# Talk:Spin-weighted spherical harmonics

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## Questions

This page consistently has both l and m as lower indices, while the spherical harmonics page uses a raised l. Is there any reason to use a different convention here? (I'm also seeing an upper l used in a paper I'm looking at right now.) --Starwed (talk) 02:37, 11 April 2008 (UTC)

In the expression

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

it is not immediately clear what s stands for. I believe that it refers to the existing spin weighting of eta, is that right? --Starwed (talk) 03:55, 11 April 2008 (UTC)

(Answering self) Yes, from ref cited in the article I see that s is defined to be the spin weighting of the function acted on by the operator. --Starwed (talk) 08:39, 12 April 2008 (UTC)

## Relation to vector harmonics?

One can Clebsch-Gordan couple a spinfunction and a spherical harmonic to a vector harmonic. What is the relation to Spin-weighted spherical harmonics ?--Virginia fried chicken (talk) 16:20, 11 April 2008 (UTC)

I found a paper ]which mentions this:
The complete multipole expansion of the electromagnetic field using vector spherical harmonics is treated in some textbooks in classical electrodynamics. The subject is generally considered difficult and hard to understand. It has long been known to relativists, however, that one can also expand the electromagnetic field in another set of basis functions, the spin-weighted spherical harmonics. The spin-weighted harmonics are a spherical analog of the vector spherical harmonics and are a more natural set of expansion functions for radiation problems with finite sources since the boundary conditions at infinity are spherical in nature.
Not sure if that helps answer your question or not, though.--Starwed (talk) 08:38, 12 April 2008 (UTC)

## Diagrams requested

It would be nice to see what the first few of these eigenfunctions actually look like on a sphere, showing their modulus and phase. Anyone fancy plotting up any pix? Jheald (talk) 15:22, 27 July 2011 (UTC)