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Correspondence between tensorial spin-s and spin-weighted spherical harmonics

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Abstract

The tensorial spin-s and the spin weighted spherical harmonics are functions on the sphere which are extremely useful for many calculations in general relativity. These sets of functions constitute two orthonormal bases and can be used to expand spin-s fields in General Relativity and other fields. Particularly, we are interested in the Weyl scalar \(\psi _4\) which is related to the gravitational radiation. In this article, we build a catalog where the correspondence between tensorial spin-s and the spin weighted spherical harmonics is shown. Also, as a simple application, we use the resulting transformations to link the quadrupole and octupole shear of a tensorial spin-s expansion with the gravitational wave functions expressed in terms of the spin weighted spherical harmonics.

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Acknowledgements

The authors would like to thank Dr. Carlos Kozameh for his suggestions that improved the quality of this article. This research has been supported by grants from CONICET-Argentina.

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Authors and Affiliations

  1. Instituto de Física Enrique Gaviola, FaMAF, Universidad Nacional de Córdoba, C. P. 5000, Córdoba, Argentina

    P. A. Mandrilli, J. I. Nieva & G. D. Quiroga

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  1. P. A. Mandrilli
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  2. J. I. Nieva
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  3. G. D. Quiroga
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Correspondence to G. D. Quiroga.

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A harmonics transformation list for \(s>0\)

A harmonics transformation list for \(s>0\)

In this appendix, we include the transformation between both harmonics assuming spin \(s>0\).

l = 1; s = 1:

$$\begin{aligned} Y^1_{1x}&=\sqrt{\frac{2\pi }{3}}\Big ({}_{(1)}Y_{11}-{}_{(1)}Y_{1-1}\Big ) \end{aligned}$$
(96)
$$\begin{aligned} Y^1_{1y}&=-i\sqrt{\frac{2\pi }{3}}\Big ({}_{(1)}Y_{1-1}+{}_{(1)}Y_{11}\Big ) \end{aligned}$$
(97)
$$\begin{aligned} Y^1_{1z}&=-2\sqrt{\frac{\pi }{3}}{}_{(1)}Y_{10} \end{aligned}$$
(98)

l = 2 ; s = 1:

$$\begin{aligned} Y^1_{2xx}&=-2\sqrt{\frac{\pi }{5}}\Big ({}_{(1)}Y_{2-2}+{}_{(1)}Y_{22}\Big )+2\sqrt{\frac{2\pi }{15}}{}_{(1)}Y_{20} \end{aligned}$$
(99)
$$\begin{aligned} Y^1_{2xy}&=2i\sqrt{\frac{\pi }{5}}\Big ({}_{(1)}Y_{22}-{}_{(1)}Y_{2-2}\Big ) \end{aligned}$$
(100)
$$\begin{aligned} Y^1_{2xz}&=-2\sqrt{\frac{\pi }{5}}\Big ({}_{(1)}Y_{2-1}-{}_{(1)}Y_{21}\Big ) \end{aligned}$$
(101)
$$\begin{aligned} Y^1_{2yy}&=2\sqrt{\frac{\pi }{5}}\Big ({}_{(1)}Y_{2-2}+{}_{(1)}Y_{22}\Big )+2\sqrt{\frac{2\pi }{15}}{}_{(1)}Y_{20} \end{aligned}$$
(102)
$$\begin{aligned} Y^1_{2yz}&=-2i\sqrt{\frac{\pi }{5}}\Big ({}_{(1)}Y_{2-1}+{}_{(1)}Y_{21}\Big ) \end{aligned}$$
(103)

l = 2 ; s = 2:

