Black hole perturbation in nondynamical and dynamical Chern-Simons gravity

Abstract: Chern-Simons gravitational theories are extensions of general relativity in which the parity is violated due to the Chern-Simons term. We study linear perturbations on the static and spherically symmetric background spacetime both for nondynamical and dynamical Chern-Simons theories. We do not make an assumption that the background Chern-Simons scalar field vanishes, which has been adopted in the literature. By eliminating nondynamical variables using their constraint equations, we derive the reduced second order action from which a set of closed evolution equations containing only dynamical variables are immediately obtained and therefore the number of propagating degrees of freedom as well. It is found that ghost is present both for the nondynamical case and for the dynamical case unless the background Chern-Simons scalar field vanishes. It is also found that if the background scalar field vanishes, ghost degrees of freedom are killed and all the modes propagate at the speed of light.