Gravitational collapse to extremal black holes and the third law of black hole thermodynamics

TL;DR

The authors present a rigorous, nonperturbative framework for gluing null cones to black-hole horizons in the Einstein–Maxwell–charged scalar field system under spherical symmetry. Their $C^k$ characteristic gluing produces regular initial data whose evolutions yield spacetimes that are exactly Schwarzschild or RN in the exterior but become extremal RN along the horizon in finite advanced time, thereby countering the traditional third law of black hole thermodynamics. Central to the construction is a large-data, nonperturbative gluing scheme anchored by a finite set of scalar-field pulses and a topological Borsuk–Ulam argument that achieves $k$-th order matching. These results illuminate the global structure of dynamical black holes, permit controlled formation of extremal horizons, and demonstrate the existence of gravitational collapses with disjoint apparent horizons and even Cauchy horizons that can close off spacetime. The methods open new avenues for exploring critical behavior in gravitational collapse and the stability of extremal configurations within a precise, gauge-theoretic setting.

Abstract

We construct examples of black hole formation from regular, one-ended asymptotically flat Cauchy data for the Einstein-Maxwell-charged scalar field system in spherical symmetry which are exactly isometric to extremal Reissner-Nordström after a finite advanced time along the event horizon. Moreover, in each of these examples the apparent horizon of the black hole coincides with that of a Schwarzschild solution at earlier advanced times. In particular, our result can be viewed as a definitive disproof of the "third law of black hole thermodynamics." The main step in the construction is a novel $C^k$ characteristic gluing procedure, which interpolates between a light cone in Minkowski space and a Reissner-Nordström event horizon with specified charge to mass ratio $e/M$. Our setup is inspired by the recent work of Aretakis-Czimek-Rodnianski on perturbative characteristic gluing for the Einstein vacuum equations. However, our construction is fundamentally nonperturbative and is based on a finite collection of scalar field pulses which are modulated by the Borsuk-Ulam theorem.

Figures (14)

• Figure 1: Penrose diagram of our counterexample to the third law arising from regular initial data on $\Sigma$. The northwest edge of the Schwarzschild region is exactly isometric to a section of the $r=2M$ hypersurface in Schwarzschild. The outermost apparent horizon $\mathcal{A}'$ is initially indistinguishable from Schwarzschild and then jumps out in finite time to be exactly isometric to the event horizon of extremal Reissner--Nordström. For speculations about the future boundary of the interior, see already \ref{['figure-1-comment']}. The behavior of our solutions can be modified to be subextremal near $i^0$, see already \ref{['rk:Leonhard']}.
• Figure 2: Setup of \ref{['thm-informal-statement']}.
• Figure 3: Penrose diagram for \ref{['main-corollary']}. The textured line segment is where the data constructed in \ref{['thm-informal-statement']} live.
• Figure 4: Schematic illustration of the pulses.
• Figure 5: Illustration of the contrapositive of \ref{['Israel-observation']}. The outermost apparent horizon $\mathcal{A}'=\mathcal{A}'_1\cup \mathcal{A}_2'$ becomes disconnected when a black hole with trapped surfaces "becomes extremal," while the spacetime and matter fields remain regular. The trapped region begins to the north of $\mathcal{A}_1'$ and persists for all advanced time.

Theorems & Definitions (99)

• Conjecture : The third law of black hole thermodynamics
• Theorem 1
• Remark 1.1
• Remark 1.2
• Conjecture
• Theorem 2: Rough version
• Remark 1.3
• Remark 1.4
• Corollary 1: Exact Reissner--Nordström arising from gravitational collapse
• Remark 1.5