Non-linear instability of the Kerr Cauchy horizon near $i_+$

Abstract

We consider solutions of the Einstein vacuum equations which arise from smooth initial data on a hypersurface slightly inside a dynamical black hole settling down to a subextremal Kerr black hole, and satisfying a precise non-linear Price's law-type estimate (which we expect to hold generically). We prove that the corresponding maximal globally hyperbolic development admits a non-trivial piece of future null boundary - the Cauchy horizon - emanating from timelike infinity $i_+$, which exhibits a kind of curvature blow-up, and across which the spacetime metric is Lipschitz-inextendible. Our results thus imply a Lipschitz version of Strong Cosmic Censorship for Kerr spacetimes near timelike infinity under this Price's law-type assumption. The analysis relies on the proof of the $C^0$ stability of the Kerr Cauchy horizon by Dafermos and Luk, on the non-integrable formalism of Giorgi-Klainerman-Szeftel and principal temporal gauge of Klainerman and Szeftel used in the proof of the exterior stability of slowly rotating Kerr black holes, on the linearized analysis for the Teukolsky equation inside subextremal Kerr black holes by the author, and on Sbierski's criterion for Lipschitz inextendibility. More precisely, we proceed by decomposing the black hole interior into different regions equipped with appropriate gauges, allowing for a proof of stability estimates and a thorough analysis of the non-linear analog of the Teukolsky equation, from which we infer our instability results.

Figures (6)

• Figure 1: The Penrose diagram of Kerr spacetimes with $0<|a|<M$, with the exterior region in white, the black hole interior in the darker shaded grey region, and the region with infinitely many extensions in the lighter shaded grey region.
• Figure 2: The blueshift effect. Observer $B$ lives outside of the black hole and reaches timelike infinity in infinite proper time, while sending signals to observer $A$ that crosses the event horizon and then the Cauchy horizon in finite proper time. The waves are sent periodically by $B$, but as $A$ approaches $\mathcal{CH_+}$, the perceived frequency of the waves becomes infinite.
• Figure 3: The Penrose diagram of the spacetime $(\mathcal{M},{\mathbf{g}})$ constructed in Theorem \ref{['thm:roughversion']}.
• Figure 4: Region $\mathbf{I}[\underline{w}_1,\underline{w}_2]$ in grey, for $\underline{w}_1\geq\underline{w}_f$.
• Figure 5: The intersection region $\mathbf{I}\cap\mathbf{II}$ in grey.

Theorems & Definitions (550)

• Conjecture : SCC conjecture
• Conjecture : Instability of the Kerr Cauchy horizon
• Theorem 1.5.1: Main theorem, rough version
• Remark 1.5.2
• Remark 1.5.3
• Definition 2.1.1
• Definition 2.1.2
• Definition 2.1.3
• Definition 2.1.4
• Definition 2.1.5