In special functions, a topic in mathematics, 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 Ylm, and are typically denoted by sYlm, 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:
0 Y l m = Y l m . {\displaystyle {}_{0}Y_{lm}=Y_{lm}\ .}
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.
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
x ⋅ a = x ⋅ b = 0 a ⋅ a = b ⋅ b = 1 a ⋅ b = 0 x ⋅ ( a × b ) > 0 , {\displaystyle {\begin{aligned}\mathbf {x} \cdot \mathbf {a} =\mathbf {x} \cdot \mathbf {b} &=0\\\mathbf {a} \cdot \mathbf {a} =\mathbf {b} \cdot \mathbf {b} &=1\\\mathbf {a} \cdot \mathbf {b} &=0\\\mathbf {x} \cdot (\mathbf {a} \times \mathbf {b} )&>0,\end{aligned}}}
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
f ( x , ( cos θ ) a − ( sin θ ) b , ( sin θ ) a + ( cos θ ) b ) = e i s θ f ( x , a , b ) {\displaystyle f{\bigl (}\mathbf {x} ,(\cos \theta )\mathbf {a} -(\sin \theta )\mathbf {b} ,(\sin \theta )\mathbf {a} +(\cos \theta )\mathbf {b} {\bigr )}=e^{is\theta }f(\mathbf {x} ,\mathbf {a} ,\mathbf {b} )}
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
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