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Handbuch der Geodäsie pp 1–113Cite as

A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy

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Abstract

The spin-weighted spherical harmonics (by Newman and Penrose) form an orthonormal basis of 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 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 spin-weighted spherical harmonics are the eigenfunctions of a differential operator. In this context, we found Green’s second surface identity for this differential operator and the fact that the spin-weighted spherical harmonics are the only eigenfunctions of this differential operator.

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Authors and Affiliations

  1. Geomathematics Group, University of Siegen, Siegen, Germany

    Volker Michel & Katrin Seibert

Authors
  1. Volker Michel
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  2. Katrin Seibert
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Correspondence to Volker Michel .

Editor information

Editors and Affiliations

  1. Lehr- und Forschungsgebiet "Geomathemati, TU Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Physikalische Geodäsie, TU München Inst. Astronomische und, München, Germany

    Reiner Rummel

Section Editor information

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

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Michel, V., Seibert, K. (2018). A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy. In: Freeden, W., Rummel, R. (eds) Handbuch der Geodäsie. Springer Reference Naturwissenschaften . Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46900-2_102-1

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  • DOI: https://doi.org/10.1007/978-3-662-46900-2_102-1

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  • Publisher Name: Springer Spektrum, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46900-2

  • Online ISBN: 978-3-662-46900-2

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