A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

TY - JOUR

T1 - A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation

AU - Rodnianski, Igor

AU - Speck, Jared

N1 - Publisher Copyright: © 2018 Department of Mathematics, Princeton University.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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;∞) × T3. 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) ≃ T3 and study the linear solution's behavior in the collapsing direction t ↓ 0. Our first main result is the proof of an approximate L2 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-Lemaĭtre-Robertson-Walker (FLRW) solution. In particular, we show that as t ↓ 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 para-bolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction t ↓ 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↓ 0 under small perturbations of its data at (t = 1).

AB - 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;∞) × T3. 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) ≃ T3 and study the linear solution's behavior in the collapsing direction t ↓ 0. Our first main result is the proof of an approximate L2 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-Lemaĭtre-Robertson-Walker (FLRW) solution. In particular, we show that as t ↓ 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 para-bolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction t ↓ 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↓ 0 under small perturbations of its data at (t = 1).

KW - BKL conjectures

KW - Constant mean curvature

KW - FLRW

KW - Kasner solution

KW - Monotonicity

KW - Parabolic gauge

KW - Quiescent cosmology

KW - Spatial harmonic coordinates

KW - Stable blowup

KW - Strong cosmic censorship

KW - Transported spatial coordinates

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U2 - 10.4007/annals.2018.187.1.2

DO - 10.4007/annals.2018.187.1.2

M3 - Article

SN - 0003-486X

VL - 187

SP - 65

EP - 156

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -