Zel'dovich approximation and General Relativity
  We show how the Zel'dovich approximation and the second order displacement
field of Lagrangian perturbation theory can be obtained from a general
relativistic gradient expansion in \Lambda{}CDM cosmology. The displacement
field arises as a result of a second order non-local coordinate transformation
which brings the synchronous/comoving metric into a Newtonian form. We find
that, with a small modification, the Zel'dovich approximation holds even on
scales comparable to the horizon. The corresponding density perturbation is not
related to the Newtonian potential via the usual Poisson equation but via a
modified Helmholtz equation. This is a consequence of causality not present in
the Newtonian theory. The second order displacement field receives relativistic
corrections that are subdominant on short scales but are comparable to the
second order Newtonian result on scales approaching the horizon. The
corrections are easy to include when setting up initial conditions in large
N-body simulations.
