Gravity and large black holes in Randall-Sundrum II braneworlds

TL;DR

Confronts the problem of black hole solutions in Randall–Sundrum II by linking low-energy brane gravity to the $AdS_5$-$CFT_4$ framework. It derives a perturbative RSII solution in the small-curvature limit using the Fefferman–Graham expansion and Israel junction conditions, showing that leading brane corrections are the local four-derivative term $b_{\mu\nu}$ and the CFT stress tensor $\langle T^{CFT}_{\mu\nu}\rangle$. The authors then numerically construct large 5D RSII black holes by modifying the $AdS_5$-$CFT_4$ boundary to include an RSII brane and solving the Einstein–DeTurck equations to radii $R_4/\ell \approx 20$, finding that large holes closely approach the associated AdS5-CFT4 solution with vanishing local-curvature corrections and a CFT-dominated correction. They report a pancake-like horizon geometry, a smooth transition to 4D Schwarzschild on the brane at large radius, and a single negative mode for axisymmetric perturbations, indicating dynamical stability. Collectively, the work validates the RSII/CFT correspondence for black holes and provides concrete, numerically stable large-brane solutions.

Abstract

We show how to construct low energy solutions to the Randall Sundrum II (RSII) model using an associated AdS_5-CFT_4 problem. The RSII solution is given in terms of a perturbation of the AdS-CFT solution, with the perturbation parameter being the radius of curvature of the brane metric compared to the AdS length \ell. The brane metric is then a specific perturbation of the AdS-CFT boundary metric. For low curvatures the RSII solution reproduces 4d GR on the brane. The leading correction is from local higher derivative curvature terms. The subleading correction is derived from similar terms and also the dual CFT stress tensor. Recently AdS-CFT solutions with 4d Schwarzschild boundary metric were numerically constructed. We modify the boundary conditions to introduce the RSII brane, and use elliptic numerical methods to solve the resulting boundary value problem. We construct large RSII static black holes with radius up to ~ 20 \ell. For large radius the RSII solutions are indeed close to the associated AdS-CFT solution. In this case the local curvature corrections vanish, and we confirm the leading correction is given by the AdS-CFT solution stress tensor. We also follow the black holes to small radius << \ell, where as expected they transition to a 5d behaviour. Our numerical solutions indicate the RSII black holes are dynamically stable for axisymmetric perturbations for all radii.

Figures (4)

• Figure 1: Area of the black hole as a function of the radius of the horizon on the brane (black dots), and the same quantity for an asymptotically flat Schwarzschild black hole in 5$d$ (red) and in 4$d$ (blue). Note the log scale of both axes.
• Figure 2: Embedding of the spatial cross sections of the horizon into $\mathbb H^4$ (red). The black curve corresponds to the embedding of the horizon of the $AdS_5$-$CFT_4$ solution of PFLuciettiTW, with 4$d$ Schwarzschild as the conformal boundary metric.
• Figure 3: $R_4^{~4} \, G_\tau^{~\tau}\,\ell^{-2}$ computed from the induced geometry on the brane against proper distance for braneworld black holes of sizes $R_4/\ell\sim 1.24-6.70$ (red). In black, r.h.s. of \ref{['eq:vac']} computed from the solution of PFLuciettiTW using standard holographic renormalisation deHaro:2000xn. The red curves approach the black one as the black hole size is increased. For large black holes, the actual value of $G_\tau^{~\tau}$ on the brane is so small as to be comparable to the numerical error, and we see some noise in this quantity.
• Figure 4: Maximum value of $\phi$, $\phi_\textrm{max}$, in the whole domain (including the brane) for braneworld black holes with $R_4/\ell=13.35, 5.04, 1.13, 0.07$, as a function of the number of grid points $N$ for the pseudospectral code. Lines are drawn to guide the eye. As this figure shows, $\phi_\textrm{max}\to 0$ in the continuum limit, which provides evidence that our solutions are not Ricci solitons.