Jürgen Renn (ed.).

The Genesis of General Relativity,

Vol. 3

Gravitation in the Twilight of Classical Physics: Between Mechanics, Field Theory, and Astronomy.

HENDRIK A. LORENTZ

OLD AND NEW QUESTIONS IN PHYSICS(EXCERPT)

Originally published as “Alte und neue Fragen der Physik” in PhysikalischeZeitschrift, 11, 1910, pp. 1234–1257. Based on the Wolfskehl lectures given inGöttingen, 24–29 October 1910. The second and third lectures are translated here(pp. 1236–1244). Lorentz made use of Born’s notes from the lecture and submittedthe paper on 2 November 1910.

SECOND LECTURE

To discuss Einstein’s

principle of relativity

here at Göttingen, where Minkowski wasactive, seems to me a particularly welcome task.

The significance of this principle can be illuminated from several different angles.We will not speak here of the mathematical aspect of the question, which was givensuch a splendid treatment by Minkowski, and which was further developed by Abra-ham, Sommerfeld and others. Rather, after some epistemological remarks about theconcepts of space and time, the physical phenomena that may contribute to an exper-imental test of the principle shall be discussed.

The principle of relativity claims the following: If a physical phenomenon isdescribed by certain equations in the system of reference then a phenome-non will also exist that can be described by the same equations in another system ofreference Here the two systems of reference are connected by relationscontaining the speed of light and expressing the motion of one system with a uni-form velocity relative to the other.

If observer is located in the first, and in the second system of reference, andeach is supplied with measuring rods and clocks at rest in his system, then willmeasure the values of and the values of where it should benoted that and can also use one and the same measuring rod and the same clock.We have to assume that when the first observer somehow hands his rod and clockover to the second observer, they automatically assume the proper length and theproper rate so that arrives at the values from his measurements. Eitherone will then find the same value for the speed of light, and will quite generally beable to make the same observations.

x y z t ,, , ,

x' y' z' t'., , ,c

A BA

x y z t ,, , , B x' y' z' t',, , ,A B

B x' y' z' t', , ,

© 2007 Springer.

288 H

ENDRIK

A. L

ORENTZ

Assume there is an aether; then among all systems a single one would bedistinguished by the state of rest of its coordinate axes as well as its clock in theaether. If one associates with this the idea (also held tenaciously by the speaker) thatspace and time are totally different from each other and that there is a “true time”(simultaneity would then exist independent of location, corresponding to the circum-stance that we are able to imagine infinitely large velocities) then it is easily seen thatthis true time should be shown precisely by clocks that are at rest in the aether. Now,if the principle of relativity were generally valid in nature, then we would of coursenot be in a position to determine whether the system of reference being used at themoment is the distinguished one. Thus one arrives at the same results as those foundwhen one denies the existence of the aether and of the true time, and regards all sys-tems of reference as equivalent, following Einstein and Minkowski. It is surely up toeach individual which of the two schools of thought he wishes to identify with.

In order to discuss the physical aspect of the question, we first have to establishthe transformation formulas, limiting ourselves to a special form | already used in theyear 1887 by W. Voigt in his treatment of the Doppler principle; namely,

where the constants satisfy the relation

which entails the identity The origin ofthe system moves with respect to the system in the direction withspeed which is always less than In general we have to assume that everyvelocity is less than

All state variables of any phenomenon, measured in one or the other system areconnected by certain transformation formulas. For example, for the speed of a pointthese are

where

Further we consider a system of points whose velocity is a continuous function ofthe coordinates. Let be a volume element surrounding the point at time

to this value of there corresponds according to the transformation equations apoint in time in the other system of reference, and every point lying in attime has certain [coordinates] for this fixed value of The points

fill a volume element which is related to as follows

[1]

x y z t, , ,

[1237]

x' x,= y' y,= z' az bct– ,= t' atbc---z– ; =

a 0,> b

a2 b2– 1,=

x'2 y'2 z'2+ + c2t'2– x2 y2 z2 c2t2.–+ +=x' y' z', , x y z, , z-

b a⁄( )c, c.c.

