C Tensorial Spin-s Spherical Harmonics

Some time ago, the generalization of ordinary spherical harmonics Ylm (ζ, ¯ζ) to spin-weighted functions (s)Ylm (ζ, ¯ζ) (e.g., [33, 28, 56]) was developed to allow for harmonic expansions of spin-weighted functions on the sphere. In this paper we have instead used the tensorial form of these spin-weighted harmonics, the tensorial spin-s spherical harmonics, which are formed by taking appropriate linear combinations of the (s)Ylm (ζ, ¯ζ) [59Jump To The Next Citation Point]:

s ∑ sm Yli...k = Kli...k(s)Ylm,

where the indices obey |s| ≤ l, and the number of spatial indices (i.e., i...k) is equal to l. Explicitly, these tensorial spin-weighted harmonics can be constructed directly from the parametrized Lorentzian null tetrad, Eq. (1.1View Equation) – (1.2View Equation):

√ -- 2 ( ) ˆla = -------¯-- 1 + ζ ¯ζ,ζ + ¯ζ,i¯ζ − iζ,− 1 + ζ¯ζ , (C.1 ) 2(1 +√-ζζ) a 2 ( ) ˆn = -------¯-- 1 + ζ ¯ζ,− (ζ + ζ¯),iζ − i¯ζ,1 − ζ¯ζ , 2(1 +√-ζζ) a -----2----( ¯2 ¯2 ¯) ˆm = 2(1 + ζ¯ζ) 0,1 − ζ ,− i(1 + ζ ),2ζ , P ≡ 1 + ζ¯ζ.
Taking the spatial parts of their duals, we obtain the one-forms
( ) li = √−-1- ζ + ¯ζ,− i(ζ − ¯ζ),− 1 + ζ ¯ζ , (C.2 ) 2P 1 ( ) ni = √---- ζ + ¯ζ,− i(ζ + ¯ζ),− 1 + ζ ¯ζ , 2P m = √−-1-(1 − ¯ζ2,− i(1 + ¯ζ2),2¯ζ), i 2P √-- ci = li − ni = − 2i𝜖ijkmjm¯k.
From this we define Y lli...k as [59Jump To The Next Citation Point]
Y l = m m ...m , (C.3 ) li...k i j k Y −lil...k = ¯mim¯j...m¯k.
The other harmonics are determined by the action of the ∂-operator on the forms, Eq. (C.2View Equation), (with complex conjugates) via
∂l = m, (C.4 ) ∂m = 0, ∂n = − m, ∂c = 2m, -- ∂m = n − l = − c.
Specifically, the spin-s harmonics are defined by
Y s = ¯∂l−s(Y l ), (C.5 ) li...k ( li...k ) Y −li|..s.k|= ∂l−|s| Y −lil...k .

We now present a table of the tensorial spherical harmonics up to l = 2, in terms of the tetrad. Higher harmonics can be found in [59Jump To The Next Citation Point].

l = 0
0 Y0 = 1
l = 1
1 Y1i = mi,
0 Y1i = − ci,
− 1 Y1i = ¯mi
l = 2
Y22ij = mimj, Y 12ij = − (cimj + micj), Y20ij = 3cicj − 2δij
Y−2ij2= ¯mim¯j, Y2−ij1= − (cim ¯j + ¯micj)

In addition, it is useful to give the explicit relations between these different harmonics in terms of the ∂-operator and its conjugate. Indeed, we can see generally that applying ∂ once raises the spin index by one, and applying ¯∂ lowers the index by one. This in turn means that

∂Y lli...k = 0, ¯ −l ∂Y li...k = 0.
Other relations for l ≤ 2 are given by
¯∂Y 1 = Y 0 = ∂Y −1, 1i 1i 1i ∂Y 01i = − 2Y11i, ¯∂Y 0 = − 2Y −1, 1i 1i
¯∂Y 22ij = Y21ij, ¯2 2 0 2 −2 ∂ Y2ij = Y2ij = ∂ Y2ij , ∂Y 0 = − 6Y 1 , 21ij 2i2j ∂Y 2ij = − 4Y2ij.

Finally, due to the nonlinearity of the theory, we have been forced throughout this review to consider products of the tensorial spin-s spherical harmonics while expanding nonlinear expressions. These products can be expanded as a linear combination of individual harmonics using Clebsch–Gordon expansions. The explicit expansions for products of harmonics with l = 1 or l = 2 are given below (we omit higher products due to the complexity of the expansion expressions). Further products can be found in [59Jump To The Next Citation Point, 42].

 C.1 Clebsch–Gordon expansions

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