Exploring Axial Symmetry in Modified Teleparallel Gravity

TL;DR

This paper investigates axially symmetric solutions in a broad class of teleparallel gravity theories, finding new solutions like a generalized Taub-NUT and slowly rotating Kerr spacetime, and clarifying issues with spin connection determination.

Contribution

It provides a comprehensive analysis of axial symmetry in generalized teleparallel gravities, introducing new solutions and addressing the spin connection determination problem.

Findings

Found generalized Taub-NUT and slowly rotating Kerr solutions.

Showed the 'turning off gravity' method does not always solve antisymmetric equations.

Analyzed the role of connection components in axial symmetry.

Abstract

Axially symmetric spacetimes play an important role in the relativistic description of rotating astrophysical objects like black holes, stars, etc. In gravitational theories that venture beyond the usual Riemannian geometry by allowing independent connection components, the notion of symmetry concerns, not just the metric, but also the connection. As discovered recently, in teleparallel geometries, axial symmetry can be realised in two branches, while only one of these has a continuous spherically symmetric limit. In the current paper, we consider a very generic $f(T,B,\phi,X)$ family of teleparallel gravities, whose action depends on the torsion scalar $T$ and the boundary term $B$, as well as a scalar field $\phi$ with its kinetic term $X$. As the field equations can be decomposed into symmetric and antisymmetric (spin connection) parts, we thoroughly analyse the antisymmetric equations and look for solutions of axial spacetimes which could be used as ans\"atze to tackle the symmetric part of the field equations. In particular, we find solutions corresponding to a generalisation of the Taub-NUT metric, and the slowly rotating Kerr spacetime. Since this work also concerns a wider issue of how to determine the spin connection in teleparallel gravity, we also show that the method of "turning off gravity" proposed in the literature, does not always produce a solution to the antisymmetric equations.