Finsler geometric extension of Einstein gravity

TL;DR

This paper develops a Finsler geometric extension of Einstein gravity, formulating a new gravitational theory on Finsler spacetimes that generalizes Lorentzian manifolds and reduces to Einstein gravity in the metric limit.

Contribution

It introduces a gravitational action on Finsler spacetimes, defines observer measurements, and explores symmetries, extending Einstein gravity to a broader geometric framework.

Findings

Finsler gravity reduces to Einstein gravity in the metric limit.

Defined a groupoid of observer transformations generalizing the Lorentz group.

Applied the theory to Finsler refinements of the Schwarzschild solution.

Abstract

We construct gravitational dynamics for Finsler spacetimes in terms of an action integral on the unit tangent bundle. These spacetimes are generalizations of Lorentzian metric manifolds which satisfy necessary causality properties. A coupling procedure for matter fields to Finsler gravity completes our new theory that consistently becomes equivalent to Einstein gravity in the limit of metric geometry. We provide a precise geometric definition of observers and their measurements, and show that the transformations by means of which different observers communicate form a groupoid that generalizes the usual Lorentz group. Moreover, we discuss the implementation of Finsler spacetime symmetries. We use our results to analyze a particular spacetime model that leads to Finsler geometric refinements of the linearized Schwarzschild solution.