[Paper Review] The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon

Mihalis Dafermos, Jonathan Luk|arXiv (Cornell University)|Oct 4, 2017

Black Holes and Theoretical Physics96 references104 citations

TL;DR

The authors initiate a study of the interiors of dynamical rotating vacuum black holes without symmetry and prove C0-stability of a piece of the Kerr Cauchy horizon under suitable interior data, suggesting a refined view of cosmic censorship.

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.

Motivation & Objective

• Motivate and formalize the study of the interior of dynamical rotating vacuum black holes without symmetry assumptions.
• Establish that the maximal Cauchy evolution can be extended across a non-trivial piece of the Kerr Cauchy horizon with a continuous metric.
• Lay groundwork for relating interior stability to conjectures about cosmic censorship and Kerr exterior stability.
• Set up a framework to derive the C0-stability result from geometric data inside the black hole interior.

Proposed method

• Use Einstein vacuum equations in double null gauge to describe the spacetime interior.
• Formulate a bootstrap argument with energies and weighted norms to control geometric quantities and curvature.
• Introduce reduced schematic equations to manage nonlinear interactions and null structure.
• Develop estimates for transport, elliptic, and Bianchi equations to bound metric and curvature components.
• Prove continuity of the spacetime metric up to the Cauchy horizon and relate it to Kerr geometry via C0-closeness.

Experimental results

Research questions

• RQ1 Can the maximal future development of interior data be extended across a portion of the Cauchy horizon while maintaining a continuous metric?
• RQ2 What are the precise regularity and stability properties of the Kerr Cauchy horizon under non-symmetric dynamical perturbations in vacuum?
• RQ3 How does the interior evolution relate to a potential violation or refinement of strong cosmic censorship in the C0 sense?
• RQ4 Under what data assumptions can one approximate Kerr exterior and interior regimes, and how does this influence the global causal structure?
• RQ5 What is the role of weak null singularities in the interior and their impact on higher-regularity extensions?

Key findings

• There exists an extension across a non-trivial piece of the Kerr Cauchy horizon with a continuous Lorentzian metric for the evolved interior data.
• The analysis supports a C0-formulation of strong cosmic censorship that can be false, while indicating the C0-extensions are generically singular in a weak null sense.
• The work connects interior stability to the conjectured non-linear stability of the Kerr exterior and motivates a revised cosmic censorship picture.
• A framework is established showing how interior data close to Kerr geometry influences the behavior near the Cauchy horizon.
• The results are foundational for subsequent work that will derive data assumptions from Kerr-like exterior stability and study additional horizon structures.