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

TL;DR

This paper characterizes the local structure of the black hole threshold in the moduli space of spherically symmetric Einstein-Maxwell-scalar solutions, revealing universal scaling laws and instabilities near extremality.

Contribution

It provides a detailed description of the black hole threshold, including a $C^1$ foliation by charge-to-mass ratio and the activation of Aretakis instability for near-extremal solutions.

Findings

Solutions near the threshold decay to Reissner-Nordström black holes or become superextremal.

Universal scaling laws with critical exponent 1/2 for horizon location and properties.

Activation of Aretakis instability on a dense set of threshold solutions.

Abstract

In this paper, we give a complete description of the black hole threshold, locally near the Reissner-Nordstr\"om 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-Nordstr\"om family in $\mathfrak M$, we prove the following: (i) Any solution that forms a black hole eventually decays to a Reissner-Nordstr\"om 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.