The moduli space of dynamical spherically symmetric black hole spacetimes and the extremal threshold

Abstract

In this paper, we give a complete description of the black hole threshold, locally near the Reissner-Nordström family, in the infinite-dimensional moduli space $\mathfrak M$ of dynamical spherically symmetric solutions to the Einstein-Maxwell-neutral scalar field system. In a neighborhood of the full Reissner-Nordström family in $\mathfrak M$, we prove the following: (i) Any solution that forms a black hole eventually decays to a Reissner-Nordström black hole. (ii) Any solution that fails to collapse into a black hole eventually becomes superextremal along null infinity and exists globally in the domain of dependence of the bifurcate characteristic initial data. (iii) The subset of this neighborhood consisting of black hole solutions admits a $C^1$ foliation by hypersurfaces of constant final charge-to-mass ratio, up to and including extremality. (iv) The mutual boundary between the set of black hole solutions and noncollapsing solutions, i.e., the black hole threshold, is the extremal leaf of the foliation. Black holes which are not on the threshold are asymptotically subextremal. Our quantitative control of near-threshold solutions allows us to prove "universal" scaling laws for the location of the event horizon and its final area and temperature (surface gravity), with critical exponent $\frac 12$. Moreover, we show that the celebrated Aretakis instability is activated for an open and dense set of threshold solutions and that generic near-threshold subextremal black holes experience a transient horizon instability on the timescale of their inverse final temperature.

Figures (10)

• Figure 1: Penrose diagrams depicting evolutions of characteristic data in the sets $\mathfrak M_\mathrm{non}$ and $\mathfrak M_\mathrm{black}$. Spacetimes arising from $\mathfrak M_\mathrm{non}$ have an incomplete null infinity $\mathcal{I}^+$ because the ingoing cone $\underline C{}_\mathrm{in}$ is incomplete and no black hole has formed. However, $r\to \infty$ along every outgoing cone and the solution remains regular up to and including the final outgoing cone. Every solution arising from data in $\mathfrak M$ has one of these two Penrose diagrams (for more information on the possibilities in the black hole interior, we refer to Kommemi13).
• Figure 2: Penrose diagrams depicting evolutions of characteristic data in the sets $\mathfrak M_\mathrm{non}\cap\mathfrak M_\mathrm{nbhd}$ and $\mathfrak M_\mathrm{black}\cap\mathfrak M_\mathrm{nbhd}$. In the first case, $|Q|>M_{\mathcal{I}^+}$ somewhere along $\mathcal{I}^+$. In the second case, the solution converges to a Reissner--Nordström black hole. The statement about (existence and absence of) trapped surfaces follows from C-91dafermos2005interiorAKU24.
• Figure 3: A schematic depiction of the set $\mathfrak M_\mathrm{nbhd}$ and the isologous foliation from \ref{['thm:intro-foliation']}. The moduli space $\mathfrak M$ has a natural product structure, where the ordinate is $\rho_\circ$, the charge-to-mass ratio of the bifurcation sphere of $\mathcal{C}$. The abscissa $x$ lives in a Banach space $\mathfrak X$ and accounts for the remaining degrees of freedom (such as the initial data for $\phi$). Therefore, the horizontal axis represents an infinite-dimensional space in this picture and the stable manifolds $\mathfrak M_\mathrm{stab}^{\sigma}$ are infinite-dimensional hypersurfaces. Essentially, $\mathfrak M_\mathrm{nbhd}=\{(x,\rho_\circ):\|x\|_\mathfrak{X}<\varepsilon_0\},$ where $\varepsilon_0$ is a small parameter. The dashed horizontal lines are the constant-$\rho_\circ$ coordinate hyperplanes. The black hole portion of $\mathfrak M_\mathrm{nbhd}$ is shaded dark gray and the noncollapse portion is shaded light gray. For the definition of $\mathfrak M$ and its topology, see already \ref{['sec:intro-mod-space']}.
• Figure 4: A schematic depiction of the level sets $\mathfrak H_h$ of the asymptotic Aretakis charge from \ref{['thm:intro-instability']}. One should think of this disk as the top or bottom face of the "cylinder" depicted in \ref{['fig:foliation-intro-1']}, i.e., it is a hypersurface in the moduli space $\mathfrak M$ parametrized by initial data for the scalar field, $\phi_\circ$. (We have suppressed infinitely many degrees of freedom in this picture, including $r_\circ$ and $\varpi_\circ$; see already \ref{['sec:intro-mod-space']}.) The sets $\mathfrak H_h$ are hypersurfaces in the faces, and hence codimension-two in $\mathfrak M$. The set $\mathfrak H_0$ corresponds to those initial data which evolve to extremal Reissner--Nordström but do not experience the Aretakis instability at arbitrarily large times. The tangent space of $\mathfrak H_0$ at exact extremal Reissner--Nordström, $T_\mathrm{ERN}\mathfrak H_0$, consists of those initial data for the wave equation with vanishing Aretakis charge on exact extremal Reissner--Nordström. However, because of the nonlinearity of the coupled system, the actual set $\mathfrak H_0$ is a (non-explicit) quadratic graph over $T_\mathrm{ERN}\mathfrak H_0$.
• Figure 5: A Penrose diagram of Reissner--Nordström depicting the foliations $C(\tau)$ and $\underline C(\tau)$ used in the estimates \ref{['eq:intro-T']}--\ref{['eq:intro-hp']}. The region of integration in the double integrals is shaded darker. In the left figure, the bifurcation sphere of $\mathcal{H}^+$ and $\mathcal{H}^-$ is included in the spacetime when $|e|<M$ (solid point) but not when $|e|=M$ (open point).

Theorems & Definitions (421)

• Theorem
• Theorem I: Refined dichotomy and uniform stability
• Remark 1: Comparison with AKU24 I
• Remark 2
• Remark 3
• Remark 4
• Theorem II: Isologous foliation and threshold property
• Remark 5: Physical interpretation of the isologous foliation
• Remark 6: No trapped surfaces on the threshold
• Remark 7: Asymptotically subextremal black holes