Pseudo-Finsler spaces modeled on a pseudo-Minkowski space

TL;DR

This paper develops explicit formulas for key geometric objects in pseudo-Finsler spaces based on pseudo-Minkowski models, facilitating computations and exploring properties like Berwald spaces and symmetries.

Contribution

It provides closed-form expressions for geometric objects in pseudo-Finsler spaces modeled on pseudo-Minkowski space, including applications to Lorentz-Finsler and Finslerian pp-wave metrics.

Findings

Derived explicit formulas for Berwald's curvature, Landsberg's tensor, and Ricci scalar.

Identified conditions for Berwald property in Lorentz-Finsler spaces.

Showed that non-trivial Berwald spaces have indicatrices with non-trivial symmetry groups.

Abstract

We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These expressions are particularly convenient in computations since they factor the dependence on the base and the fiber. As an illustration, we study Lorentz-Finsler spaces modeled on the Bogoslovsky Lorentz-Minkowski space, and give sufficient conditions which guarantee the Berwald property. We then specialize to a recently proposed Finslerian pp-wave metric. Finally, the paper points out that non-trivial Berwald spaces have necessarily indicatrices which admit some non-trivial linear group of symmetries.