TY - JOUR

T1 - STABLE BIG BANG FORMATION FOR EINSTEIN’S EQUATIONS

T2 - THE COMPLETE SUB-CRITICAL REGIME

AU - Fournodavlos, Grigorios

AU - Rodnianski, Igor

AU - Speck, Jared

N1 - Funding Information:
The first author was supported by the ERC grant 714408 GEOWAKI, under the European Union’s Horizon 2020 research and innovation program. The second author was supported by NSF grant # DMS 2005464. The third author was supported by NSF grant # 2054184, from NSF CAREER grant # 1914537, and from a Chancellor’s Faculty Fellowship administered by Vanderbilt University.
Funding Information:
Received by the editors March 6, 2021, and, in revised form, March 22, 2022. 2020 Mathematics Subject Classification. Primary 83C75; Secondary 35A21, 35Q76, 83C05, 83F05. Key words and phrases. Big Bang, constant mean curvature, curvature singularity, Fermi– Walker transport, geodesically incomplete, Hawking’s theorem, Kasner solutions, maximal globally hyperbolic development, singularity theorem, stable blowup, transported spatial coordinates. The first author was supported by the ERC grant 714408 GEOWAKI, under the European Union’s Horizon 2020 research and innovation program. The second author was supported by NSF grant # DMS 2005464. The third author was supported by NSF grant # 2054184, from NSF CAREER grant # 1914537, and from a Chancellor’s Faculty Fellowship administered by Vanderbilt University. 1By “cosmological solutions,” we mean ones with compact spatial topology.
Publisher Copyright:
© 2023, American Mathematical Society. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t, and to handle this difficulty, we use t-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

AB - Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t, and to handle this difficulty, we use t-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.

KW - Big Bang

KW - Fermi–Walker transport

KW - Hawking’s theorem

KW - Kasner solutions

KW - constant mean curvature

KW - curvature singularity

KW - geodesically incomplete

KW - maximal globally hyperbolic development

KW - singularity theorem

KW - stable blowup

KW - transported spatial coordinates

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U2 - 10.1090/jams/1015

DO - 10.1090/jams/1015

M3 - Article

AN - SCOPUS:85153755627

SN - 0894-0347

VL - 36

SP - 827

EP - 916

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

IS - 3

ER -