Global recovery of Lorentzian principal geometry from the hyperbolic Dirichlet-to-Neumann map
Abstract
We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric Gg,c=−c(x,t)2dt2+g(x) encoded by the principal symbol of the wave operator. The result holds in spatial dimensions n≥3 for multiplicatively separable wave speeds c(x,t)=a(x)b(t), provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric g(x) and the wave speed c(x,t) throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor b is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.
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BibTeX
@misc{chen2026global,
title = {{Global recovery of Lorentzian principal geometry from the hyperbolic Dirichlet-to-Neumann map}},
author = {Lu Chen and Yan Jiang and Hongyu Liu and Longyue Tao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29332},
eprint = {2608.29332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29332},
}
RIS
TY - UNPB
TI - Global recovery of Lorentzian principal geometry from the hyperbolic Dirichlet-to-Neumann map
AU - Lu Chen
AU - Yan Jiang
AU - Hongyu Liu
AU - Longyue Tao
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29332
UR - https://arxiv.org/abs/2608.29332
AB - We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric G_g,c=−c(x,t)^2dt^2+g(x) encoded by the principal symbol of the wave operator. The result holds in spatial dimensions n≥3 for multiplicatively separable wave speeds c(x,t)=a(x)b(t), provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric g(x) and the wave speed c(x,t) throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor b is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.
ER -