  We compute the evolution of a quasi-spherical, slowly rotating accretion flow
around a black hole, whose mass and spin evolve adequately to the mass-energy
transfer through the horizon. Our model is relevant for the central engine
driving a long gamma ray burst, that originates from the collapse of a massive
star. The computations of a GRB engine in a dynamically evolving spacetime
metric are important specifically due to the transient nature of the event, in
which a huge amount of mass is accreted and changes the fundamental black hole
parameters, its mass and spin, during the process. We discuss the results in
the context of angular momentum magnitude of the collapsing star. We also study
the possible formation and evolution of shocks in the envelope, which may
temporarily affect accretion. Our results are important for the limitations on
the mass and spin range of black holes detected independently by
electromagnetic observations of GRBs and gravitational waves. We speculate on
the possible constraints for the final masses and spins of these astrophysical
black holes. It is shown that the most massive BHs were rahter not formed in a
powerful GRB explosion, if the cores of their progenitors were only weakly
rotating.
