Translation:On the Electrodynamics of Moving Systems I

by Emil Cohn

On the Electrodynamics of Moving Systems.

By Prof. Emil Cohn.

in Straßburg i. E.

Submitted by Mr. Warburg.

Caused by a series of experimental investigations of recent years – all have shown the independence of the observed phenomena from Earth's motion – H.A. Lorentz has recently modified the foundations of the theory of electron by new hypotheses. In the following, I intend to show that by these modifications, the electrodynamic equations of H.A. Lorentz for extended bodies are in agreement with the equations which were stated by me some years ago.

§ 1. A comparison of my approach with that of H.A. Lorentz was not completely possible yet. This is caused by the fact, that both "theories" are of different kind throughout. My theory represents, by a few equations, the influence of visible motions on the electromagnetic processes in gross matter. In this field, it is directly comparable with experience. It only gives instructions for a molecular theory still to be developed; it shall be developed in a way, by which exactly those equations occur for the measurable averages.

Lorentz directly gives a rule for the electromagnetic actions, which are exerted and suffered by supposed particles of the smallest kind. From them, equations must be developed for those magnitudes "which are related to the state of visible parts of the body, and therefore are accessible to observation". Developed up to this point, the theory can be found in the Proceedings of the Amsterdam Academy of 1902.^[1] The equations read:

${\begin{cases}{\mathsf {P}}\{cH-[wE]\}=J+{\frac {\partial D}{\partial t}}+\Gamma (D)w-P[wD]\\{\mathsf {P}}(cE)=-{\frac {\partial B}{\partial t}}\\\Gamma (D)=\rho ;\ \Gamma (B)=0\\D=E+P;\ B=H+M\end{cases}}$.

(L)

Here, P = rotation, $\Gamma$ = divergence, $c$ is the speed of light in vacuum, $w$ the velocity of matter; $E$ and $H$ the electric and magnetic field intensity; $D$ and $B$ the electric polarization and magnetic induction (recently denoted as electric and magnetic excitation); $P$ and $M$ the electric and magnetic moment of unit volume (recently denoted as polarization); $J$ the electric current (by conduction).

To be able to apply equations (L), is is obviously necessary to represent J, P, M as functions of E and H. With this postulate the mentioned paper ends.

For our purposes is is only required, that one can give the form of the functions for arbitrary $w$, when they are known for $w=0$. In this connection, the papers given so far by Lorentz (including the article in the Mathematical encyclopedia) only gave assumptions that are near at hand according to the author's own opinion;^[2] those are also connected to magnitudes only, which are proportional to the first power of the ratio of the body's velocity and the speed of light. A comparison between both theories was, in a strict sense, only possible in the single case, where J, P and M have no considerable values, i.e., with respect to the propagation of light in moving gases. Here, it is actually carried out.^[3] In addition is was near at hand in the cases where only the first power of ${\tfrac {w}{c}}$ was of relevance (although it was connected with some uncertainty).

The recent paper of Lorentz^[4], however, brings a series of new assumptions on electrons, molecules, and the forces acting on them, which lead to a very specific answer to the questions stated above, as far as the whole considered system has a common velocity of translation $w$.

§ 2. We want to derive the sought relations, and introduce them into (L).^[5] Here, we want to presuppose that all bodies are "non-magnetizable", i.e., generally $B=H$. Furthermore, we want to pass to relative coordinates, and denote the corresponding derivatives with respect to time by ${\tfrac {d}{dt}}$, so that generally we have

${\frac {\partial A}{\partial t}}+\Gamma (A)w-{\mathsf {P}}[wA)={\frac {dA}{dt}}$.

If we choose the speed of light in vacuum as unity, then (L) becomes:

${\begin{cases}{\mathsf {P}}\{H-[wE]\}=J+{\frac {\partial D}{\partial t}}\\{\mathsf {P}}\{E+[wH]\}=-{\frac {\partial H}{\partial t}}\\\Gamma (D)=\rho ;\ \Gamma (H)=0\\D=E+P;\end{cases}}$

(L1)

Here, the hypotheses of Lorentz have to be supplemented. Let $w$ be parallel to $x$; then they read:

1. By a translation, every body suffers a deformation, so that length $r_{0}$ with components $x_{0}\ y_{0}\ z_{0}$ goes over to $r$ with components $x={\tfrac {x_{0}}{k}},\ y=y_{0},\ z=z_{0}$, where

$k^{2}={\frac {1}{1-w^{2}}}$

(1)

Following Lorentz, this shall be denoted by the symbol

$r=\left({\frac {1}{k}},\ 1,\ 1\right)r_{0}$.

