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We give a perturbative analysis for an infinitesimally thin cylindrical shellcomposed of counter rotating collisionless particles, originally devised byApostolatos and Thorne. They found a static solution of the shell and concludedby C-energy argument that it is stable. Recently, the present authors and Idareanalyzed this system by evaluating the C-energy on the future null infinityand found that the system has an instability, though it was not shown how thesystem is unstable. In this paper, it is shown in the framework of the linearperturbation theory that, if the constituent particles move slowly, the staticshell is unstable in the sense that the perturbation of its circumferentialradius oscillates with exponentially growing amplitude, whereas if the speed ofthe constituent particle exceeds a critical value, the shell just expands orcontracts exponentially with time.

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