A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
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;∞) × T[superscript 3]. The Kasner solutions model a spatially uniform scalar field evolving i...
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Princeton University Press
2018
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| Online Access: | http://hdl.handle.net/1721.1/115990 https://orcid.org/0000-0001-5020-3568 |
| _version_ | 1826207984625123328 |
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| author | Rodnianski, Igor Speck, Jared R. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Rodnianski, Igor Speck, Jared R. |
| author_sort | Rodnianski, Igor |
| collection | MIT |
| description | 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;∞) × T[superscript 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) ≃ T[superscript 3] and study the linear solution's behavior in the collapsing direction t ↓ 0. Our first main result is the proof of an approximate L[superscript 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-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). 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 |
| first_indexed | 2024-09-23T13:58:29Z |
| format | Article |
| id | mit-1721.1/115990 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:58:29Z |
| publishDate | 2018 |
| publisher | Princeton University Press |
| record_format | dspace |
| spelling | mit-1721.1/1159902022-10-01T18:18:59Z A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation Rodnianski, Igor Speck, Jared R. Massachusetts Institute of Technology. Department of Mathematics Speck, Jared R. 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;∞) × T[superscript 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) ≃ T[superscript 3] and study the linear solution's behavior in the collapsing direction t ↓ 0. Our first main result is the proof of an approximate L[superscript 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-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). 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 National Science Foundation (U.S.) (Grant DMS-1162211) National Science Foundation (U.S.) (CAREER Grant DMS1454419) Alfred P. Sloan Foundation. Fellowship Solomon Buchsbaum AT&T Research Fund 2018-05-30T18:40:54Z 2018-05-30T18:40:54Z 2017-12 2015-02 2018-05-30T16:41:57Z Article http://purl.org/eprint/type/JournalArticle 0003-486X http://hdl.handle.net/1721.1/115990 Rodnianski, Igor, and Jared Speck. “A Regime of Linear Stability for the Einstein-Scalar Field System with Applications to Nonlinear Big Bang Formation.” Annals of Mathematics, vol. 187, no. 1, Jan. 2018, pp. 65–156. https://orcid.org/0000-0001-5020-3568 http://dx.doi.org/10.4007/ANNALS.2018.187.1.2 Annals of Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Princeton University Press arXiv |
| spellingShingle | Rodnianski, Igor Speck, Jared R. A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title | A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title_full | A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title_fullStr | A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title_full_unstemmed | A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title_short | A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation |
| title_sort | regime of linear stability for the einstein scalar field system with applications to nonlinear big bang formation |
| url | http://hdl.handle.net/1721.1/115990 https://orcid.org/0000-0001-5020-3568 |
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