`Translation invariant' black hole: autoparallels and complete integrability

TL;DR

This paper investigates the motion of test bodies in torsion-inclusive spacetimes, proving integrability for certain geometries and deriving a novel black hole solution with torsion, revealing differences between autoparallels and geodesics.

Contribution

It introduces a class of translation-invariant geometries with complete integrability of autoparallel motion and derives a new torsionful Schwarzschild black hole solution.

Findings

Complete integrability of autoparallel motion in specified geometries.

Existence of a new Schwarzschild black hole solution with torsion.

Notable differences between autoparallels and geodesics in orbital dynamics.

Abstract

We consider the autoparallel motion of test bodies in static, spherically symmetric spacetimes with torsion. We prove complete integrability of such motion for a wide range of off-shell geometries via four commuting autoparallel Killing vectors. Since these vectors reduce to translation generators in a certain limit, we refer to these geometries as `translation invariant.' Invoking the field equations of quadratic Poincar\'e gauge gravity we re-derive an exact Schwarzschild black hole solution endowed with a non-trivial torsion field scaling as $GM/r^2$, where $M$ denotes the ADM mass of the black hole. Studying the qualitative orbital dynamics via effective potentials we find notable discrepancies between autoparallels (straightest possible paths) and geodesics (shortest possible paths).