SUPERLUMINAL VELOCITIES 59 conditions on the magnetic field. After formulating another, more general definition of signal velocity, Einstein abandons the project of defining signal velocities for ab- sorptive media, pointing out that an approach using Fourier analysis might be neces- sary.[16] He maintains, however, on the general grounds provided by his interpretation of Wiechert's result, that dispersion theory, because it recognizes no forces that are not electromagnetic in origin, precludes superluminal signal propagation. Wien was not convinced by Einstein's arguments[17] and remained concerned with the problem of superluminal velocities. Discussions with Sommerfeld on this top- ic[18] in the same summer led the latter to a systematic study, the results of which he presented at the meeting of the Gesellschaft Deutscher Naturforscher und Arzte in Dresden in September 1907.[19] The paper only gives an outline of the calculations that had led Sommerfeld to conclude that the speed of the front of a wave-signal can never exceed the speed of light in vacuum full details were only given several years later.[20] In the discussion following the lecture Wien still showed himself uncon- vinced: he stated that the physical meaning of Sommerfeld's calculation escaped him. And in a review paper on radiation theory which he completed in November 1908 he pointed out that the problem of what constitutes the group velocity in dis- persive media had by no means yet been resolved.[21] [1]The letters from Wien to Einstein from this period are not available. [2]See Heaviside 1889. [3]See Miller 1981, chap. 1, for a review of electron theories, and pp. 118-119 for more on the early debate on superluminal speeds. [4]See Sommerfeld 1904a, 1904b, 1904c, 1905. [5]See Wiechert 1905. [6]The discussion, which became rather sharp in tone, dealt with a variety of topics concern- ing radiation theory. Wien's point of view on superluminal speeds is expressed in Wien 1904a, p. 637, in response to Abraham's criticism in Abraham 1904b. [7]"physikalisch wenig wahrscheinlich" (Wien 1906, p. 35). [8]The nonphysicality of such speeds is demonstrated through several examples: the infinite contraction of objects moving at the speed of light (p. 903), the infinite intensity that a light- source would have for an observer approaching it at the speed of light (p. 912), and the infinite amount of energy that would be needed to accelerate an electron to the speed of light (p. 920). Einstein's conclusion is that superluminal velocities have "no possibility of existence" ("keine Existenzmöglichkeit") in his theory. [9]See Einstein 1907h (Vol. 2, Doc. 45), pp. 381-382. Einstein 1907j (Vol. 2, Doc. 47), sub- mitted in December of 1907, repeats the argument without significant modification (pp. 423- 424). [10]"Wenn dies Resultat auch, meiner Meinung nach, rein logisch genommen keinen Wider- spruch enthält, so widerstreitet es doch so unbedingt dem Charakter unserer gesamten Erfah- rung, daß durch dasselbe die Unmöglichkeit der Annahme W V zur Genüge erwiesen ist." (Einstein 1907h [Vol. 2, Doc. 45], p. 382). V is the speed of light, W the speed with which the signal is propagated. [11]This was in particular pointed out by Max Laue in a discussion of the propagation of ra- diation in dispersive and absorptive media (Laue 1905). Laue concluded that the group veloc- ity is the velocity with which energy is transported by radiation, except in highly absorptive media, where the concept of group velocity can no longer be used. The history of the concepts of group velocity and phase velocity goes back to the work of Rayleigh. See, e.g., Brillouin 1960, chap. 1, for a brief historical review.
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