Rotation halts cylindrical, relativistic gravitational collapse

Abstract

It is shown, in a simple analytic example, that an infinitesimal amount of rotation can halt the general relativistic gravitational collapse of a pressure-free cylindrical body. The example is a thin cylindrical shell (a shell with translational symmetry and rotation symmetry), made of counterrotating dust particles. Half of the particles rotate about the symmetry axis in one direction with (conserved) angular momentum per unit rest mass α, and the other half rotate in the opposite direction with the same α. It is shown, using C-energy arguments, that the shell can never collapse to a circumference smaller than C=8παΛ, where Λ is the shell’s nonconserved mass per unit proper length. Equivalently, if R≡‖∂/∂φ∥∂/∂z‖ is the product of the lengths of the rotational and translational Killing vectors at the shell’s location and λ is the shell’s conserved rest mass per unit Killing length z, then the shell can never collapse smaller than R=4αλ. It is also shown that after its centrifugally induced bounce, the shell will oscillate radially and will radiate gravitational waves as it oscillates, the waves will carry away C energy, and this loss of C energy will force the shell to settle down to a static, equilibrium radius.