  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.
