The $C^0$-inextendibility of Kerr spacetimes

Technical Summary: The $C^0$-Inextendibility of Kerr Spacetimes

Problem Statement
The Kerr solution describes the vacuum gravitational field of a rotating black hole. While the metric extends analytically across its regular event horizons, the curvature diverges at the singular ring defined by $\Sigma = r^2 + a^2 \cos^2 \theta = 0$. A central question in general relativity is whether the spacetime can be extended beyond this singularity as a continuous ($C^0$) Lorentzian manifold, even if the metric is not differentiable. Previous results have established inextendibility for specific regimes or under stronger regularity assumptions (e.g., $C^2$ or $C^{1,1}$), but the $C^0$-inextendibility of the full Kerr realizations—including subextremal, extremal, and overextremal cases, as well as their connected covering spacetimes—remained to be proven without global causality assumptions.

Methodology
The paper employs a geometric obstruction based on causal homotopy lengths near the singular ring. The core strategy involves three main components:

1. Local Causal Obstruction: The author constructs a specific causal homotopy near the ring where a reference timelike curve can be deformed (fixing endpoints) to include a non-constant closed causal curve. This construction relies on the existence of two independent linear combinations of the stationary ($T$) and axial ($\Phi$) Killing fields that are future timelike at one transverse point and whose negatives are future timelike at another. By traversing orbit segments of these fields, the author cancels the transverse displacement, creating a closed loop.
2. Unbounded Lengths: Repeating this closed loop within the same fixed-endpoint causal homotopy class yields curves of arbitrarily large proper time. The author proves that for any positive continuous conformal rescaling $\Omega^2 g$, the length of these curves diverges as the curve approaches the ring ($\Sigma \to 0$).
3. Global Extension Argument: The paper invokes a boundary theorem (Minguzzi and Suhr) stating that if a proper $C^0$ extension exists, there must be a future-inextendible unit-speed timelike geodesic of finite proper time ending at the boundary. The author then demonstrates that every such geodesic in the Kerr realizations must approach the ring ($\Sigma \to 0$). By combining the local obstruction (infinite homotopy length) with the global behavior of geodesics, they derive a contradiction: a curve ending at a continuous boundary must have bounded causal homotopy lengths in a coordinate chart, yet the ring construction forces these lengths to be infinite.

Key Contributions and Results

• Main Theorem ($C^0$-Inextendibility): The paper proves that every connected covering spacetime of the full Kerr realizations (subextremal, extremal, and overextremal) is $C^0$-inextendible. This holds for any continuous, nondegenerate Lorentzian extension metric, regardless of symmetry, time-orientation, or global causality properties.
• Singularity Characterization: The obstruction is localized to the singular ring. The proof shows that any finite-proper-time inextendible timelike geodesic must satisfy $\Sigma \to 0$, $r \to 0$, and $\theta \to \pi/2$ (approaching the equatorial ring).
• Robustness under Perturbations: The obstruction persists under stationary and axisymmetric perturbations $g + h$ provided the perturbation satisfies a sharp bound: $\limsup_{\Sigma \to 0} \|h\|_E / \Sigma < 1/(1+a^2)$ in a specific coordinate norm. The paper provides examples showing this bound is sharp; exceeding it allows for conformal rescalings where homotopy lengths remain finite.
• Conformal and Cone Stability: The result holds for arbitrary positive continuous conformal factors and for any continuous Lorentzian metric whose future causal cones contain the Kerr cones. This includes null approaches to the ring.
• Geometric Analysis of Terminal Geodesics:
  • The paper classifies terminal geodesics, showing that generic initial data lead to complete geodesics, while incomplete ones are confined to a codimension-two submanifold (equatorial, specific Killing momenta).
  • It derives sharp asymptotic expansions for the radial coordinate $r(s)$ and orbit coordinates near the ring, finding a leading order behavior of $r \sim u^{2/5}$ (where $u$ is the remaining proper time).
  • It analyzes curvature blow-up, showing that scalar invariants like the Kretschmann scalar diverge as $\Sigma^{-3}$ (or $u^{-12/5}$), and provides exact reductions for polar Jacobi fields.
• Perturbation Bounds: The author establishes precise thresholds for perturbations. For the leading order, the coefficient $1/(1+a^2)$ is optimal. At the next order, a critical coefficient $\kappa^* = M^2/(2a^4)$ is identified, with the obstruction persisting for $a^2 \ge 2M^2$ but failing for $a^2 < 2M^2$ under specific rank-one perturbations.

Significance and Claims
The paper claims to resolve the $C^0$-inextendibility of the full Kerr spacetime, a long-standing problem in mathematical relativity. By relying solely on local cone bounds and causal homotopy arguments, the proof avoids the need for global hyperbolicity or higher regularity assumptions (such as $C^2$ or curvature bounds) that were previously required for similar results.

The author emphasizes that the obstruction is intrinsic to the causal structure of the Kerr ring. The result implies that the singularity is not merely a coordinate artifact or a place where differentiability fails, but a genuine boundary that cannot be crossed even by a continuous metric. The work also provides a rigorous framework for understanding the stability of this inextendibility under perturbations, offering sharp quantitative bounds that distinguish between metrics that retain the singularity's "blocking" nature and those that might allow for finite-length extensions under specific conformal rescalings.

The paper concludes that the ring singularity represents a fundamental barrier to extension, reinforcing the Strong Cosmic Censorship conjecture in the context of rotating black holes, albeit within the specific class of stationary, axisymmetric, and continuous extensions considered. The geometric analysis of terminal geodesics and curvature provides a detailed map of the spacetime's behavior as it approaches this boundary.