Spin-weighted spherical harmonics

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Spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are complex functions on the sphere. These harmonics are typically denoted by sYlm, where s is the spin weight, and l and m are akin to the usual parameters familiar from the standard spherical harmonics. The spin-weighted spherical harmonics can be derived 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:

{}_0Y_{lm} = Y_{lm}\ .

Though these functions were first introduced by Ezra T. Newman and Roger Penrose to describe gravitational radiation [1], they are quite general, and can be applied to other functions on a sphere.

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[edit] Origin

The spin-weighted harmonics—like their standard relatives—are functions on a sphere. We select a point on the sphere, and rotate the sphere about that point by some angle ψ. By definition, a function η with spin weight s transforms as \eta \rightarrow e^{i s \psi}\eta.

Working in standard spherical coordinates, we can define a particular operator \eth acting on a function η as:

\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]\ .

This gives us another function of θ and φ. [The operator \eth is effectively a covariant derivative operator in the sphere.]

An important property of the new function \eth\eta is that if η had spin weight s, \eth\eta has spin weight s + 1. 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:

\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]\ .

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

{}_sY_{lm} = \sqrt{\frac{(l-s)!}{(l+s)!}}\ \eth^s Y_{lm},\ \ 0\leq s \leq l;
{}_sY_{lm} = \sqrt{\frac{(l+s)!}{(l-s)!}}\ (-1)^s \bar\eth^{-s} Y_{lm},\ \ -l\leq s \leq 0;
{}_sY_{lm} = 0,\ \ l < |s|\ .

The functions sYlm then have the property of transforming with spin weight s.

Other important properties include the following:

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

[edit] Orthogonality and completeness

The harmonics are orthogonal over the entire sphere:

\int_{S^2} {}_sY_{lm}\ {}_s\bar{Y}_{l'm'}\ dS = \delta_{ll'} \delta_{mm'},

and satisfy the completeness relation

\sum_{lm} {}_s\bar Y_{lm}(\theta',\phi') {}_s Y_{lm}(\theta,\phi) = \delta(\phi'-\phi)\delta(\cos\theta'-\cos\theta)


[edit] 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 Goldberg, et al. [2]. 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:

{}_sY_{lm} (\theta, \phi) = (-1)^m \sqrt{ \frac{(l+m)! (l-m)! (2l+1)} {4\pi (l+s)! (l-s)!} } \sin^{2l} \left( \frac{\theta}{2} \right)
\times\sum_{r=0}^{l-s} {l-s \choose r} {l+s \choose r+s-m} (-1)^{l-r-s} e^{i m \phi} \cot^{2r+s-m} \left( \frac{\theta} {2} \right)\ .

A Mathematica notebook using this formula to calculate arbitrary spin-weighted spherical harmonics can be found here.

With the phase convention here {}_s\bar Y_{lm} = (-1)^{s+m}{}_{-s}Y_{l(-m)} and sYlm(π − θ,φ + π) = ( − 1)l − sYlm(θ,φ).

[edit] First few spin-weighted spherical harmonics

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

[edit] Spin-1, degree l = 1

{}_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\varphi}


[edit] Relation to Wigner rotation matrices

D^l_{-m s}(\phi,\theta,-\psi) =(-1)^m \sqrt\frac{4\pi}{2l+1} {}_sY_{lm}(\theta,\phi) e^{is\psi}

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

[edit] References

  1. ^ E. T. Newman and R. Penrose (May 1966). "Note on the Bondi-Metzner-Sachs Group". J. Math. Phys. 7: 863--870. American Institute of Physics. doi:10.1063/1.1931221. 
  2. ^ J. N. Goldberg and A. J. Macfarlane and E. T. Newman and F. Rohrlich and E. C. G. Sudarshan (November 1967). "Spin-s Spherical Harmonics and ð". J. Math. Phys. 8: 2155--2161. American Institute of Physics. doi:10.1063/1.1705135. 

Tevian Dray (May 1985). "The relationship between monopole harmonics and spin-weighted spherical harmonics". J. Math. Phys. 26: 1030--1033. American Institute of Physics. doi:10.1063/1.526533.  A more modern and somewhat generalized treatment.