Birefringence in pseudo-Finsler spacetimes

TL;DR

This paper explores the feasibility of combining pseudo-Finsler geometries with birefringence in spacetime models, concluding that multiple metrics are more appropriate than a single unified pseudo-Finsler structure for different polarization modes.

Contribution

It investigates whether a single pseudo-Finsler metric can encompass multiple signal cones associated with birefringence, finding that separate metrics are more suitable.

Findings

Pseudo-Finsler spacetimes are useful but limited in handling multiple signal cones.

A single pseudo-Finsler metric cannot generally represent multiple birefringent signal cones.

Physics is better modeled with multiple pseudo-Finsler metrics, one per polarization mode.

Abstract

Based on the analogue spacetime programme, and many other ideas currently mooted in "quantum gravity", there is considerable ongoing speculation that the usual pseudo-Riemannian (Lorentzian) manifolds of general relativity might eventually be modified at short distances. Two specific modifications that are often advocated are the adoption of Finsler geometries (or more specifically, pseudo-Finsler spacetimes) and the possibility of birefringence (or more generally, multi-refringence). We have investigated the possibility of whether it is possible to usefully and cleanly deal with these two possibilities simultaneously. That is, given two (or more) "signal cones": Is it possible to naturally and intuitively construct a "unified" pseudo-Finsler spacetime such that the pseudo-Finsler metric is null on these "signal cones", but has no other zeros or singularities? Our results are much less encouraging than we had originally hoped, and suggest that while pseudo-Finsler spacetimes are certainly useful constructs, it is physically more appropriate to think of physics as taking place in a single topological manifold that carries several distinct pseudo-Finsler metrics, one for each polarization mode.