Velocity requirements for causality violation
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We re-examine the "Regge-Tolman paradox" with reference to some recent experimental results. It is straightforward to find a formula for the velocity v of the moving system required to produce causality violation. This formula typically yields a velocity very close to the speed of light (for instance, v/c > 0.97 for X-shaped microwaves), which raises some doubts about the real physical observability of the violations. We then compute the velocity requirement introducing a delay between the reception of the primary signal and the emission of the secondary. It turns out that in principle for any delay it is possible to find moving observers able to produce active causal violation. This is mathematically due to the singularity of the Lorentz transformations for beta to 1. For a realistic delay due to the propagation of a luminal precursor, we find that causality violations in the reported experiments are still more unlikely (v/c > 0.989), and even in the hypothesis that the superluminal propagation velocity goes to infinity, the velocity requirement is bounded by v/c > 0.62. We also prove that if two macroscopic bodies exchange energy and momentum through superluminal signals, then the swap of signal source and target is incompatible with the Lorentz transformations; therefore it is not possible to distinguish between source and target, even with reference to a definite reference frame.