How should spin-weighted spherical functions be defined?

Understand

Spin-weighted spherical functions provide a useful tool for analyzing tensor-valued functions on the sphere.

• A tensor field can be decomposed into complex-valued functions by taking contractions with tangent vectors on the sphere and the normal to the sphere.
• These component functions are usually presented as functions on the sphere itself, but this requires an implicit choice of distinguished tangent vectors with which to contract.
• Thus, we may more accurately say that spin-weighted spherical functions are functions of both a point on the sphere and a choice of frame in the tangent space at that point.