Dynamical instability in a relativistic cylindrical shell composed of counter rotating particles

TL;DR

This paper investigates the stability of a relativistic cylindrical shell made of counter rotating particles, revealing that slow-moving shells are unstable with oscillatory growth, while faster shells either expand or contract exponentially.

Contribution

It provides a linear perturbation analysis demonstrating the conditions under which the shell becomes unstable or undergoes exponential expansion or contraction.

Findings

Slow-moving shells exhibit oscillatory instability.

Shells with particle speed above a critical value expand or contract exponentially.

Contradicts previous stability conclusions based on C-energy arguments.

Abstract

We give a perturbative analysis for an infinitesimally thin cylindrical shell composed of counter rotating collisionless particles, originally devised by Apostolatos and Thorne. They found a static solution of the shell and concluded by C-energy argument that it is stable. Recently, the present authors and Ida reanalyzed this system by evaluating the C-energy on the future null infinity and found that the system has an instability, though it was not shown how the system is unstable. In this paper, it is shown in the framework of the linear perturbation theory that, if the constituent particles move slowly, the static shell is unstable in the sense that the perturbation of its circumferential radius oscillates with exponentially growing amplitude, whereas if the speed of the constituent particle exceeds a critical value, the shell just expands or contracts exponentially with time.