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 SO ( 3 ) , which has been previously studied in the literature, and extend these previous results to the full orthogonal group 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.
Highlights
By its geometric nature, the description of gravity within the theory of general relativity stands out from all other field theories
The most general metric-affine geometry with the latter kind of symmetry, to demonstrate how it follows from the spherically symmetric case which we study in detail
The starting point of our derivation is the notion of spacetime symmetry for metric-affine geometries, which is derived from a more general notion of symmetry in Cartan geometry [12], and which we briefly review here
Summary
The description of gravity within the theory of general relativity stands out from all other field theories. These parametrized geometries may directly be inserted into the field equations of any gravity theory based on the corresponding subclass of metric-affine geometries, in order to find its spherically symmetric solutions While such a parametrization has been provided already for the case of the symmetry group SO(3) of proper rotations [14], we extend this result to the full orthogonal group O(3), including reflections. The most general metric-affine geometry with the latter kind of symmetry, to demonstrate how it follows from the spherically symmetric case which we study in detail We emphasize that it is not the aim of this article to determine exact or approximate solutions to the field equations of any specific gravity theory or class of gravity theories.
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