General theories of linear gravitational perturbations to a Schwarzschild black hole

We use the covariant formulation proposed by Tattersall, Lagos, and Ferreira [Phys. Rev. D 96, 064011 (2017), 10.1103/PhysRevD.96.064011] to analyze the structure of linear perturbations about a spherically symmetric background in different families of gravity theories, and hence study how quasinormal modes of perturbed black holes may be affected by modifications to general relativity. We restrict ourselves to single-tensor, scalar-tensor and vector-tensor diffeomorphism-invariant gravity models in a Schwarzschild black hole background. We show explicitly the full covariant form of the quadratic actions in such cases, which allow us to then analyze odd parity (axial) and even parity (polar) perturbations simultaneously in a straightforward manner.