ХИМИЧЕСКАЯ ФИЗИКА, 2004, том 23, № 7, с. 80-82
ЭЛЕКТРИЧЕСКИЕ И МАГНИТНЫЕ ^^^^^^^^^^ СВОЙСТВА МАТЕРИАЛОВ
УДК 535.324
THE LORENTZ TRANSFORMATION SIGN AMBIGUITY AND ITS RELATION TO MEASURED FASTER-THAN-C PHOTON SPEEDS
© 2004 r. R. J. Buenker
Bergische Universität-Gesamthochschule Wuppertal, Fachbereich 9-Theoetische Chemie, Gaussstr. 20, D-42097 Wuppertal, Germany Received 11.12.2002
It is shown that time intervals At' measured for photons moving with speed v > c can be of the same sign for all observers according to special relativity, thereby avoiding any violation of Einstein causality. Previous assertions to the contrary have led to unnecessarily complicated interpretations of experiments which indicate that single photons do travel with faster-than-c speeds in regions of anomalous dispersion.
Shortly after the publication of Einsteins's original paper on special relativity in 1905 [1], it was pointed out by Wien [2] that a possible contradiction to the new theory might occur for light passing through a region of anomalous dispersion. Since the refractive index n can be less than unity under these conditions, in was argued that both the phase and group velocities of light can be greater than c, the speed of light in free space. Continuing to the present day, it has been widely assumed that special relativity precludes the possibility of faster-than-c speeds, so in 1907 detailed arguments were presented by Sommerfeld [3] to show that the above experimental phenomenon can be reconciled with this theory as long as the corresponding signal and energy velocities do not exceed v = c. In the last decade new experimental evidence has been resented [4, 5] which shows quite clearly that single photons do propagate with v > c speeds in the neighborhood of absorption lines, but there has been a general reluctance to accept this result at face value [4, 6], again because it is thought to be inconsistent with special relativity.
It is important to note that one of the objections to faster-than-c speeds is not applicable to the case of anomalous dispersion of light, and is in fact the only one actually mentioned by Einstein in his original work [1]. He pointed out that particles with non-zero proper mass would experience a singularity in their kinetic energy when accelerated to v = c and thus ruled out speeds at or above this value for any such entity, but by the same argument that this restriction does not apply to photons by virtue of their vanishing proper mass. There is another objection, however, namely the assertion that Einstein causality must be violated according to special relativity whenever v > c because of a consequence of the Lorentz transformation [7]. This second difficulty is potentially relevant for photons and implies that the time order of two events can be opposite for two observers moving with different relative speeds, something which is justifiably considered to be strictly unphysical and thus to stand in contradiction to special relativity.
In view of the lingering uncertainties in the interpretation of the recent experiments dealing with the anomalous dispersion phenomenon, it is well to reexamine the latter objection to faster-than-c speeds. The Lorentz transformation for space and time intervals Ax and At are given below for two observes moving at relative speed u in the x direction:
Ax '( u ) = £ x ( u )y( u )(Ax - u At ), A t'( u ) = £t ( u )y( u )^A t Axjj,
(1)
where y = (1 - u2/c2)-1/2. The quantities ex and et must be chosen to be consistent with the relativistic invariance condition for the two observers in their respective (primed and umpired) inertial systems, namely
2T-2
Ax'(u) - c At'(u) = Ax - c At
(2)
whereby the corresponding spatial intervals in the transverse directions for the two observers are taken to be equal in arriving at the above expression. This re-
2 2
quirement leads to the conditions, £ x = £t = 1, and thus to an ambiguity in the respective signs of these two quantities. This uncertainty is normally removed by noting that the above equations must reduce to the Galilean transformation for small relative speed u. Consequently, both £x and £t are assigned a value of +1, and by virtue of the physical requirement of continuous variation in both Ax'(u) and At'(u), it is argued that this result must hold quite generally for all accessible relative speeds u < c.
In the case of present interest the speed v of photons in a dispersive medium with a group index of refraction ng ng is defined as
v =
Ax At
c_
n„
(3)
THE LORENTZ TRANSFORMATION SIGN AMBIGUITY
81
Substitution in Eq.(1) gives
Ax' (u) = £x (u )y( u )Ax I 1-
V v.