vx'vx

ω----,= vy'

vy

ω----,= vz'

avz bc–

ω-------------------,=

ω abvz

c-------.–=

Sd P x y z, ,( )t; t

P t' Sdt x' y' z', , t'.

x' y' z', , S',d Sd

O

LD

AND

N

EW

Q

UESTIONS

IN

P

HYSICS

(E

XCERPT

) 289

Let us imagine some agent (matter, electricity etc.) connected with the points, andlet us assume that the observer has occasion to associate with each point the sameamount of the agent as the observer then the spatial densities must obviously be inthe inverse ratio as the volume elements, that is

All these relations are reciprocal, that is, the primed and unprimed letters may beinterchanged if at the same time is replaced by

The basic equations of the electromagnetic field retain their form under the trans-formation if the following quantities are introduced:

Thus in the system the following equations hold between these quanti-ties, the transformed space density and the transformed velocity

With this the field equations of the electron theory satisfy the principle of relativ-ity; but there is still the matter of harmonizing the equations of motion of the elec-trons themselves with this principle.

We will consider somewhat more generally the motion of an arbitrary materialpoint. Here it is useful to introduce the concept of “proper time,” Minkowski’s beau-tiful invention. According to this there belongs to each point a time of its own, as itwere, which is independent of the system of reference chosen; its differential isdefined by the equation

The expressions formed with the aid of the proper time

S'dSd

ω------.=

BA,

ρ' ωρ.=

b b.–

dx' adx bhy,–= dy' ady bhx,+= dz' dz,=

hx' ahx bdy,+= hy' ahy bdx,–= hz' hz.=

x' y' z' t', , ,ρ' v':

div d' ρ',=

div h' 0,=

curl h'1c--- d' ρ'v'+( ),=

curl d' 1c---– h' .=

τd 1 v2

c2-----– t .d=

τ,

τdd xd

τd----- ,

τdd yd

τd-----,

τdd zd

τd-----,

290 HENDRIK A. LORENTZ

which are linear homogeneous functions of the components of the ordinary accelera-tion, will be called the components of the “Minkowskian acceleration.” We describethe motion of a point by the equations:

where is a constant, which we call the “Minkowskian mass.” We designate thevector as the “Minkowskian force.”

It is then easy to derive the transformation formulas for this acceleration and thisforce; we leave unchanged. Thus we have

The essential point is the following. The principle of relativity demands that if for anactual phenomenon the Minkowskian forces depend in a certain way on the coordi-nates, velocities, etc. in one system of reference, then the transformed Minkowskianforces | depend in the same way on the transformed coordinates, velocities etc. in theother system of reference. This is a special property that must be shared by all forcesin nature if the principle of relativity is to be valid. Presupposing this we can calculatethe forces acting on moving bodies if we know them for the case of rest. For example,if an electron of charge is in motion, we consider a system of reference in which itis momentarily at rest. Then the electron in this system is under the influence of theMinkowskian force

from this it follows by application of the transformation equations for and thatthe Minkowskian force acting on the electron that moves with velocity in an arbi-trary coordinate system amounts to

This formula does not agree with the usual ansatz of the electron theory, because ofthe presence of the denominator. The difference is due to the fact that usually onedoes not operate with our Minkowskian force, but with the “Newtonian force” and we see that these two forces are related as follows for an electron:

It is to be assumed that this relation is valid for arbitrary material points.

mτd

d xdτd

----- Kx,= etc.,

mK

m

Kx' Kx,= Ky' Ky,= Kz' aKzbc--- v K⋅( )– .=

[1238]

e

K cd;=

K d

v

K cd

1c--- v h⋅[ ]+

1 v2

c2-----–

--------------------------.=

F,

F K 1 v2

c2-----– .=

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 291

Thus the phenomena of motion can be treated in two different ways, using eitherthe Minkowskian or the Newtonian force. In the latter case the equations of motiontake the form

and here means the ordinary acceleration in the direction of motion, the ordi-nary normal acceleration, and the factors

are called the “longitudinal” and the “transverse mass.”Just like the Minkowskian forces, the Newtonian forces that occur in nature must

also fulfill certain conditions in order to satisfy the relativity principle. This is thecase if, for example, a normal pressure of a constant magnitude per unit area actson a surface regardless of the state of motion; then in the transformed system a nor-mal pressure of the same magnitude acts on the corresponding moving surface ele-ment. Since we have already recognized the invariance of the field equations, thequestion of whether the motions in an electron system satisfy the relativity principleamounts merely to an experimental test of the formulas for the longitudinal and trans-verse masses although the experiments of Bucherer and Hupka seem toconfirm these formulas, one has not yet arrived at a definitive decision.

