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The $C^0$-inextendibility of a class of FLRW spacetimes

Jan Sbierski

TL;DR

The paper addresses the question of how FLRW spacetimes behave under $C^0$-level inextendibility, distinguishing the $K=+1$ and $K=-1$ cases. It develops a boundary-chart causality framework and leverages future one-connectedness and a TIF-based obstruction to prove past $C^0$-inextendibility: for $K=+1$ without horizons, the big-bang singularity blocks any past $C^0$-extension, while for $K=-1$ with $a(t) e^{\int_t^1 \frac{1}{a(t')} dt'} \to \infty$ as $t\to 0$, a geometric obstruction similarly prevents past extensions. The results are complemented by known Milne-like extendibility in certain $K=-1$ scenarios and by prior $C^2$-level singularity analyses, clarifying the low-regularity structure of FLRW cosmologies. The methods combine boundary-chart analysis, global hyperbolicity, and causality arguments to establish robust, dimension-dependent obstruction mechanisms.

Abstract

This paper studies the singularity structure of FLRW spacetimes without particle horizons at the $C^0$-level of the metric. We show that in the case of constant spatial curvature $K=+1$, and without any further assumptions on the scale factor, the big bang singularity is sufficiently strong to exclude continuous spacetime extensions to the past. On the other hand it is known that in the case of constant spatial curvature $K=-1$ continuous spacetime extensions through the big-bang exist for certain choices of scale factor [4], giving rise to Milne-like cosmologies. Complementing these results we exhibit a geometric obstruction to continuous spacetime extensions which is present for a large range of scale factors in the case $K=-1$.

The $C^0$-inextendibility of a class of FLRW spacetimes

TL;DR

The paper addresses the question of how FLRW spacetimes behave under -level inextendibility, distinguishing the and cases. It develops a boundary-chart causality framework and leverages future one-connectedness and a TIF-based obstruction to prove past -inextendibility: for without horizons, the big-bang singularity blocks any past -extension, while for with as , a geometric obstruction similarly prevents past extensions. The results are complemented by known Milne-like extendibility in certain scenarios and by prior -level singularity analyses, clarifying the low-regularity structure of FLRW cosmologies. The methods combine boundary-chart analysis, global hyperbolicity, and causality arguments to establish robust, dimension-dependent obstruction mechanisms.

Abstract

This paper studies the singularity structure of FLRW spacetimes without particle horizons at the -level of the metric. We show that in the case of constant spatial curvature , and without any further assumptions on the scale factor, the big bang singularity is sufficiently strong to exclude continuous spacetime extensions to the past. On the other hand it is known that in the case of constant spatial curvature continuous spacetime extensions through the big-bang exist for certain choices of scale factor [4], giving rise to Milne-like cosmologies. Complementing these results we exhibit a geometric obstruction to continuous spacetime extensions which is present for a large range of scale factors in the case .
Paper Structure (12 sections, 15 theorems, 68 equations, 3 figures)

This paper contains 12 sections, 15 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The class of FLRW spacetimes and the main theorems
  3. Previous results and remarks concerning the main theorems
  4. Discussion of proofs and outline of paper
  5. Properties of general $C^0$-extensions
  6. Properties of FLRW spacetimes without particle horizons
  7. The case $K=+1$
  8. The case $K=-1$
  9. Proof of the main theorems
  10. Proof of Theorem \ref{['Thm1']}: $K=+1$ without particle horizons
  11. Proof of Theorem \ref{['Thm2']}: $K=-1$ without particle horizons
  12. Lipschitz hypersurfaces and tangent cones

Key Result

Theorem 1.5

Let $d \geq 2$. Consider the class of $(d+1)$-dimensional FLRW spacetimes without particle horizons and with $K=+1$ as defined above. Each such spacetime $(M,g)$ is past $C^0$-inextendible.

Figures (3)

  • Figure 1: The proof of Proposition \ref{['PropIntNG']}.
  • Figure 2: The proof of Theorem \ref{['Thm1']}.
  • Figure 3: The set-up in the proof of Theorem \ref{['Thm2']} together with lightcone bounds.

Theorems & Definitions (32)

  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 1.8
  • Proposition 2.1
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 22 more