Calculation of a regularized Teukolsky Green function in Schwarzschild spacetime

We obtain exact expressions for various factors involved in the Hadamard form of the retarded Green function for the (Bardeen-Press-)Teukolsky equation on Schwarzschild spacetime. We use these to improve on previous results for the calculation of this Green function. We work in a spacetime $\mathcal{M}_2\times\mathbb{S}^2$ conformal to Schwarzschild, in which the metric takes a direct product form. This allows us to derive a separable form for the direct (i.e., singular) part of the Hadamard form of the retarded Green function. The angular factor in this quantity is calculated explicitly. This shows an interesting interplay between geodesics of $\mathbb{S}^2$, spin-weighted spherical harmonics, and Euler angles. The $\mathcal{M}_2$ factor equates to a spin-dependent factor that satisfies a transport equation along geodesics, times the square root of the van Vleck determinant. Both terms are calculated in an exact form for constant radius orbits (which includes the cases of circular timelike geodesics and static worldlines of Schwarzschild spacetime). This separable form also allows us to obtain the multipolar $\ell$-modes of the direct part for electromagnetic and gravitational field perturbations. We then use these $\ell$-modes to calculate, in the gravitational case, the retarded Green function minus its direct part: this is a better representation in practise of the retarded Green function for points near coincidence.