# Symmetries and conserved quantities with arbitrary torsion: A generalization of Killing's theorem

## TL;DR

This paper generalizes Killing's theorem to spacetimes with torsion by introducing T-Killing vectors, enabling the analysis of autoparallels and revealing conserved quantities in torsionful geometries.

## Contribution

It introduces T-Killing vectors, extending the concept of symmetries and conserved quantities to geometries with arbitrary torsion, and applies this to autoparallel equations.

## Key findings

- T-Killing vectors have constant contraction with autoparallel tangents.
- In static, spherically symmetric spacetimes, autoparallel equations reduce to a one-dimensional problem.
- The framework generalizes symmetry analysis to torsionful geometries.

## Abstract

When spacetime torsion is present, geodesics and autoparallels generically do not coincide. In this work, the well-known method that uses Killing vectors to solve the geodesic equations is generalized for autoparallels. The main definition is that of T-Killing vectors: vector fields such that, when their index is lowered with the metric, have vanishing symmetric derivative when acted with a torsionfull and metric-compatible derivative. The main property of T-Killing vectors is that their contraction with the autoparallels' tangents are constant along these curves. As an example, in a static and spherically symmetric situation, the autoparallel equations are reduced to an effective one-dimensional problem. Other interesting properties and extensions of T-Killing vectors are discussed.