Space-time measures for subluminal and superluminal motions

Abstract: In present work we examine the implications on both, space-time measures and causal structure, of a generalization of the local causality postulate by asserting its validity to all motion regimes, the subluminal and superluminal ones. The new principle implies the existence of a denumerable set of metrical null cone speeds, {$c_k}$, where $c_1$ is the speed of light in vacuum, and $c_k/c \simeq \epsilon^{-k+1}$ for $k\geq2$, where $\epsilon^2$ is a tiny dimensionless constant which we introduce to prevent the divergence of the $x, t$ measures in Lorentz transformations, such that their generalization keeps $c_k$ invariant and as the top speed for every regime of motion. The non divergent factor $\gamma_k$ equals $k\epsilon^{-1}$ at speed $c_k$. We speak then of $k-$timelike and $k-$null intervals and of k-timelike and k-null paths on space-time, and construct a causal structure for each regime. We discuss also the possible transition of a material particle from the subluminal to the first superluminal regime and vice versa, making discrete changes in $v^2/c^2$ around the unit in terms of $\epsilon^2$ at some event, if ponderable matter particles follow k-timelike paths, with $k=1,2$ in this case.