A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy
The spin-weighted spherical harmonics (by Newman and Penrose) form an orthonormal basis of Open image in new window on the unit sphere Ω and have a huge field of applications. Mainly, they are used in quantum mechanics and geophysics for the theory of gra
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Spin Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Definition by Newman and Penrose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Properties and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Definition by the Wigner D-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Operator ð . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Uniqueness of the Eigenfunctions of Δ∗,N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Additional Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Relation to the Scalar, Vector, and Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Abstract
The spin-weighted spherical harmonics (by Newman and Penrose) form an orthonormal basis of L2 (Ω) on the unit sphere Ω and have a huge field of applications. Mainly, they are used in quantum mechanics and geophysics for the theory of gravitation and in early universe and classical cosmology. Furthermore, they have also applications in geodesy. The quantity of formulations conditioned this huge spectrum of versatility. Formulations we use are for example given by the Wigner D-function, by a spin raising and spin lowering operator or as a function of spin weight. We present a unified mathematical theory which implies the collection of already known properties of the spin-weighted spherical harmonics. We recapitulate this in a mathematical way and connect it to the notation of the
V. Michel () · K. Seibert Geomathematics Group, University of Siegen, Siegen, Germany E-Mail: [email protected]; [email protected] © Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018 W. Freeden, R. Rummel (Hrsg.), Handbuch der Geodäsie, Springer Reference Naturwissenschaften, https://doi.org/10.1007/978-3-662-46900-2_102-1
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V. Michel and K. Seibert
theory of spherical harmonics. Here, the fact that the spherical harmonics are the spin-weighted spherical harmonics with spin weight zero is useful. Furthermore, our novel mathematical approach enables us to prove some previously unknown properties. For example, we can formulate new recursion relations and a Christoffel-Darboux formula. Moreover, it is known that the spinweighted spherical harmonics are the eigenfunct
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