Determination of structures in the space-time from local measurements: a detailed exposition

We consider inverse problems for the Einstein equation with a time-depending metric on a 4-dimensional globally hyperbolic Lorentzian manifold $(M,g)$. We formulate the concept of active measurements for relativistic models. We do this by coupling the Einstein equation with equations for scalar fields and study the system $Ein(g)=T$, $T=T(g,\phi)+F_1$, and $\square_g \phi=F_2+S(g,\phi,F_1,F_2)$. Here $F=(F_1,F_2)$ correspond to the perturbations of the physical fields which we control and $S$ is a secondary source corresponding to the adaptation of the system to the perturbation so that the conservation law $div_g(T)=0$ will be satisfied. The inverse problem we study is the question, do the observation of the solutions $(g,\phi)$ in an open subset $U\subset M$ of the space-time corresponding to sources $F$ supported in $U$ determine the properties of the metric in a larger domain $W\subset M$ containing $U$. To study this problem we define the concept of light observation sets and show that these sets determine the conformal class of the metric. This corresponds to passive observations from a distant area of the space which is filled by light sources (e.g. we see light from stars varying in time). One can apply the obtained result to solve inverse problems encountered in general relativity and in various practical imaging problems.