Geometry of Deformations of Relativistic Membranes

TL;DR

This paper develops a geometric framework for analyzing infinitesimal deformations of relativistic membranes, applicable to various actions including Nambu and extrinsic curvature models, aiding in understanding membrane dynamics.

Contribution

It introduces a kinematical description of membrane deformations that generalizes to any local geometric worldsheet scalar action, unifying classical and perturbative analyses.

Findings

Framework applicable to Nambu membranes

Extension to actions quadratic in extrinsic curvature

Provides equations for deformations around solutions

Abstract

A kinematical description of infinitesimal deformations of the worldsheet spanned in spacetime by a relativistic membrane is presented. This provides a framework for obtaining both the classical equations of motion and the equations describing infinitesimal deformations about solutions of these equations when the action describing the dynamics of this membrane is constructed using {\it any} local geometrical worldsheet scalars. As examples, we consider a Nambu membrane, and an action quadratic in the extrinsic curvature of the worldsheet.