Stable Big Bang Formation for Einstein's Equations: The Complete Sub-Critical Regime

摘要

For (t, x) ∈ (0,∞) × T~D, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as t ↓ 0, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents q_1, · · · , q_D ∈ 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 dynamically stable, that is, 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: I,J,B=1,··· ,D {q_I +q_J ?q_B} < 1. Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: 1) the Einstein-scalar field system with D = 3 and q_1 ≈ q_2 ≈ q_3 ≈ 1/3, which corresponds to the stability of the Friedmann-Lema?tre-Robertson-Walker solution's Big Bang or 2) the Einstein-vacuum equations for D ≥ 38 with I=1,··· ,D |q_I | < 1/6. In this paper, we prove that the Kasner singularity is dynamically stable for all sub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the 1 + D-dimensional Einstein-scalar field system for all D ≥ 3 and the 1 + D-dimensional Einstein-vacuum equations for D ≥ 10; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in 1+3 dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized U(1)-symmetric perturbations of their initial data. Our results hold for open se