Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature |
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Authors: | N Straumann |
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Institution: | Institute for Theoretical Physics University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland |
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Abstract: | In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses – beside the Hodge decomposition for one‐forms – an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations. |
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Keywords: | Cosmological perturbations elliptic equations harmonic analysis |
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