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The apparent distribution of large-scale structures in the universe issensitive to the velocity/potential of the sources as well as the potentialalong the line-of-sight through the mapping from real space to redshift space(redshift-space distortions, RSD). Since odd multipoles of the halocross-correlation function vanish when considering standard Doppler RSD, thedipole is a sensitive probe of relativistic and wide-angle effects. We build acatalogue of ten million haloes (Milky-Way size to galaxy-cluster size) fromthe full-sky light-cone of a new "RayGalGroupSims" N-body simulation whichcovers a volume of ($2.625~h^{-1}$Gpc)$^3$ with $4096^3$ particles. Usingray-tracing techniques, we find the null geodesics connecting all the sourcesto the observer. We then self-consistently derive all the relativisticcontributions (in the weak-field approximation) to RSD: Doppler, transverseDoppler, gravitational, lensing and integrated Sachs-Wolfe. It allows us, forthe first time, to disentangle all contributions to the dipole from linear tonon-linear scales. At large scale, we recover the linear predictions dominatedby a contribution from the divergence of neighbouring line-of-sights. While thelinear theory remains a reasonable approximation of the velocity contributionto the dipole at non-linear scales it fails to reproduce the potentialcontribution below $30-60~h^{-1}$Mpc (depending on the halo mass). At scalessmaller than $\sim 10~h^{-1}$Mpc, the dipole is dominated by the asymmetrycaused by the gravitational redshift. The transition between the two regimes ismass dependent as well. We also identify a new non-trivial contribution fromthe non-linear coupling between potential and velocity terms.

Imprints of relativistic effects on the asymmetry of the halo cross-correlation function: from linear to non-linear scales