General theories of linear gravitational perturbations to a Schwarzschild Black Hole

We use the covariant formulation proposed in Tattersall et al (2017) to analyse the structure of linear perturbations about a spherically symmetric background in different families of gravity theories, and hence study how quasi-normal 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 analyse odd parity (axial) and even parity (polar) perturbations simultaneously in a straightforward manner.