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THE PRINCIPLE OF RELATIVITY

THE PRINCIPLE OF RELATIVITY
A COLLECTION OF ORIGINAL MEMOIRS ON THB SPECIAL AND GENERAL THEORY OF RBLATIVITY
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 8RVEN DIAGRAMS
DOVER PUBLICATIONS, INC.

This new Dover edition is an unaltered and unabridged republication of the 1923 translation first published by Methuen and Company, Ltd. Reprinted
thro~h special arrangement with Methuen and Company
and Albert Einstein.
Tliis Translation First Published in r923
PRINTED AND BOUND IN THE UNITED STATES OF AMERICA

TRANSLATORS' PREFACE
T H~~ 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 ,ve must confess that Relativity is still very much of a problem-and therefore worthy of our study.
The authors of the papers are still actively at work on the subject-all 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 fro1n the text, as published in a German collection, under the title "Des Relativitatsprinzip" (Teubneri 4th ed., 1922).
V
ta

vi THE PRINCIPLE OF RELATIVITY
The second paper by Lorentz is an exception 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

I. MICHELSON'S INTERFERENCE EXPERIMENT. By H. A. Lorentz .

§ 1. The experiment. § 2. The contraction hypothesis. §§ 3-4.

The contraction in relation to molecular forces.

II. ELECTROMAGNETIC PHENOMENA IN A SYSTEM MOVING WITH ANY

VELOCITY LESS THAN THAT OF LIGHT. By H. A. Lorentz .

§ 1. Experimental evidence. § 2. Poincare's criticism of the

coutraction hypothesis. § 3. Maxwell's equations for

moving axes. § 4. The modified vectors. § 5. Retarded

potentials. § 6. Electrostatic fields. § 7. A polarized

particle. § 8. Corresponding states. § 9. Momentum of

an electron. § 10. The influence of the earth's motion on

optical phenomena. § 11. Applications. § 12. Molecular

motions. § 13. Kaufmann's experiments.

III. ON TIIE ELECTRODYNAMICS OF l\foVlNG BODIES. By A. Einstein.

KcNEMATICAL PART. § 1. Definition of simultaneity. § 2. On

the relativity of lengths and times. § 3. The transforma-

tion of co-ordinates and times. § 4. Physical meaning of

the equations. § 5. The composition of velocities.

ELECTRODYNAMICAL PART. § 6. Transformation of the Max-

well-Hertz equations. § 7. Doppler's principle and aber-

ration. § 8. The energy of light rays and the pressure of

radiation. § 9. Transformation of the equations with

convection cunents. § 10. Dynamics of the slowly accel-

erated electron.

IV. DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY·

CONTENT? By A. Einstein .

V. SPACE AND TIME. By H. Minkowski .

I. The invariance of the Newtonian equations and its represen-

tation in four dimensional space. II. The world-postulate.

III. The representation of motion in the continuum. IV.

The new mechanics. V. The motion of one and two

electrons.

Notes on this paper. By A. Sommerfeld.

.

VI. ON THE INFLUENCE OF GRAVITATION ON THE PROPAGATION OF

LIGHT. By A. Einstein .
8 1 l"f1'hA 1'\YIVQl/lA.1 Tlnt:111".:) n-F O''Nll'ITirDHnn

.

. .

R C) rr,t,.,.,. ............,a.. H,..,,,.,

PAGES
1-7 9-34
35-65
67-71 73-91 92-96 97-108

V•l•l•l THE PRINCIPLE OF RELATIVIrry

PAGES

VII. THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY. By

A. Einstein .

. 109-164

A. FUNDAMENTAL CONSIDERATIONS ON THE POSTULATE OF HELA-

TIVITY. § 1. Observations on the special theory. § 2. The

need for au extension of the postulate of relativity. § 3.

The space-time continuum; general co-variance. § 4.

Measurement in Space and Tiine.

B. l\'.fATHEMATICAL AIDS TO THE FORMULATION OF GENERALLY

CovARIANT EQUATIONS. § 5. Contravariant and covariant

four-vectors. § 6. Tensors of the second and higher ranks.

§ 7. Multiplication of tensors. § 8. The fundamental

tensor gµ.v, § 9. The equation of the geodetic line. § 10.

The formation of tensors by differentiation. § 11. Some

cases of special importance. § 12. The Riemann-Christoffel

tensor.

C. THEORY OF THB GRAVITATIONAL FIELD. § 13. Equations

of motion of a material point. § 14. The field equations

of gravitation in the abseuce of matter. § 15. The Hamil-

tonian function for the gl'avitational field. Laws of

momentum and energy. § lG. The general form of the

field equations. ~ 17. The laws of conservation. § H3.

The laws of momentum and energy.

D. 1\!ATERIAL PHENOMENA. § 19. Euler's equations for a fluid.

§ 20. !viaxwell's equations for free space.

E. APPLICATIONS OF THE THEORY. § 21. Newton's theory as

a first approximation. § 22. Behaviour of rods and c1ocks

in a static gravitational field. Bending of light rays.

:Motion of the perihelion of a planetary orbit.
VIII. HAMILTON's PRINCIPLE AND THE GENERAL TirnonY OF RELA-

TIVITY. By A. Einstein

165-173

§ 1. The principle of variation and the field-equations. § 2.

Separate existence of the gravitational field. § 3. Pro-

perties of the field equations conditioned by the theory of

invariants.

IX. COSMOLOGICAL CONSIDERA'l'IONS ON THE GENERAL THEORY OP

RELATIVITY. By A. Einstein

175-188

§ 1. The Newtonian theory. § 2. The boundary conditions

accorcliug to the general theory of relativity. § 3. rrhe

spatially finite universe. § 4. On an additional term for

the field equations of gravitation. § 5. Calculation and

result.

X. Do GRAVITATIONAL FIELDS PLAY AN EssENTIAL PART rn THE

STRUCTURE OF THE ELEMENTARY PARTICLES OF '!\IA'T'TEH.?

By A. Einstein

. 189-198

§ 1. Defects of the present view. § 2. The field equations freed

of scalars. § 3. On the cosmological questiou. § 4. Con-

cluding remarks.

XI. GRAVITATION AND ELECTRICITY. By H, Weyl .

. 200-216

MICHELSON'S INTERFERENCE EXPERIMENT
BY
H.A.LORENTZ
Translated f1·om '' Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Korpern," Leiden, 1895, §§ 89-92.

MICHELSON'S INTERFERENCE EXPERIMENT
BY H. A. LORENTZ
1. A S Maxwell first rema.rked 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, coulcl 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 tho 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 displacement-for the sake of brevity we will call it the Maxwell displacement-conditioned 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 wa.s soon brought into question, for by an oversight Michelson had
* Michelson, American Journal of Science, 22, 1881, p. 120. 3

...,_

- ..- - - -.. - - - - - ~ · - ......~ • 1 1 1 •

- .....- - - - - -

.-T"

4

MICHELSON'S EXPERIMENT

taken the change in the phase difference, which was to be expected in accordance with the theory, at twice its proper value. If we make the necessary correction, we arrive at displacements no greater than might be masked by errors of observa.tion.
Subsequently Michelson* took up the investigation anew in collaboration with Morley, enhancing the delica.cy of the experiment by causing each pencil to be reflected to and fro between a number of mirrors, thereby obtaining the same advantage as if the arms of the earlier apparatus had been considerably lengthened. The mirrors were mounted on a massive stone disc, floating on mercury, and therefore easily revolved. Each pencil now ha.d 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 O02 of this distance, and these might well be ascribed to errors of observation.
Now, does this result entitle us to assume that the ether takes part in the motion of the Earth, and 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, a.nd I would rather try to remove the contradiction between Fresnel's theory and Michelson's result. An hypothesis which I brought forward some time ago,t and which, as I subsequently learned, has also occurred to Fitzgerald,! enables us to do this. The next paragraph will set out this hypothesis.
2. To simplify matters we will assume that we are working with apparatus a.s employed in the first experiments, and that in, the one principal position the arm P lies exactly in

* Michelson e.nd Morley, American Journal of Science, 34, 1887, p. 888;
Phil. Mag., 24, 1887, p. 4:49.
t Lorentz, Zittingsverslagen der Ake.d. v. Wet. te Amsterdam, 1892-93,
p. 74.

H. 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
Lv2

There would be this same differenee if the translation had., no influence and the arm P were longer than the arm Q by

1Lv /c2

2

•

Similarly with the second principal position.

Thus we see that the phase differences expected by the

theory might also 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 can be compensated by

contrary changes of the dimensions.

If we assume the arm which lies in the direction of the r
Earth's motion to be shorter than the other by ½Lv2/c2, and,

at the same time, that the translation has the influence wp.ich

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 disc employed in

the later experiments) through the resting ether exerts upon

the dimensions of that body an influence which varies accord-

ing to the orientation of the body with respect to the direction

o, of motion. If, for example, the dimensions parallel to this
direction were changed in the proportion of 1 to 1 + and

those p.erpendicular in the proportion of 1 to 1 + E, then we

should have the equation

. (1)

in which the va,lue of one of the quantities 8 and e would

remain undetermined. It might be that e = 0, 8 == - tv2 /ci,

o but also e =

½v2 /c2,

8 =

0,

ore =

iv /c2

2

,

and

=

-

iv /c2

2

•

3. Surprising as this hypothesis may appear at first sight,

* Cf. Lorentz, Arch. N~erl., 2, 1887, pp. 168-176.

6

MICHELSON'S EXPERIMENT

yet we shall have to admit that it is by no means far-fetched, as soon as we assume that molecular forces are also transmitted through the ether, like the electric and magnetic forces of which we are 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 be.. tween 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 di.. mensions 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/c2 = 10-s, 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 ~ 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 alterations of length is compensated by the Maxwell displacement.
4. It is worth noticing that we are 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, secondly-though to be sure, there is no reason for doing so-if 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 bewegt~n Korpern."

H.A. LORENTZ

7

electrostatic actions. For if we now understand by S1 and 82 not, as formerly, two systems of charged particles, but two systems of molecules-the second at rest and the first moving

with a velocity v in the direction of the axis of x-between the

dimensions of which the relationship subsists as previously

stated; and if we assume tha.t in both systems the x com-

ponents of the forces are the sa.me, while they and z com-

ponents differ from one another by the factor ✓1 - v'l/c"', then

it is clear that the forces in S1 will be in equilibrium whenever they are so in S2• If therefore 82 is the state of equilibrium of a solid body at rest, then the molecules in 81 have precisely those positions in which they can persist under the influence

of translation. The displacement would naturally bring about

this disposition of the molecules of its own accord, and

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 above-mentioned paragraph. This leads to the

values

8 = - ½v22, C
in agreement with (1).

0 e :a:

In reality the molecules of a body are not at rest, but in

every '' state of equilibrium " there is a stationary movement.

What influence this circumstance may have in the phe-

nomenon 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
aand e.

ELECTROMAGNETIC PHENOMENA IN

i

A SYSTEM MOVING WITH ANY VELOCITY LESS THAN THAT OF LIGHT

BY

i

ff.A.LORENTZ

i

Reprinted from the English version in Proceedings of the

1

Academy of Sciences of Amsterdam, 6, 1904.

·~>

ii

l
l

't::

•.

,~l 1~

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 and optical phenomena by a translation, such as all systems have in virtue of the Earth's annual motion, admits of a comparatively simple solution, so long as only those terms need be taken into account, which are proportional to the first power of the ratio between the velocity of translation v and the velocity of light c. Cases in which quantities of the second order, i.e. of the order v2/c2, may be perceptible, present more difficulties. The first exa)nple of this kind is Michelson's well-known interferenceexperiment, the negative result of which has led Fitzgerald and myself to the conclusion that the dimensions of solid bodies are slightly altered by their motion through the ether. Son1e ne\V 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 1night 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 Noble t have endeavoured
to detect a turning couple acting 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. Mag. (6), 4, 1902, p. 678.
t Brace, Phil. Mag. (6), 7, 1904, p. 817.
::: Trouton and 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 O, the couple in question will be determined
t by the vector product

[O. v] .

. (1)

Now, if the axis of z is chosen perpendicular to the con-
denser plates, the velocity v having any direction we like ; and if U is the energy of the condenser, calculated in the ordinary way, the components of O are given :J: by the following formulre, which are exact up to the first order,

Gx

=

2U
c2 Vx,

2U Gy = c'l. Vy,

Gz = 0.

Substituting these values in (1), we get for the components of the couple, up to terms of the second order,

2U

2U

-C2- VyVz, - -C2 VxVz,

0.

These expressions show that the axis of the couple lies in the plane of the plates, perpendicular to the translation. If a is the angle between the velocity and the normal to the plates, the moment of the couple will be U(v/c)2 sin 2a; it tends to turn the condenser into such a position that the plates are parallel to the Earth's motion.
In the apparatus of Trouton and Noble the condenser was fixed to the beam of a torsion-balance, sufficiently delicate to be deflected by a couple of the above order of magnitude. No effect could however be observed.
§ 2. The experiments of which I have spoken are not the
only rea.son for which a new examination of the problems connected with the motion of the Earth is desirable. Poin-

• A vector will be denoted by a Ola,rendon letter, its magnitude by the corresponding Latin letter.
t See my article: "Weiterbildung der Ma,xwell'schen Theorie. Electron-
entheorie," Mathem. Encyclopiidie1 V, 14, § 21, a. (This article will be quoted

-son's negative result,- the introduction of a new hypothesis

has been required, and that the same necessity may occur

each time new facts will be brought to light. Surely this

course of inventing -special hypotheses for each new experi-

mental result is somewhat artificial. It would be more

satisfactory if it were possible to show by means of certain

fundamental assumptions and without neglecting terms of

one order of magnitude or another, that many electromagnetic

actions are entirely independent of the motion of the system.

