Abstract
We study the light ray transform on Minkowski space-time and its small metric perturbations acting on scalar functions which are solutions to wave equations. We show that the light ray transform uniquely determines the function in a stable way. The problem is of particular interest because of its connection to inverse problems of the Sachs–Wolfe effect in cosmology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics, vol. 36. Cummings Publishing Company, Reading, MA (1978)
Dodelson, S.: Modern Cosmology. Academic Press, Amsterdam (2003)
Duistermaat, J.J.: Fourier Integral Operators, vol. 130. Springer, Berlin (1995)
Durrer, R.: The Cosmic Microwave Background. Cambridge University Press, Cambridge (2008)
Greenleaf, A., Uhlmann, G.: Nonlocal inversion formulas for the X-ray transform. Duke Math. J. 58(1), 205–240 (1989)
Guillemin, V.: On some results of Gelfand in integral geometry. Proc. Symp. Pure Math. 43, 149–155 (1985)
Guillemin, V.: Cosmology in \((2+ 1)\)-dimensions, cyclic models, and deformations of \(M_{2, 1}\). No. 121. Princeton University Press (1989)
Hörmander, L.: The analysis of linear partial differential operators IV: Fourier integral operators. Classics in Mathematics (2009)
Hörmander, L.: Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26. Springer, Berlin (1997)
Ilmavirta, J.: X-ray transforms in pseudo-Riemannian geometry. J. Geom. Anal. 28(1), 606–626 (2018)
Krauss, L.M., Dodelson, S., Meyer, S.: Primordial gravitational waves and cosmology. Science 328(5981), 989–992 (2010)
Lassas, M., Oksanen, L., Stefanov, P., Uhlmann, G.: On the inverse problem of finding cosmic strings and other topological defects. Commun. Math. Phys. 357(2), 569–595 (2018)
Lassas, M., Oksanen, L., Stefanov, P., Uhlmann, G.: The light ray transform on Lorentzian manifolds. Commun. Math. Phys. 377, 1–31 (2020)
Manzotti, A., Dodelson, S.: Mapping the integrated Sachs-Wolfe effect. Phys. Rev. D 90(12), 123009 (2014)
Melrose, R.B.: Geometric Scattering Theory, vol. 1. Cambridge University Press, Cambridge (1995)
Mukhanov, V.F., Feldman, H.A., Brandenberger, R.H.: Theory of cosmological perturbations. Phys. Rep. 215(5–6), 203–333 (1992)
Stefanov, P., Uhlmann, G.: Microlocal analysis and integral geometry. Book in progress
Stefanov, P., Uhlmann, G.: Inverse backscattering for the acoustic equation. SIAM J. Math. Anal. 28(5), 1191–1204 (1997)
Sachs, R.K., Wolfe, A.M.: Perturbations of a cosmological model and angular variations of the microwave background. Astrophys. J. 147, 73 (1967)
Sharafutdinov, V.A.: Integral Geometry of Tensor Fields, vol. 1. Walter de Gruyter, Berlin (2012)
Trèves, F.: Introduction to pseudodifferential and Fourier integral operators. Volume 2: Fourier integral operators. Springer, Berlin (1980)
Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov). Inventiones Mathematicae 194(2), 381–513 (2013)
Wang, Y.: Parametrices for the light ray transform on Minkowski spacetime. Inverse Probl. Imaging 12(1), 229–237 (2018)
Acknowledgements
The authors thank Plamen Stefanov and Gunther Uhlmann for helpful discussions. They also thank the anonymous referees for valuable comments. A.V. acknowledges support from the National Science Foundation under Grant No. DMS-1664683.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P.Chrusciel.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Vasy, A., Wang, Y. On the Light Ray Transform of Wave Equation Solutions. Commun. Math. Phys. 384, 503–532 (2021). https://doi.org/10.1007/s00220-021-04045-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-021-04045-7