spin-weighted spherical harmonics

Regard the sphere {{math|S²}} as embedded into the three-dimensional Euclidean space {{math|R³}}. At a point {{math|x}} on the sphere, a positively oriented orthonormal basis of tangent vectors at {{math|x}} is a pair {{math|a, b}} of vectors such that

:$$

\begin{align}

\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{align}

where the first pair of equations states that {{math|a}} and {{math|b}} are tangent at {{math|x}}, the second pair states that {{math|a}} and {{math|b}} are unit vectors, the penultimate equation that {{math|a}} and {{math|b}} are orthogonal, and the final equation that {{math|(x, a, b)}} is a right-handed basis of {{math|R³}}.

A spin-weight {{mvar|s}} function {{mvar|f}} is a function accepting as input a point {{math|x}} of {{math|S²}} and a positively oriented orthonormal basis of tangent vectors at {{math|x}}, such that

:$f\backslash bigl(\backslash mathbf\; x,(\backslash cos\backslash theta)\backslash mathbf\{a\}-(\backslash sin\backslash theta)\backslash mathbf\{b\},\; (\backslash sin\backslash theta)\backslash mathbf\{a\}\; +\; (\backslash cos\backslash theta)\backslash mathbf\{b\}\backslash bigr)\; =\; e^\{is\backslash theta\}f(\backslash mathbf\; x,\backslash mathbf\{a\},\backslash mathbf\{b\})$

for every rotation angle {{mvar|θ}}.

Following {{harvtxt|Eastwood|Tod|1982}}, denote the collection of all spin-weight {{mvar|s}} functions by {{math|B(s)}}. Concretely, these are understood as functions {{mvar|f}} on {{math|C²\{0}}} satisfying the following homogeneity law under complex scaling

:$f\backslash left(\backslash lambda\; z,\backslash overline\{\backslash lambda\}\backslash bar\{z\}\backslash right)\; =\; \backslash left(\backslash frac\{\backslash overline\{\backslash lambda\}\}\{\backslash lambda\}\backslash right)^s\; f\backslash left(z,\backslash bar\{z\}\backslash right).$

This makes sense provided {{mvar|s}} is a half-integer.

Abstractly, {{math|B(s)}} is isomorphic to the smooth vector bundle underlying the antiholomorphic vector bundle {{math|{{overline|O(2s)}}}} of the Serre twist on the complex projective line {{math|CP¹}}. A section of the latter bundle is a function {{mvar|g}} on {{math|C²\{0}}} satisfying

:$g\backslash left(\backslash lambda\; z,\backslash overline\{\backslash lambda\}\backslash bar\{z\}\backslash right)\; =\; \backslash overline\{\backslash lambda\}^\{2s\}\; g\backslash left(z,\backslash bar\{z\}\backslash right).$

Given such a {{mvar|g}}, we may produce a spin-weight {{mvar|s}} function by multiplying by a suitable power of the hermitian form

:$P\backslash left(z,\backslash bar\{z\}\backslash right)\; =\; z\backslash cdot\backslash bar\{z\}.$

Specifically, {{math|f {{=}} P^−sg}} is a spin-weight {{mvar|s}} function. The association of a spin-weighted function to an ordinary homogeneous function is an isomorphism.

The spin weight bundles {{math|B(s)}} are equipped with a differential operator {{mvar|ð}} (eth). This operator is essentially the Dolbeault operator, after suitable identifications have been made,

:$\backslash partial\; :\; \backslash overline\{\backslash mathbf\; O(2s)\}\; \backslash to\; \backslash mathcal\{E\}^\{1,0\}\backslash otimes\; \backslash overline\{\backslash mathbf\; O(2s)\}\; \backslash cong\; \backslash overline\{\backslash mathbf\; O(2s)\}\backslash otimes\backslash mathbf\{O\}(-2).$

Thus for {{math|f ∈ B(s)}},

:$\backslash eth\; f\; \backslash \; \backslash stackrel\{\backslash text\{def\}\}\{=\}\backslash \; P^\{-s+1\}\backslash partial\; \backslash left(P^s\; f\backslash right)$

defines a function of spin-weight {{math|s + 1}}.

