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

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
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[1] Ames, Ellery; Beyer, Florian; Isenberg, James; LeFloch, Philippe G., Quasilinear hyperbolic {F}uchsian systems and {AVTD} behavior in {\(T^2\)}-symmetric vacuum spacetimes, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 14, 1445-1523, (2013) · Zbl 1272.83009 · doi:10.1007/s00023-012-0228-2
[2] Isenberg, James; Moncrief, Vincent, Asymptotic behavior of the gravitational field and the nature of singularities in {G}owdy spacetimes, Ann. Physics. Annals of Physics, 199, 84-122, (1990) · Zbl 0723.53061 · doi:10.1016/0003-4916(90)90369-Y
[3] Ames, Ellery; Beyer, Florian; Isenberg, James; LeFloch, Philippe G., Quasilinear symmetric hyperbolic {F}uchsian systems in several space dimensions. Complex Analysis and Dynamical Systems {V}, Contemp. Math., 591, 25-43, (2013) · Zbl 1320.35193 · doi:10.1090/conm/591/11824
[4] Andersson, Lars; Moncrief, Vincent, Elliptic-hyperbolic systems and the {E}instein equations, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 4, 1-34, (2003) · Zbl 1138.81306 · doi:10.1007/s00023-003-0120-1
[5] Andersson, Lars; Moncrief, Vincent, Future complete vacuum spacetimes. The {E}instein Equations and the Large Scale Behavior of Gravitational Fields, 299-330, (2004) · Zbl 1105.83001 · doi:10.1007/978-3-0348-7953-8_8
[6] Andersson, Lars; Moncrief, Vincent, Einstein spaces as attractors for the {E}instein flow, J. Differential Geom.. Journal of Differential Geometry, 89, 1-47, (2011) · Zbl 1256.53035 · doi:10.4310/jdg/1324476750
[7] Andersson, Lars; Moncrief, Vincent; Tromba, Anthony J., On the global evolution problem in {\(2+1\)} gravity, J. Geom. Phys.. Journal of Geometry and Physics, 23, 191-205, (1997) · Zbl 0898.58003 · doi:10.1016/S0393-0440(97)87804-7
[8] Anguige, Keith, A class of plane symmetric perfect-fluid cosmologies with a {K}asner-like singularity, Classical Quantum Gravity. Classical and Quantum Gravity, 17, 2117-2128, (2000) · Zbl 0967.83041 · doi:10.1088/0264-9381/17/10/306
[9] Andersson, Lars; Rendall, Alan D., Quiescent cosmological singularities, Comm. Math. Phys.. Communications in Mathematical Physics, 218, 479-511, (2001) · Zbl 0979.83036 · doi:10.1007/s002200100406
[10] Anguige, Keith; Tod, K. P., Isotropic cosmological singularities. {I}. {P}olytropic perfect fluid spacetimes, Ann. Physics. Annals of Physics, 276, 257-293, (1999) · Zbl 1003.83027 · doi:10.1006/aphy.1999.5946
[11] Barrow, John D., Quiescent cosmology, Nature. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 272, 211-215, (1978) · doi:10.1038/272211a0
[12] Bartnik, Robert, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 39, 661-693, (1986) · Zbl 0598.53045 · doi:10.1002/cpa.3160390505
[13] Balakrishna, Jayashree; Daues, Gregory; Seidel, Edward; Suen, Wai-Mo; Tobias, Malcolm; Wang, Edward, Coordinate conditions in three-dimensional numerical relativity, Classical Quantum Gravity. Classical and Quantum Gravity, 13, L135-L142, (1996) · Zbl 0875.83013 · doi:10.1088/0264-9381/13/12/001
