Lagrangian Formulation of the Raychaudhuri Equation in Non-Riemannian Geometry

Abstract

The Raychaudhuri equation for a congruence of curves in a general non-Riemannian geometry is derived. A formal connection is established between the expansion scalar and the cross-sectional volume of the congruence. It is found that the expansion scalar is equal to the fractional rate of change of volume, weighted by a scalar factor that depends on the non-Riemannian features of the geometry. Treating the congruence of curves as a dynamical system, an appropriate Lagrangian is derived such that the corresponding Euler-Lagrange equation is the Raychaudhuri equation. A Hamiltonian formulation and Poisson brackets are also presented.