Renormalization group approach to Einstein-Rosen waves

TL;DR

This paper applies a renormalization group method to Einstein-Rosen waves, revealing stable and unstable self-similar solutions that describe the asymptotic behavior of gravitational waves in cylindrical symmetry.

Contribution

It introduces a renormalization group framework to analyze Einstein-Rosen waves, extending the similarity hypothesis beyond spherical symmetry.

Findings

Self-similar solutions appear as fixed points in the RG analysis.

Explosive gravitational waves are stable under certain conditions.

Collapsing gravitational waves are weakly unstable.

Abstract

We present a renormalization group analysis to Einstein-Rosen waves or vacuum spacetimes with whole-cylinder symmetry. It is found that self-similar solutions appear as fixed points in the renormalization group transformation. These solutions correspond to the explosive gravitational waves and the collapsing gravitational waves at late times and early times, respectively. Based on the linear perturbation analysis of the self-similar solutions, we conclude that the self-similar evolution is stable as explosive gravitational waves under the condition of no incoming waves, while it is weakly unstable as collapsing gravitational waves. The result implies that self-similar solutions can describe the asymptotic behavior of more general solutions for exploding gravitational waves and thus extends the similarity hypothesis in general relativity from spherical symmetry to cylindrical symmetry.