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An extended solution space for Chern-Simons gravity: the slowly rotating Kerr black hole

In the Einstein-Cartan formulation, an iterative procedure to find solutions in non-dynamical Chern-Simons (CS) gravity in vacuum is proposed. The iterations, in powers of a small parameter $β$ which codifies the CS coupling, start from an arbitrary torsionless solution of Einstein equations. With Schwarzschild as the zeroth-order choice, we derive a second-order differential equation for the $\mathcal{O}(β)$ corrections to the metric, for an arbitrary zeroth-order embedding parameter. In particular, the slowly rotating Kerr metric is an $\mathcal{O}(β)$ solution in either the canonical or the axial embeddings.