(2)

2. When the distribution of electricity $e$ upon the material element is invariantly given, then all forces $F_{0}$ upon given particles are suffering a change by the translation, which is represented by the same symbolism

$F=\left(1,\ {\frac {1}{k}},\ {\frac {1}{k}}\right)F_{0}$

(3)

3. The motions which are carried out by a material particle of the progressing system under the action of forces $F$ in space $r$, are different from the motions, which the same particle in the case of rest is carrying out under the forces $F_{0}$ in space $r_{0}$, only by the fact that the process is slowed down in a constant ratio. Corresponding distances are traversed in times $t$ and $t_{0}$, which are connected by the equation

$t=kt_{0}\,$

(4)

If (with respect to a certain particle) $r_{0}\ t_{0}\ F_{0}$ forms a connected system of distances, times, forces in the case of rest, then $rtF$ – being in agreement with equations (1) to (4) – also belong together as a system of values that represent a possible state in the case of translation.

We apply the following theorems: in the theory of electrons it is given by definition, that

the "electric force" $E+[wH]$ is the spatial average of the force upon a particle, charged by the quantity of electricity 1;

the "electric moment of unit volume" $P={\tfrac {\Sigma (er)}{\tau }}$, where $r$ is the relative displacement of $e$, $\tau$ denotes a volume, and the sum is to be extended over all $e$ in this volume;

the "convection current" $J={\tfrac {\Sigma (eu)}{\tau }}$, where $u$ is the relative velocity of $e$.

Now, let $E_{0}\ P_{0}\ J_{0}$ be the connected values in the case of rest. According to (3),

$E+[wH)=\left(1,\ {\frac {1}{k}},\ {\frac {1}{k}}\right)E_{0}$

are corresponding to them for $e=e_{0}$ in the case of translation; furthermore, since $\tau ={\tfrac {\tau _{0}}{k}}$ according to (2), and $u=\left({\frac {1}{k^{2}}},\ {\frac {1}{k}},\ {\frac {1}{k}}\right)u_{0}$ according to (2) and (4), we have:

$P=(1,\ k,\ k)P_{0}$

$J=\left({\frac {1}{k}},1,1\right)J_{0}$.

Thus, if it is given for the state of rest:

$P_{0}=\eta E_{0}\,$ $J_{0}=\sigma E_{0}\,$

then consequently it follows for the translation:

${\begin{cases}P=(1,k^{2},k^{2})\eta \{E+[wH]\}\\J=\left({\frac {1}{k}},k,k\right)\sigma \{E+[wH]\}\end{cases}}$

(L2)

or written in a different manner:

$\left.{\begin{matrix}P=(\eta )\{E+[wH]\}&&(\eta )=\left(1,\ k^{2},\ k^{2}\right)\eta \\&where\\P=(\sigma )\{E+[wH]\}&&(\sigma )=\left(1,\ k,\ k\right)\sigma \end{matrix}}\right\}$

(L'2)

The values from (L'₂) are to be included into (L'₁). The differential equations for $E$ and $H$ with given coefficients, will follow. These equations have, as independent variables, the coordinates $xyz$ and the time $t$.

Instead of them, we will introduce new variables $x_{0}\dots t_{0}$ by the equations

$x_{0}=kx;\ y_{0}=y;\ z_{0}=z;\ t_{0}={\frac {t}{k}}.$

(L3)

Furthermore, we denote by $\rho _{0}$ the magnitude ${\tfrac {\rho }{k}}={\tfrac {de}{dx_{0}\cdot dy_{0}\cdot dz_{0}}}$, as well as by $P_{0}$ and $\Gamma _{0}$ the operators, which formally correspond to $P$ and $\Gamma$ in the system of the new variables. Eventually, we define the two new vectors E and M by:

${\begin{cases}{\mathsf {E}}=(1,k,k)\{E+[wH]\}\\{\mathsf {M}}=(1,k,k)\{H-[wE]\}\end{cases}}$

(L4)

from which it conversely follows due to (1)

${\begin{cases}E=(1,k,k)\{{\mathsf {E}}-[wM]\}\\H=(1,k,k)\{{\mathsf {M}}-[wE]\}\end{cases}}$

(L'4)

Then it arises:

${\begin{cases}&{\mathsf {P}}_{0}(M)=\sigma {\mathsf {E}}+{\frac {d{\mathfrak {E}}}{dt_{0}}}\\&{\mathsf {P}}_{0}({\mathsf {E}})=-{\frac {d{\mathfrak {M}}}{dt_{0}}}\\&\Gamma _{0}({\mathfrak {E}})=\rho _{0;}\ \Gamma _{0}({\mathfrak {M}})=0\\\\\mathrm {where} \\&{\mathfrak {E}}=(1+\eta ){\mathsf {E}}-[w{\mathsf {M}}]\\&{\mathfrak {M}}={\mathsf {M}}+[w{\mathsf {E}}]\end{cases}}.$.

