal of the additive constant mc 2, the expression !mv2 of Newtonian mechanics down to magnitudes of the order 1/c2• It comes out very clearly in this way, how the energy depends on the system of reference. But as the a.xis of t may be laid in the direction of any timelike vector, the law of energy, framed for all possible systems of reference, already contains, on the other hand, the whole system of the equations of motion. At the limiting transition which we have discussed, to c = oo, this fact retains its importance for * H. Minkowski, loo. cit., p 107. Cf. also M. Planck, Verhe.ndlunge.n der physike.lisohen Gesellsche.ft, 4, 1906, p. 186.
88
SPACE AND TIME
the axiomatic structure of Newtonian mechanics as well, and has already been appreciated in this sense by I. R. Schlitz.* We can determine the ratio of the units of length and time beforehand in such a way that the natural limit of velocity becomes c = 1. If we then introduce, further, ~::f t = s in place of t, the quadratic differential expression di' =  dx2  dy 2  dz 2  ds 2 thus becomes perfectly symmetrical in x, y, z, s; and this symmetry is communicated to any law which does not contradict the worldpostulate. Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 3 . 105 km =
"=I
secs.
v The advantages afforded by the worldpostulate will perhaps be most strikingly exemplified by indicating the effects proceeding from a point charge in any kind of motion according to the MaxwellLorentz theory. e, e Let us imagine the worldline of such a point electron with the charge e, and introduce upon it the proper time r Y from any initial point. In order to find the field caused by the electron at any worldpoint P 1, we construct the front cone belonging to P 1 (Fig. 4). The cone evidently meets the worldline, since the directions of the line a.re evefywhere those of timelike vectors, at the single point P. We draw the tangent to the worldline at P, and construct through FIG. 4. P 1 the normal P 1Q to this tangent. Let the length of P 1Q be r. Then, by the definition of a front cone, the length of PQ must be r/c. Now the vector in the direction PQ of magnitude e/r repre"I. R. Schiitz, "Das Prinzip der absoluten Erha.ltung der Energia," Gottinger Nachr., 1897, p.110.
H. MINKOWSKI
89
sents by its components along the axes of x, y, z, the vector potential multiplied by c, and by the component along the axis of t, the scalar potential of the field excited by e at the worldpoint P. Herein lie the elementary laws formulated by A. Lienard and E. Wiechert.* Then in the description of the field produced by the electron we see that the separation of the field into electric and magnetic force is a relative one with regard to the underlying time axis ; the most perspicuous way of describing the two forces together is on a certain analogy with the wrench in mechanics, though the analogy is not complete. I will now describe the ponderomotive action of a moving point charge on another moving point charge. Let us imagine the worldline of a second point electron of the charge' ei, passing through the worldpoint P 1• We define P, Q, r as before, then construct (Fig. ~the centre M of the hyperbola of curvature at P, and finally the normal MN from M to a. stra.ight line imagined through P pa.rallel to QP1 • With P a.s sta.rtingpoint we now determine a system of reference a.s follows :The axis of t in the direction PQ, the axis of x in direction QPi, the a.xis of y in direction MN, whereby finally the direction of the axis of z is also defined as normal to the axes of t, x, y. Let the acceleration vector at P be x, y, z, t, the velocity vector a.t P 1 be 1, fh, 1, t1• The motive force vector exerted at P 1 by the first moving electron e on the second moving electron e1 now takes the form
x
 ee1
z
(t ~)S?, 1 
where the components S?a:, $?11 , S?z, S?t of the vector S? satisfy the three rela.tions
cS?t  S?a: = r.!.2 ,
S?y = c·r ~ , S?z = 0,
and where, fourthly, this vector S? is normal to the velocity vector at Pi, and through this circumstance alone stands in dependence on the latter velocity vector. * A. Liens.rd, " Cha.mp electrique et ma.gnetique produit pa.r une charge concentree en un point et a.nimee d'un mouvement quelconque," L'Ecla.ira.ge Electrique, 16, 1898, pp. 5, 58, 106 ; E. Wiechert, " Elektrodyna.misclle Elementa.rgesetze," Arch. Neer!. (2), 5, 1900, p. 549.
