Some generalizations of the Raychaudhuri equation

Details Some generalizations of the Raychaudhuri equation Some generalizations of the Raychaudhuri equation

2010-12-21

arxiv.org/abs/1012.4806v1

Phys.Rev.D83:104016,2011

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

8 pages

Scientific paper

10.1103/PhysRevD.83.104016

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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