Algebraic classification of the gravitational field in general metric-affine geometries

TL;DR

This paper develops a comprehensive algebraic classification of the gravitational field in four-dimensional metric-affine geometries, extending previous classifications to include traceless nonmetricity and applying it to black hole solutions.

Contribution

It introduces a new algebraic classification scheme for the curvature tensor in metric-affine geometries, incorporating traceless nonmetricity and identifying sixteen main algebraic types.

Findings

Identified sixteen main algebraic types of the curvature tensor.

Extended classification to include traceless nonmetricity tensor.

Applied classification to static, spherically symmetric black hole solutions.

Abstract

We present the algebraic classification of the gravitational field in four-dimensional general metric-affine geometries, thus extending the current results of the literature in the particular framework of Weyl-Cartan geometry by the presence of the traceless nonmetricity tensor. This quantity switches on four of the eleven fundamental parts of the irreducible representation of the curvature tensor under the pseudo-orthogonal group, in such a way that three of them present similar algebraic types as the ones obtained in Weyl-Cartan geometry, whereas the remaining one includes thirty independent components and gives rise to a new algebraic classification. The latter is derived by means of its principal null directions and their levels of alignment, obtaining a total number of sixteen main algebraic types, which can be split into many subtypes. As an immediate application, we determine the algebraic types of the broadest family of static and spherically symmetric black hole solutions with spin, dilation and shear charges in Metric-Affine Gravity.