Some addition theorems for spin-weighted spherical harmonics
Alessandro Monteverdi, Elizabeth Winstanley
TL;DR
The paper addresses extending the scalar addition theorem to spin-weighted spherical harmonics by applying spin-raising and spin-lowering operators to the standard addition theorem to obtain closed-form sums over the azimuthal index with derivatives acting on one or both harmonics. The resulting expressions relate to rotated spin-weighted harmonics with shifted spins, e.g. ${}_{s\pm1}Y_\ell^{-s'}$, accompanied by phase factors $e^{-i(s\pm1)\alpha}$, thereby generalizing the scalar case. These results enable more efficient Green-function expansions and stress-energy tensor computations in quantum field theory on curved spacetimes and black hole backgrounds. The work provides a set of tools with broad applicability to CMB analyses, gravitational-wave perturbations, and spherical data analysis.
Abstract
We present some addition theorems for spin-weighted spherical harmonics, generalizing previous results for scalar (spin-zero) spherical harmonics. These addition theorems involve sums over the azimuthal quantum number of products of two spin-weighted spherical harmonics at different points on the two-sphere, either (or both) of which are differentiated with respect to one of their arguments.