Superluminal velocity

One of the reasons for the studies on the dynamical Casimir effect was Schwinger s hypothesis [153-157] that this effect could explain the sonolumi-nescence phenomenon, specifically, the emission of bright short pulses of the visible light from the gas bubbles in the water, when the bubbles pulsate because of the pressure oscillations in a strong standing acoustic wave. (Several reviews and numerous references related to this effect are available, [121,326-328].) There are several publications [329-331], whose authors considered the models giving tremendous numbers of photons that could be produced even in the visible range as a result of the fast motion of the boundaries. However, analysis of these models shows that they are based on such laws of motion of the boundaries that imply the superluminal velocities, so they are not realistic. 

Fig. 14.1 Discussion of the effect of the superluminal velocity of laser pulses in a nonlinear amphfier with Professor C. H. Townes (left) during his visit to the USSR (P. N. Lebedev Physical Institute) in the middle 1960s (the author is on the right).

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

The original aim of the Florence group was just to repeat the experiment by T. K. Ishii and G. C. Giakos, who claimed in 1991 to have observed, by a similar device, superluminal propagation in free space. Such an experiment was basically flawed by the confusion made by the authors between phase and group velocity. 

Figure 6. The shape of a spherical particle as its appears in (a) its rest frame, (b) in a frame moving with subluminal (but relativistic) velocity, (c) a frame moving at the light velocity (an unphysical case ), and (d) a superluminal frame. This last figure clearly shows that, according to ER, tachyons are X-shaped objects (waves). A spatial dimension has been dropped for simplicity of representation. (From Barut et al. [50].)...

There is presently no serious doubt about the actual evidence for superluminal group velocities, provided by the experiments on photon tunneling and X-shaped waves. On the contrary, it is still a subject of debates and controversies the question whether such a superluminal behavior does imply violation... 

The mysterious phase velocity of the de Broglie wave and the group velocity of the amplitude wave, c2/ > c, refer to the, by now familiar superluminal motion in the interior of the electron. As many authors noted and Molski(1998) recently reviewed [86] an attractive mechanism for construction of dispersion-free wave packets is provided in terms of a free bradyon4 and a free tachyon that trap each other in a relativistically invariant way. It is demonstrated in particular how an electromagnetic spherical cavity may be... 

Re Entry [86], Ref. [2]) Special Relativity permits arbitrarily fast superluminal phenomena that transmit no mass-energy or information, as well as mutual velocities up to 2c see pp. 56 and 70 of Ref. [2], Section 2.10 of Ref. [2] states that the speed U of transmission of information must not exceed c if violation of causality is to be prevented in Special Relativity. But Eqs. (2.21) and (2.22) in Sect. 2.10 of Ref. [2] at least suggest the possibility that Special Relativity may be consistent with a somewhat less conservative limit, namely U < c2/v, where v is the relative velocity between the transmitter and receiver. Of course to guarantee causality Nature must then have a method to checkmate any attempt by the transmitter and/or receiver to "cheat" by increasing v while a signal is en route. 

The surprising implication is that Dirac s equation does not allow of a self-consistent single-particle interpretation, although it has been used to calculate approximate relativistic corrections to the Schrodinger energy spectrum of hydrogen. The obvious reason is that a 4D point particle is without duration and hence undefined. An alternative description of elementary units of matter becomes unavoidable. Prompted by such observation, Dirac [3] re-examined the classical point model of the electron only to find that it has three-dimensional size, with an interior that allows superluminal signals. It all points at a wave structure with phase velocity > c. 

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