The Raychaudhuri equations: a brief review

TL;DR

This paper provides a concise review of the Raychaudhuri equations, exploring their foundational aspects, extensions to non-Riemannian spacetimes, and recent applications across various theories of gravity and physics.

Contribution

It offers a comprehensive overview of the Raychaudhuri equations, including their generalizations and recent applications in multiple areas of theoretical physics.

Findings

Discussion of Raychaudhuri equations in non-Riemannian spacetimes

Overview of applications in quantum field theory and string theory

Insights into future research directions

Abstract

We present a brief review on the Raychaudhuri equations. Beginning with a summary of the essential features of the original article by Raychaudhuri and subsequent work of numerous authors, we move on to a discussion of the equations in the context of alternate non--Riemannian spacetimes as well as other theories of gravity, with a special mention on the equations in spacetimes with torsion (Einstein--Cartan--Sciama--Kibble theory). Finally, we give an overview of some recent applications of these equations in General Relativity, Quantum Field Theory, String Theory and the theory of relativisitic membranes. We conclude with a summary and provide our own perspectives on directions of future research.