Gravitational collapse in the vicinity of the extremal black hole critical point

TL;DR

This work investigates the threshold of black-hole formation in the Einstein-Maxwell-Vlasov system for charged, self-gravitating matter in spherical symmetry. It constructs unstable static shell solutions and employs a time-reversal perturbation approach to study near-threshold dynamics, identifying a critical point at $Q/M\to1^-$ where the threshold switches from horizonless shells to extremal black holes. The study reveals two distinct scaling behaviors: near threshold, the time to formation or dispersal scales as $T= -\tau \log|Q-Q_*|$ with an instability timescale $\tau$ that diverges as $\tau \sim M[2(1-Q_*/M)]^{-1/2}$; in the extremal regime, dispersal times scale as $T\sim M(Q/M-1)^{-1/2}$. These results illuminate a phase-transition-like structure in gravitational collapse and suggest a route to extending extremal critical collapse concepts to rotating black holes, including potential counterexamples to the rotating third law.

Abstract

We study the threshold of gravitational collapse in spherically symmetric spacetimes governed by the Einstein-Maxwell-Vlasov equations. We numerically construct solutions describing a collapsing distribution of charged matter that either forms a charged black hole or eventually disperses. We first consider a region of parameter space where the solutions at the threshold of black hole formation are stationary, horizonless shells. These solutions terminate at a critical point, with their charge-to-mass ratio approaching unity from below, and the instability timescale diverging. Beyond the critical point, we find a new region of parameter space where the threshold solution is an extremal black hole. We measure the scaling of the dynamical time period of the near threshold solutions and discuss how they are connected in the two regimes. If a similar picture to the one found here holds for known families of stationary solutions of rotating matter that approach the exterior of an extremal Kerr spacetime, they could provide a route to forming an extremal spinning black hole.

Figures (6)

• Figure 1: Top: The phase diagram showing the regions of parameter space that result in a black hole (shaded region) versus those that result in a dispersing charged shell (white region). The x-axis is the total charge of the spacetime (and also the charge of the black hole when it forms) and the y-axis indicates the particle angular momentum. For higher values of the particle angular momentum, the threshold solutions at the boundary between the two regions are static shell solutions (solid black line) which terminate at a critical point (red dot) as $Q/M\rightarrow1$. For lower values, the threshold solution is an extremal black hole (dashed blue line). Bottom: The outer (areal) radius of the threshold solutions, beyond which the spacetime is electrovacuum, as a function the particle angular momentum. For the extremal black hole cases (also shown in the inset), this is the minimum radius the matter reaches inside the horizon. The crosses and diamonds show the specific parameters evolved. The green curve indicates the radius of a Reissner-Nordström black hole with the same charge as the threshold solution.
• Figure 2: Example evolutions as a function of advanced time just above (solid lines) and below (dashed lines) the threshold for black hole formation in the two regimes, where the threshold solution is either a horizonless shell (black and red curves) or an extremal black hole (blue and green curves). Top: The minimum value of the metric function $a(r)=1-2m(r)/r$ at each time. The circles indicate when a horizon first appears for the dashed lines ($a=0$). Bottom: The outer radius of the region containing the particles as a function of time. The inset shows a zoom-in to illustrate that $R_{\rm out}$ reaches a minimum slightly below $M$ for the case where $1-Q/M\approx10^{-12}$.
• Figure 3: Timescales governing near threshold behavior above (left and center panels) and below (right) the critical particle angular momentum $\ell_c$. Left: The time it takes for a black hole apparent horizon to form (black) or for the charged shell to disperse (blue; defined as $R_{\rm out}/M=10$) as a function of the difference in the total charge from its threshold value. Time is measured as the difference in the ingoing Eddington-Finkelstein coordinate from when this occurs when $|1-Q/Q_*|=10^{-3}$. This approximately matches the dependence expected from measuring the instability timescale of the static threshold solution (dashed red curve). Center: The inverse instability timescale (i.e. instability rate) for the static charged shell solutions, as a function of their total charge. The dotted red line shows an analytic prediction from the surface gravity of a near extremal black hole with the same mass and spin. Right: The time it takes for the charged shell to disperse (defined as $R_{\rm out}/M=10$) as a function of how much the total charge is above the threshold value of $Q_*=M$ (for $\ell/M\approx0.52$). This matches well with a $-1/2$ power law scaling (dotted red line).
• Figure 4: Charge density as function of areal radius. Top: Several static shell solutions approaching $Q/M \rightarrow 1$. Bottom: Charge density obtained after letting the intermediate case in the top panel disperse. This is used for initial data for the evolutions where the threshold solution is an extremal black hole.
• Figure 5: Resolution study of an unstable static solution (with $Q/M\approx 0.9994$) dispersing when evolved in outgoing coordinates. We show the difference in the minimum value of the metric function $a$ from its initial value. The times have been shifted to align the curves when $\min(a)=0.02$