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 nonlinear. We apply coupled Einstein-scalar field equations and study question of whether structure space-time can be determined by making active measurements near world line an observer. show that such determine topological, differential, conformal in optimal chronological diamond-type set containing line. In case when unknown part is vacuum, we also metric itself. exploit nonlinearity equation obtain rich propagating singularities, produced nonlinear interaction singularities propagate initially as linear wave equations. This effect then used tool problem system. works even cases corresponding remain open, it potentially applied large class encountered practical imaging problems.