  The evolution of the spin and tilt of black holes in compact black hole -
neutron star and black hole - black hole binary systems is investigated within
the framework of the coalescing compact star binary model for short gamma ray
bursts via the population synthesis method. Based on recent results on
accretion at super critical rates in slim disk models, estimates of natal
kicks, and the results regarding fallback in supernova models, we obtain the
black hole spin and misalignment. It is found that the spin parameter, a_spin},
is less than 0.5 for initially non rotating black holes and the tilt angle,
i_tilt, is less than 45 deg for 50% of the systems in black hole - neutron star
binaries. Upon comparison with the results of black hole - neutron star merger
calculations we estimate that only a small fraction (~ 0.01) of these systems
can lead to the formation of a torus surrounding the coalesced binary
potentially producing a short-hard gamma ray burst. On the other hand, for high
initial black hole spin parameters (a_spin>0.6) this fraction can be
significant (~ 0.4). It is found that the predicted gravitational radiation
signal for our simulated population does not significantly differ from that for
non rotating black holes. Due to the (i) insensitivity of signal detection
techniques to the black hole spin and the (ii) predicted overall low
contribution of black hole binaries to the signal we find that the detection of
gravitational waves are not greatly inhibited by current searches with non
spinning templates. It is pointed out that the detection of a black hole -
black hole binary inspiral system with LIGO or VIRGO may provide a direct
measurement of the initial spin of a black hole.
