Metric-affine Geometries With Spherical Symmetry

TL;DR

This paper classifies all spherically symmetric metric-affine geometries, including torsion, curvature, and nonmetricity, providing a foundation for solving gravity field equations with these geometries in various formulations.

Contribution

It offers a comprehensive classification of spherically symmetric metric-affine geometries, extending previous work to include reflections and analyzing their properties for gravity theories.

Findings

Classification of all subclasses based on geometric quantities

Extension from rotation group SO(3) to full O(3) including reflections

Application to autoparallel circular orbits

Abstract

We provide a comprehensive overview of metric-affine geometries with spherical symmetry, which may be used in order to solve the field equations for generic gravity theories which employ these geometries as their field variables. We discuss the most general class of such geometries, which we display both in the metric-Palatini formulation and in the tetrad / spin connection formulation, and show its characteristic properties: torsion, curvature and nonmetricity. We then use these properties to derive a classification of all possible subclasses of spherically symmetric metric-affine geometries, depending on which of the aforementioned quantities are vanishing or non-vanishing. We discuss both the cases of the pure rotation group $\mathrm{SO}(3)$, which has been previously studied in the literature, and extend these previous results to the full orthogonal group $\mathrm{O}(3)$, which also includes reflections. As an example for a potential physical application of the results we present here, we study circular orbits arising from autoparallel motion. Finally, we mention how these results can be extended to cosmological symmetry.