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The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry

Kai Dong

TL;DR

The paper proves a past stability result for vacuum Kasner spacetimes under polarized $U(1)$-symmetry in $(3+1)$ dimensions by reducing the Einstein vacuum equations to a $(2+1)$-dimensional Einstein–scalar system via a novel $(2+1)$ orthonormal-frame decomposition. It then reformulates the evolution as a symmetric hyperbolic system, converts it to a Fuchsian form with a projection identifying decaying and converging components, and analyzes constraint propagation to obtain global existence up to the Big Bang singularity. The main outcome is that small polarized perturbations yield spacetimes that are asymptotically Kasner and AVTD at $t\to 0^+$, with a crushing singularity and curvature blow-up, and with precise asymptotics for rescaled variables. The approach combines a new frame-based reduction, careful symmetrization, and Fuchsian techniques to handle the non-subcritical Kasner exponents in $(3+1)$ dimensions, contributing a robust method for past stability results in general relativity.

Abstract

In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the $(3+1)$-dimensional Einstein vacuum equations under polarized $U(1)$-symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'ı}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed $(2+1)$ orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the $(3+1)$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities.

The past stability of Kasner singularities for the $(3+1)$-dimensional Einstein vacuum spacetime under polarized $U(1)$-symmetry

TL;DR

The paper proves a past stability result for vacuum Kasner spacetimes under polarized -symmetry in dimensions by reducing the Einstein vacuum equations to a -dimensional Einstein–scalar system via a novel orthonormal-frame decomposition. It then reformulates the evolution as a symmetric hyperbolic system, converts it to a Fuchsian form with a projection identifying decaying and converging components, and analyzes constraint propagation to obtain global existence up to the Big Bang singularity. The main outcome is that small polarized perturbations yield spacetimes that are asymptotically Kasner and AVTD at , with a crushing singularity and curvature blow-up, and with precise asymptotics for rescaled variables. The approach combines a new frame-based reduction, careful symmetrization, and Fuchsian techniques to handle the non-subcritical Kasner exponents in dimensions, contributing a robust method for past stability results in general relativity.

Abstract

In this paper, we give a new proof to a past stability result established in Fournodavlos-Rodnianski-Speck (arXiv:2012.05888), for Kasner solutions of the -dimensional Einstein vacuum equations under polarized -symmetry. Our method, inspired by Beyer-Oliynyk-Olvera-Santamar{\'ı}a-Zheng (arXiv:1907.04071, arXiv:2502.09210), relies on a newly developed orthonormal-frame decomposition and a careful symmetrization argument, after which the Fuchsian techniques can be applied. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete and crushing at the Big Bang singularity. They are achieved by reducing the Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of constraints propagation, we obtain global existence and precise asymptotics of the solutions up to the singularities.
Paper Structure (21 sections, 9 theorems, 144 equations)

This paper contains 21 sections, 9 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Einstein field equations and Kasner spacetimes
  3. The spatial topology and symmetry assumption of spacetimes
  4. Rough version of the main result
  5. Prior and Related results
  6. Proof strategy and New ingredients
  7. Acknowledgments
  8. Formulation of the equations
  9. The relativistic $(2+1)$-dimensional orthonormal frame decomposition
  10. The choice of the background metric
  11. The gauge freedom
  12. The derivation of equations
  13. Symmetrization of evolution system
  14. The Fuchsian system
  15. Propagation of the constraints
  16. ...and 6 more sections

Key Result

Theorem 1.1

Consider the $(3+1)$-dimensional Einstein Vacuum Equations under polarized $U(1)$-symmetry. Then every nontrivial vacuum Kasner solution (g_Kas) with $B=0$ and $D=3$ is dynamically stable under small polarized $U(1)$-symmetric perturbations near their Big Bang singularity. More specifically, for suf

Theorems & Definitions (21)

  • Theorem 1.1: Rough version
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5: Key symmetrization
  • Lemma 4.1
  • ...and 11 more