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Weakly regular \(T^2\)-symmetric spacetimes. The global geometry of future Cauchy developments. (English) Zbl 1321.83010

In this paper, the authors examine \({\text{ U}}(1)\times{\text{ U}}(1)\) symmetric weakly regular \(3\) dimensional initial data sets for the vacuum Einstein’s equation. The authors prove a well-posedness theorem for initial data sets of this kind (Theorem 1.1) and an existence theorem for their solvability (Theorem 1.2). They also establish an appropriate uniqueness result (Theorem 8.3).
\({\text{ U}}(1)\times{\text{ U}}(1)\) symmetry means that the underlying smooth \(3\)-manifold admits a smooth fixed-point-free \(2\)-torus action and the initial data are invariant under this action. From this it is easy to show, and is assumed in the paper, that if the \(3\)-manifold is compact then it must be diffeomorphic to the \(3\)-torus. Weak regularity here roughly means that the function on the underlying \(3\)-manifold which assigns to a \(2\)-torus-orbit its volume is Lipschitz continuous moreover the coefficients of the tensor fields appearing in the initial data set belong to the Sobolev space \(H^1\) or have even lower regularity (for a precise definition of “weak regularity” in this case, cf. Section 2 of the paper).
The necessity of considering only weakly regular initial data sets (compared e.g. to smooth ones) is also motivated from a physical viewpoint in the text. By the non-linear and hyperbolic nature of the Einstein’s field equation, regular initial data often evolve into less regular ones (cf. the presence of singularities in impulsive gravitational wave space-times or the formation of shock waves in the case of relativistic fluids).

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35L05 Wave equation
83C40 Gravitational energy and conservation laws; groups of motions
53Z05 Applications of differential geometry to physics
83C35 Gravitational waves
83C75 Space-time singularities, cosmic censorship, etc.
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[1] Barnes, A. P., LeFloch, P. G., Schmidt, B. G., Stewart, J. M.: The Glimm scheme for perfect fluids on plane-symmetric Gowdy spacetimes. Class. Quantum Grav. 21, 5043-5074 (2004) · Zbl 1080.83011 · doi:10.1088/0264-9381/21/22/003
[2] Berger, B. K., Chruściel, P., Moncrief, V.: On asymptotically flat spacetimes with G2-invariant Cauchy surfaces. Ann. Phys. 237, 322-354 (1995) · Zbl 0967.83507 · doi:10.1006/aphy.1995.1012
[3] Berger, B. K., Chruściel, P., Isenberg, J., Moncrief, V.: Global foliations of vacuum spacetimes with T 2 isometry. Ann. Phys. 260, 117-148 (1997) · Zbl 0929.58013 · doi:10.1006/aphy.1997.5707
[4] Burtscher, A., LeFloch, P. G.: The formation of trapped surfaces in spherically-symmetric Einstein-Euler spacetimes with bounded variation. J. Math. Pures Appl. 102, 1164-1217 (2014) · Zbl 1302.83004 · doi:10.1016/j.matpur.2014.10.003
[5] Choquet-Bruhat, Y.: General Relativity and the Einstein Equations. Oxford Math. Mono- graphs, Oxford Univ. Press (2009) · Zbl 1157.83002
[6] Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14, 329-335 (1969) · Zbl 0182.59901 · doi:10.1007/BF01645389
[7] Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein- scalar field equations. Comm. Pure Appl. Math. 46, 1131-1220 (1992) · Zbl 0853.35122 · doi:10.1002/cpa.3160460803
[8] Chruściel, P.: On spacetimes with U (1) \times U (1) symmetric compact Cauchy surfaces. Ann. Phys. 202, 100-150 (1990) · Zbl 0727.53078 · doi:10.1016/0003-4916(90)90341-K
[9] Chruściel, P., Isenberg, J., Moncrief, V.: Strong cosmic censorship in polarized Gowdy space- times. Class. Quantum Grav. 7, 1671-1680 (1990) · Zbl 0703.53081 · doi:10.1088/0264-9381/7/10/003
[10] Eardley, D., Moncrief, V.: The global existence problem and cosmic censorship in general relativity. Gen. Relat. Grav. 13, 887-892 (1981)
