Stable Big Bang formation for Einstein's equations: The complete sub-critical regime

Abstract

For $(t,x)\in (0,\infty )\times {\mathbb{T}}^D$, the generalized Kasner solutions are a family of explicit solutions to various Einstein-matter systems that start out smooth but then develop a Big Bang singularity as $t\downarrow 0$, i.e., curvature blowup along a spacelike hypersurface. The family is parameterized by the Kasner exponents ${\tilde{q}}_1,\cdots ,{\tilde{q}}_D\in \mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, given say at $\{t=1\}$, as long as the exponents are "sub-critical" in the following sense: ${{\max }_{I,J,B=1,\cdots ,D}}_{I<J}\{{\tilde{q}}_I+{\tilde{q}}_J-{\tilde{q}}_B\}<1$. Previous works have shown the stability of the singularity under stronger assumptions: 1) the Einstein-scalar field system with $D=3$ and ${\tilde{q}}_1\approx {\tilde{q}}_2\approx {\tilde{q}}_3\approx 1/3$ or 2) the Einstein-vacuum equations for $D\ge 39$ with ${\max }_{I=1,\cdots ,D}|{\tilde{q}}_I|<1/6$. We prove that the Kasner singularity is dynamically stable for \emph{all} sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the $1+D$-dimensional Einstein-scalar field system for $D\ge 3$ and the $1+D$ dimensional Einstein-vacuum equations for $D\ge 10$. Moreover, for the Einstein-vacuum equations in $1+3$ dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have stable Big Bangs under polarized $U(1)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U(1)$-symmetric solutions.

72 pages. This version is equivalent to the published version. Features additional details on some proofs and corrections of minor errors

References in corpus (22)

• Quiescent cosmological singularities
• Numerical Approaches to Spacetime Singularities
• The Bianchi IX attractor
• Analytic description of singularities in Gowdy spacetimes
• Black holes without spacelike singularities
• Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
• Asymptotic Behavior of Polarized and Half-Polarized U(1) symmetric Vacuum Spacetimes
• Asymptotic Behavior in Polarized {\bf T}${}^2$-symmetric Vacuum Spacetimes
• A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
• Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes
• Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes
• A new proof of the Bianchi type IX attractor theorem
• Fuchsian analysis of S^2xS^1 and S^3 Gowdy spacetimes
• The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology ${\mathbb{S}}^3$
• Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles
• A class of solutions to the Einstein equations with AVTD behavior in generalized wave gauges
• Cauchy horizons in Gowdy space times
• On the backward stability of the Schwarzschild black hole singularity
• A New Class of Asymptotically Non-Chaotic Vacuum Singularities
• Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors
• Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes
• Kasner-like regions near crushing singularities

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