Regularization of Gauss-Bonnet Gravity in Riemann-Cartan Geometry

TL;DR

This paper introduces a conformal, dimensional-derivative regularization of the Gauss–Bonnet term in four-dimensional Riemann–Cartan geometry, yielding a regularized Einstein–Cartan–Gauss–Bonnet action whose torsionless limit matches the known scalar–tensor realization of 4D EGB gravity. By varying the action with respect to the scalar field, tetrad, and spin connection, the authors derive a complete, self-consistent set of second-order field equations, thereby avoiding Ostrogradsky instabilities. They show that the regularized GB term can act as an intrinsic geometric source of torsion in four dimensions and construct static, spherically symmetric black-hole solutions carrying torsion hair under a special coupling condition. These results provide a minimal four-dimensional mechanism to sustain torsion without extra dimensions and reveal new curvature–torsion–scalar couplings unique to RC geometry.

Abstract

We extend the conformal dimensional-derivative regularization of four-dimensional Gauss- Bonnet gravity to Riemann-Cartan geometry, obtaining a regularized action whose torsionless limit equals the well-known regularized four-dimensional Einstein-Gauss-Bonnet model. Varying independently with respect to the scalar, tetrad, and spin connection yields field equations that remain strictly second order in covariant derivatives, thereby avoiding Ostrogradsky-type instabil- ities. Within this framework we obtain static, spherically symmetric black holes carrying torsion hair, showing that the regularized Gauss-Bonnet interaction can support long-range torsion hair without invoking extra dimensions.