Generalised Raychaudhuri Equations for Strings and Membranes

Abstract

A recent generalisation of the Raychaudhuri equations for timelike geodesic congruences to families of $D$ dimensional extremal, timelike, Nambu--Goto surfaces embedded in an $N$ dimensional Lorentzian background is reviewed. Specialising to $D=2$ (i.e the case of string worldsheets) we reduce the equation for the generalised expansion $\theta _{a}, (a =\sigma,\tau)$ to a second order, linear, hyperbolic partial differential equation which resembles a variable--mass wave equation in $1+1$ dimensions. Consequences, such as a generalisation of geodesic focussing to families of worldsheets as well as exactly solvable cases are explored and analysed in some detail. Several possible directions of future research are also pointed out.