Mathematical foundations for field theories on Finsler spacetimes

TL;DR

This paper develops a mathematical framework for formulating and analyzing field theories on Finsler spacetimes, extending variational calculus and conservation laws to this generalized geometric setting.

Contribution

It introduces configuration bundles for homogeneous fields on Finsler spacetimes and proves a generalized energy-momentum conservation law.

Findings

Established a coordinate-free variational calculus for Finsler fields

Derived a generalized energy-momentum conservation law

Connected Finsler conservation laws to Lorentzian case

Abstract

The paper introduces a general mathematical framework for action based field theories on Finsler spacetimes. As most often fields on Finsler spacetime (e.g., the Finsler fundamental function or the resulting metric tensor) have a homogeneous dependence on the tangent directions of spacetime, we construct the appropriate configuration bundles whose sections are such homogeneous fields; on these configuration bundles, the tools of coordinate free calculus of variations can be consistently applied to obtain field equations. Moreover, we prove that general covariance of natural Finsler field Lagrangians leads to an averaged energy-momentum conservation law which, in the particular case of Lorentzian spacetimes, is equivalent to the usual, pointwise energy-momentum covariant conservation law.