$$\begin{aligned} Y^2_{2xx}&=\sqrt{\frac{\pi }{5}}\Big ({}_{(2)}Y_{2-2}+{}_{(2)}Y_{22}\Big )-\sqrt{\frac{2\pi }{15}}{}_{(2)}Y_{20} \end{aligned}$$
(104)
$$\begin{aligned} Y^2_{2xy}&=i\sqrt{\frac{\pi }{5}}\Big ({}_{(2)}Y_{2-2}-{}_{(2)}Y_{22}\Big ) \end{aligned}$$
(105)
$$\begin{aligned} Y^2_{2xz}&=\sqrt{\frac{\pi }{5}}\Big ({}_{(2)}Y_{2-1}-{}_{(2)}Y_{21}\Big ) \end{aligned}$$
(106)
$$\begin{aligned} Y^2_{2yy}&=-\sqrt{\frac{\pi }{5}}\Big ({}_{(2)}Y_{2-2}+{}_{(2)}Y_{22}\Big )-\sqrt{\frac{2\pi }{15}}{}_{(2)}Y_{20} \end{aligned}$$
(107)
$$\begin{aligned} Y^2_{2yz}&=i\sqrt{\frac{\pi }{5}}\Big ({}_{(2)}Y_{2-1}+{}_{(2)}Y_{21}\Big ) \end{aligned}$$
(108)

l = 3 ; s = 1:

$$\begin{aligned} Y^1_{3xxx}&=-2\sqrt{\frac{\pi }{14}}\Big (3{}_{(1)}Y_{31}-\sqrt{15}{}_{(1)}Y_{33}-3{}_{0}Y_{3-1}+\sqrt{15}{}_{(1)}Y_{3-3}\Big ) \end{aligned}$$
(109)
$$\begin{aligned} Y^1_{3xxy}&=2i\sqrt{\frac{\pi }{14}}\Big ({}_{(1)}Y_{31}-\sqrt{15}{}_{(1)}Y_{33}+{}_{(1)}Y_{3-1}-\sqrt{15}{}_{(1)}Y_{3-3}\Big ) \end{aligned}$$
(110)
$$\begin{aligned} Y^1_{3xxz}&=\frac{1}{3}\Big (6\sqrt{\frac{6\pi }{7}}{}_{(1)}Y_{30}-\sqrt{\frac{15\pi }{7}}\Big (2\sqrt{3}{}_{(1)}Y_{32}+2\sqrt{3}{}_{(1)}Y_{3-2}\Big )\Big ) \end{aligned}$$
(111)
$$\begin{aligned} Y^1_{3xyy}&=-2\sqrt{\frac{\pi }{14}}\Big ({}_{(1)}Y_{31}+\sqrt{15}{}_{(1)}Y_{33}-{}_{(1)}Y_{3-1}-\sqrt{15}{}_{(1)}Y_{3-3}\Big ) \end{aligned}$$
(112)
$$\begin{aligned} Y^1_{3xyz}&=2i\sqrt{\frac{15\pi }{21}}\Big ({}_{(1)}Y_{32}-{}_{(1)}Y_{3-2}\Big ) \end{aligned}$$
(113)
$$\begin{aligned} Y^1_{3yyy}&=2i\sqrt{\frac{\pi }{14}}\Big (3{}_{(1)}Y_{31}+\sqrt{15}{}_{(1)}Y_{33}+3{}_{(1)}Y_{3-1}+\sqrt{15}{}_{(1)}Y_{3-3}\Big ) \end{aligned}$$
(114)
$$\begin{aligned} Y^1_{3yyz}&=\frac{2}{21}\sqrt{\pi }\Big (3\sqrt{42}{}_{(1)}Y_{30}+\sqrt{315}\Big ({}_{(1)}Y_{32}+{}_{(1)}Y_{3-2}\Big )\Big ) \end{aligned}$$
(115)

l = 3 ; s = 2:

