Static Vacuum Solutions on Curved Spacetimes with Torsion

TL;DR

This paper explores higher-order curvature and torsion terms in Einstein-Cartan gravity, revealing the possibility of static vacuum solutions with black holes and naked singularities, extending classical GR results.

Contribution

It introduces fourth-order scalar invariants of curvature and torsion into the Lagrangian, leading to new static vacuum solutions with nontrivial torsion effects.

Findings

Existence of static vacuum solutions with black holes.

Presence of naked singularities in these solutions.

Torsion can propagate in vacuum due to higher-order terms.

Abstract

The Einstein-Cartan-Kibble-Sciama ({\sf ECKS}) theory of gravity naturally extends Einstein\rq{}s general relativity ({\sf GR}) to include intrinsic angular momentum (spin) of matter. The main feature of this theory consists of an algebraic relation between spacetime torsion and spin of matter which indeed deprives the torsion of its dynamical content. The Lagrangian of {\sf ECKS} gravity is proportional to the Ricci curvature scalar constructed out of a general affine connection so that owing to the influence of matter energy-momentum and spin, curvature and torsion are produced and interact only through the spacetime metric. In the absence of spin, the spacetime torsion vanishes and the theory reduces to {\sf GR}. It is however possible to have torsion propagation in vacuum by resorting to a model endowed with a non-minimal coupling between curvature and torsion. In the present work we try to investigate possible effects of the higher order terms that can be constructed from spacetime curvature and torsion, as the two basic constituents of Riemann-Cartan geometry. We consider Lagrangians that include fourth-order scalar invariants from curvature and torsion and then investigate the resulted field equations. The solutions that we find show that there could exist, even in vacuum, nontrivial static spacetimes that admit both black holes and naked singularities.