Dynamical Instability in a Relativistic Cylindrical Shell Composed of Counter-Rotating Particles

We perform a perturbative analysis of an infinitesimally thin cylindrical shell composed of counter-rotating collisionless particles, originally devised by Apostolatos and Thorne, who found a static solution of the shell and concluded by C-energy argument that it is stable. Recently, the present authors and Ida have reanalyzed this system by evaluating the C-energy on the future null infinity and found that the system shows instability, although 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 constituent particles move slowly, the static shell is unstable in the sense that the perturbation of its circumferential radius oscillates with an exponentially growing amplitude, whereas if the speed of the constituent particle exceeds a critical value, unstable modes exhibiting exponential expansion or contraction of the shell appear. The present result disproves the evidence for the conjecture of Apostolatos and Thorne, which states that rotation halts the spindle gravitational collapse, and seems to support the numerical results on the spindle collapse obtained by Shapiro and Teukolsky. It is still controversial whether their numerical results imply the formation of naked singularities. However, if the spindle gravitational collapse really results in the formation of naked singularities, our result implies a violation of the cosmic censorship conjecture. Subject Index: 451