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A note on Lorentz-like transformations and superluminal motion

In this extended note a critical discussion of an extension of the Lorentz transformations for velocities faster than the speed of light given recently by Hill and Cox is provided. The presented approach reveals the connection between faster-than-light speeds and the issue of isotropy of space. It is shown if the relative speed between the two inertial frames $v$ is greater than the speed of light, the condition of isotropy of space cannot be retained. It further specifies the respective transformations applying to $-\infty<v<-c$ and $c<v<+\infty$. It is proved that such Lorentz-like transformations are improper transformations since the Jacobian is negative. As a consequence, the wave operator, the light-cone and the volume element are not invariant under such Lorentz-like transformations. Also it is shown that such Lorentz-like transformations are not new and already known in the literature.