Approaching a dynamical extreme black hole horizon

TL;DR

This work analytically characterizes the late-time approach to dynamical extreme Reissner–Nordström horizons by reducing the near-horizon dynamics to Jackiw–Teitelboim gravity on an ${\rm AdS}_2$ throat. By imposing boundary conditions that encode linear Aretakis behavior on ${\rm AdS}_2$ and drive the dilaton toward the extremal exterior, the authors derive explicit, singularity-free dilaton profiles that describe the late-time near-horizon region of a DERN, including a final burst of outward scalar flux. The analysis connects three RN scaling limits to ${\rm AdS}_2$ throats, clarifying how the invariant ${\mu}$ classifies the backreaction as ERN, sub-ERN, or super-ERN, and shows how DERN sits at the threshold of black hole formation. Overall, the work provides a complete, solvable framework for nonlinearly evolving extremal horizons and highlights the critical role of boundary data and energy leakage in sustaining horizon dynamics.

Abstract

We give an explicit closed form description of the late-time near-horizon approach to dynamical extreme Reissner-Nordstrom (DERN) black holes. These are spherically symmetric dynamical solutions of Einstein-Maxwell theory coupled to a neutral scalar that feature: (i) a spacetime metric which tends to that of a static extreme Reissner-Nordstrom (RN), and (ii) a scalar field which displays the linear Aretakis instability ad infinitum in the non-linear theory. We employ the two-dimensional Jackiw-Teitelboim (JT) gravity to solve explicitly for the non-linear s-wave dynamics of the four-dimensional theory near an ${\rm AdS}_2\times {\rm S}^2$ throat. For a teleologically defined black hole horizon, we impose boundary conditions on JT's dilaton field (which encodes the gravitational dynamics) and the scalar matter as follows: (i) the JT dilaton decays at late times on the ${\rm AdS}_2$ boundary to a value that corresponds to a static extreme RN in the exterior, and (ii) the scalar obeys boundary conditions characteristic of linear Aretakis behavior on ${\rm AdS}_2$. We ensure our DERN solutions are singularity-free and we note that our approach to DERN is accompanied by a final burst of outgoing scalar matter flux leaking out of the ${\rm AdS}_2$ throat. The boundary conditions we impose on the JT dilaton place its late-time boundary profile on the threshold of black hole formation with sub-extreme and super-extreme RN on either side of our DERNs.

Figures (5)

• Figure 1: Penrose diagrams showing: (a) DERN arising from maximal development of characteristic data on $\Sigma_1\cup\Sigma_2$ (the data on $\Sigma_1$ ends in the solid point where it is incomplete). The shaded blue region at late times near the horizon is given by a dynamical ${\rm AdS}_2\times {\rm S}^2$ throat. (b) The ${\rm AdS}_2\times {\rm S}^2$ throat with dynamics described by the JT theory. A Poincare horizon is identified with the black hole horizon $\mathcal{H}^+$ and appropriate boundary conditions are imposed on the dilaton and matter fields in JT so as to obtain a solution in the shaded blue region that matches DERN. There is matter flux leaking out of ${\rm AdS}_2\times {\rm S}^2$.
• Figure 2: Penrose diagrams showing the emergence of ${\rm AdS}_2\times {\rm S}^2$ from scalings of RN. Top row: from left to right, we have extreme RN (ERN), sub-extreme RN (sub-ERN), and super-extreme RN (super-ERN). In each case, assuming (near)-extremality, the regions having an ${\rm AdS}_2$ geometry are displayed hatched. Bottom row: a global ${\rm AdS}_2$ is drawn and in each case the patch covered by the corresponding scaling coordinates is displayed shaded, that is from left to right, the shaded blue regions are covered by Poincare, Rindler, and global coordinates.
• Figure 3: Penrose diagram of ${\rm AdS}_2$ marked with global coordinates $(U,V)$. The left boundary is $U=V+\pi$, while the right boundary $\mathcal{B}$ is $U=V$. The asymptotically flat region of ERN is glued onto the Poincare patch, given by $-\pi/2<V<U<\pi/2$, at the right boundary $\mathcal{B}$. (See also panels (a) and (d) in Fig. \ref{['Fig:ThreeLimits']})
• Figure 4: Penrose diagram of ${\rm AdS}_2$ with the Poincare horizon $U=\pi/2$ marked in the global coordinates $(U,V)$. There is matter leaking at early times through the right boundary $\mathcal{B}$ at $U=V$. At late times on $\mathcal{B}$, that is to say, along the thick blue line, the dilaton is $\Phi\approx\Phi_{\rm vac}$. The asymptotically flat region of DERN is glued onto the Poincare patch at the right boundary $\mathcal{B}$, starting at late times and we can trust the gluing for as long as $\Phi\ll 1$ (i.e. before the singularity hits $\mathcal{B}$). In particular, on the late-time thick blue portion of $\mathcal{B}$ this reduces to gluing in a static ERN exterior. (See also Fig. \ref{['Fig:main']})
• Figure 5: Plots of the singularity locus $1+\Phi=0$ on a global ${\rm AdS}_2$ strip showing the Poincare patch $-\pi/2<V<U<\pi/2$ bounded by the solid $45^\circ$ black lines. In each case, we plot the singularity curve for three different amplitudes $A$ of the corresponding net matter flux $\delta\mathcal{T}(U)<0$ leaking through the right boundary at $U=V$. Dot-dashed magenta: $A=A_{\rm min}/10$, Dashed orange: $A=A_{\rm min}$, Solid red: $A=10A_{\rm min}$. The minimum amplitude $A_{\rm min}$ is determined in each case by the requirement that the singularity barely grazes the future horizon at $U=\pi/2$. For $A>A_{\rm min}$ we get a dilaton $\Phi$ that accurately describes the late-time near-horizon dynamics of a DERN. Both panels assume $a=2\,, b=c=0$. Left panel (a) corresponding to the dilaton \ref{['PhiSolnExampleH']}: $f(U) =-2U$, $\delta\mathcal{T}(U) = -A(\pi/2-U)^2$, $A_{\rm min}=2.76$. Right panel (b) corresponding to \ref{['PhiSolnExampleNoH']}: $f(U) = (\pi/2-U)^2$, $\delta\mathcal{T}(U) = -A(\pi/2-U)^4$ , $A_{\rm min}=1.73$.