Concerning the mass of the electron, one should remember that this is electro-magnetic in nature; so it will depend on the distribution of charge within the electron.Therefore the formulas for the mass can be correct only if the charge distribution, andhence also the shape of the electron, vary in a definite way with the velocity. Onemust assume that an electron, which is a sphere when at rest, becomes an ellipsoidthat is flattened in the direction of motion as a result of translation; the amount of flat-tening is

If we assume that the shape and size of the electron are regulated by internal forces,then to agree with the relativity principle these forces must have properties such thatthis flattening occurs by itself when in motion. Regarding this Poincaré has made thefollowing hypothesis. The electron is a charged, expandible skin, and the electricalrepulsion of the different points of the electron is balanced by an inner normal tensionof unchangeable magnitude. Indeed, according to the above such normal tensions sat-isfy the relativity principle.

In the same way all molecular forces acting within ponderable matter, as well asthe quasi-elastic and resistive forces acting on the electron, have to satisfy certainconditions in order to be in accord with the relativity principle. Then every moving

F m1j1 m2j2,+=

j1 j2

m1m

1 v2

c2-----–⎝ ⎠

⎛ ⎞3

--------------------------,= m2m

1 v2

c2-----–

------------------,=

p

m1, m2;

1 v2

c2-----– .

292 HENDRIK A. LORENTZ

body will be unchanged for a co-moving observer, but for an observer at rest it willexperience a change in dimensions, which is a consequence of the change in molecu-lar forces demanded by these conditions. This also leads automatically to the contrac-tion of bodies, which was already | devised earlier to explain the negative outcome ofMichelson’s interferometer experiment and of all similar experiments that were todetermine an influence of the Earth’s motion on optical phenomena.

Concerning rigid bodies, as investigated by Born, Herglotz, Noether, and Levi-Civita, the difficulties occurring in the consideration of rotation can surely be relievedby ascribing their rigidity to the action of particularly intense molecular forces.

Finally let us turn to gravitation. The relativity principle demands a modificationof Newton’s law, foremost a propagation of the effect with the speed of light. Thepossibility of a finite speed of propagation of gravity was already discussed byLaplace, who imagined as the cause of gravity a fluid streaming toward the Sun,which pushes the planets toward the Sun. He found that the speed of this fluid mustbe assumed to be at least 100 million times larger than that of light, so that the calcu-lations remain in agreement with the astronomical observations. The necessity ofsuch a large value of is due to the occurrence of to the first power in his finalformulas, where is the speed of a planet. But if the propagation speed of gravityis to have the value of the speed of light, as demanded by the relativity principle, thena contradiction with the observations can only be avoided if only quantities of second(and higher) order in occur in the expression for the modified law of gravitation.

Restricting oneself to quantities of second order, one can, on the basis of a sug-gestive electron-theoretic analogy, easily give a condition that determines the modi-fied law in a unique way. Namely, if one considers the force acting on an electron thatmoves with a velocity

then the vectors and depend, in addition, on the velocities of the electrons thatproduce the field; in the vector product products of the form do occur,but not the square of the speed of the electron under consideration. Accordinglylet us assume that in the expression for the attraction acting on the point 1 due topoint 2 there is no term in the square of the velocity of point 1. Then all velocitieswhatsoever must drop out in a system of reference in which point 2 is at rest

therefore the law will reduce to the usual Newtonian one in this system.Now making the transition by transforming to an arbitrary coordinate system, onefinds that the force acting on point 1 is composed of two parts, the first, an attractionin the direction of the line connecting them of magnitude

the second, a force in the direction of of magnitude

[1239]

c

c v c⁄v c

v c⁄

v,

e d1c--- v h⋅[ ]+⎝ ⎠

⎛ ⎞ ,

d h v'v h⋅[ ], vv'

v2

v12

v2 0=( );

