Abstract
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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Andersson, 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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DOI: https://doi.org/10.1007/978-3-0348-7953-8_8
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