Spin-weighted Spherical Harmonics - Spin-weighted Functions

Spin-weighted Functions

Regard the sphere S2 as embedded into the three-dimensional Euclidean space R3. At a point x on the sphere, a positively oriented orthonormal basis of tangent vectors at x is a pair a, b of vectors such that

where the first pair of equations states that a and b are tangent at x, the second pair states that a and b are unit vectors, the penultimate equation that a and b are orthogonal, and the final equation that (x, a, b) is a right-handed basis of R3.

A spin-weight s function f is a function accepting as input a point x of S2 and a positively oriented orthonormal basis of tangent vectors at x, such that

for every rotation angle θ.

Following Eastwood & Tod (1982), denote the collection of all spin-weight s functions by B(s). Concretely, these are understood as functions f on C2\{0} satisfying the following homogeneity law under complex scaling

This makes sense provided s is a half-integer.

Abstractly, B(s) is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle of the Serre twist on the complex projective line CP1. A section of the latter bundle is a function g on C2\{0} satisfying

Given such a g, we may produce a spin-weight s function by multiplying by a suitable power of the hermitian form

Specifically, f = P−sg is a spin-weight s function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.