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:

Passive observations in spacetime: Consider observations in a neighborhood $V \subset M$ of a time-like geodesic $μ$ . Under natural causality conditions, we reconstruct the conformal type of the unknown open, relatively compact set $W \subset M$ , when we are given $V$ , the conformal class of $g ∣_{V}$ , and the light observations sets $P_{V} (q)$ corresponding to all source points $q$ in $W$ . The light observation set $P_{V} (q)$ is the intersection of $V$ and the light-cone emanating from the point $q$ , i.e., the points in the set $V$ where light from a point source at $q$ is observed.
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 geodesic $μ$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$ . 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 $μ$ and return back to $μ$ .

We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M, g)$ . The problems are:

Passive observations in spacetime: Consider observations in a neighborhood $V \subset M$ of a time-like geodesic $μ$ . Under natural causality conditions, we reconstruct the conformal type of the unknown open, relatively compact set $W \subset M$ , when we are given $V$ , the conformal class of $g ∣_{V}$ , and the light observations sets $P_{V} (q)$ corresponding to all source points $q$ in $W$ . The light observation set $P_{V} (q)$ is the intersection of $V$ and the light-cone emanating from the point $q$ , i.e., the points in the set $V$ where light from a point source at $q$ is observed.
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 geodesic $μ$ and the source-to-solution operator that maps the source supported on $V$ to the restriction of the solution of the wave equation in $V$ . 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 $μ$ and return back to $μ$ .

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