R1c2----- 1

2---v2

2R12---v2r

2 rRdrd

------ R–⎝ ⎠⎛ ⎞ v1 v2⋅( )R–+

⎩ ⎭⎨ ⎬⎧ ⎫

,+

v2

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 293

here means the distance between two simultaneous positions of the two points, the component of along the connecting line drawn from 1 to 2, and that functionof which represents the law of attraction in the case of rest for Newto-nian attraction, for quasielastic forces). Note that “force” is always under-stood to be the “Newtonian force,” not the “Minkowskian” one. Minkowski, by theway, has given a somewhat different expression for the law of gravity. The latter aswell as the one described above can be found in Poincaré.

THIRD LECTURE

At the end of the previous lecture a modified law of gravitation was given, whichis in agreement with the relativity principle. Concerning this one should note that theprinciple of equality of action and reaction is not satisfied.

Now the perturbations that can arise due to those additional second order termswill be discussed. Besides many short-period perturbations, which have no signifi-cance, there is a secular motion of the planets’ perihelia. Prof. de Sitter computes

per century for Mercury’s perturbations. Since Laplace, it has been knownthat Mercury has an anomalous perihelion motion of per century; although thisanomaly has the right sign, it is much too large to be explained by those additionalterms. Instead, Seeliger attributes it to a perturbation by the carrier of the zodiacallight, | whose mass one can suitably determine in a plausible way. So, from this onecan arrive at no decision, as long as the accuracy of astronomical measurements is notsignificantly increased. To be absolutely accurate one would also have to take intoaccount the difference between the Earth’s “proper time” and the time of the solarsystem.

A different method to test the validity of the modified law of gravitation can bebased on a procedure suggested by Maxwell to decide whether the solar systemmoves through the aether. If this were the case, then the eclipses of Jupiter’s moonsshould be advanced or delayed depending on Jupiter’s position with respect to theEarth.

For if the Jupiter-Earth distance is and the component of the solar system’svelocity in the aether in the direction of the line connecting Jupiter to Earth is thenthe time required to cover the distance in the case of rest, would be changedto thus the motion brings about an advance or delay, which amounts to

up to terms of second order, and which takes on different values according tothe value of the velocity component which of course depends on the position ofthe two planets. Now it is clear that such a dependence of the phenomena on themotion through the aether contradicts the relativity principle.

1c2-----v1r Rv2 ;

r vrv R

r (R k r2⁄=R kr=

6.69″44″

[1240]

av,

a a c⁄ ,a c v+( )⁄ ;

av c2⁄v,

294 HENDRIK A. LORENTZ

In order to clear up this contradiction let us simplify the situation schematically.We suppose that the Sun has a mass that is infinitely large compared to that of theplanet. Let the velocity of the solar system coincide with the axis, which we laythrough the Sun. The intersection points of the planet’s orbit with the axis aredenoted as the upper resp. lower transit, resp. (Fig. 1)

We place the observer at the Sun. At each transit of the planet through the axisa signal will propagate towards the Sun. The period of revolution shall be Whenthe Sun is at rest the time between an upper and lower transit will be for theassumed circular motion; the same is true for the time between the arrivals of the twolight signals. By contrast, if the Sun is in motion in the direction, the signal fromthe upper transit must suffer an advance that from the lower transit a delayof the same amount; if the uniform orbital motion (assumed as self-evident by Max-well) is preserved without perturbation, the time interval between the arrivals of thelight signals of two successive passes would appear alternately increased anddecreased by Preservation of the uniform circular motion during a transla-tion through the aether, as is assumed above, is, however, impossible according to therelativity principle. For if we describe the process in a coordinate system that doesnot take part in the motion, the modified law of gravitation will have to be applied,and this results in a non-uniform planetary motion, due to which the difference intime intervals between the arrivals of the light signals exactly cancels.

Therefore the determination of whether an advance or delay of the eclipses in factoccurs can be used to decide in favor or against the relativity principle. However, thenumerical situation is again rather unfavorable. Thus Mr. Burton, who has access to330 photometric observations of eclipses of Jupiter’s first moon made at the Harvardobservatory, estimates the probable error of the final result for as 50 km/sec; on theother hand, one has observed speeds of stars of 70 km/sec, and the speed of the solarsystem with respect to the fixed stars is estimated at 20 km/sec. The relativity princi-ple is therefore hardly supported by Burton’s calculations; at best they could invali-date it, namely if, for example, the final result were a value exceeding 100 km/sec.

