A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation. (English) Zbl 1384.83005

Rodnianski, Igor; Speck, Jared

Ann. Math. (2) 187, No. 1, 65-156 (2018).

Summary: We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on \((0,\infty)\times\mathbb T^3\). The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a “Big Bang” singularity at \(\{t=0\}\). We place initial data for the linearized system along \(\{t=1\}\simeq\mathbb T^3\) and study the linear solution’s behavior in the collapsing direction \(t\downarrow 0\). Our first main result is the proof of an approximate \(L^2\) monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lemaître-Robertson-Walker (FLRW) solution. In particular, we show that as \(t\downarrow 0\), various time-rescaled components of the linear solution converge to regular functions defined along \(\{t=0\}\). In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction \(t\downarrow 0\). Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction \(t\downarrow 0\) under small perturbations of its data at \(\{t=1\}\).

MSC:

83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C75	Space-time singularities, cosmic censorship, etc.
35A20	Analyticity in context of PDEs
35Q76	Einstein equations
83F05	Relativistic cosmology
83C25	Approximation procedures, weak fields in general relativity and gravitational theory
53Z05	Applications of differential geometry to physics
83C57	Black holes

Keywords:

BKL conjectures; constant mean curvature; FLRW; Kasner solution; monotonicity; parabolic gauge; quiescent cosmology; spatial harmonic coordinates; stable blowup; strong cosmic censorship; transported spatial coordinates

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