General relativistic corrections to $N$-body simulations and the
  Zel'dovich approximation
  The initial conditions for Newtonian $N$-body simulations are usually
generated by applying the Zel'dovich approximation to the initial displacements
of the particles using an initial power spectrum of density fluctuations
generated by an Einstein-Boltzmann solver. We show that in most gauges the
initial displacements generated in this way receive a first-order relativistic
correction. We define a new gauge, the $N$-body gauge, in which this
relativistic correction vanishes and show that a conventional Newtonian
$N$-body simulation includes all first-order relativistic contributions (in the
absence of radiation) if we identify the coordinates in Newtonian simulations
with those in the relativistic $N$-body gauge.
