  We study the time evolution in system of $N$ bosons with a relativistic
dispersion law interacting through an attractive Coulomb potential with
coupling constant $G$. We consider the mean field scaling where $N$ tends to
infinity, $G$ tends to zero and $\lambda = G N$ remains fixed. We investigate
the relation between the many body quantum dynamics governed by the
Schr\"odinger equation and the effective evolution described by a
(semi-relativistic) Hartree equation. In particular, we are interested in the
super-critical regime of large $\lambda$ (the sub-critical case has been
studied in \cite{ES,KP}), where the nonlinear Hartree equation is known to have
solutions which blow up in finite time. To inspect this regime, we need to
regularize the Coulomb interaction in the many body Hamiltonian with an $N$
dependent cutoff that vanishes in the limit $N\to \infty$. We show, first, that
if the solution of the nonlinear equation does not blow up in the time interval
$[-T,T]$, then the many body Schr\"odinger dynamics (on the level of the
reduced density matrices) can be approximated by the nonlinear Hartree
dynamics, just as in the sub-critical regime. Moreover, we prove that if the
solution of the nonlinear Hartree equation blows up at time $T$ (in the sense
that the $H^{1/2}$ norm of the solution diverges as time approaches $T$), then
also the solution of the linear Schr\"odinger equation collapses (in the sense
that the kinetic energy per particle diverges) if $t \to T$ and,
simultaneously, $N \to \infty$ sufficiently fast. This gives the first
dynamical description of the phenomenon of gravitational collapse as observed
directly on the many body level.
