Stable Big Bang Formation in NearFLRW Solutions to the EinsteinScalar Field and EinsteinStiff Fluid Systems
Abstract
We prove a stable singularity formation result for solutions to the Einsteinscalar field and Einsteinstiff fluid systems. Our results apply to small perturbations of the spatially flat FLRW solution with topology $(0,\infty) \times \mathbb{T}^3.$ The FLRW solution models a spatially uniform scalarfield/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. We place data on a Cauchy hypersurface $\Sigma_1'$ that are close to the FLRW data induced on $\lbrace 1 \rbrace \times \mathbb{T}^3.$ We study the perturbed solution in the collapsing direction and prove that its basic features closely resemble those of the FLRW solution. In particular, we construct constant mean curvaturetransported spatial coordinates for the perturbed solution covering $(t,x) \in (0,1] \times \mathbb{T}^3$ and show that it also has a Big Bang at $\lbrace 0 \rbrace \times \mathbb{T}^3,$ where its curvature blows up. The blowup confirms Penrose's Strong Cosmic Censorship hypothesis for the "pasthalf" of nearFLRW solutions. The most difficult aspect of the proof is showing that the solution exists for $(t,x) \in (0,1] \times \mathbb{T}^3,$ and to this end, we derive energy estimates that are allowed to mildly blowup as $t \downarrow 0.$ To close these estimates, we use the most important ingredient in our analysis: an $L^2$type energy approximate monotonicity inequality that holds for nearFLRW solutions. In the companion article "A regime of linear stability for the Einsteinscalar field system with applications to nonlinear Big Bang formation," we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.6298
 Bibcode:
 2014arXiv1407.6298R
 Keywords:

 Mathematics  Analysis of PDEs;
 General Relativity and Quantum Cosmology;
 Primary: 35A01;
 Secondary: 35L51;
 35Q31;
 35Q76;
 83C05;
 83C75;
 83F05