  We study inverse problems for the Einstein equations with source fields in a
general form. Under a microlocal linearization stability condition, we show
that by generating small gravitational perturbations and measuring the
responses near a freely falling observer, one can uniquely determine the
background Lorentzian metric up to isometries in a region where the
gravitational perturbations can travel to and return. We apply the result to
two concrete examples when the source fields are scalar fields (i.e.
Einstein-scalar field equations) and electromagnetic fields (i.e.
Einstein-Maxwell equations).
