Spin-weighted spherical harmonics
Encyclopedia
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
. They were subsequently and independently rediscovered by and applied to describe gravitational radiation, and again by as so-called "monopole harmonics" in the study of Dirac monopoles.
R3. At a point x on the sphere, a positively oriented orthonormal basis
of tangent vector
s 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 ƒ 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 , denote the collection of all spin-weight s functions by B(s). Concretely, these are understood as functions ƒ 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, ƒ = P−sg is a spin-weight s function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.
(eth
). This operator is essentially the Dolbeault operator, after suitable identifications have been made,
Thus for ƒ ∈ B(s),
defines a function of spin-weight s + 1.
s of the Laplace-Beltrami operator
on the sphere, the spin-weight s harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles
of spin-weight s functions.
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:
and satisfy the completeness relation
The more useful of the Goldberg, et al., formulas is the following:
A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found here.
With the phase convention here
and 
.
Spin-1, degree
This relation allows the spin harmonics to be calculated using recursion relations for the
D-matrices
.
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
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
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
. 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
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the Lorentz group
Lorentz group
In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
. They were subsequently and independently rediscovered by and applied to describe gravitational radiation, and again by 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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
R3. At a point x on the sphere, a positively oriented orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of tangent vector
Tangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
s 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 ƒ 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 , denote the collection of all spin-weight s functions by B(s). Concretely, these are understood as functions ƒ 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} satisfyingGiven such a g, we may produce a spin-weight s function by multiplying by a suitable power of the hermitian form
Specifically, ƒ = P−sg is a spin-weight s function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.
Eth
The spin weight bundles B(s) are equipped with a differential operatorDifferential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
(ethEth
Eth is a letter used in Old English, Icelandic, Faroese , and Elfdalian. It was also used in Scandinavia during the Middle Ages, but was subsequently replaced with dh and later d. The capital eth resembles a D with a line through the vertical stroke...
). This operator is essentially the Dolbeault operator, after suitable identifications have been made,
Thus for ƒ ∈ B(s),
defines a function of spin-weight s + 1.
Spin-weighted harmonics
Just as conventional spherical harmonics are the eigenfunctionEigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the Laplace-Beltrami operator
Laplace-Beltrami operator
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami...
on the sphere, the spin-weight s harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles
of spin-weight s functions.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 derivativeCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
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
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
as:
The functions
then have the property of transforming with spin weight s.Other important properties include the following:
Orthogonality and completeness
The harmonics are orthogonal over the entire sphere:and satisfy the completeness relation
Calculating
These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulas for direct calculation were derived by . Note that their formulas use an old choice for the Condon-Shortley phase. The convention chosen below is in agreement with Mathematica, for instance.The more useful of the Goldberg, et al., formulas is the following:
A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found here.
With the phase convention here
and .First few spin-weighted spherical harmonics
Analytic expressions for the first few orthonormalized spin-weighted spherical harmonics : Spin-1, degree 
Relation to Wigner rotation matrices
This relation allows the spin harmonics to be calculated using recursion relations for the
D-matrices
Wigner D-matrix
The Wigner D-matrix is a matrix in an irreducible representation of the groups SU and SO. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner.- Definition Wigner D-matrix :Let j_x,...
.