= £ x( u )Y( u )Ax V1 -nir), At' = £ (u )Y( u )A t V1 -^pr) =
= £t (u )Y( u )A 11 1 -V '
(4)
ngc
where again £x and £t are shown explicitly as in Eq.(1). The conventional Einstein causality argument [7] then goes as follows. If ng < 1, the quantity in parentheses in the expression for At' can be negative for c > u > ngc. By assuming as above that et = 1, it is concluded that in this range of uAt' must have the opposite sign as At, a clear violation of Einstein causality.
The latter argument overlooks an important point, however. Since At' vanishes in Eq.(4) for u = ngc, it is no longer required on continuity grounds that et = +1 for this value of u. All that is required physically is that At' itself be a continuous function of u, and this objective can be accomplished with either choice of sign for £t(u) at the critical speed u = ngc.
In Fig. 1 both Ax' and cAt' are given as a function of P = u/c for a dispersive medium with ng = 0.5. In this diagram ex(u) always has a value of+1 but et(u) changes discontinuously from +1 to -1 at P = 0.5 (motion away from the light source). The relativistic invariance condition of Eq.(2) is everywhere satisfied and a violation of Einstein causality does not occur, i.e. At' is never opposite in sign to At. Most importantly, both Ax' and At' are continuous functions of u with the above choices for £x(u) and £t(u), so that all physical constraints are satisfied.
One other consequence of the discontinuity in £t at P = ng is that the derivative dAt'/dfi is not defined at this relative speed, but there in no compelling argument against this result. The observer simply measures a monotonically decreasing time interval as P is gradually increased up to the critical value of ng, at which point At' vanishes (Fig. 1). As P is further increased, the measured time interval begins to increase monotonically, always with the same sign as At. The spatial interval Ax' also reaches a minimum value at P = ng and retains the same sign over the entire range of accessible relative speeds.
There remains one other conceivable objection to faster-than-c speeds, namely the fact that the measured velocity V = Ax'/At' has a singularity at P = ng by virtue of the vanishing of At' at this relative speed (see Fig. 1). It is only a point singularity, however, which means that the observer would have to be moving at exactly the critical relative speed P = ng to measure other than finite values for V according to the present theory. It is important to recall that the arguments under discussion
20 15 10
1 0 -1
-5
1 1 1 i Ax'/ cAt I j l i l i
l i 1 \ 1
vs. 1
1 1 1
-1.0
-0.5
0.5
ß
Schematic diagram showing the variation of the spatial and time intervals Ax' and c At' as a function of the relative speed P = u/c of the observer for a photon traveling at twice the speed of light in free space (ng = 0.5) in the inertial system with P = 0. Their ratio (measured velocity) is also shown. Note that At' is always positive in this diagram by virtue of the choice of sign £t(u) in the corresponding Lorentz transformation of Eq.(4).
only apply to photons moving through an anomalously dispersive medium. This implies that the range over which unlimited speeds can be observed is quite restricted, since it is a practical impossibility to construct such a medium with a uniform group index of refraction which extends over more than a short distance.
The latter considerations are significant in another context, however. There has been speculation [8] that particles with imaginary proper masses (tachyons) might exist which would necessarily travel with faster-than-c speeds through free space. If such a particle were observed, special relativity again demands that there be a critical relative speed u(c2/v) for which At' vanishes and thus the measured speed of the tachyon would be infinite in the corresponding inertial system. Since the particle could attain this speed in free space, there is nothing in special relativity to prevent its being observed simultaneously in parts of the universe which are many light years apart. Although one can't fully rule out such a prospect, it is far less difficult to imagine that an observer moving at close to the speed of light relative to a laser source would measure a null value for the elapsed time taken by a single photon to cover a distance of only a few ^m [4, 5]. It also should be noted that the same argument that has been used above for space-time intervals can also be applied to energy-momentum four-vectors. Hence, by proper choice of the sign employed in the corresponding Lorentz transformation it is also possible to avoid the conclusion [8] that negative-energy states are the inevitable consequence of faster-than-c tachyon speeds.
With the above choice of signs for the Lorentz transformation it is seen that the velocity of the faster-than-
5
0
82
BUENKER
c photon is in the same direction for all observers regardless of their relative speed u. This result is thus in sharp contrast to what is obtained when At' is allowed to change sign as the critical speed u = ngc is surpassed, in which case Ax'/At' changes abruptly from to at this point. Since the relative speed of the observer is always less than the photon's speed c/ng for ng < 1 it must be expected that the perceived direct
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