Perturbations of Schwarzschild Black Holes in Chern-Simons Modified Gravity

TL;DR

This paper investigates how perturbations of Schwarzschild black holes behave in Chern-Simons modified gravity, revealing that the Pontryagin constraint suppresses generic oscillations and affects the coupling of modes.

Contribution

It demonstrates that in Chern-Simons gravity, the Pontryagin constraint couples axial and polar modes, leading to overconstrained equations that suppress black hole oscillations.

Findings

Birkhoff's theorem holds for many Chern-Simons couplings.

Axial and polar modes are coupled, unlike in GR.

Pontryagin constraint suppresses generic black hole oscillations.

Abstract

We study perturbations of a Schwarzschild black hole in Chern-Simons modified gravity. We begin by showing that Birkhoff's theorem holds for a wide family of Chern-Simons coupling functions, a scalar field present in the theory that controls the strength of the Chern-Simons correction to the Einstein-Hilbert action. After decomposing the perturbations in spherical harmonics, we study the linearized modified field equations and find that axial and polar modes are coupled, in contrast to general relativity. The divergence of the modified equations leads to the Pontryagin constraint, which forces the vanishing of the Cunningham-Price-Moncrief master function associated with axial modes. We analyze the structure of these equations and find that the appearance of the Pontryagin constraint yields an overconstrained system that does not allow for generic black hole oscillations. We illustrate this situation by studying the case characterized by a canonical choice of the coupling function and pure-parity perturbative modes. We end with a discussion of how to extend Chern-Simons modified gravity to bypass the Pontryagin constraint and the suppression of perturbations.