[14] Belinski\u\i, V. A.; Khalatnikov, I. M., Effect of scalar and vector fields on the nature of the cosmological singularity, \v Z. \`Eksper. Teoret. Fiz.. \v Z. \`Eksper. Teoret. Fiz., 63, 1121-1134, (1972)
[15] Lifshitz, E. M.; Khalatnikov, I. M., Investigations in relativistic cosmology, Advances in Phys.. Advances in Physics, 12, 185-249, (1963) · Zbl 0112.44306
[16] Beyer, Florian; LeFloch, Philippe G., Second-order hyperbolic {F}uchsian systems. {G}eneral theory, (2010) · Zbl 1206.83025
[17] Beyer, Florian; LeFloch, Philippe G., Second-order hyperbolic {F}uchsian systems and applications, Classical Quantum Gravity. Classical and Quantum Gravity, 27, 245012-33, (2010) · Zbl 1206.83025 · doi:10.1088/0264-9381/27/24/245012
[18] Four\`“es-Bruhat, Y., Th\'”eor\`“eme d”existence pour certains syst\`“emes d”\'equations aux d\'eriv\'ees partielles non lin\'eaires, Acta Math.. Acta Mathematica, 88, 141-225, (1952) · Zbl 0049.19201 · doi:10.1007/BF02392131
[19] Choquet-Bruhat, Yvonne; Geroch, Robert, Global aspects of the {C}auchy problem in general relativity, Comm. Math. Phys.. Communications in Mathematical Physics, 14, 329-335, (1969) · Zbl 0182.59901 · doi:10.1007/BF01645389
[20] Choquet-Bruhat, Y.; Isenberg, J.; Moncrief, V., Topologically general {U}(1) symmetric vacuum space-times with {AVTD} behavior, Nuovo Cimento Soc. Ital. Fis. B. Il Nuovo Cimento della Societ\`a Italiana di Fisica B, 119, 625-638, (2004) · doi:10.1393/ncb/i2004-10174-x
[21] Chru\'sciel, Piotr T.; Galloway, Gregory J.; Pollack, Daniel, Mathematical general relativity: {A} sampler, Bull. Amer. Math. Soc. (N.S.). American Mathematical Society. Bulletin. New Series, 47, 567-638, (2010) · Zbl 1205.83002 · doi:10.1090/S0273-0979-2010-01304-5
[22] Christodoulou, Demetrios, Global existence of generalized solutions of the spherically symmetric {E}instein-scalar equations in the large, Comm. Math. Phys.. Communications in Mathematical Physics, 106, 587-621, (1986) · Zbl 0613.53047 · doi:10.1007/BF01463398
[23] Christodoulou, Demetrios, The problem of a self-gravitating scalar field, Comm. Math. Phys., 105, 337-361, (1986) · Zbl 0608.35039
[24] Christodoulou, Demetrios, The structure and uniqueness of generalized solutions of the spherically symmetric {E}instein-scalar equations, Comm. Math. Phys.. Communications in Mathematical Physics, 109, 591-611, (1987) · Zbl 0613.53048 · doi:10.1007/BF01208959
[25] Christodoulou, Demetrios, The formation of black holes and singularities in spherically symmetric gravitational collapse, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 44, 339-373, (1991) · Zbl 0728.53061 · doi:10.1002/cpa.3160440305
[26] Christodoulou, Demetrios, Bounded variation solutions of the spherically symmetric {E}instein-scalar field equations, Comm. Pure Appl. Math.. Communications on Pure and Applied Mathematics, 46, 1131-1220, (1993) · Zbl 0853.35122 · doi:10.1002/cpa.3160460803
[27] Christodoulou, Demetrios, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. (2). Annals of Mathematics. Second Series, 149, 183-217, (1999) · Zbl 1126.83305 · doi:10.2307/121023
[28] Chru\'sciel, Piotr T.; Isenberg, James; Moncrief, Vincent, Strong cosmic censorship in polarised {G}owdy spacetimes, Classical Quantum Gravity. Classical and Quantum Gravity, 7, 1671-1680, (1990) · Zbl 0703.53081
