  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.
