Some time ago, the generalization of ordinary spherical harmonics to spin-weighted functions (e.g., [21, 17, 40]) 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 :
where the indices obey , and the number of spatial indices (i.e., ) is equal to . Explicitly, these tensorial spin-weighted harmonics can be constructed directly from the parameterized Lorentzian null tetrad, Equation (74):
We now present a table of the tensorial spherical harmonics up to , in terms of the tetrad. Higher harmonics can be found in .
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
Finally, due to the nonlinearity of the theory, we have been forced throughout this review to consider products of the tensorial spin- 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 or are given below (we omit higher products due to the complexity of the expansion expressions). Further products can be found in [43, 29].
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