[29] Claudel, Clarissa M.; Newman, Kerri P., The {C}auchy problem for quasi-linear hyperbolic evolution problems with a singularity in the time, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.. The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences, 454, 1073-1107, (1998) · Zbl 0916.35064 · doi:10.1098/rspa.1998.0197
[30] Damour, T.; Henneaux, M.; Rendall, A. D.; Weaver, M., Kasner-like behaviour for subcritical {E}instein-matter systems, Ann. Henri Poincar\'e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 3, 1049-1111, (2002) · Zbl 1011.83038 · doi:10.1007/s000230200000
[31] Demaret, Jacques; Henneaux, Marc; Spindel, Philippe, Non-oscillatory behaviour in vacuum {K}aluza-{K}lein cosmologies, Phys. Lett. B. Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, 164, 27-30, (1985) · doi:10.1016/0370-2693(85)90024-3
[32] Fischer, Arthur E.; Moncrief, Vincent, Reducing {E}instein’s equations to an unconstrained {H}amiltonian system on the cotangent bundle of {T}eichm\"uller space. Physics on Manifolds, Math. Phys. Stud., 15, 111-151, (1994) · Zbl 0846.58013
[33] Fischer, Arthur E.; Moncrief, Vincent, Hamiltonian reduction of {E}instein’s equations of general relativity, Nuclear Phys. B Proc. Suppl.. Nuclear Physics B. Proceedings Supplement, 57, 142-161, (1997) · Zbl 0976.83500 · doi:10.1016/S0920-5632(97)00363-0
[34] Fischer, Arthur E.; Moncrief, Vincent, Hamiltonian reduction of {E}instein’s equations and the geometrization of three-manifolds. International {C}onference on {D}ifferential {E}quations, {V}ol. 1, 2, 279-282, (2000) · Zbl 0966.37047
[35] Fischer, Arthur E.; Moncrief, Vincent, The reduced {H}amiltonian of general relativity and the {\( \sigma \)}-constant of conformal geometry. Mathematical and Quantum Aspects of Relativity and Cosmology, Lecture Notes in Phys., 537, 70-101, (2000) · Zbl 0996.83011 · doi:10.1007/3-540-46671-1_4
[36] Fischer, Arthur E.; Moncrief, Vincent, The reduced {E}instein equations and the conformal volume collapse of 3-manifolds, Classical Quantum Gravity. Classical and Quantum Gravity, 18, 4493-4515, (2001) · Zbl 0998.83008 · doi:10.1088/0264-9381/18/21/308
[37] Fischer, Arthur E.; Moncrief, Vincent, Hamiltonian reduction and perturbations of continuously self-similar {\((n+1)\)}-dimensional {E}instein vacuum spacetimes, Classical Quantum Gravity. Classical and Quantum Gravity, 19, 5557-5589, (2002) · Zbl 1028.83009 · doi:10.1088/0264-9381/19/21/318
[38] Friedrich, Helmut, The conformal structure of {E}instein’s field equations. Conformal Groups and Related Symmetries: Physical Results and Mathematical Background, Lecture Notes in Phys., 261, 152-161, (1986) · doi:10.1007/3540171630_78
[39] Friedrich, Helmut, Conformal {E}instein evolution. The Conformal Structure of Space-Time, Lecture Notes in Phys., 604, 1-50, (2002) · Zbl 1054.83006 · doi:10.1007/3-540-45818-2_1
[40] Garfinkle, David; Gundlach, Carsten, Well-posedness of the scale-invariant tetrad formulation of the vacuum {E}instein equations, Classical Quantum Gravity. Classical and Quantum Gravity, 22, 2679-2686, (2005) · Zbl 1074.83005 · doi:10.1088/0264-9381/22/13/011