Some years ago, I already sought to irame a theory of this
kind. t I believe it is now possible to treat the subject with

a better result. The only restriction as regards the velocity

will be that it be less than that of light.

§ 3. I shall start from the fundamental equations of the

theory of electrons.t Let D be the dielectric displacement in

the ether, H the magnetic force, p the volume-density of the

charge of an electron, v the velocity of a point of such a

particle, and F the ponderomotive force, i.e. the force,

reckoned per unit charge, which is exerted by the ether on a

l volume-element of an electron. Then, if we use a fixed
system of co-ordinates,
div D = p, div H = 0,

curlH=1- (~-D+pv),

C c)t
l~H
curlD=-c; -~t-'

j

. (2)

F = D + ! [v. H]. C

I shall now suppose that the system as a whole moves in the direction of a: with a constant velocity v, and I shall denote by u any velocity which a point of an electron ma.y ha.vein addition to this, so that
Vx = V + Uz, Vy = Uy, 'Vz = u~.

* Poinoe.r6, Rapports du Oongres de physique de 1900, Paris, 1, pp. 22, 28.
t Lorentz, Zittingsverslag Ake.d. v. Wet., 7, 1899, p. 507 ; Amsterdam
. Proo., 1898-99, p. 427.
+.... "M. E.,'' § 2

14 ELECTROMAGNE1,IC PHENOMENA

If the equations (2) are at the same time referred to axes moving with the system, they become

div D = p, div H = 0,

!(~ - ~Hz
"'i)y

-

~~Y <)Z

=

C

ot

V -~)Dx + !p(v + Ux)

oX

C

'

!(~ - c)Hx - oH~ =

V -~)Dy + !puy

~z

().V C "?Ji QX

C '

- "Dm- -()D= z

<)Z

<)X

"?JDy ()Dx
- ox- --a-y =

§ 4. We shall further transform these formul~ by a change of variables. Putting

c2
-- 1.)2
c2 - v2 - tJ ' •

. (3)

and understanding by l another numerical quantity, to be determined further on, I take as new independent variables

x' = f3lx, y' = ly, z' = lz, .

. (4)

t, = 1l3t - t1J.>clv2x, .

. (5)

and I define two new vectors D' and H' by the formulm

H. A. LORENTZ

15

i( !( D'x = f.nx, D'11 = Dy - ~H.), D', = Dz + ~Hy),

i(n. - H'x = iR., H'y = ~(H11 + ~D ), H'z =

~n11),

t for which, on account of (3), we may also write
Dx = l2D',,, Dy = ,Bz2(D'11 + ~H'z), D,= .Bl2(D'. -~H'11) 6)
J H.,, = l2H'.,,, Hy= ,Bl'(H'u - ~D',), H,=,Bl2 ( H',+~D'11)

As to the coefficient l, it is to be considered as a function

of v, whose value is 1 for v = 0, and which, for small values

of v, differs from unity no more than by a quantity of the

second order.

The variable t' may be cal1ed the" local time" ; indeed,
for /3 = 1, l = 1 it becomes identical with what I formerly

denoted by this name.

If, finally, we put

1 {3la P = P

(7)

8 2Ux = 'll, 'x, JQJUy = UIy, JQJUz = U Iz,

,

. (8)

these latter quantities being considered as the components of
a. new vector u', the equations take the following form:-

(i x) , l d1•v, D' =

-

vu· c2 -

d. , , p , 1v H -

O,1

c cur11 H' = 1( c-);D,t, + p uI , ) ,

(9)

curI' D, =- - -c1 ·~"Ha-t'' '

)

1 Fx = l'{D'x + ~(u'11H'z - u',H'y) + ;,;(u'yD'y + u'zD'z) },

Fy= ~{D'11 + ~(u',H',,, - u'xH'z)- ; 2u'xD'11},

r (10)

F

Z

_,

z2{D' /-3 Z

+

-1·\rU
C

,
X

H

' y

-

U

,1JH

'
CC

)

-

cv-2U,XD 'Z} •

)

The meaning of the symbols div' and curl' in (9) is similar

16 ELECTROMAGNETIC PHENOMENA

to that of div and curl in (2); only, the differentiations with respect to x, y, z a.re to be replaced by the corresponding ones with respect to x ', y', z'.
§ 5. The equations (9) lead to the conclusion that the
vectors D' and H' may be represented by means of a scalar potential cf,' and a vector potential A'. These potentials
satisfy the equations*

'2,1,.' - 1 c)2cp' --

,

"v "' c2 <)t'2

-p

. (11)

'2A' 1 <)2A'

1' '

,"v - c2 "at'2 = - cp u'

. (12)

and in terms of them D' and H' are given by

- c D' =

1 <c)A)t'' -

grad'

"',1,.'

+

V 0

gra d'

A' x

. (13)

H' = curl' A'

. (14)

The symbo1 '1'2 1•s

an

abbrev1.at·~on

for

7J2
c)a;' 2

+

7J2
"i)y'2

+

c<)'z)2'2,

and grad' cf/ denotes a vector whose components are

3cp' -act,' -acp'
~x'' ";)y' ' 7Jz' •

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' as the co-ordinates of a point P' in a space S', and ascribe to this point, for each 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' at the point P of the"'system or at the
corresponding point P' of the space S', are given by t

iJ::las· . </>' =

• . (15)

A' = _!__J[p'~']dS'
4,rc r

. (16)

'M.E.," I§ 4 a.nd 10.

t Ibid., §§ 5 and 10.

H. A. I~ORENTZ

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, taking into account (4) and (7),

. (17)

(18)

the integrations now extending over the electromagnetic system itself. It should be kept in mind that in these formuloo 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 (xi, y1, z1), we must take
r' = l✓131.(x - x 1)t + (y - y 1Y, + (z - z1)2.

It is also to be remembered that, if we wish to determine
¢' and 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. Tt will suffice for our purpose to consider two special
cases. The first is that of an electrostatic system, i.e, a
systen1 having no other motion but the translation with the
velocity v. In this case u' = 0, and therefore, by (12), A' = 0.
Also, cp' is independent of t', so that the equations (11), (13),
and (14) reduce to

'v r7'2,.'J.t..' ' - - p '

}

D' - - g;ad' cp',

H' = 0

. (19)

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 formuloo
(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 ~l, and the dimensions which have the direction of y or that of z, by l-a deformation for which (/3l, l, l) is an appropriate symbol. In this new system, which we may suppose to be placed in the above-mentioned space S', we shall give to the density the value p', determined by (7), so that the charges of corresponding elements of volu1ne and of
corresponding electrons are the same in I and '!'. Then we
shall obtain the forces acting on the electrons of the moving
system '!, if we first determine the corresponding forces in '!', and next mu]tiply their components in the direction of
the axis of x by l2, and their components perpendicular to
z2
that axis by 13. This is conveniently expressed by the formula
. (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 1notion. Indeed, the formula

shows that

0 = H(D. H)dS Gx = H(DyH• - DzHy)dS.

Therefore, by (6), since H' = 0
. e:~ J G., = 4vJ(Dy'2 + D;2)dS = ~~v (D1/ 2 + Dz'2)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
JpdS = 0, but with such a distribution of density that the

I-I. A. LORENTZ

19

J integrals JpxdS, JpydS, pzdS have values differing from 0.

Let ~' µ,, , be the co-ordinates, 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 com-
ponents are
P., = Jp~dS, Py = J,,.,as, P. ~ JptdS . . (23)

Then

Of course, if E, 'f/, tare treated as infinitely small, Ux, Uy, 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 ¢' 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'/c, 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, 'f/, t), at the instant t0 at which the local time of A is
t' - r0/c. It may be seen from (5) that this instant precedes that
for which we have to take the numerator in (17) by

p2vf + ~(r'o -

c'J.

le

1·')

=

132~f
c2

+

~ (E~r'
le ()a;

+

~
'YJ '3y

+

>~ "c")zi)r')

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

t [p]

+

/32vc2E[""i)Pt ]

+

/3
le

(

E"<i))xr'

+

"i)r' 11 "i>y

+

"~iz)r')['(3) tp]

(25)

where[!:] relates a.gain to the time t0• Now, the value of t' for which the calculations are to be performed having been

20 ELECTROMAGNETIC PHENOMENA

chosen, this time t0 will be a function of the co-ordinates x, y, z of the exterior point P. The value of [p] will therefore depend on these co-ordinates in such a way that

()[p] == - ~ <)r'[~P] etc

~x

le ~x ~t-' •

by which (25) becomes

[p] + ~2vE["P] _ (E~[p} + 'l~[p] + ,~[p]).

c2 c>t

c)x

"i)y

"i)z

Again, if henceforth we understand by r' what has above

been called r'0, the factor r1, must be replaced by

1 (1 ) (1 ) (1 ) ~
r' - E-ax r'

i, - TJ'i)y r'

- ~~"cz) r' '

so that after all, in the integral (17), the element dS is multiplied by

J ~ [p

2vf["PJ _ c> f[p] c) 11[p] <> ~p]

r' + c2r' ~t 7 ()X - "3y 7 - <)z 7 •

This is simpler than the primitive form, because neither r', nor the time for which the quantities enclosed in brackets
f are to be taken, depend on x, y, z. Using (23) and re-
membering that pdS = 0, we get

cf,' = ~ 2v [~] _ _! { ~[Px] + <) [Py] + ~[P:.]}
4pc2r' ~t 4p ~x r' ay r' c,z r' '

a formula in which all the enclosed quantities are to be taken for the instant at which the local time of the centre of the particle is t' - r'/c.
We shall conclude these calculations by introducing a new vector P', whose components are
P'x = ~lPx, P'y = lPy, P'z = lPz, . . (26)
passing at the same time to x', y', z', t' as independent variables. The final result is

rormaHuu us 1t~~ vvJ.ut'uvu,u...,'"'"'' ~--- __ _
finitely small vector u'. Having regard to (8), (24), (26), and (5), I find
A' = __! ___ ()[P']
47rcr' ot' •

The field produced by the polarized particle is now wh_olly determined. The formula (13) leads to

D' -_

-

471r~C

c> 2 'ut'\?

[P'J -r- ,

+

417-r

gra d'{ :vO;X,

[Pr'x' ]

+

:u;0;y;,

[Pr'?,I]

+

",(,)z[P-r'-z,]}

(o~r-'1)

and the vector H' is given by (14). We may further use the equations (20), instead of the original formulro (10), if we wish to consider the forces exerted by the polarized particle on a similar one placed at some distance. Indeed, in the second particle, as we11 as in the first, the velocities u may be held to be infinitely small.
It is to be remarked that the formulffi for a system without translation are implied in what precedes. For such a system the quantities with accents become identical to the corresponding ones without accents ; also ~ = 1 and
l = 1. The components of (27) are at the same time those
of the electric force which is exerted by one polarized particle on another.
§ 8. Thus far we have used only the fundamental
equations without any new assumptions. I shall now suppose that the electrons, which I take to be spheres of radius R in the state of rest, have their dimensions changed by the effect of a translation, the dimensions in, the direction of 1notion becoming ~l thnes and those in perpendicular directions l times smaller.
In this deforrnation, which may be represented by
(tz• }, f), each element of volume is understood to preserve
its charge. Our assumption amounts to saying that in an electro-
static system ~' moving with a velocity v, all electrons are flattened ellipsoids with their smaller axes in the direction of

22 ELECTROMAGNETIC PHENOMENA

motion. If now, in order to apply the theorem of § 6, we

subject the system to the deformation ({3l, l, l), we shall
have again spherical electrons of radius R. Hence, if we
alter the relative position of the centres of the electrons in I

by applying the deformation (/3l, l, l), and if, in the points
thus obtained, we place the centres of electrons that remain
at rest, we shall get a syste1n, identical to the imaginary
system I', of which we have spoken in § 6. The forces in
this system and those in I w1ll bear to each other the rela..

tion expressed by (21). In the second place I shall suppose that the forces be-

tween uncharged particles, as well as those between such

particles and electrons, are influenced by a translation in

quite the same way as the electric forces in an electrostatic system. In other terms, whatever be the nature of the particles composing a ponderable body, so long as they do

not move relatively to each other, we shall have between the

forces acting in a system (~') wHhout, and the same system

(~) with a translation, the relation specified in (21), if, as re-
gards the relative position of the particles, I' is got from I by the deformation (/3l, l, l}, or I from t' by the deformation

(

1 f.3l'

1 l'

1) [ •

We see by this that, as soon as the resulting force is zero
for a particle in I', the same must be true for the corresponding particle in t. Consequently, if, neglecting the effects of
molecular motion, we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted by its neighbours, and if we take for

granted that there is but one configuration of equilibrium, we
may draw the conclusion that the system I', if the velocity v is imparted to it, will of itself change into the system I. In

other terms, the translation will produce the deformation

(

1 /3l'

1
l '

1)
l •

The case of molecular motion will be considered in § 12.

It will easily be seen that the hypothesis which was

formerly advanced in connexion with Michelson's experi-

ment, is implied in what has now been said. However, the present hypothesis is more general, because the only

H. A. LORENTZ

23

limitation imposed on the motion is that its velocity be less

than that of light.

