Dual universe and hyperboloidal relative velocity surface arising from extended special relativity

Abstract

In two recent and completely independent articles (Hill and Cox in Proc R Soc A 468:4174, 2012; Vieira in Rev Bras Ensino F’s 34(3):1–15, 2012), the authors propose an extension of the Lorentz transformations of special relativity which are applicable to velocities in excess of the speed of light and do not involve the need to introduce imaginary quantities which are difficult to reconcile with everyday experience. These independent derivations, obtained from entirely distinct perspectives, mean that there is now some commonality of agreement in the basic equations underlying superluminal motion. One consequence of the new theories is that the standard special relativity formula for the addition of relative velocities also applies for velocities in excess of the speed of light. The new theories are based on the assumption that for any two inertial frames separated by an infinite relative velocity, the product of the two measured velocities for the same particle must necessarily be the square of the speed of light. Here, we amplify the major physical consequences embodied in the theory, including the surprising and novel idea of the co-existence of two “worlds”, such that in a subluminal world, everything is travelling with speeds less than the speed of light, while in the superluminal world, everything is travelling with speeds greater than the speed of light. We also establish the remarkable result that the reciprocal surface for the relative velocity formula can be re-orientated as a fully axially symmetric hyperboloidal surface, the full physical implications of which are not altogether transparent.