Stability analysis of Schwarzschild black holes in dynamical Chern-Simons gravity

Dynamical Chern-Simons gravity has an interesting feature that the parity violating term exists, and the coupling is determined by a dynamical scalar field. When the spacetime has spherical symmetry, the parity violating term vanishes, and then the metric of the Schwarzschild spacetime with vanishing scalar field is an exact solution of dynamical Chern-Simons gravity. The effect of the Chern-Simons coupling appears in the study of perturbation around the Schwarzschild spacetime. Due to the parity violating term, the odd parity metric perturbation and the perturbed scalar field are coupled, and the perturbed field equations take the form of the coupled system of the Schr\"odinger equations. We prove linear mode stability for a generic massive scalar.