Spherically symmetric inextendibility of weak null singularities with Christoffel symbols in $L^s_{\text{loc}}$

Technical Summary: Spherically Symmetric Inextendibility of Weak Null Singularities with Christoffel Symbols in $L^s_{loc}$

Problem Statement
The paper addresses the Strong Cosmic Censor Conjecture (SCCC) within the context of general relativity, specifically focusing on the inextendibility of weak null singularities found in the interiors of spherically symmetric black holes. While previous results established the Lipschitz ($C^{0,1}_{loc}$) inextendibility of such singularities (e.g., in Dafermos-Luk-Oh spacetimes and Reissner-Nordström-Vaidya models), no geometric inextendibility results existed for extensions possessing Christoffel symbols in $L^s_{loc}$ for $s > 1$. The central problem is to determine whether these spacetimes can be extended as Lorentzian manifolds with a continuous metric and Christoffel symbols in $L^s_{loc}$, assuming the extension respects spherical symmetry.

Methodology
The authors employ a proof by contradiction, assuming the existence of a "strongly spherically symmetric" extension. This restriction requires the extension to be of the form $\iota = \iota_Q \times \text{id}_{S^2}$, where the extension of the 2-dimensional quotient manifold $(Q, g_Q)$ respects the $SO(3)$ orbits.

The methodology relies on three main components:

1. Regularity Control: The authors utilize a rigidity result from [1] regarding the $C^0$-structure of extensions. They restrict their analysis to extensions where the embedding $\iota_Q$ is locally $C^0$-equivalent to a reference extension $Q$. This allows them to control the $C^1$-differentiable structure of the extension, bounding certain derivatives of the embedding map in terms of others using near-Minkowski coordinates (Lemma 3.1).
2. Integral Blow-up Condition: The proof hinges on a specific blow-up condition on the derivative of the area-radius function $r$ transverse to the singularity. Specifically, the assumption is that $\int |\partial_v r|^s$ diverges as the singularity is approached.
3. Contradiction via $L^s$ Norms: By assuming the extension exists with Christoffel symbols in $L^s_{loc}$, the authors show that the derivatives of the metric components (specifically related to $\partial r$) must satisfy certain integrability conditions. Using the control bounds derived from the $C^0$-equivalence and the assumed blow-up of $\partial_v r$, they demonstrate that the $L^s$ norm of specific terms must diverge, contradicting the assumption that the extension is in $W^{1,s}_{loc}$.

Key Contributions and Results
The main result is Theorem 1.1, which states that for a spherically symmetric weak null singularity $(M_{wns}, g_{wns})$ where $\Omega$ and $r$ extend continuously, if the integral condition $\int_{-1}^v \inf_{u} |\partial_v r(u, v')|^s dv' \to \infty$ holds as $v \to 0$, then no strongly spherically symmetric $C^0 \cap W^{1,s}_{loc}$ extension exists across the singularity, provided the extension is locally $C^0$-equivalent to the reference extension.

The paper applies this theorem to two specific classes of spacetimes:

1. Reissner-Nordström-Vaidya (RNV) Spacetimes: The authors prove that these spacetimes, modeling null dust influx into a subextremal Reissner-Nordström black hole, are strongly spherically symmetric $C^0 \cap W^{1,s}_{loc}$-inextendible. They utilize a global volume argument (Proposition 2.5) to rule out "corner" extensions (where the extension is not $C^0$-equivalent to the reference), thereby satisfying the conditions of Theorem 1.1.
2. Dafermos-Luk-Oh (DLO) Spacetimes: These arise from generic small perturbations of subextremal Reissner-Nordström initial data under the Einstein-Maxwell-scalar field system. The authors show that for these spacetimes, the area-radius function $r$ is strictly monotonically decreasing along the weak null singularity. This local property rules out corner extensions, allowing the direct application of Theorem 1.1 to conclude $C^0 \cap W^{1,s}_{loc}$-inextendibility.

The paper also provides Example 3.3, demonstrating that the assumption of local $C^0$-equivalence in Theorem 1.1 is necessary. They construct a counter-example where the blow-up condition is met, yet a $C^0 \cap W^{1,s}_{loc}$ extension exists because the extension is not $C^0$-equivalent to the reference (a "corner" extension).

Significance
The paper claims to provide the first geometric inextendibility results for weak null singularities at the level of Christoffel symbols in $L^s_{loc}$ for $s > 1$. By strengthening previous $C^{0,1}_{loc}$ (Lipschitz) inextendibility results to $C^0 \cap W^{1,s}_{loc}$, the authors show that the regularity of the extension constructed by Dafermos [4] (where Christoffel symbols are in $L^1_{loc}$ but not $L^s_{loc}$ for $s>1$) is optimal among extensions that respect spherical symmetry.

The work relies on the interplay between the blow-up of the derivative of the area-radius function and the structural rigidity of spherically symmetric $C^0$ extensions. It confirms that for the RNV and DLO spacetimes, the weak null singularity represents a true boundary beyond which the spacetime cannot be extended as a Lorentzian manifold with a continuous metric and $L^s_{loc}$ Christoffel symbols, provided the extension maintains the spherical symmetry of the original spacetime.