(C')

From (C') the following can be derived: E and M are functions of $x_{0}\dots t_{0}w$, which contain $t_{0}$ and the translation velocity $w$ only in connection with:

$t_{1}=t_{0}-(w\cdot r_{0})$

Therefore, in the stationary state they do not depend on the translation at all, and with respect to radiation processes only in so far, as for every point a corresponding displacement of time takes place: the "local time" $t_{1}$ replaces the "general" quantity of time $t_{0}$.

Equations (C') were derived by us from the modified theory of electrons, which was recently given by Lorentz, to adapt them to the results of the experiments. These equations are now completely identical to those, which follow from my general equations for the special case of uniform velocity, which is treated here. They can be found in my treatise "On the Equations of the Electromagnetic Field for Moving Bodies"^[6] under (B₂) (C₃). When they are compared, it is only to be noticed that (with respect to the denotations employed there) we actually presupposed: $\mu =\mu _{0}$, $K=0$, and we have arbitrarily written:

$\sigma ,\ 1-\eta ,\ 1,\ 1,\ w$

instead of

$\lambda ,\ \epsilon ,\ \epsilon _{0},\ \mu _{0},\ p.$

§ 3. However, the interpretation of these equations by Lorentz and by me is different. In the case of Lorentz, is is about the representation of the two field strengths $E$ and $H$ as functions of $xyzt$ that define location and time. This is done by (L₁) and (L₂). The system, to which these equations apply, has the following properties: is is deformed in consequence of the motion in agreement with (2); it simultaneously changes its dynamical properties in agreement with the unified meaning of equations (2) (3) (4); it becomes anisotropic in the sense of a crystal of one axis, as it is indicated by equations (L'₃). In order to vividly present the electrodynamics of this system, the substitutions (L₃) and (L₄) were introduced. The $x_{0}\dots t_{0}$ are mere calculation quantities at the moment. However, at the same time they have a simple physical meaning. According to (2), $x_{0}\ y_{0}\ z_{0}$ are those measuring numbers being read at an "initially correct" measuring-rod (initially = when at rest), after it was introduced into the system and was accordingly deformed. And according to (2) (3) (4), $t_{0}\!$ are those time intervals indicated by an "initially correctly ticking" clock, after it was inserted into the system and accordingly has changed its rate.

In my view of equations (C'), the quantities E, M mean the field strengths, $x_{0}\dots t_{0}$ true coordinates and true times. They are identical with the measured coordinates and times. The moving system is not deformed, doesn't possess a specific measure of time, and hasn't become isotropic by motion.

§ 4. The Lorentzian interpretation requires from us, to distinguish between measured lengths and times $x_{0}\dots t_{0}$ and true ones $x\dots t$. But it doesn't give us the means to solve the task experimentally - even under presupposition of ideal measuring instruments. The Lorentzian electrodynamics and mechanics is only developed for $w=const$. Thus we have no other means at all, than to measure the distances with "false" co-moving measuring sticks, and to measure times with "false ticking" co-moving clocks. In order to use "correct" ones, that is, stationary instruments for measurements with respect to our moving system, a mechanics and optics must be given to us, which not only applies to the two ranges $w=const.$ and $w=0$, but which passes from one to the other through the range of variable $w$. - For the time being, the only meaning for the "true" lengths and times $x\dots t$ lies entirely in the fact, that the electrodynamics of equations (L₁)(L₂) applies to it (simultaneously with mechanics), which finds its expression in the hypotheses 1, 2, 3. They cannot be defined by any independent experience.

Thus, concerning Lorentz and me, it's only about two different kinds of expressing the same facts: either by (L₁)(L₂) and mechanics of theorems 1, 2, 3 – or by (C') and ordinary mechanics. No imaginable observation can distinguish between the two systems of explanation.

§ 5. A generalization of equations (C') for the case of velocities arbitrarily distributed in space, is represented by my "equations of the electromagnetic field...". It replaces, without changing anything in addition, the first two of equations (C') by the following ones:^[7]

${\begin{cases}\int \limits _{\circ }{\mathsf {M}}_{s}ds={\frac {d}{dt}}\int \limits _{s}{\mathfrak {E}}_{N}dS+\int \limits _{s}\Lambda _{N}dS\\\int \limits _{\circ }{\mathsf {E}}_{s}ds=-{\frac {d}{dt}}\int \limits _{s}{\mathfrak {M}}_{n}dS\end{cases}}$

(C)

where $S$ is a surface fixed in matter, and $s$ denotes a boundary curve.

These equations give, applied to closed surfaces, the known "equations of continuity" of electricity and magnetism, and lead back to (C') when applied to non-deformable surfaces. Conversely, as far as I can see, the form of equations (C) is specified by these two requirements. The vectors arising in them, can only be changed by additional vectors which contain derivatives of the velocities, i.e. in other words: any allowed generalization of (C'), which doesn't coincide with (C), should contain derivatives of second order when written in the form of differential equations. When one wants to avoid such complications, then any such theory – whatever its starting point may be – must lead to (C) in the general case, when it once formally led to (C') in the special case $w=const.$.

