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Localizing solutions of the Einstein constraint equations

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Abstract

We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of N-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time \(t\in (0,T)\), in striking contrast with the Newtonian gravity scenario.

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Notes

  1. This relies on the main theorem in [20], due to the fact that \(\mathbb {R}^n\) is a spin manifold for any \(n\ge 3\).

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Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referess for carefully proofreading the article and for suggesting a number of changes which significantly contributed to the improvement of this final version. We are also indebted to Christos Mantoulidis for the figures that appear in the article. During the preparation of this work, the authors were partially supported by NSF Grant DMS-1105323.

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Authors and Affiliations

  1. ETH-Institute for Theoretical Studies, ETH, Zürich, Switzerland

    Alessandro Carlotto

  2. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Richard Schoen

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  1. Alessandro Carlotto
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  2. Richard Schoen
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Correspondence to Alessandro Carlotto.

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Carlotto, A., Schoen, R. Localizing solutions of the Einstein constraint equations. Invent. math. 205, 559–615 (2016). https://doi.org/10.1007/s00222-015-0642-4

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  • DOI: https://doi.org/10.1007/s00222-015-0642-4

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