Some generalizations of the Raychaudhuri equation

Gabriel Abreu (Victoria University of Wellington); Matt Visser (Victoria University of Wellington)

Some generalizations of the Raychaudhuri equation

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arxiv 1012.4806 v1 pith:VP33UTXO submitted 2010-12-21 gr-qc hep-th

Gabriel Abreu (Victoria University of Wellington) , Matt Visser (Victoria University of Wellington) This is my paper · ORCID

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keywords equationraychaudhuriversiondevelopedcongruencesdiagonalricciseveral

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The Raychaudhuri equation has seen extensive use in general relativity, most notably in the development of various singularity theorems. In this rather technical article we shall generalize the Raychaudhuri equation in several ways. First an improved version of the standard timelike Raychaudhuri equation is developed, where several key terms are lumped together as a divergence. This already has a number of interesting applications, both within the ADM formalism and elsewhere. Second, a spacelike version of the Raychaudhuri equation is briefly discussed. Third, a version of the Raychaudhuri equation is developed that does not depend on the use of normalized congruences. This leads to useful formulae for the "diagonal" part of the Ricci tensor. Fourth, a "two vector" version of the Raychaudhuri equation is developed that uses two congruences to effectively extract "off diagonal" information concerning the Ricci tensor.

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