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Determination of the spacetime from local time measurements

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Abstract

We consider an inverse problem for a Lorentzian spacetime (M, g), and show that time measurements, that is, the knowledge of the Lorentzian time separation function on a submanifold \(\Sigma \subset M\) determine the \(C^\infty \)-jet of the metric in the Fermi coordinates associated to \(\Sigma \). We use this result to study the global determination of the spacetime (M, g) when it has a real-analytic structure or is stationary and satisfies the Einstein-scalar field equations. In addition to this, we require that (M, g) is geodesically complete modulo scalar curvature singularities. The results are Lorentzian counterparts of extensively studied inverse problems in Riemannian geometry—the determination of the jet of the metric and the boundary rigidity problem. We give also counterexamples in cases when the assumptions are not valid, and discuss inverse problems in general relativity.

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Acknowledgments

The authors express their gratitude to the Mittag-Leffler Institute, where parts of this work have been done. The authors would like to thank Prof. Gunther Uhlmann for his generous support related to this work, and for suggesting the method used in the proof of Theorem 1. ML was partially supported by the Academy of Finland project 272312 and the Finnish Centre of Excellence in Inverse Problems Research 2012-2017. YY was partially supported by the NSF grants DMS 1265958 and DMS 1025372. LO was partially supported by the EPSRC grant EP/L026473/1.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Matti Lassas

  2. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

    Lauri Oksanen

  3. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Yang Yang

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  1. Matti Lassas
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  2. Lauri Oksanen
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Correspondence to Lauri Oksanen.

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Lassas, M., Oksanen, L. & Yang, Y. Determination of the spacetime from local time measurements. Math. Ann. 365, 271–307 (2016). https://doi.org/10.1007/s00208-015-1286-9

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  • DOI: https://doi.org/10.1007/s00208-015-1286-9

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