$C^0$-inextendibility of a class of warped-product black hole spacetimes

Technical Summary: $C^0$-Inextendibility of a Class of Warped-Product Black Hole Spacetimes

Problem Statement
The paper addresses the question of spacetime inextendibility at low regularity ($C^0$) for a broad class of black hole spacetimes. While the $C^0$-inextendibility of the maximal analytic extension of the Schwarzschild spacetime was established by Sbierski [Sbi18a, Sbi18b], this result relies heavily on spherical symmetry. The central problem is to determine whether the mechanism underlying this inextendibility—specifically the obstruction to continuous extensions through curvature singularities and the exterior region—holds for warped-product black hole spacetimes that lack full spherical symmetry and may include non-vacuum matter fields. This inquiry is motivated by the Strong Cosmic Censorship conjecture, which posits that generic solutions to the Einstein equations should not admit extensions beyond their singular boundaries as weak solutions.

Methodology
The author adapts the proof strategy developed by Sbierski for the Schwarzschild case to a generalized warped-product setting. The methodology involves three primary components:

1. Geometric Setup: The paper considers $(d+1)$-dimensional ($d \ge 3$) globally hyperbolic spacetimes with a static exterior region, described by the metric:
   $$g = -e^{-2\psi(r)}\left(1 - \frac{2m(r)}{r}\right)dt^2 + \left(1 - \frac{2m(r)}{r}\right)^{-1}dr^2 + r^2h$$
   where $h$ is a Riemannian metric on a codimension-two fibre $F$. The functions $m(r)$ and $\psi(r)$ generalize the mass and redshift functions of Schwarzschild. The fibre $F$ is assumed to be closed (compact without boundary), connected, homogeneous, and orientable.

2. Technical Tools and Lemmas:

   • Minkowskian Charts: The proof utilizes the existence of local charts near a future boundary point where the metric is close to Minkowski and the boundary is a Lipschitz graph (Proposition 1).
   • Future One-Connectedness: A crucial step involves proving that the interior region sufficiently close to the singularity ($r \to 0$) is "future one-connected" (Proposition 4). This property, dependent on the fibre having a positive injectivity radius, allows for the construction of timelike homotopies between curves.
   • Symmetry Adaptation: Instead of relying on the spherical symmetry of the Schwarzschild sphere $S^2$, the proof leverages the homogeneity of the fibre $F$. Lemma 4 establishes that for a homogeneous manifold, nearby points can be mapped to each other via isometries that are uniformly close to the identity. This replaces the explicit coordinate control available in the spherically symmetric case.
   • Timelike Geodesic Dichotomy: The paper proves that for these spacetimes, any future-directed, future-inextendible timelike geodesic either is future complete or approaches the central singularity ($r \to 0$). This is achieved by establishing a conserved quantity associated with the fibre metric (Lemma 11) and analyzing the geodesic equations in both the exterior and interior regions (Propositions 5 and 6).

3. Proof by Contradiction: The main argument assumes the existence of a $C^0$-extension. This implies the existence of a future-directed, future-inextendible timelike geodesic ending on the boundary of the extension. The proof then demonstrates that such a geodesic cannot exist:

   • If it remains in the exterior, it must be complete (Theorem 3), which contradicts the existence of a boundary (Theorem 1).
   • If it enters the interior and approaches $r=0$, the geometry of the singularity (specifically the divergence of the Kretschmann scalar and the behavior of Cauchy surfaces) leads to a contradiction regarding the boundedness of distances on Cauchy hypersurfaces (Theorem 2).

Key Contributions and Results

• Theorem 2 (No $C^0$-extension through the central singularity): For spacetimes where the fibre is closed, connected, homogeneous, and orientable, and where $m(r) \to m_0 > 0$ as $r \to 0$, there exists no $C^0$-extension through the central curvature singularity. The proof relies on the "spaghettification" effect (divergence of distances on Cauchy surfaces near $r=0$) and the homogeneity of the fibre to construct a separating barrier.
• Theorem 3 (Timelike Geodesic Dichotomy): Under suitable conditions on $m(r)$ and $\psi(r)$ (including bounds on their derivatives), any future-directed, future-inextendible timelike geodesic in the exterior region is either future complete or intersects the black hole horizon. If it enters the interior, it is future incomplete if and only if $r \to 0$. This generalizes O'Neill's result for Schwarzschild to warped products without spherical symmetry.
• Theorem 4 (Main $C^0$-Inextendibility Result): Combining Theorems 2 and 3 with the obstruction provided by timelike geodesic completeness (Theorem 1), the paper establishes that the class of warped-product black hole spacetimes described is future $C^0$-inextendible.
• Extension to Multiple Horizons: The results are shown to hold even if the spacetime possesses more than one regular black hole horizon (Appendix A1).

Examples and Applicability
The paper provides specific examples to demonstrate the scope of the results:

1. Non-vacuum Spherically Symmetric Black Holes: A model with a specific $m(r)$ satisfying the dominant energy condition, showing that non-vacuum models are included.
2. Non-Spherically Symmetric Black Holes: A model with a fibre $F = S^p \times S^q$, demonstrating that the results apply to geometries beyond spherical symmetry, provided the fibre is homogeneous.
3. Birmingham Black Holes (AdS): The paper notes that while Theorem 2 applies to Birmingham black holes in Anti-de Sitter space (proving inextendibility through the singularity), Theorem 3 (and thus the full Theorem 4) does not apply because the specific metric coefficients fail the derivative bounds required for the exterior geodesic dichotomy.

Significance and Claims
The paper claims to isolate the mechanism underlying the $C^0$-inextendibility of the Schwarzschild spacetime and demonstrate its robustness under weaker symmetry assumptions. By replacing spherical symmetry with the homogeneity of the fibre, the author extends the class of black hole models for which the Strong Cosmic Censorship conjecture (in its $C^0$ formulation) can be verified. The work highlights that the obstruction to continuous extension is not an artifact of spherical symmetry but a more general feature of warped-product black holes with curvature singularities and specific geometric properties of the horizon cross-sections. The results contribute to the understanding of spacetime singularities and the validity of determinism in general relativity at low regularity.