Some remarks on curvature tensor integrability in spacetimes

Details Some remarks on curvature tensor integrability in spacetimes Some remarks on curvature tensor integrability in spacetimes

Mar 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006cqgra..23.1485h&link_type=abstract

Classical and Quantum Gravity, Volume 23, Issue 5, pp. 1485-1492 (2006).

Physics

1

Scientific paper

If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.

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