Coupling Electromagnetism to Torsion: Black Holes and Spin-Charge Interactions

TL;DR

This work investigates a non-minimal coupling between electromagnetism and torsion in a Riemann–Cartan spacetime via the term $F^{\mu\nu}\tilde{R}_{\mu\nu}$, within a framework that also includes a quadratic torsion sector. The authors derive the full field equations in the tetrad/spin-connection formalism and obtain exact black-hole solutions in four dimensions (a spin–charge–modified RN-like geometry) and three dimensions (a BTZ-like geometry with a spin–charge–angular-momentum coupling under slow rotation). A key result is the emergence of an intrinsic spin charge $\kappa_s$ that couples to the electric charge $q$ through the non-minimal term, producing hair and allowing the effective charge to depend on torsion, with Maxwell equations sourced by $ abla_\mu\tilde{R}^{[\mu\nu]}$ vanishing in the solutions studied. In the 3D case, rotation introduces a new coupling between $\kappa_s$, angular momentum $J$, and magnetic charge, highlighting rich torsion-induced phenomenology that could have implications for thermodynamics, perturbations, and holographic duals.

Abstract

The coupling between matter fields and gravity, encoded in the geometry of spacetime, can be realized in various ways. Most commonly, a minimal coupling principle is employed, meaning that all matter fields, except spinors, couple only to the spacetime metric, while spinors additionally couple to the spacetime connection. Non-minimal couplings between matter fields and spacetime curvature can arise, for example, from quantum field theory on curved spacetime through renormalization corrections, in gauge theories of gravity, and in effective field theories. In this article, we consider a non-minimal coupling $F^{μν}\tilde{R}_{μν}$ between the field strength tensor of the electromagnetic field $F_{μν}$ and the antisymmetric part of the Ricci tensor $\tilde{R}_{[μν]}$ in Riemann--Cartan geometry, which is based on a general metric-compatible connection with torsion. We find an exact four-dimensional vacuum solution that generalizes the Reissner--Nordstr{ö}m black hole from Einstein--Maxwell and reveals new interactions between the intrinsic torsion-spin charge and the electric charge. Qualitatively, this solution exhibits two distinct features: the effective charge is not constrained to be positive, and the sign of the electric charge influences its gravitational effects. We also derive slowly rotating solutions in three dimensions, representing a generalized slowly rotating BTZ black hole solution with couplings among the magnetic and electric charges, the angular momentum, and the intrinsic torsion-spin charge.