If these conclusions are justified, then everything that was said about the dual interpretation of special equations (C'), must transferred to the possible different interpretations of the general equations (C): then any conceptually expressible difference between them vanishes.

§ 6. The actual difference between the Lorentzian equation and the ones by me persists, as soon as one also considers the paramagnetic and diamagnetic bodies (what we have avoided here). My equations^[8], as those of Hertz, are symmetric as regards the electric and magnetic magnitudes, while the Lorentzian ones are not.^[9] This appears as an essential feature of the theory of electrons: its starting equations already show it. Also with respect to this relation, an experimentum crucis seems to be excluded, although at this place (contrary to other differences) a deviation in expressions of first order exists. I intend to explain this more closely in the near future, and simultaneously to present the essential content of my mentioned treatise again, namely in a (as I think) more satisfactory form.

§ 7. Against the applicability of my equations – more precisely spoken of my interpretation of equations (C') – Lorentz has raised an objection.^[10] According to these equations, as soon as $x_{0}\dots t_{0}$ are considered as true coordinates and times, the absolute (i.e. estimated with respect to a resting coordinate-system) speed of light $V$ in the direction $\nu$ is defined by ${\tfrac {1}{V-w_{\nu }}}={\sqrt {\eta +1}}+w_{v}$ (l.c., equation 16); thus in a medium as air, for which it is notably $\eta =0$:

$V=1+{\frac {w_{\nu }^{2}}{1+w_{\nu }}}$

(5)

However, in vacuum it must necessarily be $V_{0}=1$. The influence of the medium thus doesn't vanish only by the fact, that its electromagnetic constants are passing into that of vacuum, but only by the fact, that its velocity $w$ simultaneously takes the value which we have assumed in the vacuum once and for all time: the value 0. That there is a finite difference between $V$ and $V_{0}$, which doesn't depend on the density of the gas anymore, appears to Lorentz as an inadmissible consequence of my equations.

In opposition to that I want you to consider: imagine that Maxwell would have carried out his observations on the friction of gases before his theoretical investigation. As an experimental result, he had to say that the coefficient of friction $\varkappa$ is independent of density $\rho$. He presumably would have added, that this law of course cannot be valid up to the most extended rarefactions, yet that the last part of curve $\varkappa =f(\rho )$ which must be directed from the constant finite value to the point $\rho =0$, is unknown in terms of form and extension.

We are in a similar position with respect to function $V=f(\rho )$. A theory, which represents the properties of a continuum, must necessarily have a gap at the place, where the concept of the continuum fails. By that, however, also the limit for the applicability of equation (5) is given: when we think of a gas as rarefied to an extent, at which it cannot be spoken about the velocity of gas as a steady function of space anymore (which is a constant in our case), then the symbol $w$ has no meaning anymore. Then the ideas by which we have operated, vanish; in this field only an atomistic theory can try to represent the phenomena.^[11]

The dilemma is: Either the equation (5) actually is the case, so that the difference $V-V_{0}$ is caused by air (C). Or in reality it is $V=V_{0}$, so that the value of (5) is feigned by the deformation of the stone console (L) in the Michelson experiment. None of both assumptions may (in my view) be rejected by us for reasons of physical experience. The decision between the theories cannot be found here.

§ 8. In the preceding discussions, we only spoke about the influence of motion on the electrodynamics of extended bodies. This is only a province of the great empire, in which the theory of electrons offers itself as a guidance. In this field lie, however, the problems with respect to which the Lorentzian theory was developed, and in which it continuously found its test and regulator.

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1. Lorentz, Amsterdam Proceedings, 27. September 1902, p. 254. The citation above stems from the introduction. The same equations are in Mathematische Encyklopädie V, 2, p. 209.
2. See Math. Enc. V. 2, p. 215 ff.
3. See § 7. below.
4. Kon. Akad. van Wetensch. te Amsterdam Dl. XII, p. 986 (23. April 1904). Only this Dutch edition was accessible to me.
5. The now following derivations are using the same mathematical operations, by which Lorentz was led to his assumptions. As to the details of the calculations, it shall be referred to Lorentz l.c.
6. Göttinger Nachrichten 1901, Heft 1; also Ann. der Physik 7, p. 29. 1902. (Mainly already in Arch. Neerland. (2) 5, p. 516, 1900.)
7. Comp. l.c. under (B).
8. see l.c.
9. See Lorentz, Math. Enc. V 2, p. 238.
10. Math. Enc. V 2, p. 275 f.
11. I already have alluded l.c. in the introduction to the things said here.