Spin-weighted Spherical Harmonics

Spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree ℓ, just like ordinary spherical harmonics, but have an additional spin weight s that reflects the additional U(1) symmetry. A special basis of harmonics can be derived from the Laplace spherical harmonics, and are typically denoted by, where ℓ and m are the usual parameters familiar from the standard Laplace spherical harmonics. In this special basis, the spin-weighted spherical harmonics appear as actual functions, because the choice of a polar axis fixes the U(1) gauge ambiguity. The spin-weighted spherical harmonics can be obtained from the standard spherical harmonics by application of spin raising and lowering operators. In particular, the spin-weighted spherical harmonics of spin weight s = 0 are simply the standard spherical harmonics:

Spaces of spin-weighted spherical harmonics were first identified in connection with the representation theory of the Lorentz group (Gelfand, Minlos & Shapiro 1958). They were subsequently and independently rediscovered by Newman & Penrose (1966) and applied to describe gravitational radiation, and again by Wu & Yang (1976) as so-called "monopole harmonics" in the study of Dirac monopoles.