[11] Foures-Bruhat, Y.: Théor‘emes d’existence pour certains syst‘emes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141-225 (1952) · Zbl 0049.19201 · doi:10.1007/BF02392131
[12] Gowdy, R.: Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: topologies and boundary conditions. Ann. Phys. 83, 203-241 (1974) · Zbl 0325.53061 · doi:10.1016/0003-4916(74)90384-4
[13] Grubic, N., LeFloch, P. G.: Weakly regular Einstein-Euler spacetimes with Gowdy symmetry. The global areal foliation. Arch. Ration. Mech. Anal. 2008, 391-428 (2013) · Zbl 1266.83027 · doi:10.1007/s00205-012-0597-1
[14] Grubic, N., LeFloch, P. G.: On the area of the symmetry orbits in weakly regular Einstein- Euler spacetimes with Gowdy symmetry. SIAM J. Math. Anal. 47, 669-683 (2015) · Zbl 1320.83022 · doi:10.1137/130950641
[15] Hughes, T. J. R., Kato, T., Marsden, J. E.: Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal. 63, 273-294 (1976) · Zbl 0361.35046 · doi:10.1007/BF00251584
[16] Isenberg, J., Moncrief, V.: Asymptotic behavior of the gravitational field and the nature of singularities in Gowdy spacetimes. Ann. Phys. 99, 84-122 (1990) · Zbl 0723.53061 · doi:10.1016/0003-4916(90)90369-Y
[17] Isenberg, J., Weaver, M.: On the area of the symmetry orbits in T 2-symmetric spacetimes. Class. Quantum Grav. 20, 3783-3796 (2003) · Zbl 1049.83004 · doi:10.1088/0264-9381/20/16/316
[18] Klainerman, S., Rodnianski, I.: Rough solutions of Einstein vacuum equations. Ann. of Math. 161, 1143-1193 (2005) · Zbl 1089.83006 · doi:10.4007/annals.2005.161.1143
[19] Klainerman, S., Rodnianski, I.: A Kirchhoff-Sobolev parametrix for the wave equation and applications. J. Hyperbolic Differential Equations 4, 401-433 (2007) · Zbl 1148.35042 · doi:10.1142/S0219891607001203
[20] Klainerman, S., Rodnianski, I., Szeftel, J.: The bounded L2 curvature conjecture. Princeton Math. Ser. 50, 224-244 (2014) · Zbl 1304.53072
[21] Lee, J. M.: Introduction to Smooth Manifolds. Grad. Texts in Math. 218, Springer, New York (2003) · Zbl 1258.53002 · doi:10.1007/978-1-4419-9982-5
[22] LeFloch, P. G., Mardare, C.: Definition and weak stability of spacetimes with distributional curvature. Portugal. Math. 64, 535-573 (2007) · Zbl 1144.35311 · doi:10.4171/PM/1794
[23] LeFloch, P. G., Rendall, A.: A global foliation of Einstein-Euler spacetimes with Gowdy symmetry on T 3. Arch. Ration. Mech. Anal. 201, 841-870 (2011) · Zbl 1256.35165 · doi:10.1007/s00205-011-0425-z
[24] LeFloch, P. G., Smulevici, J.: Global geometry of T 2-symmetric spacetimes with weak regu- larity. C. R. Math. Acad. Sci. Paris 348, 1231-1233 (2010) · Zbl 1205.83017 · doi:10.1016/j.crma.2010.09.009
[25] LeFloch, P. G., Smulevici, J.: Weakly regular T 2-symmetric spacetimes. The future causal geometry of Gowdy spaces. · Zbl 1331.35341
[26] LeFloch, P. G., Smulevici, J.: Future asymptotics and geodesic completeness of polarized T 2-symmetric spacetimes. Anal. PDE, submitted · Zbl 1331.35341
[27] LeFloch, P. G., Stewart, J. M.: Shock waves and gravitational waves in matter spacetimes with Gowdy symmetry. Portugal. Math. 62, 349-370 (2005)
[28] LeFloch, P. G., Stewart, J. M.: The characteristic initial value problem for plane- symmetric spacetimes with weak regularity. Class. Quantum Grav. 28, 145019-145035 (2011) · Zbl 1222.83028 · doi:10.1088/0264-9381/28/14/145019
[29] Moncrief, V.: Global properties of Gowdy spacetimes with T 3 \times R topology. Ann. Phys. 132, 87-107 (1981)
[30] Rendall, A. D.: Crushing singularities in spacetimes with spherical, plane, and hyperbolic symmetry. Class. Quantum Grav. 12, 1517-1533 (1995) · Zbl 0823.53072 · doi:10.1088/0264-9381/12/6/017
[31] Rendall, A. D.: Existence of constant mean curvature foliations in spacetimes with two- dimensional local symmetry. Comm. Math. Phys. 189, 145-164 (1997) · Zbl 0897.53049 · doi:10.1007/s002200050194
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