On the Finsler variational nature of autoparallels in metric-affine geometry

TL;DR

This paper investigates conditions under which autoparallels in certain metric-affine geometries can be derived from a Finsler variational principle, explicitly constructing Finsler Lagrangians where possible.

Contribution

It provides necessary and sufficient conditions for Finsler metrizability of torsion-free connections with vectorial nonmetricity and constructs explicit Finsler Lagrangians in these cases.

Findings

Identifies conditions for Finsler metrizability of specific affine connections.

Constructs explicit Finsler Lagrangians for metrizable connections.

Shows a broad class of such connections are Finsler metrizable.

Abstract

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not the extremals of any action integral. The existence of a parametrization-invariant action principle for autoparallels is a long-standing open problem, which is equivalent to the so-called Finsler metrizability of the connection -- that is, to the fact that these autoparallels can be interpreted as Finsler geodesics. In this article, we address this problem for the class of torsion-free affine connections with vectorial nonmetricity, which includes, as notable subcases, Weyl and Schr\"odinger connections. For this class, we determine the necessary and sufficient conditions for the existence of a Finsler Lagrangian that metrizes the connection (and depends only algebraically on the metric and on the nonmetricity defining vector field). In the cases where such a Finsler Lagrangian exists, we construct it explicitly. In particular, we show that a broad class of such connections is in fact Finsler metrizable, i.e., the autoparallels of these connections are Finsler geodesics.