We consider the semi-relativistic system of N gravitating Bosons with
gravitation constant G. The time evolution of the system is described by the
relativistic dispersion law, and we assume the mean-field scaling of the
interaction where N→∞ and G→0 while GN=λ fixed. In
the super-critical regime of large λ, we introduce the regularized
interaction where the cutoff vanishes as N→∞. We show that the
difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and
the corresponding semi-relativistic Hartree dynamics is at most of order
N−1 for all λ, i.e., the result covers the sub-critical regime and
the super-critical regime. The N dependence of the bound is optimal.Comment: 29 page