An extended Einstein-Cartan formulation of Chern-Simons gravity

We consider the non-dynamical Chern-Simons modification to general relativity in the framework of the Einstein-Cartan formulation, as providing a source for torsion. Since the experimental and observational bounds on torsion are very stringent, we propose a new iterative procedure to look for vacuum solutions of the system by expanding the tetrad, connection and embedding parameter, together with all derived quantities, in terms of a dimensionless small parameter β which codifies the Chern-Simons coupling. Careful consideration is given to the Bianchi identities together with the consistency conditions they impose via the equations of motion. Starting from a torsionless zeroth-order vacuum solution we derive the second order differential equation for the O(β) corrections to the metric, for an arbitrary embedding parameter. Subsequent specialization to either the canonical or the axial embedding allows us to show that a slowly rotating Kerr metric is an O(β) solution of the system.