§ 9. We are now in a position to calculate the electro-

magnetic momentum of a single electron. For simplicity's

sake I shall suppose the charge e to be uniformly distributed

over the surface, so long as the electron remains at rest.

Then a distribution of the same kind will exist in the system

I' with which we are concerned in the last integral of (22).

Hence

•

and

It must be observed that the product f3l is a function of v and that, for reasons of symmetry, the vector O has the direction of the translation. In general, representing by v the velocity of this motion, we have the vector equation

e2
0 = 671'c2R,8Zv •

. (28)

Now, every change in the motion of a system will entail a corresponding change in the electromagnetic momentum and will therefore require a certain force, which is given in direction and magnitude by

F = dO
dt

. (29)

Strictly speaking, the formula (28) may only be applied in the case of a uniform rectilinear translation. On account of this circumstance-though (29) is always true--the theory of rapidly varying motions of an electron becomes very com-
plicated, the more so, because the hypothesis of § 8 would
imply that the direction and amount of the deformation are continually changing. It is, indeed, hardly probable that the form of the electron wi 11 be determined solely by the velocity existing at the moment considered.
Neverthcless, provided the changes in the state of motion

24 ELEC1,ROMAGNETIC PHENOMENA
be sufficiently slow, we shall get a satisfactory approximation
by using (28) a.t every instant. The application of (29) to
such a quasi-stationary translation, as it has been called by Abraham,* is a very simple matter. Let, at a certain instant, a1 be the acceleration in the direction of the path, and a2 the acceleration perpendicular to it. Then the force F will consist of two components, having the directions of these accelerations and which are given by
F1 = m1a1 and F2 = m2a2,
if
. (30)
Hence, in phenomena in which there is an acceleration in the direction of motion, the electron behaves as if it had a mass m1 ; in those in which the acceleration is normal to the path, as if the mass were m2• These quantities m1
and m2 may therefore properly be called the " longitudinal "
and " transverse " electromagnetic masses of the electron. I shall suppose that there is no other, no " true " or " material " mass.
Since ~ and l differ from unity by quantities of the order v2/c2, we find for very small velocities
e'l
1n1 = 111,2 - 61rc2R.
This is the mass with which we are concerned, if there
are small vibratory motions of the electrons in a. system
without translation. If, on the contrary, motions of this kind are going on in a, body moving with the velocity v in the direction of the axis of x, we shall have to reckon with the mass mi, as given by (30), if we consider the vib• rations parallel to that axis, and with the mass m2, if we treat of those that are parallel to OY or OZ. Therefore, in short
terms, referring by the index I to a moving system and by
!' to one that remains at rest,
. (31)
* Abra.ham, Wied. Ann., 10, 1908, p. 105.

ll. A. LORENTZ

25

§ 10. We can now proceed to examine the influence of the

Earth's motion on optical phenomena in a system of trans-

parent bodies. In discussing this problem we shall fix our

attention on the variable electric moments in the particles or

" atoms " of the system. To these moments we may apply

what has been said in § 7. For the sake of simplicity we

shall suppose that, in each particle, the charge is concentrated

in a certain number of separate electrons, and that the

" elastic " forces that act on one of these, and, conjointly

with the electric forces, determine its motion, have their

origin within the bounds of the same atom.

I shall show that, if we start from any given state of

motion in a system without translation, we may deduce from

it a corresponding state that can exist in the same system

after a' translation has been imparted to it, the kind of corre-

spondence being as specified in what follows.

(a) Let A'i, A'2, A'3 , etc., be the centres of the particles in

the system without translation (I') ; neglecting molecular

motions we shall assume these points to remain at rest. The

system of points A1, A2, A3, etc., formed by the centres of the
particles in the moving system t, is obtained from A\, A'2,

(~l' ~, t). A'8, etc., by means of a. deformation

According to

what has been said in § 8, the centres will of themselves take these positions A\, A'2, A'3, etc., if originally, before there was a translation, they occupied the positions A1, A2, A3, etc.
We may conceive any point P' in the space of the
system "i' to be displaced by the above deformation, so that a definite point P of "i corresponds to it. For two corre-
sponding points P' and P we shall define corresponding
instants, the one belonging to P', the other to P, by stating
that the true time at the first instant is equal to the local time, as determined by (5) for the point P, at the second
instant. By corresponding times for two corresponding particles we shall understand times that may be said to correspond, if we fix our attention on the centres A' and A of
these particles. (b) As regards the interior state of the atoms, we shall as-
sume that the configuration of a particle A in I at a certain

26 ELEC'fROMAGNE'I'IC PHEN01\1ENA
(Jl' f,}) time may be derived by means of the deformation
from the configuration of the corresponding particle in ~', such as it is at the corresponding instant. In so far as this assumption relates to the form of the electrons them-
selves, it is implied in the first hypothesis of § 8.
Obviously, if we start from a state really existing in the
system I', we have now completely defined a state of the moving system I. The question remains, however, whether
this state will likewise be a possible one. In order to judge of this, we may remark in the first place
that the electric moments which we have supposed to exist in the moving system and which we shall denote by P, will be certain definite functions of the co-ordinates x, y, z of the centres A of the particles, or, as we shall say, of the coordinates of the particles themselves, and of the time t. The equations which express the relations between P on one hand and x, y, z, t on the other, may be replaced by other equations containing the vectors P' defined by (26) and the quantities x', y', z', t' defined by (4) and (5). Now, by the above assumptions a and b, if in a particle A of the moving system, whose co-ordinates are x, y, z, we find an electric moment P at the time t, or at the local time t', the vector P' given by (26) will be the moment which exists in the other system at the true time t' in a particle whose co-ordinates are x', y', z'. It appears in this way that the equations between P', x', y', z', t' are the same for both systems, the difference being only
this, that for the system I' without translation these symbols
indicate the moment, the co-ordinates, and the true time, whereas their meaning is different for the moving system, P', x', y', z', t' being here related to the moment P, the oo-ordin-
ates x, y, z and the general time t in the manner expressed
by (26), (4), and (5). It has already been stated that the equation (27) applies
to both systems. The vector D' will therefore be the same
in I' and I, provided we always compare corresponding
places and tin1es. However, this vector has not the sa1ne 1neaning in the two cases. In ~' it represents the electric
force, in I it is related to this force in the way expressed by
(20). We may therefore conclude that the ponderomotive

H. A. LORENTZ

27

forces acting, in I and in "£', on corresponding particles at
corresponding instants, bear to each other the relation determined by (21). In virtue of our assumption (b), taken in con-
nexion with the second hypothesis of § 8, the same relation
will exist between the "elastic" forces; consequently, the formula (21) may also be regarded as indicating the relation between the total forces, acting on corresponding electrons, at corresponding instants.
It is clear that the state we have supposed to exist in the
moving system will really be possible if, in I and :I,', the pro-
ducts of the mass m and the acceleration of an electron are to each other in the same relation as the forces, i.e. if

ma~) = ( l2, :, :)ma{t') .
Now, we have for the accelerations

. (32)

. (33)

as may be deduced from (4) and (5), and combining this with (32), we find for the masses

m(~) = (/33l, {3l, /3l)m(I').

If this is compared with (31), it appears that, whatever be the value of l, the condition is always satisfied, as regards the masses with which we have to reckon when we consider vibrations perpendicular to the translation. The only condition we have to impose on l is therefore

d(/3lv) _ f.)sz dv - /J •

But, on account of (3),

d(~v) = /.)3

dv

tJ '

so that we must put

d-d'Il) = 0, l = const.

The value of the constant must be unity, because we know already that, for ·v = 0, l = 1.

28 ELECTROMAGNETIC PHENOMENA
We are therefore led to suppose that the influence of a translation on the dimensions (of the separate electrons and of a ponderable body as a whole) is confined to those that have the direction of the motion, these becoming ~ times smaller than they are in the state of rest. If this hypothesis is added to those we have already made, we may be sure that two states, the one in the moving system, the other in the same system while at rest, corresponding as stated above, may both be possible. Moreover, this correspondence is not limited to the electric moments of the particles. In corre-' sponding points that are situated either in the ether between the particles, or in that surrounding the ponderable bodies, we shall find at corresponding times the same vector D' and, as is easily shown, the same vector H'. We may sum up by saying: If, in the system without translation, there is a state of motion in which, at a definite place, the components of P, D, and H are certain functions of the time, then the same system after it has been put in motion (and thereby deformed) can be the seat of a state of motion in which, at the corresponding place, the components of P', D', and H' are the same functions of the local time.
There is one point which requires further consideration. The values of the masses m1 and m2 having been deduced from the theory of quasi-stationary motion, the question arises, whether we are justified in reckoning with them in the case of the rapid vibrations of light. Now it is found on closer examination that the motion of an electron may be treated as quasi-stationary if it changes very little during the time a light-wave takes to travel over a distance equal to the diameter. This condition is fulfilled in optical phenomena, because the diameter of an electron is extremely small in comparison with the wave-length.
§ 11. It is easily seen that the proposed theory can
account for a large number of facts. Let us take in the first place the case of a system without
translation, in some parts of which we have continually
P = 0, D = 0, H = 0. Then, in the corresponding state for
the maying system, we shall have in corresponding parts (or, as we may say, in the sa1ne parts of the deformed system)
P' = 0, D' = 0, H' =- 0. These equations implying P = 0,

H. A. I"'ORENTZ

29

D = 0, H :::a 0, as is seen by (26) and (6), it appears that those
parts which are dark while the system is at rest, will remain so after it has been put in motion. It will therefore be impossible to detect an influence of the Earth's motion on any optical experi1nent, made with a terrestrial source of light, in which the geometrical distribution of light and darkness is observed. Many experiments on interference and diffraction belong to this class.
In the second place, if, in two points of a system, rays of light of the same state of polarization are propagated in the same direction, the ratio between the amplitudes in these points may be shown not to be altered by a translation. The latter remark applies to those experiments in which the intensities in adjacent parts of the field of view are compared.
The above conclusions confirm the results which I formerly obtained by a similar train of reasoning, in which, however, the terms of the second order were neglected. They also contain an explanation of Michelson's negative result, more general than the one previously given, and of a somewhat different form; and they show why Rayleigh and Brace could find no signs of double refraction produced by the motion of the Earth.
As to the experiments of Trouton and Noble, their negative result becomes at once clear, if we admit the hypo-
theses of § 8. It may be inferred from these and from our
last assumption (§ 10) that the only effect of the translation must have been a contraction of the whole system of electrons and other particles constituting the charged condenser and the beam and thread of the torsion-balance. Such a contraction does not give rise to a sensible change of direction.
It need hardly be said that the present theory is put forward with all due reserve. Though it seems to me that it can account for all well-established facts, it leads to some consequences that cannot as yet be put to the test of experiment. One of these is that the result of Michelson's experim~nt must remain negative, if the interfering rays of light are made to travel through some ponderable transparent body.
Our assumption about the contraction of the electrons

30 ELECTROMAGNETIC PHENOMENA
cannot in itself be pronounced to be either plausible or inadmissible. What we know about the nature of electrons is very little, and the only means of pushing our way farther will be to test such hypotheses as I have here made. Of course, there will be difficulties, e.g. as soon as we come to consider the rotation of electrons. Perhaps we shall have to suppose that in those phenomena in which, if there is no translation, spherical electrons rotate about a diameter, the points of the electrons in the moving syste1n will describe elliptic paths, corresponding, in the manner specified in § 10, to the circular paths described in the other case.
§ 12. There remain to be said a, few words about molecular
motion. We may conceive that bodies in which this bas a sensible influence or even predominates, undergo the same deformation as the systems of particles of constant relative position of which alone we have spoken till now. Indeed, in
two systems of molecules S' and~' the first without and the
second with a translation, we may imagine molecular motions corresponding to each other in such a way that, if a particle
in I' bas a certain position at a definite instant, a particle in
~ occupies at the corresponding instant the corresponding position. This being assumed, we may use the relation (33) between the accelerations in all those cases in which the
velocity of molecular motion is very small as compared with v.
In these cases the molecular forces may be taken to be determined by the relative positions, independently of the velocities of molecular motion. If, finally, we suppose these forces to be limited to such small distances that, for particles acting on each other, the difference of local times may be neglected, one of the particles, together with those which lie in its sphere of attraction or repulsion, will form a system which undergoes the often mentioned deformation. In virtue of
the second hypothesis of § 8 we may therefore apply to the
resulting molecular force acting on a particle, the equation (21). Consequently, the proper relation between the forces and the accelerations will exist in the two cases, if we sup• pose that the masses of all particles are influenced by a translation to the same degree as the electromagnetic masses of the elel]trons.
§ 13. The values (30), which I have found for the longi-

H. A. LOREN'rZ

31

tudinal and transverse masses of an electron, expressed in

terms of its velocity, are not the same as those that had

been previously obtained by Abraham. The ground for this

difference is to be sought solely in the circumstance that, in

bis theory, the electrons are treated as spheres of invariable

dimensions. Now, as regards the transverse mass, the re-

sults of Abraham have been confirmed in a most remarkable

way by Kaufmann's measurements of the deflexion of

radium-rays in electric and magnetic fields. Therefore, if

there is not to be a most serious objection to the theory I

have now proposed, it must be possible to sho,v that those

measurements agree with my values nearly as well as with

those of Abraham.

