Space-time measures for subluminal and superluminal motions

TL;DR

This paper generalizes the concept of causality and space-time measures to include both subluminal and superluminal motions, introducing a hierarchy of invariant speeds and a new causal structure.

Contribution

It proposes a novel framework extending local causality to all motion regimes, defining multiple invariant speeds and associated space-time intervals, and explores transitions between subluminal and superluminal states.

Findings

Introduces a set of denumerable invariant speeds c_k including the speed of light.

Defines k-timelike and k-null intervals and constructs a causal structure for each regime.

Analyzes possible discrete transitions between subluminal and superluminal motion regimes.

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.