90
SPACE AND TIME
When we compare this statement with previous formulations• of the same elementary la.w of the ponderomotive action of moving point charges on one another, we a.re compelled to admit that it is only in four dimensions that the relations here ta.ken under consideration reveal their inner being in full simplicity, a.nd that on a. three dimensional space forced upon us a priori they ca.st only a. very complicated projection. In mechanics a.s reformed in accordance with the worldpostulate, the disturbing lack of harmony between Newtonian mechanics and modern electrodynamics disappears of its own a.ccord. Before CO!).cluding I will just touch upon the attitude of Newton's law of attraction toward this postulate. I shall assume tha.t when two points of mass m, m1 describe their worldlines, a motive force vector is exerted by m on mi, of exactly the sa.me form as tha.t just given in the case of electrons, except that + mm1 must now take the place of  eer We now specially consider the case where the acceleration vector of m is constantly zero. Let us then introduce t in such a. way that m is to be ta.ken as at rest, and let only m1 move under the motive force vector which proceeds from m. If we now modify this given vector in the first place by adding the factor i1 = ~l  v2 /c 2, which, to the order of 1/c2 , is equal to 1, it will be seen t that for the positions :z:1, y1, z1, of mi and their variations in time, we should arrive exactly a.t Kepler's laws again, except that the proper times T 1 of m1 would take the place of the times "ti· From this simple remark it may then be seen that the proposed law of attraction combined with the new meQha.nics is no less well adapted to explain astronomical observations than the Newtonian law of attraction combined with Newtonian mechanics. The fundamental equations for electromagnetic processes in ponderable bodies also fit in completely with the worldpostulate. As I sha.11 show elsewhere, it is not even by any means necessary to abandon the derivation of these funda• K. Sohwa.rzwald, Gottinger Na.ohr., 1908, p. 182; H. A. Lorentz, Enzykl. d. math. Wissensoh., V, Art. 14, p. 199. t H. Minkowski, loo. cit., p. 110.
H. MINKOWSKI
91
mental equations from ideas of the electronic theory, as taught by Lorentz, in order to adapt them to the worldpostulate. The validity without exception of the worldpostulate, I like to think, is the true nucleus of an electromagnetic image of the world, which, discovered by Lorentz, and further re· vealed by Einstein, now lies open in the full light of day. In the development of its mathematical consequences there will be ample suggestions for experimental verifications of the postulate, which will suffice to conciliate even those to whom the abandonment of oldestablished. views is unsympathetic or painful, by the idea of a preestablished harmony between pure mathematics and physics.
NOTES by A. SOMMERFELD The following notes a.re given in an appendix so as to interfere in no way with Minkowski's text. They are by no means essential, having no other purpose than that of removing certain small formal ma.thema.tica.l difficulties which might hinder the comprehension of Minkowski's great thoughts. The bibliographical references a.re confined to the literature dee.ling expressly with the subject of his address. From the physical point of view there is nothing in what Minkowski says that must now be withdrawn, with the exception of the final remark on Newton's la.w of attraction. What will be the epistemo· logioa.l attitude towards Minkowski's conception of the timespace problem is another question, but, as it seems to me, a. question which does not essentially touch his physics. (1) Page 81, line 8. " On the other hand, the concept of rigid bodies has meaning only in mechanics satisfying the group 0 00 ." This sentence was confirmed in the widest sense in a. discussion on a paper by his disciple M. Bom, a year after Minkowski's dee.th. Born (Ann. d. Physik, 80, 1909, p. 1) had defined a relatively rigid body a.s one in which every element of volume, even in a.ccelera.ted motions, undergoes the Lorentzian contraction appropriate to its velocity. Ehrenfest (Phys. Zeitschr.: 10, 1909, p. 918) showed that such a. body cannot be set in rotation ; Herglotz (Ann. d. Phys., 81, 1910, p. 898) and F. Nother (Ann. d. Phys., 81, 1910, p. 919) that it has only three degrees of freedom of movement. The attempt was also ma.de to define a. relatively rigid body with six or nine degrees of freedom. But Planck (Phys. Zeitschr., 11, 1910, p. 294) expressed the view that the theory of relativity can operate only with more or less elastic bodies, and Laue (Phys. Zeitschr., 12, 1911, p. 48), employing Minkowski's methods, and his Fig. 2 in the text a.hove, proved that in the theory of relativity every solid body must have an infinite number of degrees of freedom. Finally Herglotz (Ann. d. Physik, 86, 1911, p. 468) developed a. relativistic theory of elasticity, according to which elastic tensions always occur if the motion of the body is not relatively rigid in Born's sense. Thus the relatively rigid body pla.ys the same pa.rt in this theory of elasticity a.s the ordinary rigid body pla.ys in the ordina.ry theory of elasticity. (2) Page 82, line 18. " If ax/at for the second band is equal to v, a.n easy oa.loula.Uon gives OD' = 00 .JI  v2/c2.'' In Fig. 1, let a. = LA'OA, fJ = LB'OA' "O'OB', in which the equality of the last two angles follows from the symmetrical position of the asymptotes with respect to the new axes of coordinates (conjugate die.meters of the hyperbola.).* Since a.+ fJ = i1r, sin 2/j = oos 2a..
