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

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 $\ell$, we adopt a linear perturbation approach in ${\ell }^2$. 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 $D>3$. 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 $D>3$. 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.

Figure 1

The picture shows a sequence of subcritical and supercritical thin brane solutions in the case when a $D=4$ dimensional brane embedded into a $N=5$ dimensional bulk. The different configurations belong to different initial conditions of $r_1$ and ${\theta }_0$, respectively. For simplicity, the bulk black hole’s horizon radius is put to be 1, and $R$ and $Z$ are the standard cylindrical coordinates.

Figure 2

The picture shows the numerical solution of the perturbation Eq. (42) with varying the parameter ${\theta }_0$ of the brane initial inclination in the case of $N=5$, $n=2$, black hole embedding.

Figure 3

The picture shows the numerical solutions of the perturbation Eq. (42) in the case of a $D=4$ dimensional ($n=2$) brane embedded into a $N=6$, 7, 8, 9 (top left, top right, bottom left, bottom right, respectively) dimensional bulk spacetime. The graphs belong to the initial conditions that are close to the minimal ${\theta }_0$, i.e. when the perturbation method reaches its limit $|\phi |\simeq \frac{\pi }{2}$.

Figure 4

The picture shows the thick (red) brane configurations together with their thin (blue) counterparts in a cylindrical coordinate system, in the case of an $N=5$, $n=2$ black hole embedding. The initial conditions are ${\theta }_0=\frac{\pi }{4}$ (bottom curves) and $\frac{\pi }{17.5}$ (top curves), and the thickness parameter $\ell$ is chosen to be large for the purpose of making the effects visible. The black curve represents the black hole’s event horizon.

Figure 5

The figure shows the numerical solutions of the perturbation Eq. (42) in the cases when a $D=3$ dimensional brane is embedded into an $N=5$ (top pictures) and an $N=6$ (bottom pictures) dimensional bulk. The pictures belong to the Minkowski embedding branch with initial parameter regions $1.001\le r_1\le 1.01$ (left hand side), and $1.1\le r_1\le 3$, (right hand side).

Figure 6

The picture shows the thick (red) brane configurations together with their thin (blue) counterparts in a cylindrical coordinate system, in the case of an $N=5$, $n=1$ Minkowski embedding. The initial parameters are $r_1=1.001$ (bottom curves) and $r_1=1.04$ (top curves), and the thickness parameter $\ell$ is chosen to be large for the purpose of making the effects visual. The black thick curve represents the black hole’s event horizon.

Figure 7

The figure shows the energy difference of a brane configuration (that quasistatically evolves from the equatorial configuration) with respect to the equatorial configuration, as a function of the parameter $Z_{\infty }$ in the case of $N=5$ bulk dimensions. The top pictures belong to the thin, while the bottom pictures to the thick system. The red (blue) curves represent the black hole (Minkowskian) embedding branch. The pictures on the right hand side are the zooms into the near region of ${\theta }_{\min }$, where the effects of the perturbations are the largest, and the difference become apparent between the thin (top) and thick (bottom) cases.

Figure 8

The pictures on this figure are the corresponding ones to Fig. 7, in the case of $N=6$ bulk dimensions.