Harmonic coordinates in the string and membrane equations

TL;DR

This paper demonstrates that solutions to relativistic string and membrane equations are diffeomorphic and shows how harmonic coordinate transformations can linearize these nonlinear equations, simplifying their analysis.

Contribution

It establishes the diffeomorphic nature of solutions and clarifies the role of harmonic coordinates in simplifying relativistic string equations.

Findings

Solutions are diffeomorphic for the considered equations.

Harmonic coordinates transform nonlinear equations into linear wave equations.

Simplification aids in the analysis of string dynamics in Minkowski space.

Abstract

In this note, we first show that the solutions to Cauchy problems for two versions of relativistic string and membrane equations are diffeomorphic. Then we investigate the coordinates transformation presented in Ref. [9] (see (2.20) in Ref. [9]) which plays an important role in the study on the dynamics of the motion of string in Minkowski space. This kind of transformed coordinates are harmonic coordinates, and the nonlinear relativistic string equations can be straightforwardly simplified into linear wave equations under this transformation.