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{{distinguish|spin spherical harmonics}}
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 field]]s rather than [[scalar field]]s: 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 <math>Y_{\ell m}</math>, and are typically denoted by <math>{}_sY_{\ell m}</math>, 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 [[Ladder operators|spin raising and lowering operators]].  In particular, the spin-weighted spherical harmonics of spin weight ''s''&nbsp;=&nbsp;0 are simply the standard spherical harmonics:
:<math>{}_0Y_{\ell m} = Y_{\ell m}\ .</math>

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

==Spin-weighted functions==
Regard the sphere ''S''<sup>2</sup> as embedded into the three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>.  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
:<math>
\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}
</math>
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 vector]]s, the penultimate equation that '''a''' and '''b''' are [[orthogonal]], and the final equation that ('''x''',&nbsp;'''a''',&nbsp;'''b''') is a right-handed basis of '''R'''<sup>3</sup>.

A spin-weight ''s'' function ''f'' is a function accepting as input a point '''x''' of ''S''<sup>2</sup> and a positively oriented orthonormal basis of tangent vectors at '''x''', such that
:<math>f(\mathbf x,\cos\theta\mathbf{a}-\sin\theta\mathbf{b}, \sin\theta\mathbf{a} + \cos\theta\mathbf{b}) = e^{is\theta}f(\mathbf x,\mathbf{a},\mathbf{b})</math>
for every rotation angle θ. 

Following {{harvtxt|Eastwood|Tod|1982}}, denote the collection of all spin-weight ''s'' functions by '''B'''(''s'').  Concretely, these are understood as functions ''f'' on '''C'''<sup>2</sup>\{0} satisfying the following homogeneity law under complex scaling
:<math>f(\lambda z,\overline{\lambda}\bar{z}) = \left(\frac{\overline{\lambda}}{\lambda}\right)^s f(z,\bar{z}).</math>
This makes sense provided ''s'' is a half-integer.

Abstractly, '''B'''(''s'') is [[isomorphic]] to the smooth [[vector bundle]] underlying the [[antiholomorphic]] vector bundle <math>\overline{\mathbf{O}(2s)}</math> of the [[Serre twist]] on the [[complex projective line]] '''CP'''<sup>1</sup>.  A section of the latter bundle is a function ''g'' on '''C'''<sup>2</sup>\{0} satisfying
:<math>g(\lambda z,\overline{\lambda}\bar{z}) = \overline{\lambda}^{2s} g(z,\bar{z}).</math>
Given such a ''g'', we may produce a spin-weight ''s'' function by multiplying by a suitable power of the hermitian form
:<math>P(z,\bar{z}) = z\cdot\bar{z}.</math>
Specifically, ''f''&nbsp;=&nbsp;''P''<sup>&minus;''s''</sup>''g'' 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 operator]] <math>\eth</math> ([[eth]]).  This operator is essentially the [[Dolbeault operator]], after suitable identifications have been made,
:<math>\partial : \overline{\mathbf O(2s)} \to \mathcal{E}^{1,0}\otimes \overline{\mathbf O(2s)} \cong \overline{\mathbf O(2s)}\otimes\mathbf{O}(-2).</math>
Thus for ''f''&nbsp;∈&nbsp;'''B'''(''s''),
:<math>\eth f \stackrel{\text{def}}{=} P^{-s+1}\partial (P^s f) </math>
defines a function of spin-weight ''s''&nbsp;+&nbsp;1.

==Spin-weighted harmonics==
Just as conventional spherical harmonics are the [[eigenfunction]]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 <math>\mathcal E(s)</math> 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 <math>\eta</math> with ''spin weight s'' transforms under rotation about the pole via <math>\eta \rightarrow e^{i s \psi}\eta</math>.

Working in standard spherical coordinates, we can define a particular operator <math>\eth</math> acting on a function <math>\eta</math> as:
:<math>\eth\eta =  - (\sin{\theta})^s \left\{ \frac{\partial}{\partial \theta} + \frac{i}{\sin{\theta}} \frac{\partial} {\partial \phi} \right\} \left[ (\sin{\theta})^{-s} \eta \right].</math>
This gives us another function of <math>\theta</math> and <math>\phi</math>. (The operator <math>\eth</math> is effectively a [[covariant derivative]] operator in the sphere.)

An important property of the new function <math>\eth\eta</math> is that if <math>\eta</math> had spin weight <math>s</math>, <math>\eth\eta</math> has spin weight <math>s+1</math>.  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:
:<math>\bar\eth\eta =  - (\sin{\theta})^{-s} \left\{ \frac{\partial}{\partial \theta} - \frac{i}{\sin{\theta}} \frac{\partial} {\partial \phi} \right\} \left[ (\sin{\theta})^{s} \eta \right].</math>

The spin-weighted spherical harmonics are then defined in terms of the usual [[spherical harmonics]] as:
:<math>{}_sY_{\ell m} = \sqrt{\frac{(\ell-s)!}{(\ell+s)!}}\ \eth^s Y_{\ell m},\ \ 0\leq s \leq \ell;</math>
:<math>{}_sY_{\ell m} = \sqrt{\frac{(\ell+s)!}{(\ell-s)!}}\ (-1)^s \bar\eth^{-s} Y_{\ell m},\ \ -\ell\leq s \leq 0;</math>
:<math>{}_sY_{\ell m} = 0,\ \ \ell < |s|.</math>
The functions <math>{}_sY_{\ell m}</math> then have the property of transforming with spin weight ''s''.