I shall begin by discussing two of the series of measure-

ments published by Kaufmann* in 1902. From each series

he has deduced two quantities ?'/ and ,, the " reduced "

electric and magnetic deflexions, which are related as follows

to the ratio ry = v/c : -
' 'Y = k1-, 7J

. (34)

Here y ('Y) is such a function, that the transverse mass is

given by

. (35)

whereas k1 and k2 are constant in each series. It appears from the second of the formulre (30) that my
theory leads likewise to an equation of the form (35); only Abraham's function ,; (ry) must be replaced by
3413 = 43(1 - ry2)-1/2.

Hence, my theory requires that, if we substitute this
+ value for (ry) in (34), these equations shall still hold. Of
course, in seeking to obtain a good agreement, we shall be justified in giving to k1 and k other values than those of
! Kaufmann, and in taking for every measurement a proper
value of the velocity v, or of the ratio 'Y· Writing sk,, k'2
* Kaufmann, Physik. Zeitschr., 4, 1902, p. 55,

32 ELECrfROMAGNE'fIC PHENOMENA

and ry' for the new values, we may put (34) in the form

ry' = sk1t

. (36)

71

and

. (37)

Kaufmann has tested his equations by choosing for k1 such a value that, calculating 'Y and k2 by means of (34), he obtained values for this latter number which, as well as might be, rernained constant in each series. This constancy was the proof of a sufficient agreement.
I have followed a similar method, using, however, some of the numbers calculated by Kauf1nann. I have computed for each measure1nent the value of the expression

k''J. m: (1 - ry'2)½t(,y)k~i, .

. (38)

that may be got from (37) combined with the second of the equations (34). The values of ,Jr (ry) and k2 have been taken from Kauf1nann's tables, and for ry' I have substituted the value he has found for ry, 1nultiplied bys, the latter coefficient being chosen with a view to obtaining a good constancy of (38). The results are contained in the tables on opposite page, corresponding to the Tables III and IV in Kaufmann' s paper.
The constancy of k'2 is seen to come out no less satisfactorily than that of k2, the more so as in each case the value of s has been determined by means of only two measurements. The coefficient has been so chosen that for these two observations, which were in Table III the first and the last but one, and in Table IV the first and the last, the values of k'2 should be proportional to those of k.2.
I shall next consider two series from a later publication
by Kaufrnann,* which have been calculated by Runge t by
rneans of the method of least squares, the coefficients k1
s, and k2 having been determined in such a way that the
values of 77, calculated, for each observed from I{aufmann's equations (3-!), agree as closely as may be with the observed
values of 77.
* Kaufmann, Gott. Nachr. 1\1:ath. phys, Kl., 1903, p. 90.
t Runge, ibid., p. 326.

H. A. LORENTZ

III. s = 0·933.

'Y•

t/1( 'Y ).

k2,

-y'.

0·851 0·766 0·727 0·6615 0·6075

2·147 1·86 1·78 1·66 1·595

I

1·721 1·736 1·725 1·727
1655

IV. s = 0·954.

'Y·

t/1(-y).

k.1.,

0·794 0·715 0·678 0·617 0·o67
-y'.

0·968 0•949 0·933 0·888 0·860 0·830 0·801 0·777 0·752
0·732

3·28 2·86
2·78 2·31 2·195 2·06 1·96 1·89 l ·83 1 ·785

8·12

0·919

7·99

0·Sm5

7·46

0·890

8·82

0·842

8·09

0·820

8·13

0·792

8•13

0·764

8·04

0·741

8·02

0·717

7·97

0·698

83
k'2·
2·246 2·258 2·256 2·256 2·175
.
",, , 2·
10·36 9·70 9·28 10·36
10·15 10·23 10·28 10·20 10·22 10·18

I have determined by the same condition, likewise using
the method of lea.st squares, the constants a and b in the
formula
r," = ar,3 + bt4'
which may be deduced from my equations (36) and (37).
_, Knowing a and b, I find ,y for each measurement by means
of the relation
'Y = ✓a 11·
For two plates on which Kaufmann had measured the electric and magnetic deflexions, the results are as follows (p. 34), the deflexions being given in centimetres.
I have not found time for calculating the other tables in Kaufmann's paper. As they begin, like the table for Plate 15 (next page) with a rather large negative difference between the values of r, which have been deduced from the observations and calculated by Runge, we may expect a satisfactory agreement with my formulm.

84 ELECTROMAGNETIC PHENOMENA
Plate No. 15. a = 0·06489, b = 0·3089.

(

Observed.

Calculated by R.

0·1495 0·.199 0·2475 0·296 0 3435 0·391 0·437 0·4 25 0·5265

0·0388 0·0548 0·0716 0·0896 0·1080 0·1290 0·1524 0·1788 0·2033

0·04:04 0·0550 0·0710 0·0887 0·1081 0·1297 0·1527 O·l 777 0·2089

1'/

Di:tf.

-- 16 2

+ 6

+---

9 1 7
3

-+ 11 6

Calculated by L.
----
0·0400 0·0552 0·0715 0·0895 0·1090 0·1805 0·1532 0·1777 0•2033

Diff.
-- 12 4
+ 1 + 1
- 10
-- 15 8
+ 11
0

'Y

Calculated by

R.

L.

0·987 0-9~4
0·930 0·889 0·847 0·804 0·763 0·~24 0·688

0·951 0·918
0·881 0·842 0·803 0·763 0·727 0·692 0·660

Plate No. 19. a = 0-05867, b = 0·2591.

,,

'Y

'

Observed.

Calculated by R.

Ditf.

Calculated by L.

Diff.

Calculated by

R.

L.

0·1495 0·199
0·247 0·296 0·3435 0·391 0·487 0·4825 0·5265

0·0404 0·0529 0·0678 0·0834 0·1019 0·1219 0·1429 0·1660 0·1916

0·0388 0·0527 0·0675 0·0842 0·1022 0·1222 0·1434 0·1665 0·1906

+ 16

+ 2

-+----

8 8 3 3 5

5

+ 10

0·0379 0·0522 0·0674 0-0544 0·1026 0·1226 0·1487 0•1664 0·1902

+ 25

+ 7

+ 4

- 10

---

7 7 8

-4

+ 14

0·990 0·969 0·939 0·902 0·862 0·822 0·782 0·744 0·709

0·954 0·923 0·888 0·849 0•811
0·773 0·786 0·702 0·671

ON THE ELECTRODYNAMICS op·
MOVING BODIES
BY
A. EINSTEIN
Translated from "Zur Elektrodynamik bewegter Karper," A.nnalen der Physik, 17, 1905.

ON THE ELECTRODYNAMICS OF MOVING BODIES
BY A. EINSTEIN
I T is known that Maxwell's electrodynamics-as usually understood at the present time-when applied to 1noving bodies, leads to asynuuetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative n1otion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the c'on·ductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where. parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise-assuming equality of relative- motion in the two cases discussed-to electric curr~nts of the same path and intensity as those produced by the electric forces in the former case.
Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of
37
~ , , , . . . ~..~ ~ . t ; " ' ~ ~ ~ - - - t l " l l f f l ~ " ,..... 1 • • - - - - -...........- - - - - - ~ - - - - - - -

38

ELECTRODYNAMICS

mechanics hold good.* We will raise this conjecture (the purport of which will hereafter be called the " Principle of Relativity '') to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynan1ics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a " luminiferous ether " will prove to be superfluous inasmuch as the view here to be developed will not require an " absolutely stationary space" provided with special properties, nor assign a velocity-vector to a point of the empty space in which electron1agnetic processes take place.
1,he theory to be developed is based-like all electrodynamics-on the kinen1atics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

I. .KINEMATICAL PART
§ 1. Definition of Simultaneity
Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.t In order to .render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the " stationary system.''
If a rnaterial point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.
If we wish to describe the motion of a material point, we
* The preceding memoir by Lorentz was not at this time known to the

A. EINSTEIN

89

give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear ~s to what we understand by "time." We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, " That train arrives here at 7 o'clock,u I mean something like this: '' The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events."*
It might appear possible to overcome all the difficulties attending the definition of "time " by substituting "the position of the small hand of my watch '' for " time." And in fact such a definition is satisfactory when we are concerned with defining a ti1ne exclusively for the place where the watch is located ; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or-what comes to the same thing-to evaluate the times of events occurring at places remote from the watch.
We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in

* We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately t,he same place, which can only be removed by an abstraction.

40

ELECTRODYNAMICS

respect of tilne, an event at A with an event at B. We have so far defined only an '' A time " and a " B time." We have not defined a common "ti1ne '' for A and B, for the latter cannot be defined at all unless we establish by definition that the "time'' required by light to travel from A to B equals the "time" it requires to travel from B to A. Let a ray of light start at the "A time" tA from A towards
B, let it at the " B time'' tB be reflected at B in the direction
of A, and arrive again at A at the " A time " tB' In accordance ,vith definition the two clocks synchronize
if iB - tA = t'A - tB,

"'\Ve assume that this definition of synchronism is free

from contradictions, and possible for any nurnber of points;

and that the following relations are universally valid :-

1. If the clock at B synchronizes with the clock at A, the

clock at A synchronizes with the clock at B.

2. If the clock at A synchronizes with the clock at B and

also with the clock at C, the clocks at B and C also syn-

chronize with each other.

Thus with the help of certain imaginary physical experi-

ments we have settled what is to be understood by synchron-

ous stationary clocks located at different places, and have

evidently obtained a definition of "simultaneous," or " syn-

chronous,'' and of "time." The " time '' of an event is

that which is given simultaneously with the event by a

stationary clock located at the place of the event, this clock

being synchronous, and indeed synchronous for all time deter-

minations, with a specified stationary clock.

In agreement with experience we further assume the

quantity

2AB

ti A -

t = c,
A

to be a universal constant-the velocity of light in empty space.
It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it "the time of the stationary system."

A. EINSTEIN

41

§ 2. On the Relativity of Lengths and Times
The foilowing reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define a,s follows:-
1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.
2. Any ray of light moves in the " stationary " system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence

l .t _ light path ve oci Y - titn e interval

where time interval is to be taken in the sense of the definition
in § 1.
Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod fying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations :-
(a) The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.
(b) By means of stationary clocks set up in the stationa.ry system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated " the length of the rod.''
In accordance with the principle of relativity the length

42

ELECTRODYNAMICS

to be discovered by the operation (a)-we will call it "the length of the rod in the moving system "-must be equal to the length l of the stationary rod.
The length to be discovered by the operation (b) we will call " the length of the (moving) rod in the stationary system." This we shall determine on the basis of our two principles, and we shall find that it differs from l.
Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equa.l, or in other words, that a 1noving rigid body at the epoch t may in geoznetrical respects be perfectly represented by the same body at rest in a definite position.
We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the "time of the stationary system" at the places where they happen to be. These clocks are therefore "synchronous in the stationary system."
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks
the criterion established in § 1 for the synchronization of two
clocks. Let a ray of light depart from A at the time* tA, let it be reflected at B at the time tB, and reach A again at the time t'A• Taking into consideration the principle of the constancy of the velocity of light we find that

tn -

tA =--= -rA-B- and t,A -
C- V

tB

=

TAB
C+ V

where rAB denotes the length of the moving rod-measured in the stationary system. Observers moving with th~ moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.
So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when en-

*"Time,, here denotes "time of the stationary system,, and also "position of hands of the moving clock situated at the place under discussion."

A. EINSTEIN

48

visaged from a system which is in motion relatively to that system.

§ 3. Theory of the Transformation of Co-ordinates and
Times from a Stationary System to another System
in Uniform Motion of Translation Relatively to the Former
Let us in '' stationary" space take two systems of coordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a, point. Let the axes of X of the two systems coincide, and their axes of Yand Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the'two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving systen1, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this " t " always denotes a time of the stationary system) parallel to the axes of the stationary system.
We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates
x, y, z, and f, "l, , respectively. Further, let the time t of
the stationary system be determined for all points thereof at which there are clocks by means of light signals in the
manner indicated in § 1 ; similarly let the time r of the
moving system be deter1nined for all points of the moving system at which there are clocks at rest relatively to that
system by applying the method, given in § 1, of light signals
between the points at which the latter clocks are located. To any system of values x, y, z, t, which completely defines
the place and time of an event in the stationary system, there

44

ELEC'l'RODYNAMICS

belongs a system of values E, .,,, t, T, determining that event
relatively to the system k, and our task is now to find the system of equations connecting these quantities.
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.
If we place x' = x - vt, it is clear that a point at rest in
the system k must have a system of values x', y, z, independent of time. We first define T as a function of x', y, z, • and t. To do this we have to express in equations that T is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the
rule given in § 1.
From the origin of system k let a ray be emitted at the time To along the X-axis to x', and at the time T1 be reflected thence to the origin of the co-ordinates, arriving there at the time T2; we then must have t (T0 + T2) = T1, or, by inserting the arguments of the function T and applying the principle of the cop.stancy of the velocity of light in the stationary system:-
_£_)] t[T(O, 0, 0, t) + T(o, 0, 0, t + c-31-__v_ + c+v = T(x', 0, 0, t + c_!-!v_).

Hence, if x' be chosen infinitesimally small,

t c~, ( 1

} )()T ~T

}

+ C + V • Z>t = <)x' + C-

or

c)r

V ~T

~t ~x' + c2 - v2 = O.

