On the Stability of Spherical Membranes in Curved Spacetimes

TL;DR

This paper investigates the existence and stability of spherical membranes in curved spacetimes, revealing unstable equilibrium solutions for Dirac membranes and stable solutions for higher order membranes, with implications for black hole horizon models.

Contribution

It demonstrates that higher order membranes can have stable equilibrium solutions in curved backgrounds, unlike Dirac membranes, and explores implications for black hole horizon modeling.

Findings

Dirac membranes have unstable equilibrium solutions outside the horizon.

Higher order membranes can have stable equilibrium solutions in curved backgrounds.

Modes with specific angular momentum cause instability in Dirac membranes.

Abstract

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 $\Lambda M^2,$ 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 $\Lambda M^2<16/243$ 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, {\it 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.