Understand
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 $\widetilde{q}_1,\cdots,\widetilde{q}_D \in \mathbb{R}$, 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 $\lbrace t = 1 \rbrace$, as long as the exponents are "sub-critical" in the following sense: $\mathop{\max_{I,J,B=1,\cdots,D}}_{I < J} \{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1$.
- Previous works have shown the stability of the singularity under stronger assumptions: 1) the Einstein-scalar field system with $D = 3$ and $\widetilde{q}_1 \approx \widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3$ or 2) the Einstein-vacuum equations for $D \geq 39$ with $\max_{I=1,\cdots,D} |\widetilde{q}_I| < 1/6$.