Let us leave it undecided whether or not the new mechanics will receive confirma-tion by astronomical observations. But we will not fail to familiarize ourselves withsome of its basic formulas.

If one defines work as the scalar product of “Newtonian force” and | displace-ment, then the equations of motion yield the energy principle in its usual form, so thatthe work done per unit of time equals the increase in energy :

Here energy is expressed by

Sz-

z-A B.

z-T .

1 2⁄( )T

z-av c2⁄ ,

2av c2⁄ .

v

[1241]

ε

Fxxdtd

----- Fyydtd

----- Fzzdtd

-----+ +εdtd

-----.=

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 295

this agrees up to second order terms with the value of the kinetic energy in customarymechanics:

Furthermore, from the equations of motion one can derive Hamilton’s principle

here is the work of the “Newtonian force” upon a virtual displacement and isthe Lagrangian, which takes the form

From Hamilton’s principle one can conversely obtain the equations of motion.The quantities

are to be identified as the components of the momentum. All these formulas can be verified by the electromagnetic equations of motion for

an electron. One then has to take the following value for the “Minkowskian mass”

and to add to the electric and magnetic energy the energy of those internal stresseswhich determine the shape of the electron, as we saw above. Thus from the generalprinciple of least action for arbitrary electromagnetic systems, discussed in the firstlecture, one can obtain Hamilton’s principle for a point mass as given above by spe-cialization to an electron, but the work of those internal stresses must be taken intoaccount.

We now go over to a discussion of the equations of the electromagnetic field forponderable bodies. These have been written down purely phenomenologically byMinkowski, then M. Born and Ph. Frank showed that they can also be derived from

ε mc2 1

1 v2

c2-----–

------------------ 1–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

;=

ε12---mv2.=

Lδ Aδ+( ) tdt1

t2

∫ 0;=

Aδ L

L mc2 1 v2

c2-----– 1–⎝ ⎠

⎛ ⎞ .–=

L∂x∂

------,L∂y∂

------,L∂z∂

------

m,

me2

6πRc2----------------,=

296 HENDRIK A. LORENTZ

the ideas of electron theory; by the latter procedure Lorentz himself also found theseequations, in a slightly different technical form.

To obtain relations between observable quantities one must smear out the detailsof the phenomena due to individual electrons by averaging over a large number ofthem. One is lead to the following equations (which are identical to those of the usualMaxwell theory):

Here is the dielectric displacement, the magnetic induction, the mag-netic force, the electric force, the electric current, and the density of theobservable electric charges. Denoting mean values by an overbar we have, for example,

where have their former meaning; further we have

where is the electric moment and the magnetization per unit volume, and denotes the velocity of matter. When deriving these formulas one divides the elec-trons into three types. The first type, the polarization electrons, generate the electricmoment by their displacement; the second type, the magnetization electrons, gen-erate the magnetic state by their orbital motion; the third type, the conductionelectrons, move freely within the matter and generate the observable charge density

and the current The latter is additionally to be divided into two parts; for if is the relative velocity of the electrons with respect to the matter, then the total veloc-ity of the electrons is hence the current carried by them is

is the observable charge is the convection current, and the conductioncurrent proper.

There are transformation formulas for all these quantities, | and we give a few ofthem below:

divD ρl,=

divB 0,=

curl H1c--- E D+( ),=

curl E 1c---– B.=

D B H

E C ρl

E d,= B h,=

d, h

D E P,+=

H B M1c--- P w⋅[ ],––=

P M w

P

M

ρl C. u

v w u,+=

E ρv ρw ρu;+= =

ρ ρl, ρw ρu

[1242]

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 297

Further, the following auxiliary vectors are useful:

Now the field equations given above must still be completed by establishing therelations that exist between the vectors and These relations can beobtained in two ways.

