Black hole perturbation in nondynamical and dynamical Chern-Simons gravity

TL;DR

This paper investigates linear perturbations in both nondynamical and dynamical Chern-Simons gravity theories, revealing the presence of ghost degrees of freedom unless the background scalar field vanishes, which restores standard propagation speeds.

Contribution

It derives the reduced second order action for perturbations without assuming a vanishing background scalar field, providing new insights into the degrees of freedom in Chern-Simons gravity.

Findings

Ghosts are present unless the background scalar field vanishes.

All modes propagate at the speed of light when the scalar field is zero.

The analysis applies to both nondynamical and dynamical Chern-Simons theories.

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.