  Discerning the likelihood of the so-called runaway instability of thick
accretion disks orbiting black holes is an important issue for most models of
cosmic gamma-ray bursts. To this aim we investigate this phenomenon by means of
time-dependent, hydrodynamical simulations of black hole plus torus systems in
general relativity. The evolution of the central black hole is assumed to be
that of a sequence of Kerr black holes of increasing mass and spin, whose
growth rate is controlled by the transfer of mass and angular momentum from the
material of the disk spiralling in through the event horizon of the black hole.
The self-gravity of the disk is neglected. We find that when the black hole
mass and spin are allowed to increase, constant angular momentum disks undergo
a runaway instability on a dynamical timescale (a few orbital periods).
However, our simulations show that a slight increase of the specific angular
momentum of the disk outwards has a dramatic stabilizing effect. Our results,
obtained in the framework of general relativity, are in broad agreement with
earlier studies based both on stationary models and on time-dependent
simulations with Newtonian and pseudo-Newtonian gravitational potentials.
