Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations

We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are: 1. Passive observations in spacetime: consider observations in an open set \(V{\subset } M\). The light observation set corresponding to a point source at \(q\in M\) is the intersection of V and the light-cone emanating from the point q. Let \(W\subset M\) be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points \(q\in W\) determine uniquely the conformal type of W. 2. Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path \(\mu \) and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from \(\mu \) and return back to \(\mu \).