Frozen Firewall: Generic Singularity Formation on an Extremal Horizon

TL;DR

The paper analyzes nonlinear perturbations of extremal planar AdS black holes and shows that horizon curvature can diverge in time, despite nonlinear dynamics often damping such instabilities. By combining near-horizon AdS$_2$ analyses, a scalar toy model, and a full nonlinear Einstein-Maxwell evolution in $D=5$ with $SO(3)$ symmetry, it demonstrates that late-time horizon growth is governed by the linear spectrum, with a dominant nonzero momentum $k_0$ driving the strongest growth. The study identifies a universal late-time tail, links horizon data to boundary holographic data via Fefferman–Graham expansion, and introduces a frozen firewall interpretation: curvature grows on the horizon while remaining confined to near-horizon regions and not visible from infinity. The results imply a robust nonlinear instability for extremal planar AdS black holes, with implications for holography, weak cosmic censorship, and the structure of extremal horizons in AdS/CFT.

Abstract

It is known that linearized perturbations of extremal black holes result in growing curvature on the horizon. However, nonlinear perturbations typically do not evolve to extremal black holes and do not have growing curvature at late times. We show that a large class of nonlinear perturbations of an extremal planar anti-de Sitter black hole does have horizon curvature that grows unbounded in time. The late time behavior of the nonlinear evolution is found to be captured by a linearized analysis. We argue that the generic nonlinear perturbation behaves similarly.

Figures (10)

• Figure 1: Modes of the rescaled horizon tidal curvature $\tilde{\Phi}_{\tilde{k}}$\ref{['eq:lineartidal']} versus rescaled time $\hat{v}$ on a $\log-\log$ scale, shown for several values of $\tilde{k}$ labeled on the right, with initial data $A_1=1/2$ and $B_1=1$ in \ref{['eq:initialdata']}. The late time evolution is consistent in all plots with $\hat{v}^{2-\alpha(\tilde{k})}$ at late times. At intermediate times, the system exhibits a ringdown phase, characteristic of the corresponding quasinormal mode spectrum.
• Figure 2: Plots of the boundary one-point functions $\langle T_{tt}\rangle$, $\langle T_{tR}\rangle$, $\langle T_{RR}\rangle$, and $\langle T_{\Omega}\rangle$ as functions of $R$ and $t$. In all cases, the quantities begin at their extreme planar black hole values, develop structure near the origin corresponding to our localized bulk deformation, and decay at late times back to the extreme planar black hole values. These plots were generated using the initial data specified in Eq. (\ref{['eq:initialdata']}), with $B_1 = 2A_1 = 10^{-2}$.
• Figure 3: Maximum of tidal force $|\Psi|$\ref{['eq:exacttidal']} on the apparent horizon $\widetilde{\mathcal{H}}$ (corresponding to $y=1$ in our coordinates) as a function of time $v$. Blue disks represent the nonlinear evolution with initial data $B_1=2 A_1=10^{-2}$ in \ref{['eq:initialdata']}, while red squares show the linear result obtained by superimposing forty linear simulations with uniformly spaced $\tilde{k}\in[0.25,10]$. At late times the two curves have equal slope in this $\log-\log$ plot, and for sufficiently small $A_1$ and $B_1$ they coincide.
• Figure 4: Maximum of tidal force $|\Psi|$\ref{['eq:exacttidal']} on the apparent horizon $\widetilde{\mathcal{H}}$ as a function of $v$. Blue disks represent the nonlinear evolution with larger initial data $B_1=2 A_1=5\times 10^{-2}$ in \ref{['eq:initialdata']}, while red squares show the linear result obtained by superimposing forty linear simulations with uniformly spaced $\tilde{k}\in[0.25,10]$. At late times the two curves have equal slope in this $\log$-$\log$ plot showing they grow at the same rate.
• Figure 5: Profiles of the tidal force $|\Psi(v, x_{\max}, \lambda)|$ along ingoing null geodesics for several values of $v$ ($\lambda$ is the affine parameter \ref{['eq:affine']}), showing that the profile becomes increasingly steep near the horizon, $\lambda=0$, as $v$ increases. This plot was generated using initial data with $B_1 = 2A_1 = 10^{-2}$ in (\ref{['eq:initialdata']}).