The Lanczos Potential for WeylCandidate Tensors Exists Only in Four Dimensions
Abstract
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 nontrivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a nontrivial 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.
 Publication:

General Relativity and Gravitation
 Pub Date:
 December 2000
 DOI:
 10.1023/A:1001951609641
 arXiv:
 arXiv:grqc/9711022
 Bibcode:
 2000GReGr..32.2307E
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 11 pages, LaTeX, Heavily revised April 2000