# Finsler gravity action from variational completion

## TL;DR

This paper uses variational methods to derive a consistent Finsler gravity field equation, clarifying the limitations of previous proposals and formulating the action on a compact domain for mathematical rigor.

## Contribution

It demonstrates that Rutz's equation cannot be derived from an action and introduces a variational completion that yields the correct Finsler gravity equations, with a rigorous formulation on the positive projective tangent bundle.

## Key findings

- Rutz's equation is not variationally derivable.
- The variational completion produces the Finsler gravity field equations.
- Formulating the action on the positive projective tangent bundle improves mathematical rigor.

## Abstract

In the attempts to apply Finsler geometry to construct an extension of general relativity, the question about a suitable generalization of the Einstein equations is still under debate. Since Finsler geometry is based on a scalar function on the tangent bundle, the field equation which determines this function should also be a scalar equation. In the literature two such equations have been suggested: the one by Rutz and the one by one of the authors. Here we employ the method of canonical variational completion to show that Rutz equation can not be obtained from a variation of an action and that its variational completion yields the latter field equations. Moreover, to improve the mathematical rigor in the derivation of the Finsler gravity field equation, we formulate the Finsler gravity action on the positive projective tangent bundle. This has the advantage of allowing us to apply the classical variational principle, by choosing the domains of integration to be compact and independent of the dynamical variable. In particular in the pseudo-Riemannian case, the vacuum field equation becomes equivalent to the vanishing of the Ricci tensor.