The Finsler spacetime condition for (\alpha,\beta)-metrics and their isometries

TL;DR

This paper characterizes when (,eta)-metrics in pseudo-Finsler spaces form Finsler spacetimes with Lorentzian signature and explores their isometry groups, relevant for gravitational physics.

Contribution

It provides necessary and sufficient conditions for (,eta)-metrics to admit Finsler spacetime structures and analyzes their isometry groups in relation to underlying pseudo-Riemannian metrics.

Findings

Identified conditions for (,eta)-metrics to have Lorentzian signature

Determined the relation between (,eta)-metric isometries and underlying metric isometries

Listed all (,eta)-metrics with non-trivial isometry groups

Abstract

For the general class of pseudo-Finsler spaces with $(\alpha,\beta)$-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future pointing timelike vectors is ensured. The identified $(\alpha,\beta)$-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an $(\alpha,\beta)$-metric and the isometries of the underlying pseudo-Riemannian metric $a$; in particular, we list all $(\alpha,\beta)$-metrics which admit isometries that are not isometries of $a$.