Are Higher Order Membranes stable in Black Hole Spacetimes?

TL;DR

This paper investigates the stability of higher order spherical membranes in black hole spacetimes, finding that while static solutions exist, they are inherently unstable due to energy considerations.

Contribution

It introduces a second-order Lagrangian for membranes including scalar curvature terms and analyzes their stability in curved spacetimes.

Findings

Static spherical membranes can be constructed but are energetically unfavorable.

No stable equilibrium configurations are found for these membranes.

The stability analysis extends to membranes with all second derivative terms.

Abstract

We continue the study of the existence and stability of static spherical membrane configurations in curved spacetimes. We first consider higher order membranes described by a Lagrangian which, besides the Dirac term, includes a term proportional to the scalar curvature of the world--volume ${}^{(3)}R$. Notably, in this case, the equations of motion can be reduced to second order ones and an effective potential analysis can be made. The conditions for stability are then explicitly derived. We find a self--consistent static spherical membrane, determining the spacetime generated by the membrane itself. In this case we find, however, that the total energy of the membrane has to be negative, and no {\it stable} equilibrium can be achieved. We then generalize the discussion to a membrane described by a Lagrangian including all possible second derivative terms. We conclude the paper with some discussion on the generality of the results obtained.