[41] Gundlach, Carsten; Mart\'{\i}n-Garc\'{\i}a, Jos\'e M., Gauge-invariant and coordinate-independent perturbations of stellar collapse: the interior, Phys. Rev. D (3). Physical Review. D. Third Series, 61, 084024-17, (2000) · doi:10.1103/PhysRevD.61.084024
[42] Hawking, S. W., The occurrence of singularities in cosmology. {II}, Proc. Roy. Soc. Ser. A, 295, 490-493, (1966) · Zbl 0148.46504 · doi:10.1098/rspa.1966.0255
[43] Isenberg, James; Kichenassamy, Satyanad, Asymptotic behavior in polarized {\(T^2\)}-symmetric vacuum space-times, J. Math. Phys.. Journal of Mathematical Physics, 40, 340-352, (1999) · Zbl 1061.83512 · doi:10.1063/1.532775
[44] Witten, Edward, A new proof of the positive energy theorem, Comm. Math. Phys.. Communications in Mathematical Physics, 80, 381-402, (1981) · Zbl 1051.83532 · doi:10.1007/BF01208277
[45] Kichenassamy, Satyanad; Rendall, Alan D., Analytic description of singularities in {G}owdy spacetimes, Classical Quantum Gravity. Classical and Quantum Gravity, 15, 1339-1355, (1998) · Zbl 0949.83050 · doi:10.1088/0264-9381/15/5/016
[46] {Luk}, J., {\it J. Amer. Math. Soc.} (2017), 63pp., published electronically: September 27, 2017
[47] Moncrief, Vincent, Reduction of the {E}instein equations in {\(2+1\)} dimensions to a {H}amiltonian system over {T}eichm\"uller space, J. Math. Phys.. Journal of Mathematical Physics, 30, 2907-2914, (1989) · Zbl 0704.53076 · doi:10.1063/1.528475
[48] Moncrief, Vincent, How solvable is {\((2+1)\)}-dimensional {E}instein gravity?, J. Math. Phys.. Journal of Mathematical Physics, 31, 2978-2982, (1990) · Zbl 0732.53069 · doi:10.1063/1.528950
[49] Newman, Richard P. A. C., On the structure of conformal singularities in classical general relativity, Proc. Roy. Soc. London Ser. A. Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 443, 473-492, (1993) · Zbl 0810.53085 · doi:10.1098/rspa.1993.0158
[50] Newman, Richard P. A. C., On the structure of conformal singularities in classical general relativity. {II}. {E}volution equations and a conjecture of {K}. {P}. {T}od, Proc. Roy. Soc. London Ser. A. Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 443, 493-515, (1993) · Zbl 0810.53086 · doi:10.1098/rspa.1993.0159
[51] Penrose, Roger, Gravitational collapse and space-time singularities, Phys. Rev. Lett.. Physical Review Letters, 14, 57-59, (1965) · Zbl 0125.21206 · doi:10.1103/PhysRevLett.14.57
[52] Rein, Gerhard, Cosmological solutions of the {V}lasov-{E}instein system with spherical, plane, and hyperbolic symmetry, Math. Proc. Cambridge Philos. Soc.. Mathematical Proceedings of the Cambridge Philosophical Society, 119, 739-762, (1996) · Zbl 0851.53058 · doi:10.1017/S0305004100074569
[53] Reiris, Martin, On the asymptotic spectrum of the reduced volume in cosmological solutions of the {E}instein equations, Gen. Relativity Gravitation. General Relativity and Gravitation, 41, 1083-1106, (2009) · Zbl 1177.83026 · doi:10.1007/s10714-008-0693-6
[54] Rendall, Alan D., Fuchsian analysis of singularities in {G}owdy spacetimes beyond analyticity, Classical Quantum Gravity. Classical and Quantum Gravity, 17, 3305-3316, (2000) · Zbl 0967.83021 · doi:10.1088/0264-9381/17/16/313
