The Dirichlet-to-Neumann map for Lorentzian Calderón problems with data on disjoint sets

TL;DR

This work addresses the Lorentzian Calderón problem with partial boundary data: recovering the boundary geometry and magnetic potential from the restricted Dirichlet-to-Neumann map $\Lambda^{U,V}_{g,A,q}$ on disjoint boundary parts. The authors develop a microlocal framework treating the restricted DN map as an elliptic Fourier integral operator associated with broken null-bicharacteristics and gliding rays, enabling construction of solutions via optics and computation of principal symbols. They introduce the weak lens relation and prove its upgrade to the full lens relation near the glancing set, from which the conformal class of the boundary metric and the 1-form $A$ on boundary tangential directions are recovered, up to gauge. Furthermore, they establish conditions under which the conformal factor is locally constant on recoverable boundary regions and show how a priori knowledge of the metric or time orientation extends reconstruction to larger boundary portions, all while clarifying gauge obstructions. Overall, the results advance partial-data boundary determination for Lorentzian metrics, shedding light on rigidity and identifiability in the Lorentzian Calderón problem. $($In particular, the restricted DN map yields recoverable sets $\widetilde{U},\widetilde{V}$, the conformal class of $\bar{g}$ on these sets, a lens relation near the glancing set, and $A$ on boundary tangential directions$.)

Abstract

We consider the restricted Dirichlet-to-Neumann map $Λ^{U,V}_{g,A,q}$ for the wave equation with magnetic potential $A$ and scalar potential $q$, on an admissible Lorentzian manifold $(M, g)$ of dimension $n \geq 3$ with boundary. Here $U$ and $V$ are disjoint open subsets of $\partial M$, where we impose the Dirichlet data on $U$ and measure the Neumann-type data on $V$. We use the gliding rays and microlocal analysis to show that, without any a priori information, one can reconstruct the conformal class of the boundary metric $g|_{T\partial M \times T\partial M}$ and the magnetic potential $A|_{T\partial M}$ at recoverable boundary points from $Λ^{U,V}_{g,A,q}$. In particular, the conformal factor and the jet of the metric at those points are determined up to gauge transformations. Moreover, if the metric and the time orientation are known on $U$ (or $V$), then the metric on a larger portion of $V$ (or $U$) can be reconstructed, up to gauge.

Figures (2)

• Figure 1: If $(x', \xi')$ is past-pointing (left), then the unique forward null-bicharacteristic leaves from its outward preimage $(x, \xi_-)$ and travels forward in time; if $(x', \xi')$ is future-pointing (right), then the unique forward null-bicharacteristic leaves from its inward preimage $(x, \xi_+)$ and travels backward in time. In conclusion, the null-bicharacteristic is forward when it lies in the casual future of $x$, regardless of the flow direction.
• Figure 2: Dimension at least 4: in $\widetilde{U}$, we use that fact that $\varphi$ would stay constant along any piecewise boundary null-geodesic curve where each piece of boundary null-geodesic when fully extended passes through $V$. We show that the union of such paths from $x'$ has non-empty interior. Same argument works for $\widetilde{V}$.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3