The Light Ray Transform on Lorentzian Manifolds
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Communications in
Mathematical Physics
The Light Ray Transform on Lorentzian Manifolds Matti Lassas1 , Lauri Oksanen2 , Plamen Stefanov3 , Gunther Uhlmann4,5 1 Department of Mathematics and Statistics, University of Helsinki, Box 68, 00014 Helsinki, Finland 2 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 3 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
E-mail: [email protected]
4 Department of Mathematics, University of Washington, Seattle, WA 98195, USA 5 IAS, HKUST, Clear Water Bay, Hong Kong, China
Received: 20 July 2019 / Accepted: 27 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We study the weighted light ray transform L of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function f from its the weighted light ray transform L f by a suitable filtered back-projection. 1. Introduction Let g be a Lorentzian metric with signature (−, +, . . . , +) on the manifold M of dimension 1 + n, n ≥ 2. We study the weighted Light Ray Transform L κ f (γ ) = κ(γ (s), γ˙ (s)) f (γ (s)) ds, (1.1) R
of functions (or distributions) over light-like geodesics γ (s), known also as null geodesics. There is no canonical unit speed parameterization as in the Riemannian case as discussed below, and we have some freedom to chose parameterizations locally by smooth changes of the variables. We are interested in microlocal invertibility of L κ , that is, the description of which part of the singularities of the function f can be reconstructed in a stable say when L κ f is given. Observe that this property does not depend on the parameterization. Here κ is a weight function, positively homogeneous in its second variable of degree zero, which makes it parameterization independent. When κ = 1, we use the notation L. This transform appears in the study of hyperbolic equations when we ML partly supported by Academy of Finland, Grants 273979, 284715, 312110, 314879 and the AtMath project of UH. LO partly supported by EPSRC Grant EP/P01593X/1 and EP/R002207/1. PS partly supported by NSF Grants DMS-1600327 and DMS-1900475. GU was partly supported by NSF a Walker Family Endowed Professorship at UW ad a Si-Yuan Professorship at HKUST.
M. Lassas, L. Oksanen, P. Stefanov, G. Uhlmann
want to recover a potential term, or other coefficients of the equation, from boundary or scattering information, see, e.g., [1,4,21–24,32,33,35,36,43,45,46] for time dependent coefficients or in Lorentzian setting, and also [2,26] for time-independent ones. This problem arises in medical ultrasound tomography (see Sect. 5 on applications for the details). The tensorial version of inverse problem for the weighted Light Ray Transform arises in the recovery of first order perturbations [43] and in linearized problem of recovery a Lorentzian metric from remote measurements [24]. The lat
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