Correspondence between tensorial spin-s and spin-weighted spherical harmonics

Abstract

The tensorial spin-s and the spin weighted spherical harmonics are functions on the sphere which are extremely useful for many calculations in general relativity. These sets of functions constitute two orthonormal bases and can be used to expand spin-s fields in General Relativity and other fields. Particularly, we are interested in the Weyl scalar ψ4 which is related to the gravitational radiation. In this article, we build a catalog where the correspondence between tensorial spin-s and the spin weighted spherical harmonics is shown. Also, as a simple application, we use the resulting transformations to link the quadrupole and octupole shear of a tensorial spin-s expansion with the gravitational wave functions expressed in terms of the spin weighted spherical harmonics.