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Determination of the spacetime from local time measurements

Lassas, M; Oksanen, L; Yang, Y; (2016) Determination of the spacetime from local time measurements. Mathematische Annalen , 365 (1) pp. 271-307. 10.1007/s00208-015-1286-9. Green open access

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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 Σ⊂M determine the C∞-jet of the metric in the Fermi coordinates associated to Σ. 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.

Type: Article
Title: Determination of the spacetime from local time measurements
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00208-015-1286-9
Publisher version: http://dx.doi.org/10.1007/s00208-015-1286-9
Language: English
Additional information: Copyright © Springer-Verlag Berlin Heidelberg 2015. The final publication is available at Springer via http://dx.doi.org/10.1007/s00208-015-1286-9
Keywords: Science & Technology, Physical Sciences, Mathematics, Killing Vector-fields, Boundary Rigidity, Riemannian-manifolds, Inverse Problem, Black-holes, Negative Curvature, Lens Rigidity, Spectral Data, Neumann Map, Equations
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/1498556
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