Inertial Transformations and the Nonexistence of Tachyons for Spacetime Dimension Greater than Two

TL;DR

This paper proves that in higher-dimensional spacetimes, inertial transformations cannot support faster-than-light (tachyonic) motions, extending the understanding of relativistic invariance beyond two dimensions.

Contribution

It demonstrates the nonexistence of tachyonic inertial transformations in spacetime dimensions greater than two, generalizing previous results and clarifying the structure of inertial transformations.

Findings

In 2D, standard Lorentz transformations and tachyonic transformations are characterized.

For dimensions greater than 2, tachyonic inertial transformations are shown to be incompatible with constant light speed.

The conformal factor in transformations is determined under continuity and differentiability assumptions.

Abstract

We consider real linear transformations between two inertial frames with constant relative speed $v$ in a $d$-dimensional spacetime where light moves with constant speed $c=1$ (for some chosen units) in all frames. For $d=2$ we show that the standard relative velocity formula holds and that any associated anisotropic conformal factor is multiplicative under composition of inertial transformations for $|v|\neq 1$. Assuming that the inertial transformation matrix is continuous in a neighbourhood of $v=0$ and differentiable at $v=0$, we determine the conformal factor for all $|v|\neq 1$. For an isotropic spacetime, the general solution reduces to the standard $d=2$ Lorentz transformation for $|v|<1$ or to a Tachyonic transformation for $|v|>1$, first described by Parker in 1969. For $d>2$ we show that no Tachyonic-like inertial transformations exist which are compatible with constant light speed.