Nonlinear stability of extremal Reissner-Nordström black holes in spherical symmetry

TL;DR

The paper proves codimension-one nonlinear stability of the extremal Reissner–Nordström spacetime in the spherically symmetric Einstein–Maxwell–neutral scalar field model by constructing a moduli submanifold $\mathfrak M_{\mathrm{stab}}$ of seed data near extremality and a dyadic, teleologically gauged bootstrap framework. It shows that data in $\mathfrak M_{\mathrm{stab}}$ yield spacetimes where the metric decays to a nearby ERN solution up to the horizon and the scalar field decays pointwise and in energy, while horizon derivatives of the scalar exhibit Aretakis-type behavior; the instability also imprints on certain Ricci components. The approach combines a discrete modulation mechanism to select the extremal limit with a robust hierarchy of energy estimates (including $r^p$-weighted and $(\bar r-M)^2$-weighted bounds) and a refined treatment of degenerate redshift, culminating in a limiting eschatological gauge and a precise description of the local moduli space near extremality. Collectively, the results illuminate the nonlinear stability landscape near extremal black holes in a highly symmetric setting, clarifying how Aretakis-type instabilities interact with the full nonlinear dynamics and setting the stage for future extensions beyond spherical symmetry.

Abstract

In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordström family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordström family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordström family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordström family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordström black holes in the full nonlinear Einstein-Maxwell-scalar field model.

Figures (11)

• Figure 1: A Penrose diagram showing the maximal development of a solution considered in \ref{['thm:stability-rough']}. The Cauchy data ends on the left in the solid point, where it is incomplete (but not singular).
• Figure 2: A Penrose diagram of extremal Reissner--Nordström depicting the foliations $C(\tau)$ and $\underline C(\tau)$ used in the estimates \ref{['eq:boundedness-intro-1']}--\ref{['eq:r-Mp-intro']}. The region of integration in \ref{['eq:rp-intro']} and \ref{['eq:r-Mp-intro']} is shaded darker.
• Figure 3: A Penrose diagram of one of the bootstrap domains $\mathcal{D}_{\tau_f}\doteq J^-(\Gamma(\tau_f))$ used in the proof of \ref{['thm:stability-rough']}. Here $\Gamma\doteq\{r=\Lambda\}$ is the timelike curve which anchors the bootstrap domains and $(u,v)$ are double null coordinates teleologically normalized as depicted.
• Figure 4: A schematic depiction of our modulation scheme. Let $p\mapsto\mathring\phi(p)$ be a one-parameter family of characteristic initial data for the scalar field with $\mathring\phi(0)=0$. We can then consider the plane in $\mathfrak M$ parametrized by $(p,\alpha)$. Each $p$ generates a line segment $\mathcal{L}$ in $\mathfrak M$ which intersects the "submanifold" $\mathfrak M_\mathrm{stab}$ at least once. The horizontal line $\mathfrak M_0$ denotes the hyperplane in $\mathfrak M$ consisting of data sets with $\varpi_0=M_0$. On the three $\mathcal{L}$'s depicted here, we have also drawn three of the nested modulation sets $\mathfrak A_i$ which converge to $\mathcal{L}\cap \mathfrak M_\mathrm{stab}$. Note that we have drawn $\mathfrak M_\mathrm{stab}$ as a smooth, connected curve here, which is in line with our conjectures in \ref{['sec:conjectures']}, but we do not prove any such fine structure of it in this paper.
• Figure 5: A cartoon depiction of the conjectured structure of a neighborhood of extremal Reissner--Nordström in the moduli space $\mathfrak M$ of initial data posed as in \ref{['fig:stability-intro']}. We have suppressed infinitely many dimensions and emphasize the codimension-one property of the submanifolds $\mathfrak M_\mathrm{stab}^\mathfrak r$. We have drawn a distinguished point, which is extremal Reissner--Nordström. We have also drawn one of the lines $\mathcal{L}$ from \ref{['fig:mod-space-lines']} with the natural orientation given by increasing modulation parameter $\alpha$. The solid point on $\mathcal{L}$ corresponds to $\alpha=\alpha_\star$ (recall \ref{['sec:intro-modulation']}). See also \ref{['fig:critical-behavior']} below.

Theorems & Definitions (167)

• Theorem I: Codimension-one nonlinear stability of ERN, rough version
• Theorem II: Dynamical horizon instability, rough version
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Conjecture 1: Regularity of $\mathfrak M_\mathrm{stab}^\mathfrak r$