Slowly rotating Kerr black hole as a solution of Einstein-Cartan gravity extended by a Chern-Simons term

We consider the nondynamical Chern-Simons (CS) modification to General Relativity (GR) in the framework of the Einstein-Cartan formulation, as providing a way to incorporate a slowly rotating Kerr black hole in the space of solutions. Our proposal lies on considering the CS term as a source of torsion and on an iterative procedure to look for vacuum solutions of the system, by expanding the tetrad, the connection and the embedding parameter, in powers of a dimensionless small parameter $\beta$ which codifies the CS coupling. Starting from a torsionless zeroth-order vacuum solution we derive the second-order differential equation for the $\mathcal{O}(\beta)$ corrections to the metric, for an arbitrary embedding parameter. Furthermore we can show that the slowly rotating Kerr metric is an $\mathcal{O}(\beta)$ solution of the system either in the canonical or the axial embeddings.