Inverse problem for Einstein-scalar field equations

The paper introduces a method to solve inverse problems for hyperbolic systems where the leading order terms are non-linear. We apply the method to the coupled Einstein-scalar field equations and study the question whether the structure of spacetime can be determined by making active measurements near the world line of an observer. We show that such measurements determine the topological, differential and conformal structure of the spacetime in the optimal chronological diamond type set containing the world line. In the case when the unknown part of the spacetime is vacuum, we can also determine the metric itself. We exploit the non-linearity of the equation to obtain a rich set of propagating singularities, produced by a non-linear interaction of singularities that propagate initially as for linear wave equations. This non-linear effect is then used as a tool to solve the inverse problem for the non-linear system. The method works even in cases where the corresponding inverse problems for linear equations remain open, and it can potentially be applied to a large class of inverse problems for non-linear hyperbolic equations encountered in practical imaging problems.