Inverse problem for Einstein-scalar field equations

Abstract

The paper introduces a method to solve inverse problems for hyperbolic systems where the leading-order terms are nonlinear. We apply the method to the coupled Einstein-scalar field equations and study the question of whether the structure of space-time 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 space-time in the optimal chronological diamond-type set containing the world line. In the case when the unknown part of the space-time is vacuum, we can also determine the metric itself. We exploit the nonlinearity of the equation to obtain a rich set of propagating singularities, produced by a nonlinear interaction of singularities that propagate initially as for linear wave equations. This nonlinear effect is then used as a tool to solve the inverse problem for the nonlinear 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 nonlinear hyperbolic equations encountered in practical imaging problems.