Some generalizations of the Raychaudhuri equation

Abstract

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.

8 pages

References in corpus (14)

• Entropy of Static Spacetimes and Microscopic Density of States
• The Raychaudhuri equations: a brief review
• The Kerr spacetime: A brief introduction
• Acoustic geometry for general relativistic barotropic irrotational fluid flow
• Tolman mass, generalized surface gravity, and entropy bounds
• Raychaudhuri's equation and aspects of relativistic charged collapse
• Role of Horizons in Semiclassical Gravity: Entropy and the Area Spectrum
• A singularity theorem based on spatial averages
• Singularity: Raychaudhuri Equation once again
• Entropy bounds for uncollapsed rotating bodies
• On the genericity of spacetime singularities
• Semiclassical quantization of gravity I: Entropy of horizons and the area spectrum
• On a Raychaudhuri equation for hot gravitating fluids
• Horizons in 2+1-dimensional collapse of particles

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