It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x', y, z.
An analogous consideration-applied to the axes of Y and Z-it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity J(c2 - v2), gives us

~T = Q <)7 = 0

"i)y_

' <>z

•

A. EINSTEIN

45

Since T is a linear function, it follows from these equations that
z') • T = a(t - V c'I. - v2

where a is a function <f,(v) at present unknown, and where for brevity it is assumed that at the origin of k, T = O, when t = 0.
With the help of this result we easily determine the quantities E, 'TJ, , by expressing in equations that light (as required by the principle of the constancy of the velocity of
light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving
system. For a ray of light emitted at the time -r = 0 in the
direction of the increasing E

E = CT or E = ac(t -

C 2

v
-

tx').
V

But the ray moves relatively to the initial point of k, when

measured in the stationary system, with the veiocity c - v,

so that

,
X = t.
C- V

If we insert this value of t in the equation for E, we obta.in

f: S"

=

a

2

C 2

,
2X.

C- V

In an analogous manner we find, by considering rays moving along the two other axes, that

when

"I = c-r =

ac(t -

a 2

-v

ll)
V

Thus

t '1J

=

a✓

(a2

C
-

v2)Y and

=

a✓(c2

C
-

v2)z.

Substituting for x' its value, we obtain

46

ELECTRODYNAMICS

where

T = cf,(v)/3(t - vx/c2),
E= cf,(v)/3(x - vt),
'Y/ = cf,(v)y,
t = cf,(v)z,

~ =

✓

<l

1 - v 2/c2) '

and cf, is an as yet unknown function of v. If no assumption

whatever be made as to the initial position of the moving

system and as to the zero point of -r, an additive constant is

to be placed on the right side of each of these equations.

We now have to prove that any ray of light, measured in

the moving system, is propagated with the velocity c, if, as

we have assumed, this is the case in the stationary system; for

we have not as yet furnished the proof that the principle of

the constancy of the velocity of light is compatible with the

principle of relativity.

At the time t = T == 0, when the origin of the co-ordinates

is common to the two systems, let a spherical wave be

emitted therefrom, and be propagated with the velocity c in

system K. If (x, y, z) be a point just attained by this wave,

then

x2 + y2 + z2 -= c2fJ.

Transforming this equation with the aid of our equations of transformation we obtain after a, simple calculation
r f + '1}2 + -= C2T2,

The wave under consideration is therefore no less a
spherical wave with velocity of propagation c when viewed
in the moving system. This shows that our two fundamental principles are compatible.*
In the equations of transformation which have been developed there enters an unknown function cf, of v, which we will now determine.
For this purpose we introduce a third system of co-ordin-

* The equations of the Lorentz transformation may be more simply de-

duced directly from the condition that in virtue of those equations the re-

lat -

ion -:.t2

+ .

y'i.

+

a2

==

c2t j

shall

have

as

its

consequence

the

second

relation

A. EINSTEIN

47

ates K', which relatively to the system k is in a state of
parallel translatory motion parallel to the a.xis of X, such that
the origin of co-ordinates of system k moves with velocity
- v on the axis of X. At the time t = 0 let all three origins coincide, and when t = x = y = z = 0 let the time t' of the
system K' be zero. We call the co-ordinates, measured in the system K; x', y', z', and by a twofold application of our
equations of transformation we obtain

t' = ¢( - v)/3( - -v)(,- + vf/c2) = <p(v)<p( - v)t,

x' = ¢( - v)/3( - v)(~ + v,-) = cp(v)cf,( - v)x,

y' = <p( - v}r1

= cp(v )<p( - v)y,

z'

=

cp(

-

v)t

=

cp(v)cp(

-

v)z.

Since the relations between x', y', z' and x, y, z do not contain the time t, the systems K and K' are at rest with respect to one another, and it is clear that the transformation from K to K' must be the identical transformation. Thus

cf,(v)cp( - v) = 1.

We now inquire into the signification of ¢(v). We give our
attention to that part of the axis of Y of system k which lies
between E= 0, '1J = 0, , == 0 and E= 0, 11 = l, t = 0. This
part of the axis of Y is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K
the co-ordinates l
x1 - vt, y1 - <J>(v)' z1 - 0

and

x2 =- vt, y2 - 0, z2 - 0.

The length of the rod measured in K is therefore l/c/J(v); and
this gives us the meaning of the function cf,(v). From
reasons of symmetry it is now evident that the length of a
given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and - v are interchanged.
Hence follows that l/<p(v) = l/cp( - v), or

<f,(v) - ¢( - v).

It follows from this relation and the one previously found

48

ELECTRODYNAMICS

that 4'(.v) == 1, so that the transformation equations which have been found become

where

T == ~(t - vx/c2),
E= ~(x - vt),
'1/ = y, ' = z,

§ 4. Physical Meaning of the Equations Obtained in Re-
spect to Moving Rigid Bodies and Moving Clocks
We envisage a rigid sphere * of radius R, at rest relatively to the moving system k, and with its centre at the origin of co-ordinates of k. The equation of the surface of this sphere moving relatively to the system K with velocity v is

ti + "Is + t2 == R2.

The equation of this surface expressed in x, y, z at the time

t == 0 is

(✓ (1

x.2
- v2/c2))2

+

2
y

+

2
z

=

R2 •

A rigid body which, measured in a state of rest, has the form

of a sphere, therefore has in a state of motion-viewed from

the stationary system-the form of an ellipsoid of revolution

with the axes

R J (1 - v2/c2), R, R.

Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio 1 : ✓<I - v2/c2), i.e. the greater the value of v, the greater the shortening. For v = call moving objects-viewed from the " stationary " system-shrivel up into plain figures. For velocities greater than that of light our deliberations become meaningless ; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physica.lly, of an infinitely great velocity.
*The.tis, a. body possessing spherical form when examined at rest.

A. EINSTEIN

49

It is clear that the same results 4old good of bodies at rest in the " stationary " system, viewed from a system in uniform motion.
Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time T when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time T, What is the rate of this clock, when viewed from the stationary system?
Between the quantities x, t, and T, which refer to the
position of the clock, we have, evidently, x = vt and

'11hereforc,
T = t✓(l - v2/c2) = t - (l - ✓(1 - v2/ rl'))t

whence it follows that the time m~rked by the clock (viewed

in the stationary system) is slow by 1 - ✓(l - v2/c2) seconds

per. second, or-neglecting magnitudes of fourth and higher

order--by tv2/c2•

From this there ensues the following peculiar consequence.

If at the points A and B of K there are stationary clocks

which, viewed in the stationary system, are synchronous ; and

if the clock at A is mov_ed with the velocity v along the line

AB to B, then on its arrival at B the two clocks no longer

synchronize, but the clock moved from A to B lags behind the other which has remained at B by ½tv2/c'l (up to magni-

tudes of fourth and higher order), t being the time occupied

in the journey from A to B.

It is at once apparent that this result still holds goqd if

the clock moves from A to B in any polygonal line, and also

when the points A and B coincide.

.

I£ we assume that the result proved for a polygonal line

is also valid for a continuously curved line1 we arrive at this

result : If one of two synchronous clocks at A is moved in a

closed curve with constant velocity until it returns to A, the

journey lasting t seconds, then by the clock which has

remained at rest the travelled clock on its arrival ~t A

will be ½tv2/c2 second slow. Thence we conclude that a

50

ELECTRODYNAMICS

balance-clock* at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.

§ 5. The Composition of Velocities

In the system k moving along the axis of X of the system

K with velocity v, let a point move in accordance with the

equations

E= wrr, ,,, = w.,,'T, , = o,

where w~ and w.,., denote constants. Required ; the motion of the point relatively to the system
K. If with the help of the equations of transformation de-
veloped in § 3 we introduce the quantities x, y, z, t into the
equations of motion of the point, we obtain

x-~t+v t
- 1 + vwic2 '
✓ <1 - v2/c2) Y = 1 + vw~/c2 w.,,t,
z = 0.
Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set

V2

=

(

dx)
dt

2

+

(dy) 2
dt '

w2 == wl + w.,,2'
a = tan - 1 wy/w:n,

a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain

V = ✓[(v2 + w2 + 2vw cos a) - (vw sin a/c2) 2]

1 + vw cos a/c'J.

•

It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get

V= v+w, 1 + vw/c2
* Not a pendulum-clock, which is physically a system to which the Earth
belongs. This case had to be excluded.

A. EINSTEIN

51

It follows from this equation that from a. composition of two
velocities which are less than c, there always results a velocity
less than c. For if we set v = c - ,e, w = c - X, ,e and X being positive and less than c, then

V

=. o2C

-

2c
"

-
-

" X +

X /
/CA C

<c.

It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain

V

=

C
1

+ W/
+WC

=

c.

We might also have obtained the formula for V, for the case

when v and w have the same direction, by compounding

two transformations in accordance with § 3. If in addition

to the systems K and k figuring in § 8 we introduce still

another system of co-ordinates k' moving parallel to k, its

initial point moving on the axis of X with the velocity w, we

obtain equations between the quantities x, y, z, t and the

corresponding quantities of k', which differ from the equations

found in § 3 only in that the place of '' v '' is taken by the

quantity

-1

V+-+-vw'W-/c

·
2'

from which we see that such parallel transformations-necessarily--form a group.
We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.

II. ELECTRODYNAMICAL PART
§ 6. Transformation of the Maxwell-Hertz Equations for
Empty Space. On the Nature of the Electromotive Forces Occurring In a Magnetic Field During Motion
Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have

52

ELECTRODYNAMICS

o c l~X ~N "c)M 1 c)L c)Y <)Z

clt = ay" - c)z '

~t = <)z - <)y'

1 clY clL 'bN 1 ~M clZ ~X

c u c 1 c)Z ~M clL 1 c)N "c)X "c)Y

== c)x - ~y '

M == "'t>y - c)x'

where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force.
If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the system of co-ordinates there introduced, moving with the velocity v, we obtain the equations

.! ()X = ~ {!3(N _ !'y)} - ~{~(M + !!z)},

C ~T c)17

C

c)'

C

- :,{J3(N - ;)Y}·

where
Now the principle of relativity requires that if the Maxwell-Hertz equations for empty space hold good in system K, they also hold good in system k ; that is to say that the vectors of the electric and the magnetic force-(X', Y', Z') and (L', M', N')-of the moving system k, which are defined by their ponderomotive effects on electric or magnetic masses respectively, satisfy the following equations:-

A. EINSTEIN

58

1 'c)X' aN' aM'

at c-

-= aT

=

-
c),,

--,

1 t',Y' aL' 'c)N'
C c)r = tc) - I af '

l t',Z' <')M' aL'
c <)r = ~E - ()71 '

1 aL' aY' az'
oTT = c)t - a,,'
1 aM' az' ax' C c)-r -= ~ - c,t'
-1c -'ca)'-TN' - =aax,,-' - -<a)YE.'

Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the Ma.xwell-Hertz equations for system K. Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a, factor ,Jr(v), which is common for all functions of the
one system of equations, and is independent of E, '1J, , and T
but depends upon v. Thus we have the relations

X' = ,fr(v)X,

L' = ,fr(v)L,

Y' = ,fr(v),e(Y - ~N), M' = t(v),e(M + ~Z),

Z' = ,fr(v),e(Z + !M), N' = ,fr(v),e(N - ~y).
If we now form the reciprocal of this system of equations, firstly by solving the equations just obtained, and secondly by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity - v, it follows, when we. consider that the two systems of equations thus ob-
tained must be identical, that ,y(v),y( - v) = 1. Further, from reasons of symmetry* ,y(v) = +( - v), and therefore
,fr(v) = 1,
and our equations assume the form
* If, for example, X = Y = Z = L = M = 0, and N =I= O, then from
reasons of symmetry lt is clear that when v changes sign without changing its numerical value, Y' must also change sign without changing its numerical value.

54

ELECTRODYNAMICS

X' = X,

L' = L,

Y' = ;3(Y - ~N), M' = J3(M + ~z),

Z' = ;3(z + :M), N' = ;3(N - ~y).
As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude "one" when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude " one " when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (X, Y, Z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (X', Y', Z'). Consequently the first three equations above allow themselves to be clothed in words in the two following ways :-
1. If a unit electric point charge is in motion in an electromagnetic field, there acts upon it, in addition to the electric force, an " electromotive force it which, if we neglect the terms multiplied by the second and higher powers of v/c, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.)
2. If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the e]ectric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)
The analogy holds with '' magnetomotive forces." We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces

A. EINSTEIN

55

do not exist independently of the state of motion of the system of co-ordinates.
Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the "seat " of electrodynamic electromotive forces (unipolar machines) now have no point.

§ 7. Theory of Doppler's Principle and of Aberration

In the system K, very far from the origin of ea-ordinates, let there be a source of electrodyna1nic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations

where

X = X0 sin cl>, L = L0 sin cl>,
Y = Y0 sin <I>, M = M0 sin <I>,
Z == Z0 sin <I>, N = NO sin cl>,

<I> = M{t - ~(lx + my + nz)}

Here (X0, Y0, Z0) and (L0, M0, N0) are the vectors defining the amplitude of the wave-train, and l, m, n the directioncosines of the wave-normals. We wish to know the consti-
tution of these waves, when they are examined by an
observer at rest in the moving system k.
Applying the equations of transformation found in § 6 for electric and magnetic forces, and those found in § 3 for the
co-ordinates and the time, we obtain directly

X' = X0 sin cl>',

L' = L 0 sin <I>',

Y' = /3(Y0 - vN0/c) sin ti>', M' = J3(M0 + vZ0/c) sin ti>', Z' = J3(Z0 + vM0/c) sin <I>', N' = /3(N0 - vY0/c) sin <I>',
n'~} <I>' = M'{ T - ~(l'~ + m•.,, +

56

ELECTRODYNAMICS

where

<iJ1 = <iJ,8(1 - lv/c),

l' = l - v/c 1 - lv/c'

m , = /3(1--m-l-v/c-)'

I

n

n xz ,8(1 - lv/c) •

From the equation for w' it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency v, in such a way that the connecting line " source-observer " makes the angle ¢ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency 11' of the light perceived by the observer is given by the equation
, 1 - cos ¢ . v/c v = v ✓ (l - v2/c2) •

This is Doppler's principle for any velocities whatever.

