The Dirichlet-to-Neumann map for Lorentzian Calder\'on problems with data on disjoint sets

We consider the restricted Dirichlet-to-Neumann map $\Lambda^{U,V}_{g,A,q}$ 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 $\Lambda^{U,V}_{g,A,q}$. 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.