Warped Spacelike Singularities and the $C^0$-Inextendibility of Birmingham-Kottler Spacetimes

Technical Summary: Warped Spacelike Singularities and the $C^0$-Inextendibility of Birmingham-Kottler Spacetimes

Problem Statement
The paper addresses the problem of $C^0$-inextendibility for spacetimes containing singularities. A spacetime is $C^0$-inextendible if it cannot be embedded as a proper open subset of a larger spacetime equipped with a continuous Lorentzian metric. At this regularity level, curvature invariants (which typically blow up at singularities) are unavailable as obstructions, as their divergence does not preclude the existence of a continuous extension. The specific focus is on the Birmingham-Kottler (BK) family of spacetimes—warped products representing black holes with a negative or zero cosmological constant ($\Lambda \le 0$) and a closed fiber $\Sigma$. The goal is to prove that the canonical one-horizon BK spacetime is $C^0$-inextendible, even when the fiber $\Sigma$ lacks symmetry (homogeneity, orientability, or simple connectivity).

Methodology
The proof is structured in two distinct phases: a local obstruction at the singularity and a global analysis of the specific BK spacetime.

1. Local Obstruction (Warped Spacelike Singularities):
   The author establishes a criterion for the non-existence of a continuous extension at a warped spacelike singularity of the form $M_R = (0, R) \times \mathbb{R} \times \Sigma$ with metric $g = -a(r)^2 dr^2 + b(r)^2 dt^2 + c(r)^2 \gamma_\Sigma$.

   • Integrability Conditions: The proof requires the integrability of the "causal budgets" $A(r) = \int_0^r \frac{a}{b} ds$ and $B(r) = \int_0^r \frac{a}{c} ds$. These ensure that time and fiber coordinates converge as $r \to 0$.
   • Monotonicity and Divergence: The quotient $b/c$ must be non-increasing, and the longitudinal factor $b(r)$ must diverge to infinity as $r \to 0$.
   • Radial Compression: A key geometric innovation replaces the rotational symmetries used in spherical cases. The author constructs a "radial compression" map $\phi_\epsilon$ that moves the limiting trace of a causal curve from $r=0$ to a positive radius $r=\epsilon$ while controlling the angular contribution to the causal inequality.
   • Contradiction Mechanism: This compression yields a compact chronological separator in an adapted boundary chart. By translating this separator along the $t$-direction, the author constructs two points on a sequence of radial slices whose intrinsic distance diverges (due to $b(r) \to \infty$) while their distance in the extension chart remains uniformly bounded. This contradiction proves that no future boundary point can be approached by a timelike curve along which $r \to 0$.

2. Global Application (Birmingham-Kottler Spacetimes):
   The paper constructs the canonical one-horizon BK spacetime using global Kruskal coordinates.

   • Geodesic Classification: Using the field equations and geodesic first integrals (energy $E$ and angular momentum $J$), the author classifies the endpoints of finite proper-time timelike geodesics. It is shown that any geodesic that eventually leaves every compact subset must enter a dynamic block and satisfy $r \to 0$.
   • Synthesis: The local obstruction is applied to the dynamic block of the BK spacetime. Since any putative $C^0$ extension would require a finite-length causal maximizer approaching the boundary (a result by Minguzzi and Suhr), and such a maximizer must correspond to a geodesic reaching $r=0$, the local obstruction rules out the existence of such an extension.

Key Contributions

• Generalization of Local Obstructions: The paper extends the $C^0$-inextendibility results for warped singularities beyond spherically symmetric settings. It removes the requirement for the fiber $\Sigma$ to be homogeneous, orientable, or simply connected.
• Radial Compression Technique: The introduction of the radial compression map (Lemma 3.6) serves as a substitute for rotational deformations, allowing the localization of divergence arguments on arbitrary closed manifolds.
• Global Inextendibility of BK Spacetimes: The paper proves that the entire canonical one-horizon Birmingham-Kottler spacetime (with $\Lambda \le 0$) is $C^0$-inextendible. This includes the Schwarzschild-Tangherlini solution as a specific case.
• Geodesic Endpoint Classification: A rigorous classification of finite proper-time ends for timelike geodesics in the global Kruskal domain is provided, demonstrating that escaping geodesics inevitably terminate at the singularity.

Results

• Theorem A (Local): No $C^0$ extension of a warped spacelike singularity exists if the warp factors satisfy the specified integrability, monotonicity, and divergence conditions.
• Theorem B (Global): The canonical one-horizon Birmingham-Kottler spacetime $M_{BK}$ is $C^0$-inextendible for any closed connected Einstein fiber satisfying the curvature condition $\text{Ric}_{\gamma_\Sigma} = (n-2)k\gamma_\Sigma$.
• Corollary: The canonical Schwarzschild-Tangherlini spacetime is $C^0$-inextendible.

Significance and Claims
The paper claims to isolate a purely metric obstruction to continuous extensions at warped singularities, independent of field equations. By combining this local result with a global analysis of the BK family, it establishes the inextendibility of the full spacetime without relying on symmetry assumptions that typically restrict such proofs. The work distinguishes itself from previous criteria based on timelike completeness or global hyperbolicity by focusing on the specific behavior of geodesics escaping compact sets in the presence of a singularity. The author notes that the Einstein condition is required only to identify the metric as a vacuum solution but is not used in the core obstruction or geodesic classification arguments. The paper concludes by identifying future directions, such as handling charged solutions, positive cosmological constants, and non-canonical horizon identifications, which would require new global inputs.