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Localized Big Bang Stability of Spacetime Dimensions $n\geq4$

Weihang Zheng

TL;DR

The paper proves localized past nonlinear stability of sub-critical Kasner-scalar field solutions to the Einstein-scalar field equations in spacetime dimensions $n\ge 4$ on a truncated cone domain. It employs a conformal formulation and a frame-based Fuchsian method, constructing rescaled variables that vanish on Kasner backgrounds and establishing global-in-space and local-in-space stability with energy and decay estimates. The results show the perturbed solutions remain asymptotically Kasner, are $C^2$-inextendible, past timelike geodesically incomplete, and exhibit curvature blow-up consistent with a crushing singularity and AVTD behavior. This extends the four-dimensional BOZ stability result to higher dimensions and demonstrates that localized big bang formation persists in the sub-critical Kasner regime, with implications for directional singularity dynamics in higher-dimensional general relativity.

Abstract

We prove the past nonlinear stability of the sub-critical Kasner-scalar field solutions to the Einstein-scalar field equations on a truncated cone domain in spacetime dimensions $n\geq4$. Our analysis demonstrates that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete in the contracting direction and terminate at quiescent and crushing singularities characterized by the blow-up of curvature invariants. This work generalizes the result of Beyer-Oliynyk-Zheng in [arXiv:2502.09210v2] to all higher dimensional spacetimes.

Localized Big Bang Stability of Spacetime Dimensions $n\geq4$

TL;DR

The paper proves localized past nonlinear stability of sub-critical Kasner-scalar field solutions to the Einstein-scalar field equations in spacetime dimensions on a truncated cone domain. It employs a conformal formulation and a frame-based Fuchsian method, constructing rescaled variables that vanish on Kasner backgrounds and establishing global-in-space and local-in-space stability with energy and decay estimates. The results show the perturbed solutions remain asymptotically Kasner, are -inextendible, past timelike geodesically incomplete, and exhibit curvature blow-up consistent with a crushing singularity and AVTD behavior. This extends the four-dimensional BOZ stability result to higher dimensions and demonstrates that localized big bang formation persists in the sub-critical Kasner regime, with implications for directional singularity dynamics in higher-dimensional general relativity.

Abstract

We prove the past nonlinear stability of the sub-critical Kasner-scalar field solutions to the Einstein-scalar field equations on a truncated cone domain in spacetime dimensions . Our analysis demonstrates that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete in the contracting direction and terminate at quiescent and crushing singularities characterized by the blow-up of curvature invariants. This work generalizes the result of Beyer-Oliynyk-Zheng in [arXiv:2502.09210v2] to all higher dimensional spacetimes.
Paper Structure (43 sections, 12 theorems, 356 equations)

This paper contains 43 sections, 12 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. The Einstein-scalar field equations
  3. Kasner-scalar field spacetimes
  4. Related works
  5. A rough version of the main theorem
  6. Overview of the proof
  7. Acknowledgments
  8. Preliminaries
  9. Data availability statement
  10. Coordinates and frames conventions
  11. Index operators and multi-indices
  12. Vectors and matrices
  13. Domains in $\mathbb{T}{}^{n-1}$
  14. Sobolev spaces and extension operators
  15. Constants and inequalities
  16. ...and 28 more sections

Key Result

Theorem 1.3

The conformal Kasner-scalar field solutions in the sub-critical regime in sapcetime dimensions $n\geq4$ are dynamically stable under small perturbations when restricted to a truncated cone domain. More precisely, given $\rho_0\in(0,L)$, $\epsilon_2\in(0,1)$ and $\rho_1>0$ satisfying condition rho1-r

Theorems & Definitions (23)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3: Localized past stability of the sub-critical Kasner metrics in dimensions $n\geq4$
  • Lemma 2.1
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Proposition 5.1
  • proof
  • ...and 13 more