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
Famous quotes containing the word functions:
“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)