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In this paper we revisit non-spherical perturbations of the Schwarzschildblack hole in the context of $f(R)$ gravity. Previous studies were able todemonstrate the stability of the $f(R)$ Schwarzschild black hole againstgravitational perturbations in both the even and odd parity sectors. Inparticular, it was seen that the Regge-Wheeler and Zerilli equations in $f(R)$gravity obey the same equations as their General Relativity counterparts. Morerecently, the 1+1+2 semi-tetrad formalism has been used to derive a set of twowave equations: one for transverse, trace-free (tensor) perturbations and onefor the additional scalar modes that characterise fourth-order theories ofgravitation. The master variable governing tensor perturbations was shown to bea modified Regge-Wheeler tensor obeying the same equation as in GeneralRelativity. However, it is well known that there is a non-uniqueness in thedefinition of the master variable. In this paper we derive a set of twoperturbation variables and their concomitant wave equations that describegravitational perturbations in a covariant and gauge invariant manner. Thesevariables can be related to the Newman-Penrose (NP) Weyl scalars as well as themaster variables from the 2+2 formalism.

Covariant Perturbations of f(R) Black Holes: The Weyl Terms