=
"Sommerfeld seems to take ct a.s a coordinate in the graph in place oft as used by Minkowski.TBANs.
92
A. SOMMERFELD
98
In the triangle OD'O' the law of sines gives OD' sin 2/:J cos 2a. 00' = COB a. = COS a. or, a.a 00' = OA', cos 2a. OD' = OA' COB a. = OA' COB a.(l  tans a.)
• (1)
If x, t a.re the coordinates of the point A' in the re, t system, and therefore re , OA and ct . 00 = ct , OA respectivelya.re the corresponding distances from
the axes of coordinates, we have
re, OA = sin a., OA', ct. OA = cos a.. OA', !t= tan a.=~ c
IJ
•
(2)
Inserting these values of re and ct in the equation of the hyperbola., we find 0A'll(cos2 a.  sin2 a.) = OA2, OA' =
COB
a.
OA .j (1  ta.n1 a.)
, (S)
therefore, on account of (1) and (2), OD'= OA.j(l  ta.n2 a.)= OA.,J(l  v2/cl). This, because OA = 00, is the formula. to be proved. Further, in the rightangled triangle OOD, OD= 00 = OA. COB
a.
COS
a.
Equation (8) may therefore be also written in this way, OA' =
/(1  ~) . c•
OD o:r OD = • .,J(l  tan2 a) OA' 'V
This, together with (4), gives the proportion, OD: OA' = OD': OA, which, a.a OA' = 00' and OA = 00, is identical with OD:00' = OD':00 employed on page 82, line 29. (3) Page 84, line 15. " Any worldpoint between the front and be.ck cones of O can be arranged, by means of the system of reference, so a.s to be simultaneous with 0, but also just a.s well so a.s to be earlier than 0, or later than 0." M. Laue (Phys. Zeitschr., 12, 1911, p. 48) traces to this observation the proof of Einstein's theorem : In the theory of relativity no process of causality can be props.gated with a velocity greater than that of light (" Signal velocity ~ c "). Assume that a.n event O ca.uses another event P, and that the worldpoint P lies in the region between the cones of 0, In this case the effect would have been conveyed from O to P with a. velocity greater than that of light, relatively to the system of referencea:, t in question, in which, of course, the effect P is assumed to be later than the cause 0, tp 0. But now, in accordance with the words quoted above, the system of reference may be changed, so that P comes to be earlier than 0, that is to say, a system re', t' may be chosen in
>
NOTES infinitely ma.ny we.ye so the.t t'p becomes X = "ax,."ax.,  {µv, T}'axT Ts ds • Since we may intercha.~ge the order of the differentiations,
184
THE GENERAL THEORY
and since by (23) and (21) {µ.v, T} is symmetrical inµ. and v, it follows that the expression in brackets is symmetrical in µ. and v. Since a geodetic line can be drawn in a.ny direction from a point of the continuum, and therefore dx,../ds is a. fourvector with the ratio of its components arbitrary, it follows from the results of § 7 that
A,,.v =
()2cf,
~
uX,,.uXv
3cf,
 {µ.v, T}:;.,.
. (25)
u:I:r
is a covariant tensor of the second rank. We have therefore come to this result: from the covariant tensor of the first rank A = 3cf, ,,.
