On the uniqueness of continuous spacetime extensions in 1+1 dimensions with applications to weak null singularities

Technical Summary: Uniqueness of Continuous Spacetime Extensions in 1+1-Dimensions

Problem Statement
The paper investigates the local uniqueness of continuous ($C^0$) spacetime extensions across a null boundary, motivated by the study of weak null singularities in the interiors of generic rotating black holes. A weak null singularity is defined as a singular null boundary where the metric extends continuously, but the Christoffel symbols fail to be square-integrable.

The central question addressed is: Given a Lorentzian manifold $(M, g)$ with a continuous metric that extends to a boundary $\partial M$, is the topological and differentiable structure of any continuous extension $\iota: M \hookrightarrow \tilde{M}$ uniquely determined up to the boundary? Specifically, if two extensions exist, are they locally $C^0$-equivalent (homeomorphic) or $C^1$-equivalent (diffeomorphic) near the boundary? This question is critical for low-regularity inextendibility proofs, where a priori control over the $C^1$-structure of a hypothetical extension is often required to demonstrate curvature blow-up or holonomy divergence.

Methodology
The authors employ a "toy model" approach, reducing the complex 3+1-dimensional problem of weak null singularities to a 1+1-dimensional Lorentzian manifold $(M_t, g_t)$ with a double-null metric $g_t = -\Omega^2(du \otimes dv + dv \otimes du)$. The metric function $\Omega$ is assumed to extend continuously and strictly positively to the boundary $v=0$.

The methodology involves:

1. Structural Analysis: Utilizing near-Minkowski coordinates and causal homotopy arguments to characterize the geometry of the boundary $\partial \iota(M_t)$ in a continuous extension.
2. Classification of $C^0$-Structures: Investigating whether the boundary can collapse multiple null curves to a single point (a "corner" structure) or must preserve the distinctness of these curves (the "reference" structure).
3. Global Constraints: Applying volume arguments to determine if local structural possibilities can be extended globally, particularly in the context of the Reissner-Nordström spacetime.
4. Counterexample Construction: Explicitly constructing extensions that share the same $C^0$-structure but possess inequivalent $C^1$-structures, utilizing specific coordinate transformations involving logarithmic singularities.

Key Contributions and Results

1. Dichotomy of $C^0$-Structures (Theorem 4.3)
The paper establishes a rigorous classification of the local $C^0$-structure of continuous extensions across $v=0$. There are only two possibilities:

• Reference Extension: The extension is locally $C^0$-equivalent to the standard extension where the boundary is a smooth null hypersurface, and distinct constant-$u$ null curves terminate at distinct boundary points.
• Corner Extension: All future-directed null curves of constant $u$ terminate at the same boundary point. In this case, the extension is not locally $C^0$-equivalent to the reference extension.

2. Rigidity and Global Obstructions (Proposition 4.2, Corollary 4.6)
While corner extensions are locally possible, the paper demonstrates a strong local-global relation:

• If a continuous extension exhibits the corner structure locally (i.e., two nearby constant-$u$ curves meet at the same point), it must exhibit this structure globally for all $u$.
• Application to Reissner-Nordström: For the sub-extremal Reissner-Nordström spacetime, the infinite spacetime volume between the event horizon and the Cauchy horizon rules out the existence of global corner extensions. Consequently, any strongly spherically symmetric $C^0$-extension of the global Reissner-Nordström spacetime across the Cauchy horizon must be locally $C^0$-equivalent to the reference extension. This implies a rigidity of the $C^0$-structure for global, spherically symmetric black hole interiors.

3. Non-Rigidity of $C^1$-Structure (Section 5)
The paper proves that the $C^0$-structure does not determine the $C^1$-structure:

• Corner Extensions: Different corner extensions (parameterized by $\beta$) are $C^0$-equivalent but $C^1$-inequivalent.
• Reference Extensions: The authors construct a continuous extension that is locally $C^0$-equivalent to the reference extension (preserving the distinctness of boundary points) but is not locally $C^1$-equivalent. This is achieved via a coordinate transformation that is a homeomorphism but fails to be a diffeomorphism at the boundary due to unbounded derivatives.
• 3+1-Dimensional Generalization: This non-rigidity result extends to weak null singularities in 3+1-dimensions without symmetry assumptions. The authors construct a $C^0$-extension of a weak null singularity that is $C^0$-equivalent to the reference extension (as defined by Dafermos and Luk) but $C^1$-inequivalent.

Significance and Claims
The paper claims to provide a foundational understanding of the uniqueness properties of continuous spacetime extensions, which is a prerequisite for proving inextendibility in regularity classes below $C^{0,1}_{loc}$ (Lipschitz).

• On $C^0$-Uniqueness: The work clarifies that while $C^0$-extensions are not unique in general (corner extensions exist), global geometric constraints (like infinite volume) can enforce uniqueness for physically relevant spacetimes like Reissner-Nordström.
• On $C^1$-Uniqueness: The paper explicitly demonstrates that the assumption of a continuous metric extension is insufficient to guarantee a unique $C^1$-structure. This challenges the notion that one can automatically assume a priori control over the differentiable structure of a continuous extension.
• Implications for Inextendibility: The results suggest that proving inextendibility for weak null singularities in regularity classes between $C^0$ and $C^{0,1}_{loc}$ is more subtle than previously thought. One cannot simply assume that a continuous extension inherits the $C^1$-structure of a reference extension; the lack of $C^1$-rigidity must be accounted for in future proofs.

The authors remain modest regarding the 3+1-dimensional case, noting that while the 1+1 toy model provides necessary insights and counterexamples, the full 3+1-dimensional uniqueness problem remains an open and intricate area of research.