Eur. Phys. J. C (2020) 80: 1157
https://doi.org/10.1140/epjc/s10052-020-08686-1

Regular Article - Theoretical Physics

Square–torsion gravity, dark matter halos and the baryonic Tully–Fisher relation

¹ Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Via C. Saldini 50, 20133, Milan, Italy
² Istituto Nazionale di Fisica Nucleare Sezione di Milano, Via Celoria 16, 20133, Milan, Italy

^a elias.megier@unimi.it

Received: 9 September 2020
Accepted: 19 November 2020
Published online: 14 December 2020

Abstract

Square–torsion gravity is applied to the long standing dark matter problem. In this context the theory reduces to General Relativity complemented by a dark stress–energy tensor due to the torsion of spacetime and is studied under the simplifying assumption of spherical symmetry. The dark stress–energy tensor satisfies an anisotropic structure equation. In vacuum this is equivalent to a wave equation with sources. A natural class of exact solutions is found which explicitly perturbs any seed spacetime metric by a conformal factor. This leads to the concept of dark coating. Static solutions are then used to construct structures that model dark matter halos surrounding baryonic bodies. In the Newtonian régime the baryonic mass $m_b$ and the flat rotation curve velocity $v_f$ are related by the baryonic Tully–Fisher relation$m_b\propto v_f^4$, a hitherto purely empirical result. The example of a dark halo on the Schwarzschild geometry is made as a toy model for a galaxy. Qualitative an quantitative features of galactic rotation curves are recovered. In this setting, a boundary of staticity is found, called torsion sphere, placed between the photon sphere and the event horizon. The phenomenon of dark radiation is briefly exposed.