$$\begin{aligned} Y^2_{3xxx}&=\sqrt{\frac{\pi }{35}}\Big (3{}_{(2)}Y_{31}-\sqrt{15}{}_{(2)}Y_{33}-3{}_{(2)}Y_{3-1}+\sqrt{15}{}_{(2)}Y_{3-3}\Big ) \end{aligned}$$
(116)
$$\begin{aligned} Y^2_{3xxy}&=-i\sqrt{\frac{\pi }{35}}\Big ({}_{(2)}Y_{31}-\sqrt{15}{}_{(2)}Y_{33}+{}_{(2)}Y_{3-1}-\sqrt{15}{}_{(2)}Y_{3-3}\Big ) \end{aligned}$$
(117)
$$\begin{aligned} Y^2_{3xxz}&=\sqrt{\frac{2\pi }{7}}\Big ({}_{(2)}Y_{32}+{}_{(2)}Y_{3-2}\Big )-2\sqrt{\frac{3\pi }{35}}{}_{(2)}Y_{30} \end{aligned}$$
(118)
$$\begin{aligned} Y^2_{3xyy}&=\sqrt{\frac{\pi }{35}}\Big ({}_{(2)}Y_{31}+\sqrt{15}{}_{(2)}Y_{33}-{}_{(2)}Y_{3-1}-\sqrt{15}{}_{(2)}Y_{3-3}\Big ) \end{aligned}$$
(119)
$$\begin{aligned} Y^2_{3xyz}&=-2i\sqrt{\frac{\pi }{42}}\Big (\sqrt{3}{}_{(2)}Y_{32}-\sqrt{3}{}_{(2)}Y_{3-2}\Big ) \end{aligned}$$
(120)
$$\begin{aligned} Y^2_{3yyy}&=-i\sqrt{\frac{\pi }{35}}\Big (3{}_{(2)}Y_{31}+\sqrt{15}{}_{(2)}Y_{33}+3{}_{(2)}Y_{3-1}+\sqrt{15}{}_{(2)}Y_{3-3}\Big ) \end{aligned}$$
(121)
$$\begin{aligned} Y^2_{3yyz}&=-\frac{2}{21}\sqrt{\frac{\pi }{10}}\Big (3\sqrt{42}{}_{(2)}Y_{30}+\sqrt{315}\Big ({}_{(2)}Y_{32}+{}_{(2)}Y_{3-2}\Big )\Big ) \end{aligned}$$
(122)

l = 3 ; s = 3:

$$\begin{aligned} Y^3_{3xxx}&=-\sqrt{\frac{\pi }{210}}(3{}_{\Big (3)}Y_{31}-\sqrt{15}{}_{(3)}Y_{33}-3{}_{(3)}Y_{3-1}+\sqrt{15}{}_{(3)}Y_{3-3}\Big ) \end{aligned}$$
(123)
$$\begin{aligned} Y^3_{3xxy}&=i\sqrt{\frac{\pi }{210}}\Big ({}_{(3)}Y_{31}-\sqrt{15}{}_{(3)}Y_{33}+{}_{(3)}Y_{3-1}-\sqrt{15}{}_{(3)}Y_{3-3}\Big ) \end{aligned}$$
(124)
$$\begin{aligned} Y^3_{3xxz}&=\sqrt{\frac{2\pi }{35}}{}_{(3)}Y_{30}-\sqrt{\frac{\pi }{21}}\Big ({}_{(3)}Y_{32}+{}_{(3)}Y_{3-2}\Big ) \end{aligned}$$
(125)
$$\begin{aligned} Y^3_{3xyy}&=-\sqrt{\frac{\pi }{210}}\Big ({}_{(3)}Y_{31}+\sqrt{15}{}_{(3)}Y_{33}-{}_{(3)}Y_{3-1}+\sqrt{15}{}_{(3)}Y_{3-3}\Big ) \end{aligned}$$
(126)
$$\begin{aligned} Y^3_{3xyz}&=\frac{1}{3}i\sqrt{\frac{3\pi }{7}}\Big ({}_{(3)}Y_{32}-{}_{(3)}Y_{3-2}\Big ) \end{aligned}$$
(127)
$$\begin{aligned} Y^3_{3yyy}&=i\sqrt{\frac{\pi }{210}}\Big (3{}_{(3)}Y_{31}+\sqrt{15}{}_{(3)}Y_{33}+3{}_{(3)}Y_{3-1}+\sqrt{15}{}_{(3)}Y_{3-3}\Big ) \end{aligned}$$
(128)
$$\begin{aligned} Y^3_{3yyz}&=\frac{1}{21}\sqrt{\frac{\pi }{15}}\Big (3\sqrt{42}{}_{(3)}Y_{30}+\sqrt{315}\Big ({}_{(3)}Y_{32}+{}_{(3)}Y_{3-2}\Big )\Big ) \end{aligned}$$
(129)

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Mandrilli, P.A., Nieva, J.I. & Quiroga, G.D. Correspondence between tensorial spin-s and spin-weighted spherical harmonics. Gen Relativ Gravit 52, 58 (2020). https://doi.org/10.1007/s10714-020-02710-1

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