Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

Details Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

2008-05-29

arxiv.org/abs/0805.4500v1

Annalen Phys.17:609-611,1997

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

4 pages, accepted for publication in Annalen der Physik

Scientific paper

10.1002/andp.200810312

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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