When <p = 0 the equation assumes the perspicuous form

,_ V

-

✓1
V l

+

v/Ic.
V C

We see that, in contrast with the customary view, when
V = - = C, 11' 00.
If we call the angle between the wave-normal (direction of the ray) in the moving system and the connecting line
" source-observer " q,', the equation for l' assumes the form
¢' cos cf, - v/c
cos = 1 - cos cf, . v/c.

This equation expresses the law of aberration in its most
general form. If q, = t 77", the equation becomes simply
cos ¢' = - v/c.

We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A' respectively, accordingly

A. EINSTEIN

57

as it is measured in the stationary system or in the moving system, we obtain
A''l = A2(1 - cos 4>. v/c) 2
1 - v2/o2
which equation, if cf, = 0, simplifies into

A'2 = A2l - v/c. 1 + v/c
It follows from these results that to an observer approaching a source of light with the velocity o, this source of light must appear of infinite intensity.

§ 8. Transformation of the Energy of Light Rays. Theory
of the Pressure of Radiation Exerted on Perfect Reflectors
Since A2/8w equals the energy of light per unit of volume, we have to regard A'2/81r, by the principle of relativity, as the energy of light in the moving system. Thus A'2/A2 would be the ratio of the " measured in motion '' to the " m,ea.s.u. red at rest" energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If l, m, n are the direction-cosines of the wave-normals of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:-
(x - lct)2 + (y - mct) 2 + (z - nct)2 = R2•
We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k.
The spherical surface-viewed in the moving system-is an ellipsoidal surface, the equation for which, at the time T = 0, is
(/3E - lf3Ev/c)2 + (11 - m/3Ev/c)2 + (' - nf3Ev/c)2 = R2•
If S is the volume of the sphere, and S' that of this ellipsoid,

58

ELECTRODYNAMICS

then by a simple calculation

s-S' = 1--✓--1-c-o-s--cvp-2/.-cv:2/-c·

Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and E' when measured in the moving system, we obtain

E' A'28' _ 1 - cos ¢ . v(c
E = A2S - ✓(1 - v2/c2) '

and this formula, when cf, = 0, simplifies into

E' = ✓l - v/c

E

1 + v/c·

It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.
Now let the co-ordinate plane ~ = 0 be a perfectly reflecting surface, at which the plane waves considered in § 7 are
reflected. We seek for the pressure of light exerted on the
retlecting surface, and for the direction, frequency, and in-
tensity of the light after reflexion. Let the incidental light be defined by the quantities A,
cos ¢, v (referred to system K). Viewed from k the corresponding quantities are
A' = A1 - cos ¢. v/c ✓ (1 - v2/c2) '

cos¢,

=

cos cf, - v/c
1 - cos cp. v/c'

, 1 - cos cp . v/c v = v ✓ (1 - I v'J. c2) •

For the reflected light, referring the process to system k, we
obtain A'' = A'
cos cp" = - cos ¢'
VII = VI

Finally, by transforming back to the stationary system K, we obtain for the reflected light

A. EINS'fEIN

59

A"' =z A''l + cos <b" • v/c = A1 - 2 cos cf, • v/c + v2/c2

✓(l - v2/c2)

1 - v2/ c2

'

cos cf>'" = cos ¢" + v/o

(1 + v2/c2) cos cf> - 2v/c

1 + cos cp" . v/c = - l - 2 cos r.f, . v/c + v2/ c2

,,, ,,1 + cos <f,"v/c 1 - 2 cos cf>. v/c + v2/ c2

v = v ✓(l - v2/c2) = v

I - v2/c2

•

The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently A2(c cos¢ - v)/8-rr. The energy leaving the unit of surface
of the mirror in the unit of time is A"'2( - c cos cp'" + v)/81T'.
The difference of these two expressions is, by the principle of energy, the work done by the pressure of light in the unit of time. If we set down this work as equal to the product Pv, where P is the pressure of light. we obtain

A2 (cos cf> - v/c)2

= p

2 . -87T' 1 - V 2/C2 •

In agreement with experiment and with other theories, we obtain to a first approximation

A2 P = 2 . 8,rcos2 cf>.

Al I problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.

§ 9. Transformation of the Maxwell-Hertz Equations when
Convection-Currents are Taken into Account

We start fron1 the equations

!{()x + Uxp} = ()N _ ,fl\f,

C ot

~y oz

-1
C

o-~Lt

= c-) Y ()Z

-ooZy,

c !{oY + uyp} = c)L _ c,N

C ot

()z

()x'

1 <lM c,Z ()X Tt = ox - ~z'

! {"~ + Uzp} z= ()M - c)L

C ot

<°)X <)y'

1 -
C

o-o--Nt--

=

(-)x
?Jy

..

o- Y
~X'

60 where

ELECTRODYNAMICS

denotes 4?T times the density of electricity, and (uz, Uy, uz) the velocity-vector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodie~ (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.
Let these equations be valid in the system K, and transform them, with the assistance of the equations of transform-
ation given in §§ 3 and 6, to the system k. We then obtain
the equations

c c l{oX'

'} ~N' <)M' 1 <)L' oY' <)Z'

oT + utp = ~11 - ot ' ~7' = <)t - <),,, '

Jo1 Y'

'} c)L' ~N' 1 c)M' oZ' ~X'

c al oT + u.,,p = ()t - of ' 'c)T = ()E - "' '

!{~ + uHJ'} - c,l\il' _ oL' ! oN' = ax' c,Y'

c aT

v

'c)E

c).,, ' c c)7"

c)'rJ - oE '

where

Ua, - V
ui == I - ua:v/c'J.

U( == /3(1 - Ua:v/c2)' and
, oX' oY' 'c)Z' p=~+~+~
= ,8(1 - UxV/ c2) p.
Since-as follows from the theorem of addition of velocities (§ 5)-the vector (u~, u.,,, uc) is nothing else than the velocity of the electric charge, measured in the system k, we have the proof that, on the basis of our kinematical principles, the • electrodynamic foundation of Lorentz's theory of the electrodynamics of moving bodies is in agreement with the principle of relativity.
In addition I may briefly remark that the following import-

A. EINSTEIN

61

ant law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of co-ordinates moving with the body, i~s charge also remains-when regarded from the '' stationary " system Kconstant.

§ 1o. Dynamics of the Slowly Accelerated Electron
Let there be in motion in an electromagnetic field an electrically charged particle (in the sequel called an " electron"), for the law of motion of which we assume as follows:-
If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations
d2x
mdt'J. = €X
d2y
rndt"'· = eY
d 2z mdt2 = eZ

where x, y, z denote the co-ordinates of the electron, and m the mass of the electron, as long as its motion is slow.
Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time.
Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the co-ordinates, and moves with the velocity v along the axis of X of the system K. It is then clear that at the given
moment (t = 0) the electron is at rest relatively to a system
of co-ordinates which is in parallel motion with velocity v along the axis of X.
From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time (for small values of t) the electron, viewed from the system k, moves in accordance with the equations

62

ELEC1,.RODYNAMICS

E , d2
nidT2

=

eX'

d2'1}

,

mdT2 = eY'

in which the symbols E, 11, ,, -r, X', Y', Z' refer to the system
k. If, further, we decide that when t = x = y = z = 0 then
T =-= E= 11 = t = 0, the transformation equations of §§ 3 and
6 hold good, so that we have

E= /B(x - vt), '1J = y, t = z, T = /B(t - vx/c2)
X' = X, Y' = fJ(Y - vN/c), Z' = /B(Z + vM/c).

With the help of these equations we transform the above equations of motion from system k to system K, and obtain

d 2x

e

dt2 == m /B3X

d2y =- _E (Y - !'N)

dt2 'nl, {:3

C

dd- 2tz2 =m- €/3 ( Z +-cvM)

. (A)

Taking the ordinary point of view we now inquire as to the " longitudinal ,, and the '' transverse " mass of the moving electron. We write the equations (A) in the form

d2 dt2 ,QS X
mt-'

=

e X

=

X'
E '

mf32~~f = e/3(Y - ~N) = EY',

and remark firstly that eX', eY', eZ' are the components of the ponderomotive force acting upon the electron, and a.re so indeed as viewed in a system moving at the moment with the electron, with the same velocity as the electron. (This force might be measured, for example, by a spring balance at rest

A. EINSTEIN

68

in the last-mentioned system.) Now if we call this force simply "the force acting upon the electron,''• and maintain the equation-mass x acceleration == force-and if we also decide that the accelerations are to be measured in the stationary system K, we derive from the above equations

Longitudinal mass

=

✓-
( 1

m
-V

2 /

2
C )

••
3

Transverse

mass

=

1

m ~ / ::! •
-V C

With a different definition of force and acceleration w., e should naturally obtain other values for the rnasses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously.
We remark that these results as to the mass are also valid for ponderable material points, because a, ponderable material point can be made into an electron (in our sense of the word) by the addition of an electric charge, no matter how small.
We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of co-ordinates of the system K along the axis of X under the action of ~n electrostatic force X, it is clear that the energy withdrawn
f from the electrostatic field has the value EXdx. As the elec-
tron is to be slowly accelerated, and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron. Bearing in mind that during the whole process of motion which we are consiclering, the first of the equations (A) applies, we therefore obtain

W = ft:Xdx = mj;f33vdv

= mc2{.;r~-i,•}c• - 1}·

Thus, when v :a c, W becomes infinite. Velocities

* The definition of force here given is not a.dvantageous, as was first shown by M. Planck. It is more to the point to define force in such a. wa.y that the laws of momentum and energy assume the simplest form.

64

ELECTRODYNAMICS

greater than that of light have-as in our previous resultsno possibility of existence.
This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well.
vVe will now enumerate the properties of the motion of the electron which result from the system of equations (A), and are accessible to experiment.
1. Fron1 the second equation of the system (A) it follows that an electric force Y a.nd a magnetic force N have an equally strong deflective action on an electron moving with
the velocity v, when Y = Nv/c. Thus we see that it is pos-
sible by our theory to determine the velocity of the electron from the ratio of the magnetic power of deflexion Am to the electric power of deflexion Ae, for any velocity, by applying the law
o Am, V
Ae =

This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e.g. by means of rapidly oscillating electric and magnetic fields.
2. From the deduction for the kinetic energy of the electron it follows that between the potential difference, P, traversed and the acquired velocity v of the electron there must be the relationship
I} p = Jxdx = 7c2{JC! vi/if -

3. We calculate the radius of curvature of the path of the electron when a magnetic force N is present (as the only deflective force), acting perpendicularly to the velocity of the electron. From the second of the equations (A) we obtain

_

d2y
dt2

=

~=
R

=

!_~N
m c

"Jf

1

_

v2 c'l.

or

R

=

mc2
7·

✓ (1

v/c - v 2 /c2)

1 ·N·

These three relationships are a complete expression for

ELECTRODYNAMICS

65

the laws according to whiqh, by the theory here advanced, the electron must move.
In conclusion I wish to say that in working at the problem here dealt with I have had the loyal assistance of my friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.

"
DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT?
BY
A. EINSTEIN
Translated from '' Is t die Trag heit eines Korpers von seinem Energiegehalt abhangig?" Annalen der Physik, 17, 1905.

DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT?