 {µ,v, T},frn
A. EINSTEIN
151
or, in view of (56), . (57) Comparison with (41b) shows that with the choice of system of coordinates which we have made, this equation predicates nothing more or less than the vanishing of divergence of the material energytensor. Physically, the occurrence of the second term on the lefthand side shows that laws of conservation of momentum and energy do not apply in the strict sense for matter alone, or else that they apply only when the g"'v are constant, i.e. when the field intensities of gravitation vanish. This second term is an expression for momentum, and for energy, as transferred per unit of' volume and time from the gravitational field to matter. This is brought out still more clearly by rewriting (57) in the sense of (41) as (57a) The right side expresses the energetic effect of the gravitational field on matter. Thus the field equations of gravitation contain four conditions which govern the course of material phenomena.. They give the equations of material phenomena completely, if the latter is capable of being characterized by four differential equations independent of one another.*
D. MATERIAL PHENOMENA The mathematical aids developed in part B enable us forthwith to generalize the physical laws of matter (hydrodynamics, Maxwell's electrodynamics), as they are formulated in the special theory of relativity, so that they will fit in with the general theory of relativity. When this is done, the general principle of relativity does not indeed afford us a further limitation of possibilities; but it makes us acquainted with the influence of the gravitational field on all processes, * On this question cf. H. Hilbert, Na.chr. d. K. Gesellsoh. d. Wiss. zu Gottingen, Ma.th.phys. Kla.sse, 1915, p. 8.
152
THE GENERAL THEORY
without our having to introduce a.ny new hypothesis whatever. Hence it comes a.bout that it is not necessary to introduce definite assumptions a.s to the physical nature of matter (in the narrower sense). In pa.rticula.r it may remain an open question whether the theory of the electromagnetic field in conjunction with that of the gra.vita.tiona.l field furnishes a. sufficient basis for the theory of matter or not. The general postulate of rela.tivity is unable on principle to tell us anything a.bout this. It must remain to be seen, during the working out of the theory, whether electroma.gnetics and the doctrine of gravitation are able in collaboration to perform what the former by itself is unable to do.
§ 19. Euler's Equations for a Frictionless Adiabatic Fluid Let p and p be two scalars, the former of which we call the "pressure," the latter the "density "of a fluid; and let an equation subsist between them. Let the contra.variant symmetrical tensor . (58) be the contrava.ria.nt energytensor of the fluid. the covariant tensor dxa. dx/3
T,.v =  g,.vp + g,.a.g,./3 ds a;;P,
To it belongs . ( 58a.)
as well a.s the mixed tensor * . (58b) Inserting the righthand side of (58b) in (57a.), we obtain the Eulerian hydrodynamical equations of the genera.I theory of relativity. They give, in theory, a. complete solution of the problem of motion, since the four equations (57a), together * For an observer using a system of reference in the sense of the specie.I theory of. relativity for an infinitely small region, and moving with it, the density of energy T! equals p  p. This gives the definition of p. Thus p is not constant for an incompressible fluid.
A. EINSTEIN
153
with the given equation between p a.nd p, a.nd the equation dx,. dxfJ _ 1 g,.fJ ds ds  '
a.re sufficient, ga.13 being given, to define the six unknowns dx1 dx 1 dx 8 dx4 p, p, ds' ds' ds' ds' If the
g,,.,,
a.re also unknown, the equations (53) a.re brought in. These are eleven equations for defining the ten functions g,,.v, so that these functions a.ppea.r overdefined. We must remember, however, tha.t the equations (57a.) a.re a.lrea.dy contained in the equations (53), so that the latter represent only seven independent equations. There is good reason for this la.ck of definition, in that the wide freedom of the choice of coordinates ca.uses the problem to remain ma.thema.tica.lly undefined to such a. degree that three of the functions of space ma.y be chosen a.t will*
§ :Jo. Maxwell's Electromagnetic Field Equations for Free Space Let tf>,, be the components of a. covariant vectorthe electromagnetic potential vector. From them we form, in accordance with (36), the components F pa of the covariant sixvector of the electromagnetic field, in accordance with the system of equations F

pp  ?,.. ?,x..
()Xp
. ( 59)
It follows from (59) that the system of equations ?,FPO' + ?,F0'1' ?,x,
?,xP
+ ?,F,p = O . ?,x..
(60)
is satisfied, its left side being, by (37), a.n a.ntisymmetrica.l tensor of the third rank. System (60) thus contains essentially four equations which a.re written out a.s follows :* On the e.ba.ndonment of the choice of coordinates with g =  1, there rems.in/our functions of spa.ca with liberty of choice, corresponding to the four e.rbitre.ry functions a.~ our dispose.I in the choice of coordine.tes.
