Global Regularity for Einstein-Klein-Gordon System with U(1) × R Isometry Group, I

Abstract

This is the first of the two papers devoted to the study of global regularity of the 3 + 1 dimensional Einstein-Klein-Gordon system with a U(1) × ℝ isometry group. In this first part, the authors reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2 + 1 dimensional system. Then, the authors will give energy estimates and construct the null coordinate system, under which the authors finally show that the first possible singularity can only occur at the axis.

Keywords

Einstein-Klein Gordon system / Cauchy problem / Energy estimates / Null coordinate / First singularity

References

[1] Alinhac S. Geometric Analysis of Hyperbolic Differential Equations: An Introduction, 2010, Cambridge: Cambridge University Press

[2] Andersson, L. Gudapati N. and Szeftel, J., Global regularity for the 2+1 dimensional equivariant Einsteinwave map system, Ann. PDE, 3, 2017, 142 pages.

[3] Berger B K, Chruściel P T, Moncrief V. On the #x201C;Asymptotically Flat” Space-Times with G2-invariant Cauchy Surfaces. Ann. Physics, 1995, 237: 322-354

[4] Choquet-Bruhat Y, Geroch R. Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys., 1969, 14: 329-335

[5] Choquet-Bruhat Y, Moncrief V. Future global in time Einsteinian spacetimes with U(1) isometry group. Ann. Henri Poincaré, 2001, 2: 1007-1064

[6] Choquet-Bruhat Y. General Relativity and the Einstein Equations, 2009, Oxford: Oxford Univeristy Press

[7] Christodoulu D. The problem of a self-gravitating scalar field. Comm. Math. Phys., 1986, 105: 337-361

[8] Christodoulou D. A mathematical theory of gravitational collapse. Comm. Math. Phys., 1987, 109: 613-647

[9] Christodoulou D. The formation of black holes and singularities in spherically symmetric gravitational collapse. Comm. Pure Appl. Math., 1991, 53(12): 1536-1602

[10] Christodoulou D. Bounded variation solutions of the spherically symmetric einstein-scalar field equations. Comm. Pure Appl. Math., 1993, 46(8): 1131-1220

[11] Christodoulu D, Tahvildar-Zadeh A S. On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math., 1993, 46: 1041-1091

[12] Christodoulou D, Klainerman S. The global nonlinear stability of the Minkowski space, 1993, Princeton: Princeton University Press

[13] Geba D, Grillakis M. An Introduction to the Theory of Wave Maps and Related Geometric Problems, 2017, New Jersey: World Scientific

[14] Grillakis, M., Classical slutions for the equivariant wave maps in 1+2 dimensions, Google scholar, 1991.

[15] Gudapati, N., A note on the dimensional reduction of axisymmetric spacetimes, arXiv:1702.07950, 2017.

[16] Huneau C. Constraint equations for 3+1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case. Ann. Henri Poincaré, 2016, 17(2): 271-299

[17] Huneau C. Constraint equations for 3+1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II. Asymptot. Anal., 2016, 96(1): 51-89

[18] Huneau, C., Stability in exponential time of Minkowski space-time with a translation space-like Killing field, Ann. PDE, 2 (1), 2016, 151 pages.

[19] Huneau, C., Stability of minkowski space-time with a translation space-like killing field, Ann. PDE, 4 (1), 2018, 147 pages.

[20] Ionescu, A. D. and Pausader, B., The Einstein-Klein-Gordon coupled system: global stability of the Minkowski solution, arXiv:1911.10652, 2019.

[21] Klainerman S, Nicolò F. The Evolution Problem in General Relativity, 2003, Boston: Birkhäuser

[22] LeFloch P, Ma Y. The global nonlinear stability of Minkowski space for self-gravitating massive fields. Comm. Math. Phys., 2016, 346(2): 603-665

[23] Lindblad H, Rodnianski I. Global Existence for the Einstein vacuum equations in wave coordinates. Comm. Math. Phys., 2005, 256: 43-110

[24] Lindblad H, Rodnianski I. The global stability of Minkowski space-time in harmonic gauge. Ann. Math. (2), 2010, 171(3): 1401-1477

[25] Luk, J., Oh, S.-J. and Yang, S., Solutions to the einstein-scalar-field system in spherical symmetry with large bounded variation norms. Ann. PDE 4(1), 2018, 59 pages.

[26] Moncrief V. Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups. Ann. Physics, 1986, 167: 118-142

[27] Alan D. Rendall, Partial Differential Equations in Genreal Relativity, 2008, Oxford: Oxford University Press

[28] Shatah J, Tahvildar-Zadeh A S. Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds. Comm. Pure Appl. Math, 1992, 45(8): 947-971

[29] Shatah J, Tahvildar-Zadeh A S. On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math., 1994, 47(5): 719-754

[30] Shatah J, Struwe M. Geometric Wave Equations, 1998, New York: Courant Institute of Mathematical Sciences

[31] Struwe M. Radially symmetric wave maps from (1+2)-dimensional Minkowski space to the sphere. Math. Z., 2002, 242: 407-414

[32] Struwe M. Equivariant wave maps in two space dimensions. Comm. Pure Appl. Math., 2003, 56: 0815-0823

[33] Struwe M. Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets. Calc. Var., 2003, 16: 431-437

[34] Wang Q. An intrinsic hyperboloid approach for Einstein Klein-Gordon equations. J. Differential Geom., 2020, 115(1): 27-109