Curvature corrections and topology change transition in brane-black hole systems: A perturbative approach

Abstract

We consider curvature corrections to static, axisymmetric Dirac-Nambu-Goto membranes embedded into a spherically symmetric black hole spacetime with arbitrary number of dimensions. Since the next to leading order corrections in the effective brane action are quadratic in the brane thickness $\ensuremath{\ell}$, we adopt a linear perturbation approach in ${\ensuremath{\ell}}^{2}$. The perturbations are general in the sense that they are not restricted to the Rindler zone nor to the near-critical solutions of the unperturbed system. As a result, an unexpected asymmetry in the perturbed system is found. In configurations, where the brane does not cross the black hole horizon, the perturbative approach does not lead to regular solutions if the number of the brane's spacetime dimensions $Dg3$. This condition, however, does not hold for the horizon crossing solutions. Consequently we argue that the presented perturbative approach breaks down for subcritical type solutions near the axis of the system for $Dg3$. Nevertheless, we can discuss topology-changing phase transitions in cases when $D=2$ or 3, i.e. when the brane is a one-dimensional string or a two-dimensional sheet, respectively. For the general case, a different, nonperturbative approach should be sought. Based on the energy properties of those branes that are quasistatically evolved from the equatorial configuration, we illustrate the results of the phase transition in the case of a $D=3$ brane. It is found that small thickness perturbations do not modify the order of the transition, i.e. it remains first order just as in the case of vanishing thickness.