Stability of a shell around a black hole

We study the mechanical stability of a static, infinitely thin, spherically symmetric massive shell surrounding a classical Schwarzschild black hole. The shell is taken to have a non-negative surface energy density, and a speed of sound not greater than the speed of light. We show that the shell is stable against radial perturbations only outside a critical radius which is always larger than the radius of the circular photon orbit. The surface energy density of a stable shell is always larger than twice the surface pressure, and thus satisfies the dominant energy condition by a wide margin. We briefly discuss the effects of Hawking radiation in view of a path-integral approach to black-hole thermodynamics developed by York and collaborators. Our results suggest that a macroscopic thermal equilibrium situation associated with the canonical ensemble in this approach may not be realizable with a thin matter shell in Lorentzian spacetime.