Determination of the spacetime from local time measurements |
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Authors: | Matti?Lassas Email authorView authors OrcID profile" target="_blank">Lauri?OksanenEmail authorView authors OrcID profile Yang?Yang |
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Institution: | 1.Department of Mathematics and Statistics,University of Helsinki,Helsinki,Finland;2.Department of Mathematics,University College London,London,UK;3.Department of Mathematics,Purdue University,West Lafayette,USA |
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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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