Initial data engineering: existence of asymptotically simple vacuum space-times, many Schwarzschild's, manifolds of solutions of constraint equations, and all that

Initial data engineering

Author: Piotr T. Chruściel, James Isenberg, Daniel Pollack
Comments: gr-qc/0403066, 19 pages
We present a local gluing construction for general relativistic initial data sets. The method applies to generic initial data, in a sense which is made precise. In particular the trace of the extrinsic curvature is not assumed to be constant near the gluing points, which was the case for previous such constructions. No global conditions on the initial data sets such as compactness, completeness, or asymptotic conditions are imposed. As an application, we prove existence of spatially compact, maximal globally hyperbolic, vacuum space-times without any closed constant mean curvature spacelike hypersurface.

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Gluing initial data sets for general relativity

Author: Piotr T. Chruściel, James Isenberg, Daniel Pollack
Comments: gr-qc/0407xxx, 9 pages, to appear in Physical Review Letters
We establish an optimal gluing construction for general relativistic initial dats sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.

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On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications

Author: Piotr T. Chruściel, Erwann Delay
Comments: gr-qc/0301073, 84 pages, Mem.Soc.Math.France 94 (2003) 1-103
Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global $\scri$; we prove existence of initial data for many black holes which are exactly Kerr -- or exactly Schwarzschild -- both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{-m}$ terms, with arbitrary $m$, and with multipole moments freely prescribable within certain ranges.

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On solutions of the vacuum Einstein equation in the radiation regime

Author: Piotr T. Chruściel

Comments: gr-qc/0210092, 6 pages, to appear in the proceedings of the Elba Conference in memory of A.Lichnerowicz, June 2002
We review recent results by the author, in collaboration with Erwann Delay, Olivier Lengard, and Rafe Mazzeo (the papers below or here), on existence and properties of space-times with controlled asymptotic behavior at null infinity.

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Existence of non-trivial, vacuum, asymptotically simple space-times

Authors: Piotr T. Chruściel, Erwann Delay
Comments: gr-qc/0203053, 11 pages, Latex2e, various style files

We construct non-trivial vacuum space-times with a global Scri. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon.

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Version published in Classical and Quantum Gravity 19, L71-L79 (2002) [ from the CQG web site] [local file]

[Erratum to the CQG published version]

Manifold structures for sets of solutions of the general relativistic constraint equations

Author: Piotr T. Chruściel, Erwann Delay
Comments: gr-qc/0309001 , 31 pages
We construct manifold structures on various sets of solutions of the general relativistic initial data sets.

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KIDs are non-generic

Authors: Robert Beig, Piotr T. Chruściel and Richard Schoen
Comments:[gr-qc/0403042], 37 pages

We prove that generic solutions of the vacuum constraint Einstein equations do not possess any global or local space-time Killing vectors, on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-existence of global symmetries implies, generically, non-existence of local ones. As part of the argument, we prove that generic metrics do not possess any local or global conformal Killing vectors.

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On "many black hole" vacuum space-times

Authors: Piotr T. Chruściel, Rafe Mazzeo
Comments:[gr-qc/0210103], 29 pages, Latex2e, various style files, several figures

We analyze the horizon structure of families of space times obtained by evolving initial data sets containing apparent horizons with several connected components. We show that under certain smallness conditions the outermost apparent horizons will also have several connected components. We further show that, again under a smallness condition, the maximal globally hyperbolic development of the many black hole initial data constructed in the letter with Erwann Delay above, or of glued hyperboloidal data of Isenberg, Mazzeo and Pollack, will have an event horizon, the intersection of which with the initial data hypersurface is not connected. This justifies the "many black hole" character of those space-times.

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