  We use gauge-invariant cosmological perturbation theory to calculate the
displacement field that sets the initial conditions for $N$-body simulations.
Using first and second-order fully relativistic perturbation theory in the
synchronous-comoving gauge, allows us to go beyond the Newtonian predictions
and to calculate relativistic corrections to it. We use an Einstein--de Sitter
model, including both growing and decaying modes in our solutions. The impact
of our results should be assessed through the implementation of the featured
displacement in cosmological $N$-body simulations.
