Covariant perturbations of black holes: the Weyl terms

Abstract

In this paper we revisit non-spherical perturbations of the Schwarzschild black hole in the context of f(R) gravity. Previous studies were able to demonstrate the stability of the f(R) Schwarzschild black hole against gravitational perturbations in both the even and odd parity sectors. In particular, it was seen that the Regge–Wheeler (RW) and Zerilli equations in f(R) gravity obey the same equations as their general relativity (GR) counterparts. More recently, the 1+1+2 semi-tetrad formalism has been used to derive a set of two wave equations: one for transverse, trace-free (tensor) perturbations and one for the additional scalar modes that characterize fourth-order theories of gravitation. The master variable governing tensor perturbations was shown to be a modified RW tensor obeying the same equation as in GR. However, it is well known that there is a non-uniqueness in the definition of the master variable. In this paper we derive a set of two perturbation variables and their concomitant wave equations that describe gravitational perturbations in a covariant and gauge invariant manner. These variables can be related to the Newman–Penrose (NP) Weyl scalars as well as the master variables from the 2+2 formalism. As a byproduct of this study, we also derive a set of useful results relating the NP formalism to the 1+1+2 formalism valid for LRS-II spacetimes.