The Lanczos potential for Weyl-candidate tensors exists only in four dimensions

Details The Lanczos potential for Weyl-candidate tensors exists only in four dimensions The Lanczos potential for Weyl-candidate tensors exists only in four dimensions

1997-11-06

arxiv.org/abs/gr-qc/9711022v2

Gen.Rel.Grav. 32 (2000) 2307-2318

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

11 pages, LaTeX, Heavily revised April 2000

Scientific paper

We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W_abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L_abc.

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