The Gross-Pitaevskii equations of a static and spherically symmetric
  condensate of gravitons
  In this paper we consider the Dvali and G\'omez assumption that the end state
of a gravitational collapse is a Bose-Einstein condensate of gravitons. We then
construct the two Gross-Pitaevskii equations for a static and spherically
symmetric configuration of the condensate. These two equations correspond to
the constrained minimisation of the gravitational Hamiltonian with respect to
the redshift and the Newtonian potential, per given number of gravitons. We
find that the effective geometry of the condensate is the one of a gravastar (a
DeSitter star) with a sub-Planckian cosmological constant, for masses larger
than the Planck scale. Thus, a condensate corresponding to a semiclassical
black hole, is always quantum and weakly coupled. Finally, we obtain that the
boundary of our gravastar, although it is not the location of a horizon,
corresponds to the Schwarzschild radius.
