Toy model for topology change transitions: Role of curvature corrections

We consider properties of near-critical solutions describing a test static axisymmetric $D$-dimensional brane interacting with a bulk $N$-dimensional black hole $(N>D)$. We focus our attention on the effects connected with curvature corrections to the brane action. Namely, we demonstrate that the second order phase transition in such a system is modified and becomes first order. We discuss possible consequences of these results for merger transitions between caged black holes and black strings.

Figure 1

Three possible types of configurations—the subcritical embedding (left) where the brane does not touch the black hole horizon, the supercritical (right) where we have an induced black hole on the brane (BBH), and the critical (center) which is singular at its tip.

Figure 2

This figure schematically shows a configuration of a supercritical brane in the regime when it is close to the critical one. $Z$ is a proper distance from the horizon as a function of the radius $R$. $R_0$ is the radius of the surface of the intersection of the brane with the horizon. In the region where $Z\ll r_g$ the curvature surface of the horizon can be neglected (the Rindler domain).

Figure 3

${\ddot{Z}}_0$ as a function of $Z_0$ for $n=1$ and $B=0$. The symmetry of the action in this case implies that the same graph is valid to the supercritical configuration as well—$R^{\prime \prime }(0)$ as a function of $R_0$.

Figure 4

${\ddot{Z}}_0$ as a function of $Z_0$ for $n=2$. The dashed line is the same function for DNG branes (without stiffness terms).

Figure 5

The energy density integrated for $0\le R\le 5$ as a function of $Z_0$ comparing two branches in the segment ($1\lesssim Z_0\lesssim 1.25$). Note that the minimal energy is obtained at the point which corresponds approximately to ${\ddot{Z}}_0=0$.

Figure 6

${\ddot{Z}}_0$ as a function of $Z_0$ for $n=2$. $B=100$, 0.01.

Figure 7

${\ddot{Z}}_0$ as a function of $Z_0$ for $n=4$, $B=1$.

Figure 8

$R^{\prime \prime }(0)$ as a function of $R_0$ (supercritical) for $n=2$, $B=1$.

Figure 9

$R^{\prime \prime }(0)$ as a function of $R_0$ (supercritical) for $n=2$, $B=0.5$.