Birkhoff Theorem for Berwald Finsler spacetimes

TL;DR

This paper extends the classical Birkhoff theorem to Berwald Finsler spacetimes, showing that Ricci-flat, spherically symmetric cases are either flat or pseudo-Riemannian, thus bridging Finsler and Riemannian geometries.

Contribution

It proves a Birkhoff-type theorem for Berwald Finsler spacetimes, revealing their structure under Ricci-flat and spherical symmetry conditions.

Findings

Ricci-flat, spherically symmetric Berwald spacetimes are either flat or pseudo-Riemannian.

The Birkhoff theorem is generalized to Berwald Finsler spacetimes.

Berwald spacetimes closest to Riemannian geometry under specified conditions.

Abstract

Finsler spacetime geometry is a canonical extension of Riemannian spacetime geometry. It is based on a general length measure for curves (which does not necessarily arise from a spacetime metric) and it is used as an effective description of spacetime in quantum gravity phenomenology as well as in extensions of general relativity aiming to provide a geometric explanation of dark energy. A particular interesting subclass of Finsler spacetimes are those of Berwald type, for which the geometry is defined in terms of a canonical affine connection that uniquely generalizes the Levi-Civita connection of a spacetime metric. In this sense, Berwald Finsler spacetimes are Finsler spacetimes closest to pseudo-Riemannian ones. We prove that all Ricci-flat, spatially spherically symmetric Berwald spacetime structures are either pseudo-Riemannian (Lorentzian), or flat. This insight enables us to generalize the Jebsen-Birkhoff theorem to Berwald spacetimes.