Small localized black holes in a braneworld: Formulation and numerical method

TL;DR

The paper addresses static black holes localized on a brane in the Randall-Sundrum infinite braneworld by formulating the problem as a boundary-value PDE system in five dimensions. Using a conformal transformation and a relaxation-based numerical method, it finds small localized black hole solutions with horizon radius $\rho_h$ small compared to the bulk curvature scale $\ell$, exhibiting a transition from 5D Schwarzschild in the small-horizon limit to horizon flattening as the warp factor grows. The work provides detailed boundary conditions (axis regularity, horizon regularity via surface gravity, brane Israel conditions, and asymptotic AdS behavior) and demonstrates constraint checks and convergence on a fixed grid, validating the existence of these localized holes. However, it does not obtain large-horizon solutions due to numerical instabilities, highlighting the need for improved formulations and stability analyses, and pointing to future directions such as exploring large localized holes and extensions to TeV-brane contexts.

Abstract

No realistic black holes localized on a 3-brane in the Randall-Sundrum infinite braneworld have been found so far. The problem of finding a static black hole solution is reduced to a boundary value problem. We solve it by means of a numerical method, and show numerical examples of a localized black hole whose horizon radius is small compared to the bulk curvature scale. The sequence of small localized black holes exhibits a smooth transition from a five-dimensional Schwarzschild black hole, which is a solution in the limit of small horizon radius. The localized black hole tends to flatten as its horizon radius increases. However, it becomes difficult to find black hole solutions as its horizon radius increases.

Figures (6)

• Figure 1: An illustration of the metric functions $T$, $R$, and $C$ for $L=15$ in the coordinates $\{r,z \}$. This is a typical example of our numerical solutions for small black holes ($L \gtrsim 5$). In these figures the center of the black hole is placed at the bottom left corner ($r=0$ and $z-\ell=0$), and the brane is at $z-\ell=0$. The numerical calculation has been performed in the polar coordinates $\{ \rho, \xi \}$, and hence the inner region of the black hole $\rho<\rho_h$ is outside the area of our computation.
• Figure 2: An illustration of the solution for $L=30$ in polar coordinates $\{\rho,\chi\}$.
• Figure 3: An illustration of the metric functions $T$, $R$, and $C$ for $L=10$ in the coordinates $\{\rho, \chi\}$.
• Figure 4: An illustration of $\sqrt{A_4}/A_5^{1/3}$. This plot shows a degree of deformation of a black hole. One sees that as $L$ decreases the ratio of a mean radius in four-dimension ${A_4}^{1/2}$ (on the brane) to that in five-dimension $A_5^{1/3}$ increases. It indicates that the black hole tends to flatten as its horizon radius increases. The dashed line shows the same quantity for the 5D Schwarzschild black hole. Note that $L$ is expressed in natural logarithm.
• Figure 5: An illustration of the constraint equations for $L=15$. The figures that plot the absolute values of two constraint equations, $|\Theta_1|$ and $|\Theta_2|$, show the absolute errors of the constraint equations. The other two figures plotting $|\Theta_1(\rho,\chi)|/N_1 (\rho,\chi)$ and $|\Theta_2(\rho,\chi)|/N_2 (\rho,\chi)$ show the relative accuracy of the constraint equations since $N_1$ and $N_2$ are norms of respective constraint equations (see Table \ref{['table:data']}). As expected, the absolute errors of the constraint equations are observed mainly near the axis of polar coordinates and around the horizon. However, the relative accuracy is not significantly low around the horizon, but it is worse near the axis and in the asymptotic region.