In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of spatially compact variants of the k = -1 Friedmann—Robertson—Walker vacuum spacetime. We use a special gauge defined by constant mean curvature slicing and a spatial harmonic coordinate condition, and develop energy estimates through the use of the Be-Robinson energy and its higher-order generalizations. In addition to the smallness condition on the data, we need a topological constraint on the spatial manifold to exclude the possibility of a non-trivial moduli space of flat spacetime perturbations, since the latter could not be controlled by curvature-based energies such as those of Bel—Robinson type. Our results also demonstrate causal geodesic completeness of the perturbed spacetimes (in the expanding direction) and establish precise rates of decay towards the background solution which serves as an attractor asymptotically.
Supported in part by the Swedish Natural Sciences Research Council (SNSRC), contract no. R-RA 4873–307 and NSF, contract no. DMS 0104402.
Supported in part by the NSF, with grants PHY-9732629 and PHY-0098084 to Yale University.
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Similar content being viewed by others ReferencesLars Andersson and Vincent Moncrief, Elliptic-hyperbolic systems and the Einstein equations, submitted to Ann. Henri Poincaré, 2001.
Lars Andersson, Vincent Moncrief, and Anthony J. Tromba, On the global evolution problem in 2 + 1 gravity, J. Geom. Phys. 23 (1997), no. 3–4, 191–205.
Yvonne Choquet-Bruhat, Future complete Einsteinian space times with U(1) symmetry, the unpolarized case, article in this volume.
Yvonne Choquet-Bruhat and Vincent Moncrief, Future global in time Einsteinian spacetimes with U(1) isometry group, Ann. Henri Poincaré 2 (2001), no. 6, 1007–1064.
Demetrios Christodoulou and Sergiu Klainerman, Asymptotic properties of linear field equations in Minkowski space, Comm Pure Appl. Math. 43 (1990), no. 2, 137–199.
Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton, NJ, 1993.
James Eells and Luc Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), no. 1, 1–68.
Arthur E. Fischer and Vincent Moncrief, Hamiltonian reduction of Einstein’s equations of general relativity, Nuclear Phys. B Proc. Suppl. 57 (1997), 142–161, Constrained dynamics and quantum gravity 1996 (Santa Margherita Ligure).
Arthur E., Hamiltonian reduction and perturbations of continuously self-similar (n 1)-dimensional Einstein vacuum spacetimes, Classical Quantum Gravity 19 (2002), no. 21, 5557–5589.
Helmut Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm Math. Phys. 107 (1986), no. 4, 587–609.
Michael Kapovich, Deformations of representations of discrete subgroups of SO(3, 1), Math. Ann. 299 (1994), no. 2, 341–354.
Jacques Lafontaine, Modules de structures conforrnes plates et cohomologie de groupes discrets, C.R. Acad. Sci. Paris Ser. I Math. 297 (1983), no. 13, 655–658.
Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA
Lars Andersson
Department of Physics, Yale University, P.O. Box 208120, New Haven, CT, 06520, USA
Vincent Moncrief
Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Avenue Grammont, 37200, Tours, France
Piotr T. Chruściel
Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14467, Golm, Germany
Helmut Friedrich
© 2004 Springer Basel AG
About this paper Cite this paperAndersson, L., Moncrief, V. (2004). Future Complete Vacuum Spacetimes. In: Chruściel, P.T., Friedrich, H. (eds) The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7953-8_8
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