  Newtonian simulations are routinely used to examine the matter dynamics on
non-linear scales. However, even on these scales, Newtonian gravity is not a
complete description of gravitational effects. A post-Friedmann approach shows
that the leading order correction to Newtonian theory is a vector potential in
the metric. This vector potential can be calculated from N-body simulations,
requiring a method for extracting the velocity field. Here, we present the full
details of our calculation of the post-Friedmann vector potential, using the
Delauney Tesselation Field Estimator (DTFE) code. We include a detailed
examination of the robustness of our numerical result, including the effects of
box size and mass resolution on the extracted fields. We present the power
spectrum of the vector potential and find that the power spectrum of the vector
potential is $\sim 10^5$ times smaller than the power spectrum of the fully
non-linear scalar gravitational potential at redshift zero. Comparing our
numerical results to perturbative estimates, we find that the fully non-linear
result can be more than an order of magnitude larger than the perturbative
estimate on small scales. We extend the analysis of the vector potential to
multiple redshifts, showing that this ratio persists over a range of scales and
redshifts. We also comment on the implications of our results for the validity
and interpretation of Newtonian simulations.
