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 $$\psi _4$$ ψ 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.

中文翻译：

---

张量自旋s和自旋加权球谐函数之间的对应关系

张量自旋-s 和自旋加权球谐函数是球面上的函数，它们对于广义相对论中的许多计算非常有用。这些函数集构成了两个正交基，可用于扩展广义相对论和其他领域中的自旋-s 场。特别是，我们对与引力辐射有关的外尔标量 $$\psi _4$$ ψ 4 感兴趣。在本文中，我们构建了一个目录，其中显示了张量自旋 s 和自旋加权球谐函数之间的对应关系。此外，作为一个简单的应用，我们使用由此产生的变换将张量自旋 s 膨胀的四极和八极剪切与用自旋加权球谐函数表示的引力波函数联系起来。