We consider inverse problems for non-linear wave equations on Lorentzian manifolds. In addition, we study the related inverse problems for the coupled Einstein equations and matter field equations. We show that observations of the solutions and sources in an open subset $U$ of the space-time $M$ determine the properties of the metric in a larger domain $W\subset M$ containing the set $U$.

To study these problems we define the concept of light observation sets and show that these sets determine the conformal class of the metric.

The results have been done in collaboration with Yaroslav Kurylev, Lauri Oksanen, Gunther Uhlmann, and Yiran Wang.