Small Black Holes in Randall-Sundrum type I Scenario

TL;DR

The study addresses small black holes on the TeV brane in the Randall-Sundrum I scenario by introducing an $\epsilon$-expansion with $\epsilon=\mu/\ell$, where $\mu$ is the 5D Schwarzschild radius and $\ell$ is the AdS curvature length. Zeroth order recovers the five-dimensional Myers-Perry solution, while first order incorporates brane tension and the corresponding junction conditions; second order brings in the bulk cosmological constant. The near-horizon geometry is constructed via a master wave function $H(\rho,\psi)$ obeying a separable PDE, with a detailed mode expansion used to enforce no-black-string behavior and to match the linearized gravity solution at large distances, fixing several coefficients ($a_0=-\frac{2}{3\pi}$, $b_2=\frac{\pi}{3}$) and leaving some higher-order coefficients undetermined. The horizon analysis reveals a logarithmic divergence near the horizon and a non-constant surface gravity, suggesting potential horizon singularity or breakdown of the $O(\epsilon)$ expansion at the horizon, necessitating further study. Overall, the work provides a concrete, systematically improvable framework for probing TeV-scale black holes in RSI and demonstrates how near-horizon solutions can be constrained by asymptotic linearized gravity.

Abstract

An approximation method to study the properties of a small black hole located on the TeV brane in the Randall-Sundrum type I scenario is presented. The method enables us to find the form of the metric close to the matter distribution when its asymptotic form is given. The short range solution is found as an expansion in the ratio between the Schwarzschild radius of the black hole and the curvature length of the bulk. Long range properties are introduced using the linearized gravity solution as an asymptotic boundary condition. The solution is found up to first order. It is valid in the region close to the horizon but is not valid on the horizon. The regularity of the horizon is still under study.