Variational principle and free falling in a space-time with torsion

TL;DR

This paper compares two variational principles for describing free-falling particles in space-times with torsion, highlighting a new approach using a modified metric that reproduces autoparallel trajectories without torsion restrictions.

Contribution

It introduces a novel world-line action with a modified metric to derive autoparallel trajectories in torsioned space-times, expanding on traditional variational methods.

Findings

A fixed-ends variational principle reproduces autoparallel trajectories.

The approach works without restrictions on space-time torsion.

Illustrated with perturbative Weitzenböck space-time.

Abstract

A comparison between the two possible variational principles for the study of a free falling spinless particle in a space-time with torsion is noted. It is well known that the autoparallel trajectories can be obtained from a variational principle based on a non-holonomic mapping, starting with the standard world-line action. In a contrast, we explore a world-line action with a modified metric, thinking about the old idea of contorsion (torsion) potentials. A fixed-ends variational principle can reproduce autoparallel trajectories without restrictions on space-time torsion. As an illustration we have considered a perturbative Weitzenb$\ddot{o}$ck space-time. The non-perturbative problem is stablished at the end.