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Quiescent Big Bang formation in $2+1$ dimensions

Liam Urban

Abstract

In this paper, we study the past asymptotics of $(2+1)$-dimensional solutions to the Einstein scalar-field Vlasov system which are close to Friedman-Lemaître-Robertson-Walker spacetimes on an initial hypersurface diffeomorphic to a closed orientable surface $M$ of arbitrary genus. We prove that such solutions are past causally geodesically incomplete, form a curvature singularity and exhibit stable Kretschmann scalar blow-up in the contracting direction. In particular, the spacetime is $C^2$-inextendible towards the past where causal geodesics become incomplete. Moreover, we show that geometry and matter are asymptotically velocity term dominated toward the past, remaining close to their background counterparts. When viewed on the co-mass shell, the Vlasov distribution in particular converges to a limiting distribution on the Big Bang hypersurface, while asymptotics of the spatial metric lead to slight degeneracies when trying to control the components of the Vlasov energy-momentum tensor. These also manifest when viewing the distribution function on the mass shell, where velocities generally approach a smooth one-dimensional subbundle of the tangent bundle, leading the distribution to become highly anisotropic, when viewed in terms of the geometry induced by the constant curvature spatial reference metric. Compared to previous results in higher dimensions, in particular [FU25b], inhomogeneous terms in the wave and Vlasov equations factor in more strongly in our setting, which a priori creates additional hurdles at high orders while largely keeping the quiescent system intact. As a corollary, our main result shows that the Strong Cosmic Censorship conjecture holds for certain polarized $U(1)$-symmetric solutions to the Einstein vacuum equations that emanate from a spatial hypersurface diffeomorphic to $M\times\mathbb{S}^1$.

Quiescent Big Bang formation in $2+1$ dimensions

Abstract

In this paper, we study the past asymptotics of -dimensional solutions to the Einstein scalar-field Vlasov system which are close to Friedman-Lemaître-Robertson-Walker spacetimes on an initial hypersurface diffeomorphic to a closed orientable surface of arbitrary genus. We prove that such solutions are past causally geodesically incomplete, form a curvature singularity and exhibit stable Kretschmann scalar blow-up in the contracting direction. In particular, the spacetime is -inextendible towards the past where causal geodesics become incomplete. Moreover, we show that geometry and matter are asymptotically velocity term dominated toward the past, remaining close to their background counterparts. When viewed on the co-mass shell, the Vlasov distribution in particular converges to a limiting distribution on the Big Bang hypersurface, while asymptotics of the spatial metric lead to slight degeneracies when trying to control the components of the Vlasov energy-momentum tensor. These also manifest when viewing the distribution function on the mass shell, where velocities generally approach a smooth one-dimensional subbundle of the tangent bundle, leading the distribution to become highly anisotropic, when viewed in terms of the geometry induced by the constant curvature spatial reference metric. Compared to previous results in higher dimensions, in particular [FU25b], inhomogeneous terms in the wave and Vlasov equations factor in more strongly in our setting, which a priori creates additional hurdles at high orders while largely keeping the quiescent system intact. As a corollary, our main result shows that the Strong Cosmic Censorship conjecture holds for certain polarized -symmetric solutions to the Einstein vacuum equations that emanate from a spatial hypersurface diffeomorphic to .

Paper Structure

This paper contains 28 sections, 21 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Previous work
  3. Comments on the proof and outline
  4. Further applications
  5. Preliminaries
  6. Notation
  7. The reference solutions
  8. Rescaled variables and Sasaki metrics
  9. The expansion normalised $2+1$ system in CMCTC gauge
  10. Initial data, local well-posedness and bootstrap assumptions
  11. Solution norms and energies
  12. Initial data
  13. Local well-posedness
  14. Bootstrap assumptions
  15. A priori estimates
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(M,\mathring{g},\mathring{k},\mathring{\pi},\mathring{\psi},\mathring{f})$ be initial datasee Section subsec:init-2+1 to the ESFV system that is close to FLRW initial data such that the momentum support of $\mathring{f}$ is compact. In the massless case, suppose in addition that one has Then, the past maximal globally hyperbolic development of said data in the ESFV system is causally geodesi

Theorems & Definitions (50)

  • Theorem 1.1: Stability of FLRW solutions to the ESFV system in $2+1$ dimensions
  • Corollary 1.2: Cosmic Censorship and stable curvature blow-up for polarized $U(1)$-symmetric vacuum solutions
  • Remark 1.3: Polarized $U(1)$-symmetric Einstein-Vlasov solutions
  • Remark 1.4: Theorem \ref{['thm:main-intro-2+1']} and global stability
  • Lemma 2.1: FLRW solutions to the ESFV system
  • Lemma 2.2: Properties of the scale factor
  • Remark 2.3: Reference solutions for Corollary \ref{['cor:u1-intro']}
  • Definition 2.4: Expansion normalised variables
  • Definition 3.1: Solution norms
  • Definition 3.2: Energies
  • ...and 40 more