LARGE DEFORMATIONS OF RELATIVISTIC MEMBRANES - A GENERALIZATION OF THE RAYCHAUDHURI EQUATIONS

A coupled system of nonlinear partial differential equations is presen ted which describes nonperturbatively the evolution of deformations of a relativistic membrane of arbitrary dimension D in an arbitrary back ground spacetime. These equations can be considered from a formal poin t of view as higher dimensional analogues of the Raychaudhuri equation s for point particles to which they are shown to reduce when D=1. For D=1 or D=2 (a string), there are no constraints on the initial data. I f D >2, however, there will be constraints with a corresponding compli cation of the evolution problem. The consistent evolution of the const raints is guaranteed by an integrability condition which is satisfied when the equations of motion are satisfied. Explicit calculations are performed for membranes described by the Nambu action.