Representation As Functions
The spin-weighted harmonics can be represented as functions on a sphere once a point on the sphere has been selected to serve as the North pole. By definition, a function with spin weight s transforms under rotation about the pole via .
Working in standard spherical coordinates, we can define a particular operator acting on a function as:
This gives us another function of and . (The operator is effectively a covariant derivative operator in the sphere.)
An important property of the new function is that if had spin weight, has spin weight . Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator which will lower the spin weight of a function by 1:
The spin-weighted spherical harmonics are then defined in terms of the usual spherical harmonics as:
The functions then have the property of transforming with spin weight s.
Other important properties include the following:
Read more about this topic: Spin-weighted Spherical Harmonics
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