T

BY A. EINSTEIN
HE results of the previous investigation lead to a very interesting conclusion, which is here to be

deduced. I based that investigation on the Maxwell-Hertz equations for empty space, together with the Maxwellian

expression for the electromagnetic energy of space, and in addition the principle that :-

The laws by which the states of physical systems alter alf'e

independent of the alternative, to which of two systems of co-

ordinates, in uniform motion of parallel translation relatively

to each other, these alterations of state are refe1·'fed (principle

of relativity).
With these principles* as my basis I deduced inter alia the following result (§ 8) :-
Let a system of plane waves of light, referred to the system of co-ordinates (x, y, z), possess the energy l; let the direction of the ray (the wave-normal) make an angle <f, with the axis of x of the system. If we introduce a new system of co-ordinates (E, 71, ,) moving in uniform para.llel translation with respect to the system (x, y, z), and having its origin of

co-ordinates in motion along the axis of z with the velocity v, then this quantity of light-measured in the system (E, 11, t') -possesses the energy

1 - V- cos cJ,

l * = l•--::=c=:;:::;:;;:;:
✓l - v2/c2

* The principle of the constancy of the velocity of light is ot course
contained iJJ. Maxwell's equations.
69

70

INERTIA AND ENERGY

where c denotes the velocity of light. We shall make use of
this result in what follows. Let there be a stationary body in the system (x, y, z),
and let its energy-referred to the system (x, y, z)-be E 0• Let the energy of the body relative to the system (E, 11, t), moving as above with the velocity v, be H0•
Let this body send out, in a direction making an angle cp with the axis of x, plane waves of light, of energy ½L
measured relatively to (x, y, z), and simultaneously an equal
quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z). The
principle of energy must apply to this process, and in fact (by the principle of relativity) with respect to both systems of co-ordinates. If we call the energy of the body after the
emission of light E1 or H1 respectively, measured relatively to
the system (x, y, z) or (~, 71, t) respectively, then by employ-
ing the relation given above we obtain

E 0 = E1 + tL + ½L,

1 - -V cos ¢

1 + -V cos ¢

Ho = H1 + tL c

+ ½L--;::=c===-

✓1 - v2/c 2

✓ 1 - v2/c 2

= H1

+

✓ 1

L - 'l>2/C2·

By subtraction we obtain from these equations

= 1}. Ho - Eu - (H1 - E1) = Lt/1 v2/c2 -

The two differences of the form H - E occurring in this expression have simple physical significations. H and E are energy values of the same body referred to two systems of co•ordinates which are in motion relatively to each other, the body being at rest in one of the two systems (system (x, y, z)). Thus it is clear that the difference H - E can differ from the kinetic energy K of the body, with respect to the other system (~, 17, t), only by an additive constant C, which depends on the choice of the arbitrary additive constants of the energies H and E. Thus we may place

A. EINSTEIN

71

H 0 - Eo = Ko + C, H 1 - E 1 == K1 + C,
since C does not change during the emission of light. have
Ko - K1 = L{✓l ~ v2/c2 - 1},

So we

The kinetic energy of the body with respect to (E, 'TJ, ~) diminishes as a result of the emission of light, and the amount of diminution is independent of the properties of the body. Moreover, the difference K 0 - Ki, like the kinetic energy of the electron (§ 10), depends on the velocity.
Neglecting magnitudes of fourth and higher orders we may place

Fro1n this equation it directly follows that:-

// a body gives off the energy L in the form of radiation,

its mass diminishes by L/c2• The fact that the energy with-

drawn from the body becomes energy of radiation evidently

makes no difference, so that we are led to the more general

conclusion that

The mass of a body is a measure of its energy-content ; if

the energy changes by L, the mass changes in the same sense

by L/ mass

91. n

x 10~0, the grammes.

energy

being

measured

in

ergs,

and

the

It is not impossible that with bodies whose energy-con-

tent is variable to a high degree (e.g. with radium salts) the

theory may be successfully put to the test.

If the theory corresponds to the facts, radiation conveys

inertia between the emitting and n,bsorbing bodies.

SPACE AND TIME
BY
H. MINKOWSKI
A T1·anslation of an Address delivered at the 80th Assembly of German Natura.l Scientists and Physicians, at Cologne, 21 September, 1908.

SPACE AND TIME

T

BY H. MINKOWSKI
HE views of space and time which I wish to lay before you have sprung from the soil of experimental

physics, and therein lies their strength. They are

radical. Henceforth space by itself, and time by itself, are

doomed to fade away into 1nere shadows, and only a kind

of union of the two will preserve an independent reality.

I

First of all I should like to show how it might be possible, setting out from the accepted mechanics of the present day, along a purely mathematical line of thought, to arrive at changed ideas of space and time. The equations of Ne,vton's mechanics exhibit a two-fold invariance. Their form re1nains unaltered, firstly, if we subject the underlying system of spatial co-ordinates to any arbitrary change of position; secondly, if we change its state of 1notion, na1nely, by imparting to it any uniform translatory motion,· furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of 1nechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un. troubled 1ninds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a
75

....,_A:+4 ,# Q

76

SPACE AND TIME

state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think.
We will try to visualize the state of things by the graphic method. Let x, y, z be rectangular co-ordinates for space, and let t denote time. The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. But I still respect the dogma that both space and time have independent significance. A point of space at a
point of time, that is, a system of values x, y, z, t, I will call
a world-point. The multiplicity of all thinkable x, y, z, t systems of values we will christen the world. With this most valiant piece of chalk I might project upon the blackboard four world-axes. Since merely one chalky axis, as it is, consists of molecules all a-thrill, and moreover is taking part in the earth's travels in the universe, it already affords us ample scope for abstraction ; the somewhat greater abstraction associated with the number four is for the mathematician no infliction. Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying " matter '' or " electricity " I will use for this something the word " substance." We fix our attention on the substantial point which is at the worldpoint x, y, z, t, and imagine that we are able to recognize this substantial point at any other time. Let the variations dx, dy, dz of the space co-ordinates of this substantial point correspond to a time element dt. Then we obtain, as an image, so to speak, of the everlasting career of the substantial point, a curve in the world, a world-line, the points
of which can be referred unequivocally to the parameter t from - oo to + oo. The whole universe is seen to resolve
itself into similar world-lines, and I would fain anticipate myself by saying that in my opinion physical laws might find their most perfect expression as reciprocal relations between these world-lines.
The concepts, space and time, cause the x, y, z-manifold
t = 0 and its two sides t > 0 and t < 0 to fall asunder. If,

H. MINKOWSKI

77

for simplicity, we retain the same zero point of space and
time, the first-mentioned group signifies in mechanics that
we may subject the axes of x, y, z at t = 0 to any rotation we
choose about the origin, corresponding to the homogeneous
linear transformations of the expression

x2 + y2 + z2.

But the second group means that we may-also without changing the expression of the laws of mechanics-replace x, y, z, t by a; - at, y - {3t, z - ryt, t with any constant values of a, /3, ry. Hence we may give to the time axis whatever direction we choose towards the upper half of the world,
t > 0. Now what has the requirement of orthogonality in
space to do with this perfect freedom of the time axis in an upward direction ?
To establish the connexion, let us take a, positive parameter c, and consider the graphical representation of

c2t2 - x2 - y2 - z2 = 1.

It consists of two surfaces separated by t = 0, on the analogy
of a hyperboloid of two sheets. Vv.,.e consider the sheet in
the region t > 0, and now take those homogeneous linear
transformations of x, y, z, t into four new variables a;', y', z', t',
for which the expression for this sheet in the new variables is of the same form. It is evident that the rotations of space about the origin pertain to these transformations. Thus we gain full comprehension of the rest of the transformations simply by taking into consideration one among them, such that y and z remain unchanged. We draw (Fig. 1) the section of this sheet by the plane of the axes of x
and t-the upper branch of the hyperbola c2t2 - x2 = 1, with
its asymptotes. From the origin O we draw any radius vector QA' of this branch of the hyperbola; draw the tangent to the hyperbola at A' to cut the asymptote on the right at B'; complete the parallelogram OA'B'C'; and finally, for subsequent use, produce B'O' to cut the axis of x at D'. Now if we take OC' and OA' as axes of oblique co-ordinates x', t', with the measures OC' = 1, OA' = 1/c, then that branch of
the hyperbola again acquires the expression c2t'2 - x'2 = 1,
t'> 0, and the transition from x, y, z, t to x', y', z', t' is one of

78

SPACE AND TIME

the transformations in question. With these transformations we now associate the arbitrary displacements of the zero point of space and time, and thereby constitute a group of transformations, which is also, evidently, dependent on the parameter c. This group I denote by Ge.
If we now allow c to increase to infinity, and 1/c therefore to converge towards zero, we see from the figure that the

t

0

D1 C

X

,,, p.',

p

p

FIG. 1.

branch of the hyperbola bends more and more towards the
axis of a:, the angle of the asymptotes becomes more and more
obtuse, and that in the limit this special transformation changes into one in which the axis of t' may have any upward direction whatever, while x' approaches more and more exactly to x. In view of this it is clear that group Go in the limit when c = oo , that is the group G00 , becomes no other than that complete group which is appropriate to Newtonian

H. MINKOWSKI

79

mechanics. This being so, and since Ge is mathematically more intelligible than GOC), it looks as though the thought might have struck some mathe1natician, fancy-free, that after all, as a matter of fact, natural phenomena do not possess an invariance with the group Gro, but rather with a group Ge, c being finite and detarminate, but in ordinary units of measure, extremely great. Such a premonition would have been an extraordinary triumph for pure mathematics. Well, mathematics, though it now can display only staircase-wit, has the satisfaction of being wise after the event, and is able, thanks to its happy antecedents, with its senses sharpened by an unhampered outlook to far horizons, to grasp forthwith the far-reaching consequences of such a metamorphosis of our concept of nature.
I will state at once what is the value of c with which we shall finally be dealing. It is the velocity of the propagation of light in empty space. To avoid speaking either of space or of emptiness, we may define this magnitude in another way, as the ratio of the electromagnetic to the electrostatic unit of electricity.
The existence of the invariance of natural laws for the relevant group Ge would have to be taken, then, in this way:-
From the totality of natural phenomena it is possible, by successively enhanced approximations, to derive more and
more exactly a system of reference x, y, z, t, space and time,
by means of which these phenomena then present themselves in agreement with definite laws. But when this is done, this system of reference is by no means unequivocally determined by the phenomena. It is still possible to make any change in the system of reference that is in conformity with the tra,nsformations of the group Ge, and leave the expression of the laws of nature unaltered.
For example, in correspondence with the figure described
above, we may also designate time t', but then must of neces-
sity, in connexion therewith, define space by the manifold of the three parameters x', y, z, in which case physical laws would be expressed in exactly the same way by means of
x', y, z, t' as by means of x, y, z, t. We should then have in
the world no longer space, but an infinite number of spaces,

80

SPACE AND TIME

analogously as there are in three-dimensional space an infinite number of planes. Three-dimensional geometry becomes a chapter in four-dimensional physics. Now you know why I said at the outset that space and time are to fade away into shadows, and only a world in itself will subsist.

II
The question now is, what are the circumstances which force this changed conception of space and time upon us? Does it actually never contradict experience? And finally, is it advantageous for describing phenomena?
B~fore going into these questions, I must make an important remark. If we have in any way individualized space and time, we have, as a world-line corresponding to a stationary substantial point, a straight line parallel to the axis of t; corresponding to a substantial point in uniform motion, a straight line at an angle to the axis of t; to a substantial point in varying motion, a world-line in some form of a curve. If at any world-point x, y, z, t we take the world-line passing through that point, and find it parallel to any radius vector OA.' of the above-mentioned hyperboloidal sheet, we can introduce OA' as a new axis of time, and with the new concepts of space and time thus given, the substance at the world-point concerned appears as at rest. We will now introduce this fundamental axiom:-
The substance at any world-point may always, with the appropriate determination of space and time, be looked upon as at rest.
The axiom signifies that at any world-point the expression
c2dt2 - dx2 - dy2 - dz2
always has a positive value, or, what comes to the same thing, that any velocity v always proves less than c. Accordingly c would stand as the upper limit for all substantial velocities, and that is precisely what would reveal the deeper significance of the magnitude c. In this second form the first impression made by the axiom is not altogether pleasing. But we must bear in mind that a modified form of mechanics, in which the square root of this quadratic differential expression appears,

H. MINKOWSKI

81

will now 1nake its way, so that cases with a velocity greater than that of light will henceforward play only some such part as that of figures with imaginary co-ordinates in geometry.
Now the impulse and true motive for assuming the group Ge came from the fact that the differential equation for the propagation of light in empty space possesses that group Ge.* On. the other hand, the concept of rigid bodies has meaning only in mechanics satisfying the group G00 • If we have a theory of optics with Ge, and if on the other hand there were rigid bodies, it is easy to see that one and the same direction of t would be distinguished by the two hyperboloidal sheets appropriate to Ge and G00 , and this would have the furtJ1er consequence, that we should be able, by employing suitable rigid optical instruments in the laboratory, to perceive some alteration in the phenomena when the orientation with respect to the direction of the earth's motion is changed. But all efforts directed towards this goal, in particular the fam.ous interference experiment of Michelson, have had a negative result. To explain this failure, H. A. Lorentz set up an hypothesis, the success of which lies in this very invariance in optics for the group Ge. According to Lorentz any moving body must have undergone a contraction in the directioµ of its motion, and in fact with a velocity v, a contraction in the ratio
1: ✓1 - v2/c2•
This hypothesis sounds extremely fantastical, for the contraction is not to be looked upon as a consequence of resist.. ances in the ether, or anything of that kind, but simply as a gift from above,-as an accompanying circumstance of the circumstance of motion.
I will now show by our figure that the Lorentzian hypothesis is completely equivalent to the new conception of space and time, which, indeed, makes the hypothesis much more intelligible. If for simplicity we disregard y and z, and imagine a world of one spatia.l dimension, then a parallel band, upright like the axis of t, and another inclining to the axis of t (see Fig. 1)

* An application of this fact in its essentials has already been given by W. Voigt, Got..inger Nachrichten, 1887, p. 41.

