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 - Funding Information:
The authors thank Mihalis Dafermos for offering enlightening comments on an earlier version of this work. They also thank the anonymous referees for providing valuable feedback that helped improve the exposition. IR gratefully acknowledges support from NSF grant # DMS-1001500. JS gratefully acknowledges support from NSF grant # DMS-1162211, from NSF CAREER grant # DMS-1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology
Funding Information:
The authors thank Mihalis Dafermos for offering enlightening comments on an earlier version of this work. They also thank the anonymous referees for providing valuable feedback that helped improve the exposition. IR gratefully acknowledges support from NSF grant # DMS-1001500. JS gratefully acknowledges support from NSF grant # DMS-1162211, from NSF CAREER grant # DMS-1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology.
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

AN - SCOPUS:85039933695

SN - 0003-486X

VL - 187

SP - 65

EP - 156

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -