Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation II

Linear gravitational perturbations of Kerr black holes in general relativity are most efficiently treated by the Teukolsky formalism, which leads to single decoupled equations for Weyl scalars Ψ₀ and Ψ₄. These equations are further separable into radial and angular equations. The standard derivation of the Teukolsky equation in general relativity required the background spacetime to be algebraically special (Petrov type D). In beyond-General-Relativity (bGR) theories, for example, dynamical Chern-Simons (dCS) and Einstein-dilaton Gauss-Bonnet (EdGB) theories, spacetimes of rotating black holes are not type-D, but type-I instead. This lack of symmetry creates potential difficulties computing gravitational waveforms in bGR theories. In this work, for any stationary background spacetime with an order epsilon deviation from a Petrov type-D spacetime, we obtain a single decoupled modified Teukolsky equation for the perturbative Ψ₀ (and Ψ₄) of that spacetime --- accurate up to linear order in epsilon. This equation may also have a source term on the right-hand side due to matter (including, e.g., dCS and EdGB scalar-field) perturbations. Our derivation is an extension to Chandrasekhar's alternative derivation of the Teukolsky equation and his metric reconstruction procedure (both originally formulated for the Kerr spacetime). For demonstration, we apply our formalism to perturbations of slowly-rotating black holes in dCS gravity. In this case, Ψ₀ (and Ψ₄) are decoupled from all other space-time degrees of freedom, and only couple to the dCS scalar field, which arises due to the source term of the modified Teukolsky equation.