  We describe the irrotational dust component of the universe in terms of a
relativistic gradient expansion and transform the resulting synchronous metric
to a Newtonian coordinate system. The two metrics are connected via a
space-like displacement field and a time-like perturbation, providing a
relativistic generalization of the transformation from Lagrangian to Eulerian
coordinates. The relativistic part of the displacement field generates already
at initial time a non-local density perturbation at second order. This is a
purely relativistic effect since it originates from space-time mixing. We give
two options, the passive and the active approach, on how to include the
relativistic corrections for example in N-body simulations. In the passive
approach we treat the corrections as a non-Gaussian modification of the initial
Gaussian field (primordial non-Gaussianity could be incorporated as well). The
induced non-Gaussianity depends on scale and the redshift at which initial
conditions are set, with f_NL ~ few for small enough scales and redshifts. In
the active approach we show how to use the relativistic trajectory to obtain
the initial displacement and velocity of particles for N-body simulations
without modifying the initial Gaussian field.
