Translation:An Objection Against the Theory of Relativity and its Removal

by Arnold Sommerfeld

An Objection Against the Theory of Relativity of Electrodynamics and its Removal.

by A. Sommerfeld.

In electrodynamics, which was founded by Einstein on the principle of relativity, superluminal velocities are impossible under any circumstances, both as convection velocity of a charge or as propagation velocity of an electrodynamic disturbance.^[1] The conditions within anomalous dispersive media apparently speak against the latter conclusion, as it was emphasized by W. Wien in conversation. Here we have, at least for certain wavelengths, in any case phase velocities > c. We want to speak about signal velocity - following Einstein's terminology, and to avoid the word group velocity that is not easily related to individual processes, but rather to natural processes that are statistically given - and understand it as the velocity at which a suddenly starting signal, i.e. a sequence of electrodynamic oscillations of a specific wavelength, is perceived by an observer inside the dispersive medium. According to Einstein, it would be an objection against the relativity theory, when this signal velocity may be under any circumstances > c. Now we shall see that it is exactly equal to c, if we think of the observer as equipped with an ideal, i.e. infinitely sensitive detector, no matter whether the medium is dispersive in a normal or anomalous way, whether it is isotropic or anisotropic, whether it contains conduction electrons or not. The signal velocity and the process of propagation has nothing to do with the phase velocity. Therefore nothing stands in the way of the relativity theory from that side.

The proof can be provided as follows: we define the signal that is perpendicularly striking on the medium, as a function of

$t\left(f(t)=0\ {\mathsf {for}}\ t<-T\ {\mathsf {and\ for}}\ t>+T,\qquad f(t)=\sin 2\pi /\tau \ {\mathsf {for}}\ -T<t<+T\right)$

by a Fourier integral; then one also knows its propagation in the dispersive medium into its normal x as a function $f(t,x)$ of t and x. By treating the thus formed integral by Cauchy's methods of complex plane, it is seen, 1. that $f(t,x)=0$ if $t-x/c<-T$^[2], 2. that in the interval $-T<t-x/c<+T$ the disturbance consists of two parts, a forced time-undamped oscillation, in accordance with the phase shifted signal, and a damped free oscillation of the electrons, which were excited by the arrival of the signal, 3. that there remains for the following interval $+T<t-x/c$ a residue of free electron oscillation consisting also of two parts, the one excited by the arrival, the other by the end of the signal.

Discussion.

Wien (Würzburg): I'm thanking Sommerfeld for the readiness with which he has done this calculation, and I must bow to the power of his mathematical analysis, but I can not say that this case is now physically clear to me. If I think of the number of waves as more and more enlarged, then I get a finite wave train that I can consider as a signal. An actually infinite wave is traveling with superluminal velocity (Sommerfeld: only the phase, not the energy). If I have a very long wave train that is limited, then it should travel only with the speed of light and that is not clear to me. I want to ask after the propagation velocity of the wave that is in the middle. If the train is actually infinite in length, then it travels with superluminal velocity; if it is only very long and is limited, it should travel with the speed of light or less then the speed of light. If this is what the mathematical analysis gives I can say nothing against it, but it is not clear to me. There must be an effect of the wave front on all of the following waves, no matter how distant they are.

Braun (Strasbourg): If the matter is carried out mathematically, then it should be possible to express the essence of this without mathematics.

Voigt (Göttingen): I think that in the light of electron theory, an idea of the results obtained by Sommerfeld can be formed as it was wished by Braun, at least to a certain extent.

The modern theory of dispersion and absorption operates with the assumption of extensionless but inertial masses of the electrons embedded in the aether (or they have an extension and the aether penetrates them with unaltered properties). From the assumption of inertia, however, follows directly that these masses cannot affect the beginning of the wave in the aether. Only in the course of it, they start to move and act backwards on the wave. Accordingly, Sommerfeld's result that the wave front propagates in all circumstances with the speed of light of empty space, is not at all surprising for me. This propagation happens just exactly as in empty space.

If afterwards in the continuing course of the oscillations, the formulas indicate the individual propagation velocities for substance and color, then this means only: the movement during the evolution of the wave differs in such a way from the exact periodic, that ultimately the phase occurs that corresponds to that velocity. Now, a detailed examination should show that our methods of determining the velocity of light are certainly based on the phase, as are the deflection observations, because the wave plane is indeed the level of constant phase. Thus we arrive at measurements of light velocities, which vary with the substance and the color of light. We will never notice the propagation of a single wave crest. We even have hardly any means to denote it and to find it again in the course of propagation.

Wien: I would like to know the propagation velocity of the center.

Des Coudres: The mathematical image of a signal can never be a sine series, but only a definite integral with infinite limits. Even with a (beginning at a certain moment of time) infinitely long duration signal, the asymmetry in the formulas persists with respect to the disturbances that are on both sides infinitely extended in time. Terms as phase, refractive index and propagation velocity have only a meaning for the latter. According to the integral formulas, the part of the disturbance which was described in the discussion as the tail already begins to physically appear together with the appearance of the head.

(Received October 30, 1907.)

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1. Ann. d. Phys. 23, 381, 1907.
2. In accordance to the considerations of M. Laue, Ann. d. Phys. 18, 550, 1905.