Speculative Superluminal Regime for Relativistic Addition of Velocities

Using Einstein's theory of special relativity, suppose that two observers (red points at the top) are in relative motion with velocities and relative to an observer at rest, with velocities comparable with the speed of light. Each observer sees that the distance to the other changes, but not at the rate [1], and they never move apart by more than the velocity of light. In [2] Hill and Cox have extended the mathematical framework of special relativity to describe motion at speeds faster than light, but this is not necessarily physically meaningful.

The plots show the curve for a fixed value , and the red dot shows a particular value of .

[2] J. M. Hill and B. J. Cox, "Einstein's Special Relativity beyond the Speed of Light," Proceedings of the Royal Society A, 468, 2012 pp. 4174–4192. doi:10.1098/rspa.2012.0340.