Talk:Spin-weighted spherical harmonics

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)

Inconsistency between rotation around poles and explicit examples

In the section "Representation as functions" there is "By definition, a function $\eta$ with spin weight s transforms under rotation about the pole via $\eta \rightarrow e^{is\psi }\eta$." but that is not the case for the explicit examples in "First few spin-weighted spherical harmonics", specifically for ${}_{1}Y_{1,-1}$. — Preceding unsigned comment added by 140.180.245.240 (talk) 15:39, 26 April 2017 (UTC)

Similarly, the statement in the section "Spin-weighted functions" that "A spin-weight s function f is a function accepting as input a point x of S² and a positively oriented orthonormal basis of tangent vectors at x, such that ..." and then showing a function of x, a and b (with a and b degenerate) is inconsistent with the entire discussion of the spin-weighted spherical harmonics lower on the page, where the functions that are shown depend only on x, the position on the sphere; they are not functions of anything else. How does one reconcile the absence of any parameters other than $\theta$ and $\phi$ in the explicit forms of the spin-weighted spherical harmonics with the claim that they should somehow depend on the orientation of a basis? — Preceding unsigned comment added by 133.11.21.95 (talk) 10:21, 6 November 2017 (UTC)

Calculation

First of all, I find this article extremely useful. Second, the linked Mathematica-Script fails to evaluate the spin-weight spherical harmonics at the poles due to a removable singularity. This can be fixed by inserting the numerical value for theta after simplifying. The new definition of YY in the Mathematica script is

YY[s_, l_, m_, \[Theta]_, \[Phi]_] := E^(I s \[Zeta]) (-1)^m Simplify[ Sqrt[((l + m)! (l - m)! (2 l + 1))/((l + s)! (l - s)! 4 \[Pi])] (Sin[theta/2])^(2 l)       Sum[Binomial[l - s, t] Binomial[l + s, t + s - m] (-1)^(l - t - s) E^(I m \[Phi]) (Cot[theta/2])^(2 t + s - m), {t, 0, l - s}], Assumptions -> {\[Phi] \[Element] Reals, \[Theta] \[Element] Reals, s \[Element] Integers, m \[Element] Integers, l \[Element] Integers}] /. {theta -> \[Theta]};

Rappel1 (talk) 11:09, 25 March 2020 (UTC)