Just as conventional spherical harmonics are the eigenfunctions of the Laplace-Beltrami operator on the sphere, the spin-weight {{mvar|s}} harmonics are the eigensections for the Laplace-Beltrami operator acting on the bundles {{math|{{mathcal|E}}(s)}} of spin-weight {{mvar|s}} 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 {{mvar|η}} with spin weight {{mvar|s}} transforms under rotation about the pole via

:$\backslash eta\; \backslash rightarrow\; e^\{i\; s\; \backslash psi\}\backslash eta.$

Working in standard spherical coordinates, we can define a particular operator {{mvar|ð}} acting on a function {{mvar|η}} as:

:$\backslash eth\backslash eta\; =\; -\; \backslash left(\backslash sin\{\backslash theta\}\backslash right)^s\; \backslash left\backslash \{\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\; +\; \backslash frac\{i\}\{\backslash sin\{\backslash theta\}\}\; \backslash frac\{\backslash partial\}\; \{\backslash partial\; \backslash phi\}\; \backslash right\backslash \}\; \backslash left[\; \backslash left(\backslash sin\{\backslash theta\}\backslash right)^\{-s\}\; \backslash eta\; \backslash right].$

This gives us another function of {{mvar|θ}} and {{mvar|φ}}. (The operator {{mvar|ð}} is effectively a covariant derivative operator in the sphere.)

An important property of the new function {{mvar|ðη}} is that if {{mvar|η}} had spin weight {{mvar|s}}, {{mvar|ðη}} has spin weight {{math|s + 1}}. Thus, the operator raises the spin weight of a function by 1. Similarly, we can define an operator {{mvar|{{overline|ð}}}} which will lower the spin weight of a function by 1:

:$\backslash bar\backslash eth\backslash eta\; =\; -\; \backslash left(\backslash sin\{\backslash theta\}\backslash right)^\{-s\}\; \backslash left\backslash \{\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\; -\; \backslash frac\{i\}\{\backslash sin\{\backslash theta\}\}\; \backslash frac\{\backslash partial\}\; \{\backslash partial\; \backslash phi\}\; \backslash right\backslash \}\; \backslash left[\; \backslash left(\backslash sin\{\backslash theta\}\backslash right)^\{s\}\; \backslash eta\; \backslash right].$

The spin-weighted spherical harmonics are then defined in terms of the usual spherical harmonics as:

:$$

{}_sY_{l m} = \begin{cases}

\sqrt{\frac{(l-s)!}{(l+s)!}}\ \eth^s Y_{l m},&& 0\leq s \leq l; \\

\sqrt{\frac{(l+s)!}{(l-s)!}}\ \left(-1\right)^s \bar\eth^{-s} Y_{l m},&& -l\leq s \leq 0; \\

0,&&l < |s|.\end{cases}

The functions {{math|_sY_lm}} then have the property of transforming with spin weight {{mvar|s}}.

Other important properties include the following:

:$\backslash begin\{align\}$

\eth\left({}_sY_{l m}\right) &= +\sqrt{(l-s)(l+s+1)}\, {}_{s+1}Y_{l m};\\

\bar\eth\left({}_sY_{l m}\right) &= -\sqrt{(l+s)(l-s+1)}\, {}_{s-1}Y_{l m};

\end{align}

The harmonics are orthogonal over the entire sphere:

:$\backslash int\_\{S^2\}\; \{\}\_sY\_\{l\; m\}\backslash ,\; \{\}\_s\backslash bar\{Y\}\_\{l\text{'}m\text{'}\}\backslash ,\; dS\; =\; \backslash delta\_\{ll\text{'}\}\; \backslash delta\_\{mm\text{'}\},$

and satisfy the completeness relation

:$\backslash sum\_\{l\; m\}\; \{\}\_s\backslash bar\; Y\_\{l\; m\}\backslash left(\backslash theta\text{'},\backslash phi\text{'}\backslash right)\; \{\}\_s\; Y\_\{l\; m\}(\backslash theta,\backslash phi)\; =\; \backslash delta\backslash left(\backslash phi\text{'}-\backslash phi\backslash right)\backslash delta\backslash left(\backslash cos\backslash theta\text{'}-\backslash cos\backslash theta\backslash right)$