82

SPACE AND TIME

represent, respectively, the career of a body at rest or in uniform motion, preserving in each case a constant spatial extent. If OA' is parallel to the second band, we can introduce t' as the time, and x' as the space co-ordinate, and then the second body appears at rest, the first in uniform motion. We now assume that the first body, envisaged as at rest, has the length l, that is, the cross section PP of the first band on the axis of x is equal to l . OC, where OC denotes the unit of measure on the axis of x ; and on the other band, that the second body, envisaged as at rest, has the same length l, which then means that the cross section Q'Q' of the second band, measured parallel to the axis of x', is equal to l. 00'. We now have in these two bodies images of two equal Lorentzian electrons, one at rest and one in uniform 1notion.
But if we retain the original co-ordinates x, t, we must give
as the extent of the second electron the cross section of its appropriate band parallel to the axis of x. Now since Q'Q' == l. OC', it is evident that QQ = l . OD'. If dx/dt for the second band is equal to v, an easy calculation gives
OD' = OC,-./1 - v2/c2,
therefore also PP: QQ = 1: ✓1 - v2/c2• But this is the meaning of Lorentz's hypothesis of the contraction of electrons in motion. If on the other hand we envisage the second electron as at rest, and there£ore adopt the system of reference x' t', the length of the first must be denoted by the cross section P'P' of its band parallel to 00', and we should find the first electron in co1nparison with the second to be contracted in exactly the same proportion ; for in the figure
P'P' : Q'Q' = OD : OC' = OD' : OC = QQ : PP.
Lorentz called the t' combination of x and t the local time of the electron in uniform motion, and applied a physical construction of this concept, for the better understanding of the hypothesis of contraction. But the credit of first recognizing clearly that the time of the one electron is just as good as that of the other, that is to say, that t and t' are to be treated identically, belongs to A. Einstein.* Thus time, as a

* A. Einstein, Ann. d. Phys., 17, 1905, p. 891; Jahrb. d. Radioaktivitat
und Elektronik, 4, 1907, p. 411.

H. l\'1INKOWSKI

88

concept unequivocally determined by phenomena, was first deposed from its high seat. Neither Einstein nor Lorentz made any attack on the concept of space, perhaps because in the above-mentioned special transformation, where the plane of x', t' coincides with the plane of x, t, an interpretation is possible by saying that the x-axis of space maintains its position. One may expect to find a corresponding violation of the concept of space appraised as another act of audacity on the part of the higher mathematics. Nevertheless, this further step is indispensable for the true understanding of the group Ge, and when it has been taken, the word relativity.. postulate for the requirement of an invariance with the group Ge seems to me very feeble. Since the postulate comes to mean that only the four-dimensional world in space and time is given by phenomena, but that the projection in space and in time may still be undertaken with a certain degree of freedom, I prefer to call it the postulate of the absolute world (or briefly, the world-postulate).

III
The world-postulate permits identical treatment of the
four co-ordinates x, y, z, t. By this means, as I shall now
show, the forms in which the laws of physics are displayed gain in intelligibility. In particular the idea of acceleration acquires a clear-cut character.
I will use a geometrical manner of expression, which sug.. gests itself at once if we tacitly disregard z in the triplex x, y, z. I take any world-point O as the zero-point of space-
time. The cone c2t2 - w2 - y2 - z2 = 0 with apex 0 (Fig. 2)
consists of two parts, one with values t < 0, the other with values t > 0. The former, the front cone of 0, consists, let
us say, of all the world-points which " send light to O," the latter, the back cone of 0, of all the world-points 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 = ,l·t2 - ~ 2 - y2 - z2 = 1, t > 0
lies after 0. The territory between the cones is filled by the

84

SPACE AND TIME

one-sheeted hyperboloidal figures
- F = x2 + y2 + z2 - = c2t2 k2
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 world-line, would repre-
sent 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, t a vector, we have
to distinguish between the time-like vectors with directions
from O to the sheet + F = 1, t > 0, and the space-like vectors
r

Fm. 2.
with directions from Oto - F = 1. The time axis may run parallel to any vector of the former kind. Any world-point 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 world-point within the front cone of O is necessarily always before O ; any world-point 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 wedge-shaped 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
x, y, z, t, into the four components re, y, z, t. If the directions

H. MINKOWSKI

85

of two vectors are, respectively, that of a radius vector OR
from Oto one of the surfaces+ F = 1, and that of a tangent
RS at the point R of the same surface, the vectors are said to be normal to one another. Thus the condition that the
vectors with components x, y, z, t and x1, y1, Zi, t1 may be
normal to each other is
c2tt1 - xx1 - yy1 - zz1 = 0.
For the measurement of vectors in different directions the units of measure are to be fixed by assigning to .a space-like vector from O to - F = 1 always the magnitude 1, and to a
time-like vector from O to + F = 1, t > 0 always the magni-
tude 1/c. If we imagine at a world-point P (x, y, z, t) the world-
line of a substantial point running through that point, the magnitude corresponding to the time-like vector dx, dy, dz, dt laid off along the line is therefore

! dT = ✓(J°l'dt2 - dx2 - dy2 - dz 2. C
The integral ~dr = T of this amount, taken along the world-
line from any fixed starting-point P O to the variable endpoint P, we call the proper time of the substantial point at P. On the world-line we regard x, y, z, t-the components of the vector OP-as functions of the proper time r; denote their
first differential coefficients with respect to 7' by x, fl, z, i;
z, their second differential coefficients with respect to ,- by
!i, y, t; 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 to T the acceleration vector at P. Hence, since

we have

= c2, c2i2 - x2 - iJ2 - z2
c2t't - x?i - yy - zz = 0,

i.e. the velocity vector is the time-like vector of unit magnitude in the direction of the world-line at P, and the acceleration vector at Pis normal to the velocity vector at P, and is therefore in any case a space-like vector.
Now, as is readily seen, there is a definite hyperbola

86

SPACE AND TIME

which has three infinitely proximate points in common with

the world-line at P, and whose asymptotes are generators of

I

' I
\
\

I•
I I
' .I •
:' ' '

' \

I' •'

I

I

\

I p

\

I

\

I I

' \

II , I I'

\ lip I
'I,' I

M I"I" I'
' ',· \ I

I •I

I

\ I

' \'

I
I '

' '
' \
'\ \

'

.'\ \ ''''\ \ \ • \

FIG. 3.

a " front cone " and a " back cone " (Fig. 3). Let this hyperbola be caUe·d the hyperbola, of curvature at P. If Mis the centre of this hyperbola, we here have to do with an internal hyperbola with centre M. Let p be the magnitude of the vector MP ; then we
recognize the acceleration vector at P as the
vector in the direction MP of magnitude
x, z, c2/p.
If fl, t are all zero, the hyperbola of
curvature reduces to the straight line touching the world-line in P, and we must put
p = 00.
IV
To show tha.t 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 or" physics on the basis of this assumption. This

revision has to some extent already been successfully carried

out for questions of thermodynamics and heat radiation,* for

electromagnetic processes, and finally, with the retention of
t the concept of mass, for mechanics.

For this last branch of physics it is of prime importance

to raise the question-When a, force with the components

x, X, Y, Z parallel to the axes of space acts at a ~orld-point P
(x, y, z, t), where the velocity vector is iJ, z, t, what must

we take this force to be when the system of reference is in

any way changed ? Now there exist certain approved state-

ments as to the ponderomotive force in the electromagnetic

field in the cases where the group Ge is undoubtedly admis-

sible. These statements lead up to the simple rule :-When

the system of reference is changed, the force in question

transforms into a force in the new space co-ordinates in such

a way that the appropriate vector with the components tX,

* M. Planck," Zur Dyne.mik bewegter Systeme," Berliner Berichte, 1907,

n

Al~n ;n Ann ;1_ -Phvr:z

n

n49.

•

9.R

lQOR_

1

_

H. MINKOWSKI

87

tY, tZ, tT, where

T

==

1
-
c2

(.xX t

+

~t• y

+

~• Z ) 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 world-line 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

rnc2d~T! = nic2/,.j 1 - v2/ c'!.'
i.e., after removal of the additive constant mc2, the expression imv2 of Newtonian mechanics down to magnitudes of the order 1/c2• It comes out very clearly in this way, how the energy de:pends on the system of reference. But as the axis of t may be laid in the direction of any time-like 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 = co, this fact retains its importance for
* H. Minkowski, loc. cit., p 107. Cf. also M. Planck, Verha.ndlunge.n
der physika,lischen Gesellschaft, 4, 1906, p. 136.

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,
J - .1 t = s in place of t, the quadratic differential ex-
press1on dr = - 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 world-postulate. Thus the essence of this postulate may be clothed mathematically in a very pregnant 1nanner in the mystic formula
3. 105 km = ✓ - 1 secs.

V

The advantages afforded by the world-postulate will per-

haps be most strikingly exe1nplified by indicating the effects

proceeding from a point charge in any kind of motion accord-

ing to the Maxwell-Lorentz theory.

e, e t

Let us imagine the world-line of such

a point electron with the charge e, and

introduce upon it the proper time T

~ from any initial point. In order to find

the field caused by the electron at any

world-point P 1, we construct the front

....... cone belonging to P 1 (Fig. 4). The cone

M

evidently meets the world-line, since the directions of the line are everywhere

those of time-like vectors, at the single

point P. We draw the tangent to the

world-line 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, u Das Prinzip der absoluten Erhe.ltung der Energie," 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
world-point 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 under-
lying 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 world-line of a second point electron of the
charge· e1, passing through the world-point P 1. We define P,
Q, r as before, then construct (Fig. 4) the centre M of the
hyperbola of curvature at P, and finally the normal MN from M to a straight line imagined through P parallel to QPJ.' With P as starting-point we now determine a system of reference
as follows :-The axis of t in the direction PQ, the axis of x
in direction QPi, the axis 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
t, x x, ii, z, the velocity vector at P 1 be i/ z 1, 1, 1, ti- The motive
force vector exerted at P 1 by the first moving electron e on the second moving electron e1 now takes the form

- ee1( i1 - : 1).re,

where the components 5?x, 5?11, S?'z, 5?t of the vector .re satisfy

! , the three relations

1

..

cS?t - Rx = r2 , Ry = er Rz = 0,

and where, fourthly, this vector j? is normal to the velocity vector at P 1, and through this circumstance alone stands in dependence on the latter velocity vector.

* A. Lienard, "Champ electrique et magnetique produit par une charge concentree en un point et a.nimee d'un mouvement quelconque," L'Ecla.ire.ge Electrique, 16, 1898, pp. 5, 53, 106; E. Wiechert, '' Elektrodynamiscbe Elementa.rgesetzo," Arch. Neerl. (2), 5, 1900, p. 549.

90

SPACE AND TIME

When we compare this statement with previous formulations• of the same elementary law of the ponderomotive action of moving point charges on one another, we are compelled to admit that it is only in four dimensions that the relations here taken under consideration reveal their inner being in full simplicity, and that on a three dimensional space forced upon us a priori they cast only a very complicated projection.

In mechanics as reformed in accordance with the worldpostulate, the disturbing lack of harmony between Newtonian mechanics and modern electrodynamics disappears of its own accord. Before cop.eluding I will just touch upon the attitude of Newton's law of attraction toward this postulate. I shall assume that when two points of mass m, m1 describe their world-lines, a motive force vector is exerted by m on m1, of exactly the same form as that just given in the case of electrons, except that + mm1 must now take the place of - ee1. 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 taken 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 i - 1 = ✓l - v2/c 2, which, to the
order of 1/c2, is equal to 1, it will be seen t that for the posi-
tions x1, y1, z1, of m1 and their variations in time, we should arrive exactly at Kepler's laws again, except that the proper times r1 of m1 would take the place of the times t1. From this simple remark it may then be seen that the proposed law of attraction combined with the new mec;hanics 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 shall show elsewhere, it is not even by any means necessary to abandon the derivation of these funda-
* K. Schwarzwa.ld, Gottinger Nachr., 1908, p. 182; H. A. Lorentz, Enzykl. d. me.th. Wissensch., V, Art. 14, p. 199.
t H. Minkowski, loc. 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 world-postulate, I like to think, is the true nucleus of an electromagnetic image of the world, which, discovered by Lorentz, and further revealed 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 wi1l suffice to conciliate even those to whom the abandonment of old-established views is unsympathetic or painful, by the idea of a pre-established harmony between pure mathematics and physics.

NOTES
by
A. SOMMERFELD
The following notes are 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 mathematical difficulties which might hinder the comprehension of Minkowski's great thoughts. The bibliographical references are confined to the literature dealing 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 law of attraction. What will be the epistemological attitude towards Minkowski's conception of the time-space 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 G00 ." This sentence was confirmed in the widest sense in a discussion on a paper by his disciple M. Born, e. year after Minkowski's death. Born (Ann. d. Physik, 30, 1909, p. 1) had defined a relatively rigid body as one in which every element of volume, even in accelerated motions, undergoes the Lorentzian contraction appropriate to its velocity. Ehrenfest (Phys. Zei-tschr.~ 10, 1909, p. 918) showed that such a body cannot be set in rotation; Herglotz (Ann. d. Phys., 31, 1910, p. 898) and F. Nether (Ann. d. Phys., 81, 1910, p. 919) that it has only three degrees of freedom of movement. The attempt was also made 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 above, 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. 453) 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 plays the same pa.rt in this theory of elasticity as the ordinary rigid body plays in the ordinary theory of elasticity.
(2) Page 82, line 18. '' If dx/dt for the second band is equal to v, an easy
caJculaiion gives OD' = 00 Jl - v2/c'A." In Fig. 1, let ci = LA'OA, {3 = LB'OA'
= LC'OB', in which the equality of the last two angles follows from the sym-
metrical position of the asymptotes with respect to the new axes of co-ordin-
ates (conjugate diameters of the hyperbola).* Since ci + f3 = }1r,
sin 2,8 = cos 2ci.
* Sommerfeld seems to take ct a.s a co-ordinate in the graph in place oft a.s
used by Minkowski.-TRANS. 92