Large Deformations of Relativistic Membranes: A Generalization of the Raychaudhuri Equations

TL;DR

This paper develops a set of non-linear PDEs generalizing Raychaudhuri equations to describe the non-perturbative evolution of relativistic membranes of arbitrary dimension in any background spacetime, with explicit calculations for Nambu membranes.

Contribution

It introduces a new coupled PDE system for membrane deformations, extending Raychaudhuri equations to higher dimensions and analyzing constraint evolution.

Findings

Equations reduce to Raychaudhuri equations for D=1

Constraints are manageable for D≤2, complex for D>2

Explicit solutions provided for Nambu action membranes

Abstract

A coupled system of non-linear partial differential equations is presented which describes non-perturbatively the evolution of deformations of a relativistic membrane of arbitrary dimension, $D$, in an arbitrary background spacetime. These equations can be considered from a formal point of view as higher dimensional analogs of the Raychaudhuri equations 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. If $D>2$, however, there will be constraints with a corresponding complication of the evolution problem. The consistent evolution of the constraints 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.