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

Lehel Csillag, Nicoleta Voicu, Salah Elgendi, Christian Pfeifer

Abstract

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not extremals of any action integral. The existence of a parameter-invariant action principle for autoparallels is a longstanding open problem, which is equivalent to the so-called Finsler metrizability of the connection, i.e., 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 Schrödinger 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 it. 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 Finsler metrizable.

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

Abstract

In metric-affine geometry, autoparallels are generically non-variational, i.e., they are not extremals of any action integral. The existence of a parameter-invariant action principle for autoparallels is a longstanding open problem, which is equivalent to the so-called Finsler metrizability of the connection, i.e., 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 Schrödinger 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 it. 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 Finsler metrizable.
Paper Structure (10 sections, 9 theorems, 115 equations, 1 table)

This paper contains 10 sections, 9 theorems, 115 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symmetric affine connections with vectorial nonmetricity
  4. Finsler geometry
  5. Particular case: alphabeta-metrizability
  6. Generalized generalizedalphabeta- metrizability
  7. Main statement
  8. Some lemmas on necessary conditions
  9. Proof of the main statement
  10. Discussion

Key Result

Theorem 3.1

A connection with vectorial nonmetricity, locally given by with $c_1,c_2,c_3$ not all zero is pseudo-Finsler metrizable by a Berwald-type $(\alpha,\beta)$-metric $L=A \Phi(s)$ if and only if one of the following happens:

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 12 more