Self-similar motion of a Nambu-Goto string

TL;DR

This paper introduces the concept of self-similar strings in self-similar spacetimes, reduces their equations of motion to geodesic equations under certain conditions, and explores explicit solutions in an expanding universe, including stability analysis.

Contribution

It defines self-similar strings within Nambu-Goto theory and simplifies their equations of motion to geodesic equations, providing explicit solutions in cosmological backgrounds.

Findings

Self-similar strings reduce to geodesics under certain conditions.

Explicit solutions for open and closed strings in FLRW universe.

Circular string solutions exhibit linear evolution and stability analysis.

Abstract

We study the self-similar motion of a string in a self-similar spacetime by introducing the concept of a self-similar string, which is defined as the world sheet to which a homothetic vector field is tangent. It is shown that in Nambu-Goto theory, the equations of motion for a self-similar string reduce to those for a particle. Moreover, under certain conditions such as the hypersurface orthogonality of the homothetic vector field, the equations of motion for a self-similar string simplify to the geodesic equations on a (pseudo) Riemannian space. As a concrete example, we investigate a self-similar Nambu-Goto string in a spatially flat Friedmann-Lema\^itre-Robertson-Walker expanding universe with self-similarity and obtain solutions of open and closed strings, which have various nontrivial configurations depending on the rate of the cosmic expansion. For instance, we obtain a circular solution that evolves linearly in the cosmic time while keeping its configuration by the balance between the effects of the cosmic expansion and string tension. We also show the instability for linear radial perturbation of the circular solutions.