  We study the power spectrum dipole of an N-body simulation which includes
relativistic effects through ray-tracing and covers the low redshift Universe
up to $z_{\rm max} = 0.465$ (RayGalGroup simulation). We model relativistic
corrections as well as wide-angle, evolution, window and lightcone effects. Our
model includes all relativistic corrections up to third-order including
third-order bias expansion. We consider all terms which depend linearly on
$\mathcal{H}/k$ (weak field approximation). We also study the impact of 1-loop
corrections to the matter power spectrum for the gravitational redshift and
transverse Doppler effect. We found wide-angle and window function effects to
significantly contribute to the dipole signal. When accounting for all
contributions, our dipole model can accurately capture the gravitational
redshift and Doppler terms up to the smallest scales included in our comparison
($k=0.48\,h{\rm Mpc}^{-1}$), while our model for the transverse Doppler term is
less accurate. We find the Doppler term to be the dominant signal for this low
redshift sample. We use Fisher matrix forecasts to study the potential for the
future Dark Energy Spectroscopic Instrument (DESI) to detect relativistic
contributions to the power spectrum dipole. A conservative estimate suggests
that the DESI-BGS sample should be able to have a detection of at least
$4.4\sigma$, while more optimistic estimates find detections of up to
$10\sigma$. Detecting these effects in the galaxy distribution allows new tests
of gravity on the largest scales, providing an interesting additional science
case for galaxy survey experiments.
