  The physical interpretation of cold dark matter perturbations is clarified by
associating Bertschinger's Poisson gauge with a Eulerian/observer's frame of
reference. We obtain such an association by using a Lagrangian approach to
relativistic cosmological structure formation. Explicitly, we begin with the
second-order solution of the Einstein equations in a synchronous/comoving
coordinate system---which defines the Lagrangian frame, and transform it to a
Poissonian coordinate system. The generating vector of this coordinate/gauge
transformation is found to be the relativistic displacement field. The metric
perturbations in the Poissonian coordinate system contain known results from
standard/Eulerian Newtonian perturbation theory, but contain also purely
relativistic corrections. On sub-horizon scales these relativistic corrections
are dominated by the Newtonian bulk part. These corrections however set up
non-linear constraints for the density and for the velocity which become
important on scales close to the horizon. Furthermore, we report the occurence
of a transverse component in the displacement field, and find that it induces a
non-linear frame dragging as seen in the observer's frame, which is
sub-dominant at late-times and sub-horizon scales. Finally, we find two other
gauges which can be associated with a Eulerian frame. We argue that the Poisson
gauge is to be preferred because it comes with the simplest physical
interpretation.