Other important properties include the following:
:<math>\eth\left({}_sY_{\ell m}\right)     = +\sqrt{(\ell-s)(\ell+s+1)}\ {}_{s+1}Y_{\ell m};</math>
:<math>\bar\eth\left({}_sY_{\ell m}\right) = -\sqrt{(\ell+s)(\ell-s+1)}\ {}_{s-1}Y_{\ell m};</math>

== Orthogonality and completeness ==
The harmonics are orthogonal over the entire sphere:
:<math>\int_{S^2} {}_sY_{\ell m}\ {}_s\bar{Y}_{\ell'm'}\ dS = \delta_{\ell\ell'} \delta_{mm'}, </math>
and satisfy the completeness relation
:<math>\sum_{\ell m} {}_s\bar Y_{\ell m}(\theta',\phi') {}_s Y_{\ell m}(\theta,\phi) = \delta(\phi'-\phi)\delta(\cos\theta'-\cos\theta)</math>

==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 {{harvtxt|Goldberg|Macfarlane|Newman|Rohlich|1967}}.  Note that their formulas 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., formulas is the following:
:<math>{}_sY_{\ell m} (\theta, \phi) = (-1)^m \sqrt{ \frac{(\ell+m)! (\ell-m)! (2\ell+1)} {4\pi (\ell+s)! (\ell-s)!} } \sin^{2\ell} \left( \frac{\theta}{2} \right) </math>
:::::: <math>\times\sum_{r=0}^{\ell-s} {\ell-s \choose r} {\ell+s \choose r+s-m} (-1)^{\ell-r-s} e^{i m \phi} \cot^{2r+s-m} \left( \frac{\theta} {2} \right)\ .</math>

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 <math>{}_s\bar Y_{\ell m} = (-1)^{s+m}{}_{-s}Y_{\ell(-m)}</math> and <math>{}_sY_{\ell m}(\pi-\theta,\phi+\pi) = (-1)^\ell {}_{-s}Y_{\ell m}(\theta,\phi)</math>.

== First few spin-weighted spherical harmonics ==

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

=== Spin-1, degree <math>l=1</math> ===

:<math>{}_1 Y_{10}(\theta,\phi) = \sqrt{\frac{3}{8\pi}}\,\sin\theta</math>
:<math>{}_1 Y_{1\pm 1}(\theta,\phi) =  -\sqrt{\frac{3}{16\pi}}(1 \mp \cos\theta)\,e^{\pm i\phi}</math>

== Relation to Wigner rotation matrices ==

<math>
D^\ell_{-m s}(\phi,\theta,-\psi) =(-1)^m \sqrt\frac{4\pi}{2\ell+1} {}_sY_{\ell m}(\theta,\phi) e^{is\psi}
</math>

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

==See also==

* [[Spherical basis]]

==References==
<references/>
*{{citation | title = The relationship between monopole harmonics and spin-weighted spherical harmonics | first = Tevian | last=Dray | journal = J. Math. Phys. | volume = 26 | pages = 1030–1033 |date=May 1985 | doi = 10.1063/1.526533 | publisher = American Institute of Physics | url = http://link.aip.org/link/?JMP/26/1030/1 | issue = 5|bibcode = 1985JMP....26.1030D }}.
*{{citation|doi=10.1017/S0305004100059971|title=Edth-a differential operator on the sphere|last1=Eastwood|first1=Michael|last2=Tod|first2=Paul|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=92|issue=2|year=1982|pages=317–330|bibcode = 1982MPCPS..92..317E }}.
*{{Citation | last1=Gelfand | first1=I. M. | authorlink1=Israel Gelfand|last2=Minlos | first2=Robert A. | author2-link=Robert A. Minlos | last3=Shapiro | first3=Z. Ja. | title=Predstavleniya gruppy vrashcheni i gruppy Lorentsa, ikh primeneniya | publisher=Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow | id={{MathSciNet | id = 0114876}} | year=1958}}; (1963) Representations of the rotation and Lorentz groups and their applications (translation).  MacMillan Publishers.
*{{citation | title = Spin-s Spherical Harmonics and ð | first1 = J. N. | last1=Goldberg |first2=A. J. |last2=Macfarlane |first3=E. T.|last3=Newman |first4= F. |last4=Rohrlich |first5=E. C. G. |last5=Sudarshan | journal = J. Math. Phys. | volume = 8 | pages = 2155–2161 |date=November 1967 | doi = 10.1063/1.1705135 | publisher = American Institute of Physics | url = http://link.aip.org/link/?JMP/8/2155/1 | issue = 11|bibcode = 1967JMP.....8.2155G }} ''(Note: As mentioned above, this paper uses a choice for the Condon-Shortley phase that is no longer standard.)''
*{{citation | title = Note on the Bondi-Metzner-Sachs Group | first1 = E. T.|last1=Newman |authorlink1=Ezra T. Newman|first2=R.|last2=Penrose |authorlink2=Roger Penrose| journal = J. Math. Phys. | volume = 7 | pages = 863–870 |date=May 1966 | publisher = American Institute of Physics | url = http://link.aip.org/link/?JMP/7/863/1 | doi = 10.1063/1.1931221 | issue = 5|bibcode = 1966JMP.....7..863N }}.
* {{Citation | doi=10.1016/0550-3213(76)90143-7 | last1=Wu | first1=Tai Tsun | last2=Yang | first2=Chen Ning | title=Dirac monopole without strings: monopole harmonics | id={{MathSciNet | id = 0471791}} | year=1976 | journal=Nuclear Physics. B | issn=0550-3213 | volume=107 | issue=3 | pages=365–380|bibcode = 1976NuPhB.107..365W }}.

[[Category:Fourier analysis]]
[[Category:Rotational symmetry]]
[[Category:Special functions]]