The first, phenomenological method proceeds as follows: One considers an arbi-trarily moving point of matter and introduces a system of reference in which it is atrest; if the element of volume surrounding the point is isotropic in the rest system, theequation appropriate for bodies at rest (between and for example)

holds; or equally well

because the auxiliary vectors are identical with when But and transform in the same way, and this implies that also in the original systemthe equation

and correspondingly

remains valid. Concerning the conduction current we remark only that it dependson

The second method has its roots in the mechanics of electrons. Just as the equa-tion for bodies at rest turns out to be a consequence of the assumption ofquasielastic forces, which restore the electrons to their rest positions, so one willobtain the equation for moving bodies if one ascribes to the quasi-elasticforces the properties demanded by the relativity principle. The latter will be satisfied

Cx' Cx,= Cy' Cy,= Cz' aCz bcρl– ,=

ρl' aρlbc---Cz– ,=

Px' aPxbc--- wzPx wxPz–( ) bMy,+–=

Py' aPybc--- wzPy wyPz–( ) bMx,+–=

Pz' Pz.=

H1 H1c--- w D⋅[ ],–= B1 B

1c--- w E⋅[ ],–=

E1 E1c--- w B⋅[ ]+ ,= D1 D

1c--- w H⋅[ ]+ .=

E H, D B, .

E D,

D εE=

D1 εE1,=

D1 E1, D E, w 0= D1E1

D1 εE1,=

B1 µH1,=

E1.

D εE=

D1 εE1=

298 HENDRIK A. LORENTZ

if one takes for these forces the expression of the generalized law of attraction, where must be taken proportional to

The explanation of the resistance to conduction proceeds similarly. A satisfactoryelectron-theoretic explanation of the magnetic properties of matter is presently not athand.

Finally the significance of the above equations shall be elucidated in threeremarkable cases.

The first remark is connected with the equation

According to this, can vanish without having if a current is present;that is, an observer will declare a body to be charged that must be treated asuncharged by moving relative to him. One can understand this by noting that everybody contains an equal number of positive and negative electrons, which compensatein uncharged bodies. When the body moves with velocity in the presence of a con-duction current, the two types of electrons will attain different total velocities, there-fore the quantity will also have different values for the two types.When an observer moving with the body calculates the mean charge density

for both types of electrons he can obtain zero for the sum, even when foran observer in whose reference frame the body is moving the mean values of thepositive and negative electrons do not compensate.

This circumstance calls forth the memory of an old question. Around the year1880 there was a great discussion among physicists about Clausius’ fundamental lawof electrodynamics. One attempted to derive a contradiction between this law andobservations by concluding that according to the law a current-carrying conductor onthe Earth would have to exert an influence on a co-moving charge due to themotion of the Earth, which could have been observed. That the law actually does notdemand this influence was noted by Budde; this is because the current due to theEarth’s motion acts on itself and causes a “compensating charge” in the current-car-rying conductor, which exactly cancels the first influence. The electron theory leadsto similar conclusions, and Lorentz finds |

for the density of the compensating charge, if the velocity is in the direction of theaxis; this must be assumed by an observer who does not take part in the motion

of the Earth, whereas it does not exist for a co-moving observer The value givenabove agrees exactly with the formula derived from the relativity principle; for if

one finds from this formula

R r .

ρl' aρlbc---Cz– .=

ρl' ρl 0= C

AB

w

ω a b vz c⁄( )–=B

ρ' ωρ=A ρ

e

[1243]

1c2-----wz Cz ;

z- AB.

ρl' 0,=

ρlb

ac------Cz,=

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 299

and since, according to what was said in the second lecture on p. 288 [p. 1237 in theoriginal], is the speed of the two systems of reference with respect toeach other, one indeed finds

The second remark starts from the transformation equations for the electricmoment p. 296 [p. 1241 in the original] in which the presence of the magnetiza-tion lets us recognize the impossibility of differentiating precisely between polar-ization- and magnetization electrons. Rather, in a magnetized body

can vanish when judged from one system of reference, whereas in another differs from zero. This will now be applied to a special case, where we confine

attention to quantities of first order. The body we consider (such as a steel magnet)shall contain only conduction electrons and electrons that produce an but no when the body is at rest; it shall have the shape of an infinitely extended plate,bounded by two planes the middle plane shall be the plane. (Fig. 2) When itis at rest a constant magnetization shall be present, whereas When thebody is given a speed in the direction an observer who does not take part in themotion will observe the electric polarization

