Some generalizations of the Raychaudhuri equation

TL;DR

This paper extends the Raychaudhuri equation in multiple ways, including improved, spacelike, non-normalized, and two-vector formulations, enhancing its applicability in general relativity and related formalisms.

Contribution

It introduces several novel generalizations of the Raychaudhuri equation, broadening its theoretical framework and potential applications in gravitational physics.

Findings

Improved timelike Raychaudhuri equation with divergence terms

Brief discussion of spacelike Raychaudhuri equation

Development of non-normalized congruence version

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.