154
THE GENERAL THEORY ()F23 +()FM+ ()F42
()x2
c)x4
()F84
+
()F,1
()x 1
()x 8
()F41 +
()F12
()x 2 ()F1~ ().xa
()xa
+
()R.1a ()x1
()F1a
=
o
()x 4
+
()x4
+
= 0
()FH
=0
. (60a)
()x 1
+
()Fa1
=0
()x2
This system corresponds to the second of Maxwell's systems of equations. We recognize this at once by setting (61) Then in place of (60a) we may set, in the usual notation of threedimensional vector analysis,
 ():: = curl E} div H = 0
(60b)
We obtain Maxwell's first system by generalizing the form given by Minkowski. We introduce the contra.variant sixvector associated with F 11li (62) and also the contra.variant vector J" of the density of the electric current. Then, taking (40) into consideration, the following equations will be invariant for any substitution whose invariant is unity (in agreement with the cliosen coordinates) : . (63) Let
F23 = H'a:, FH =  E'a:} = H'11, F24 =  E'11 F 12 = H'z, F 34 =  E'z
Fa1
. (64)
which quantities are equal to the quantities Ha: ... Ez in
A. EINSTEIN
155
the special case of the restricted theory of relativity; a.nd in addition J1 = jz, J2 = iu, J 3 = jz, J 4 = P, we obtain in place of (63)
i)E' + J. = curlH'} ;;r
.
(63a.)
div E' = p The equations (60), (62), a.nd (63) thus form the generalization of Maxwell's field equations for free space, with the convention which we have established with respect to the choice of coordinates. The Energycomponents of the Electromagnetic Field.We form the inner product . (65) By (61) its components, written in the threedimensional manner, a.re
•:
~ P~·_+:U ·
IC4
=  (jE)
?} .
. (65a)
"a
is a covariant vector the components of which a.re equal to the negative momentum, or, respectively, the energy, which is transferred from the electric masses to the electromagnetic field per unit of time and volume. If the electric masses are free, that is, under the sole influence of the electromagnetic field, the covariant vector "a will vanish. To obtain the energycomponents T; of the electromagnetic field, we need only give to equation "a = 0 the form of equation (57). From (63) and (65) we have in the first place
= F i)F"'v = 2(F F"'')  F"'Pi)F""' ""
""' i')xv
i')xv
""'
i')xv •
The second term of the righthand side, by reason of (60), permits the transformation
F"'v~j)~:"' =  iF"''j)i)!:v =  ig"'"g·~F../i)!:v.
156
THE GENERAL THEORY
which latter expression may, for res.sons of symmetry, a.lso be written in the form
 i[g,,."g"tJFa./F,,.. + g,,."g"iFa.fJF,,..].
t Herein is to be found the res.son why the genera.I postulate of relativity leads to a very definite theory of gravitation. ::: By performing partial integration we obtain CM* = ,J  g 9""[{µ.o., S} {11S, o.}  {µ.11, o.} {o.'3, S}].
A. EINSTEIN
171
where for brevity we have set 3@•
3@*
s; = 23g,.trgJJ.11 + 23g:trg:" + @*~; 
3@*
3g~"g':·
(14)
From these two equations we dra.w two inferences which are We know that ,® is an in""  g variant with respect to a.ny substitution, but we do not know
important for what follows. this of
~* .
It is easy to demonstrate, however, that the
""  g
latter quantity is an invariant with respect to any linear substitutions of the coordinates. Hence it follows that the right side of (13) must always vanish if all ~2
~.;,r
uXvuX11
vanish.
Consequently @• must satisfy the identity
s;=o
. (15)
If, further, we choose the !::,.xv so that they differ from
zero only in the interior of a given domain, but in infinitesimal proximity to the boundary they vanish, then, with the transformation in question, the value of the boundary integral occurring in equation (2) does not change. Therefore ~F = 0, and, in consequence, t
But the lefthand side of the equation must vanish, since both , ® and ../  g dr are invariants. Consequently the ""  g righthand side also vanishes. Thus, taking (14), (15), and (16) into consideration, we obtain, in the first place, the equation
J
3@• ,..,32(~Xtr) dT
3g:trg 3x.,3xu
=
O •
(16)
Transforming this equation by two partial integrations, and having regard to the liberty of choice of the l::;.x,r, we obtain
t By the introduction of _the que.ntities@ and@• instead of 9 e.nd 9•.
172
HAMILTON'S PRINCIPLE
the identity .
(17)
From the two identities (16) and (17), which result from the inva.ria.nce of
@ , a.nd therefore from the postulate of ,../ g general relativity, we now have to dra.w conclusions. We first transform the field equa.tions (7) of gravitation by mixed multiplication by g'"'. We then obtain (by interchanging the indices tT and v ), as equivalents of the field equations (7), the equations
(gl'J'
c) . c)fJ' c)@*) c)x,. :." =

(