Curvature form of Raychaudhuri equation and its consequences: A geometric approach

TL;DR

This paper derives a curvature form of the Raychaudhuri equation, providing new geometric insights and transforming it into a harmonic oscillator-like equation with an analytic solution.

Contribution

It introduces a curvature form of the Raychaudhuri equation and transforms it into a second-order differential equation resembling a harmonic oscillator, offering new analytical tools.

Findings

Derived a curvature form of the Raychaudhuri equation.

Transformed the RE into a harmonic oscillator-like equation.

Provided an analytic solution and Lagrangian for the congruence.

Abstract

The paper aims at deriving a curvature form of the famous Raychaudhuri equation (RE) and the associated criteria for focusing of a hyper-surface orthogonal congruence of time-like geodesic. Moreover, the paper identifies a transformation of variable related to the metric scalar of the hyper-surface which converts the first order RE into a second order differential equation that resembles an equation of a Harmonic oscillator and also gives a first integral that yields the analytic solution of the RE and Lagrangian of the dynamical system representing the congruence.