Inertial frames without the relativity principle: breaking Lorentz symmetry

TL;DR

This paper explores the concept of inertial frames without Lorentz invariance, proposing a framework where transformations form a groupoid rather than a group, and analyzing the implications for energy-momentum transformations.

Contribution

It introduces a novel approach to inertial frames without the relativity principle, replacing group structures with groupoids and examining affine energy-momentum transformations.

Findings

Transformations between inertial frames form a groupoid/pseudogroup.

Energy and momentum transformation rules are generally affine.

A specific model implementing minimal Lorentz invariance violation is analyzed.

Abstract

We investigate inertial frames in the absence of Lorentz invariance, reconsidering the usual group structure implied by the relativity principle. We abandon the relativity principle, discarding the group structure for the transformations between inertial frames, while requiring these transformations to be at least linear (to preserve homogeneity). In theories with a preferred frame (aether), the set of transformations between inertial frames forms a groupoid/pseudogroup instead of a group, a characteristic essential to evading the von Ignatowsky theorems. In order to understand the dynamics, we also demonstrate that the transformation rules for energy and momentum are in general affine. We finally focus on one specific and compelling model implementing a minimalist violation of Lorentz invariance.