Now we imagine at either side of the body two conductors which form togetherwith it two equal condensers, and these shall be shorted out by a wire (from to When there is motion, charges will be created on and which can be calculatedas follows. Since a current is clearly impossible in the direction, we have

or Since the process is stationary we have thenthe existence of a potential follows from If is the thickness of theslab one has

From the symmetry of the arrangement it clearly follows that

and because the plates are shorted out, we must have

this implies

wz bc a⁄=

ρl1c2-----wz Cz.=

P

M

M 0≠( )P 0,=P'

M P

a b;, yz-My P 0.=

v z-

Pxvc--My.–=

c d ,,c d).

c d ,x-

Elx 0= Ex v c⁄( )By.= B 0;=ϕ curlE 0.= ∆

ϕa ϕb–vc--∆By.=

ϕd ϕa– ϕb ϕc ,–=

c d,

ϕd ϕc ;=

ϕd ϕa–v

2c------∆By– .=

300 HENDRIK A. LORENTZ

If is the capacity of one of the two condensers, the charge of plate becomes

and receives the equal and opposite amount. Now we compare this process with the inverse case, that the magnet is at rest

and the plates move with the opposite velocity. According to the relativity prin-ciple everything would have to be the same as in the first case. Indeed one finds atonce from the usual law of induction exactly the amount of charge on plate givenabove. But this charge on must now induce an equal and opposite one on the plane

of the magnet at rest, and corresponding statements must hold for and Sinceno current can flow there must be the same charges on the magnet,whether the magnet is moving and the plates are at rest or conversely. So we have tothink how it happens that in the first case the opposite charge appears on the plane of the moving magnet as on the plate this becomes possible only due to the polar-ization | produced by the motion. For one has

since here is to be neglected to first order in the velocity, that is the term ,we have

But is zero because we assume the plate to be infinitely extended. This implies

i.e., in the moving plate there is no dielectric displacement, so the charge on corre-sponds to that on as the relativity principle demands.

The last remark concerns again the circumstance that according to the relativityprinciple the motion of the Earth cannot have any influence on electromagnetic pro-cesses. But Liénard has pointed out a phenomenon where such an influence is to beexpected, an influence of first order in magnitude; Poincaré has also discussed thiscase in his book Electricité et Optique. It concerns the ponderomotive force on a con-ductor. To determine this force one may make the suggestive ansatz for the force act-ing on the conduction electrons per unit charge:

then this results in the force caused by the Earth’s motion on the conductor in thedirection of the motion by an amount

γ d

v2c------γ∆By– ,

ca b,

c d,

dd

a b c.C 0=( ),

ad ;

Px v c⁄( )My–=[1244]

Dx Ex Px+vc--By

vc--My;–= =

P P w⋅[ ]

B M– H,=

H

Dx 0,=

ad ,

E1 E1c--- v B⋅[ ];+=

1c2----- Cl E⋅( )wz;

OLD AND NEW QUESTIONS IN PHYSICS (EXCERPT) 301

since is the heat generated by the conduction current this expression iseasily calculated numerically (which admittedly results in a value inaccessible toobservation).

If one now asks oneself, how this result that contradicts the relativity principle cancome about, one finds that indeed one has not calculated the force acting on the mat-ter of the conductor, but on the electrons moving inside the conductor. The latterforce must first be transferred to the matter by forces, which are unknown to us indetail, and that happens without change of magnitude only if action equals reactionfor the forces between matter and electrons. But for moving bodies action does notequal reaction in this case according to the relativity principle, and this circumstanceexactly compensates Liénard’s force.

In summary, one can say that there is little prospect for an experimental confirma-tion of the principle of relativity; except for a few astronomical observations, onlymeasurements of the electron mass are worth considering. But one must not forgetthat the outcome of the negative experiments, such as Michelson’s interference exper-iment and the experiments to find a double refraction caused by the Earth’s motion,can only be explained by the relativity principle.

EDITORIAL NOTE

[1] In the original, the denominator is missing from the right-hand side.

Cl E⋅( ) C