On the Stability of Spherical Membranes in Curved Spacetimes

We study the existence and stability of spherical membranes in curved spacetimes. For Dirac membranes in the Schwarzschild–de Sitter background we find that there exists an equilibrium solution. By fine–tuning the dimensionless parameter Λ M 2 , the static membrane can be at any position outside the black hole event horizon, even at the stretched horizon, but the solution is unstable. We show that modes having l = 0 (and for Λ M 2 < 16 / 243 also l = 1) are responsible for the instability. We also find that spherical higher order membranes (membranes with extrinsic curvature corrections), contrary to what happens in flat Minkowski space, do have equilibrium solutions in a general curved background and, in particular, also in the “plain” Schwarzschild geometry (while Dirac membranes do not have equilibrium solutions there). These solutions, however, are also unstable. We shall discuss a way of by– passing these instability problems, and we also relate our results to the recent ideas of representing the black hole event horizon as a relativistic bosonic membrane.