[55] Rendall, Alan D., Theorems on Existence and Global Dynamics for the {E}instein Equations, Living Reviews in Relativity, 8, 6 pp. pp., (2005) · Zbl 1316.83008 · doi:10.12942/lrr-2005-6
[56] Ringstr\`“om, Hans, The {B}ianchi {IX} attractor, Ann. Henri Poincar\'”e. Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics, 2, 405-500, (2001) · Zbl 0985.83002 · doi:10.1007/PL00001041
[57] Ringstr{\"o}m, Hans, Strong cosmic censorship in {\(T^3\)}-{G}owdy spacetimes, Ann. of Math. (2). Annals of Mathematics. Second Series, 170, 1181-1240, (2009) · Zbl 1206.83115 · doi:10.4007/annals.2009.170.1181
[58] Hans Ringstr{\"o}m, Cosmic Censorship for {G}owdy Spacetimes, Living Reviews in Relativity, 13, 1215-83008, (2010) · Zbl 1215.83008 · doi:10.12942/lrr-2010-2
[59] Rodnianski, Igor; Speck, Jared, The nonlinear future stability of the {FLRW} family of solutions to the irrotational {E}uler-{E}instein system with a positive cosmological constant, J. Eur. Math. Soc. (JEMS). Journal of the European Mathematical Society (JEMS), 15, 2369-2462, (2013) · Zbl 1294.35164 · doi:10.4171/JEMS/424
[60] Igor Rodnianski; Jared Speck, Stable {Big} {Bang} Formation in Near-{FLRW} Solutions to the {Einstein}-Scalar Field and {Einstein}-Stiff Fluid Systems, (2014) · Zbl 1402.83120
[61] Shao, Arick, Breakdown criteria for nonvacuum {E}instein equations, 262 pp., (2010) · Zbl 1215.83020
[62] St{\aa}hl, Fredrik, Fuchsian analysis of {\(S^2\times S^1\)} and {\(S^3\)} {G}owdy spacetimes, Classical Quantum Gravity. Classical and Quantum Gravity, 19, 4483-4504, (2002) · Zbl 1028.83014 · doi:10.1088/0264-9381/19/17/301
[63] Schoen, Richard; Yau, Shing Tung, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys.. Communications in Mathematical Physics, 65, 45-76, (1979) · Zbl 0405.53045 · doi:10.1007/BF01940959
[64] Schoen, Richard; Yau, Shing Tung, Proof of the positive mass theorem. {II}, Comm. Math. Phys.. Communications in Mathematical Physics, 79, 231-260, (1981) · Zbl 0494.53028 · doi:10.1007/BF01942062
[65] Taylor, Michael E., Partial Differential Equations. {III}. Nonlinear Equations, Appl. Math. Sci., 117, xxii+608 pp., (1997) · Zbl 1206.35004 · doi:10.1007/978-1-4419-7049-7
[66] Tod, K. P., Isotropic singularities and the {\( \gamma=2\)} equation of state, Classical Quantum Gravity. Classical and Quantum Gravity, 7, L13-L16, (1990) · Zbl 0681.53062
[67] Tod, K. P., Isotropic singularities and the polytropic equation of state, Classical Quantum Gravity. Classical and Quantum Gravity, 8, L77-L82, (1991) · Zbl 0718.53062
[68] Tod, K. Paul, Isotropic cosmological singularities. The Conformal Structure of Space-Time, Lecture Notes in Phys., 604, 123-134, (2002) · Zbl 1042.83029 · doi:10.1007/3-540-45818-2_6
[69] Uggla, Claes; van Elst, Henk; Wainwright, John; Ellis, George F. R., Past attractor in inhomogeneous cosmology, Phys. Rev. D, 68, 10, 22 pp., (2003) · doi:10.1103/PhysRevD.68.103502
[70] Wald, Robert M., General Relativity, xiii+491 pp., (1984) · Zbl 0549.53001 · doi:10.7208/chicago/9780226870373.001.0001
[71] Dynamical Systems in Cosmology, 343 pp. pp., (1997)
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