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

Peter Cameron, Jan Sbierski

Abstract

Motivated by weak null singularities in black hole interiors, we study 1+1 dimensional Lorentzian manifolds $(M,g)$ which admit a continuous spacetime extension across a null boundary $v=0$, where $v<0$ is a null coordinate. We study the degree to which such extensions are unique up to the boundary. Firstly, we find that in general not even the $C^0$-structure of the extension is uniquely determined by the assumption that the metric extends continuously. However, we exhibit an interesting local-global relation regarding the $C^0$-structure which in particular entails its rigidity for ''strongly spherically symmetric'' continuous extensions across the Cauchy horizon of the Reissner-Nordström spacetime. Secondly, we construct continuous extensions which have the same $C^0$-structure, but do not have equivalent $C^1$-structures. This construction also carries over to weak null singularities in 3+1 dimensions. Understanding the uniqueness properties of continuous spacetime extensions to the boundary is of importance for the study of low-regularity inextendibility problems.

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

Abstract

Motivated by weak null singularities in black hole interiors, we study 1+1 dimensional Lorentzian manifolds which admit a continuous spacetime extension across a null boundary , where is a null coordinate. We study the degree to which such extensions are unique up to the boundary. Firstly, we find that in general not even the -structure of the extension is uniquely determined by the assumption that the metric extends continuously. However, we exhibit an interesting local-global relation regarding the -structure which in particular entails its rigidity for ''strongly spherically symmetric'' continuous extensions across the Cauchy horizon of the Reissner-Nordström spacetime. Secondly, we construct continuous extensions which have the same -structure, but do not have equivalent -structures. This construction also carries over to weak null singularities in 3+1 dimensions. Understanding the uniqueness properties of continuous spacetime extensions to the boundary is of importance for the study of low-regularity inextendibility problems.

Paper Structure

This paper contains 14 sections, 12 theorems, 100 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Previous results
  3. A 1+1-dimensional toy model
  4. Main results
  5. Outline of paper
  6. Definitions and Background Results
  7. First Structural Properties of Extensions in 1+1 Dimensions
  8. The $C^0$-Structure of Extensions in 1+1 Dimensions
  9. Dichotomy of the $C^0$-structure of local extensions: the corner and reference extensions
  10. Ruling out corner extensions from global structure
  11. Implications for strongly spherically symmetric extensions across the Reissner-Nordström Cauchy horizon
  12. The $C^1$-Structure of Extensions in 1+1 Dimensions
  13. Non-rigidity of the $C^1$-structure
  14. Application to weak null singularities in $3+1$-dimensions

Key Result

Lemma 2.1

Let $(M,g)$ be a $(d+1)$-dimensional time-oriented Lorentzian manifold with a continuous metric $g$, let $\gamma : [0,1] \to M$ be a future (or past) directed locally Lipschitz causal curve, and let $t$ be a smooth time function on $M$. Then $\gamma$ can be reparameterised by $t$ and we denote this

Figures (18)

  • Figure 1: We extend $\gamma$ as a curve of constant $x_0$ so that the range of $x_1$ is now $(-\epsilon_1,\epsilon_1)$. This figure illustrates the claim in the case where $\frac{d\overset{(1)}{\gamma}_1}{dx_0}>0$ and $\sigma(0)\in \text{Im}(\gamma)$.
  • Figure 2: Figure showing the split of $B(\frac{\epsilon'}{\sqrt{5}})$ into the regions $A$, $B$, $C_-$ and $C_+$ as described in the proof of Proposition \ref{['prop:C1nullcurve']}, where we show that $\gamma\subset C_+$.
  • Figure 3: Construction (in the case $u(p)\leq\gamma_u(\lambda_*)$) of $\Gamma(s;\lambda)$, a causal homotopy of $\gamma$ with fixed endpoints such that $\Gamma(s_*;\lambda_*)=p$ for some $\lambda_*\in[0,1]$ and some $s_*\in[\frac{\lambda_*}{2},\lambda_*]$. $\Gamma(s;\lambda)$ is shown as a dotted line for $s\in[0,\lambda)$. For $s\in[\lambda,1]$, points $\Gamma(s;\lambda)$ lie on $\gamma$.
  • Figure 4: The homotopy argument used to show $\iota\left([u_*-\epsilon,u_*]\times[v_0,0)\right)\subset\joinrel\subset\tilde{U}$.
  • Figure 5: The contradiction argument used to show that $\tilde{\gamma}^{u'}([v_*,0)) \subset K(v_0)$ for $u'\in(u_*,u_*+\epsilon]$.
  • ...and 13 more figures

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 23 more