These harmonics can be explicitly calculated by several methods. The obvious recursion relation results from repeatedly applying the raising or lowering operators. Formulae for direct calculation were derived by {{harvtxt|Goldberg|Macfarlane|Newman|Rohrlich|1967}}. Note that their formulae use an old choice for the [http://mathworld.wolfram.com/Condon-ShortleyPhase.html Condon–Shortley phase]. The convention chosen below is in agreement with Mathematica, for instance.

The more useful of the Goldberg, et al., formulae is the following:

:$\{\}\_sY\_\{l\; m\}\; (\backslash theta,\; \backslash phi)\; =\; \backslash left(-1\backslash right)^\{l+m-s\}\; \backslash sqrt\{\; \backslash frac\{(l+m)!\; (l-m)!\; (2l+1)\}\; \{4\backslash pi\; (l+s)!\; (l-s)!\}\; \}\; \backslash sin^\{2l\}\; \backslash left(\; \backslash frac\{\backslash theta\}\{2\}\; \backslash right)\; e^\{i\; m\; \backslash phi\}\; \backslash times\backslash sum\_\{r=0\}^\{l-s\}\; \backslash left(-1\backslash right)^\{r\}\; \{l-s\; \backslash choose\; r\}\; \{l+s\; \backslash choose\; r+s-m\}\; \backslash cot^\{2r+s-m\}\; \backslash left(\; \backslash frac\{\backslash theta\}\; \{2\}\; \backslash right)\backslash ,\; .$

A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found [http://www.black-holes.org/SpinWeightedSphericalHarmonics.nb here].

With the phase convention here:

:$\backslash begin\{align\}$

{}_s\bar Y_{l m} &= \left(-1\right)^{s+m}{}_{-s}Y_{l(-m)}\\

{}_sY_{l m}(\pi-\theta,\phi+\pi) &= \left(-1\right)^l {}_{-s}Y_{l m}(\theta,\phi).

\end{align}

Analytic expressions for the first few orthonormalized spin-weighted spherical harmonics:

:$\backslash begin\{align\}$

{}_1 Y_{10}(\theta,\phi) &= \sqrt{\frac{3}{8\pi}}\,\sin\theta \\

{}_1 Y_{1\pm 1}(\theta,\phi) &= -\sqrt{\frac{3}{16\pi}}(1 \mp \cos\theta)\,e^{\pm i\phi}

\end{align}

:$D^l\_\{-m\; s\}(\backslash phi,\backslash theta,-\backslash psi)\; =\backslash left(-1\backslash right)^m\; \backslash sqrt\backslash frac\{4\backslash pi\}\{2l+1\}\; \{\}\_sY\_\{l\; m\}(\backslash theta,\backslash phi)\; e^\{is\backslash psi\}$

This relation allows the spin harmonics to be calculated using recursion relations for the Wigner D-matrix.

The triple integral in the case that {{math|s₁ + s₂ + s₃ {{=}} 0}} is given in terms of the 3-j symbol:

:$\backslash int\_\{S^2\}\; \backslash ,\{\}\_\{s\_1\}\; Y\_\{j\_1\; m\_1\}$

\,{}_{s_2} Y_{j_2m_2}\, {}_{s_3} Y_{j_3m_3} = \sqrt{\frac{\left(2j_1+1\right)\left(2j_2+1\right)\left(2j_3+1\right)}{4\pi}}

\begin{pmatrix}

j_1 & j_2 & j_3\\

m_1 & m_2 & m_3

\end{pmatrix}

\begin{pmatrix}

j_1 & j_2 & j_3\\

-s_1 & -s_2 & -s_3

\end{pmatrix}

• Spherical basis

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Category:Fourier analysis

Category:Rotational symmetry

Category:Special functions