Dynamic black holes through gravitational collapse: Analysis of
  multipole moment of the curvatures on the horizon
  We have investigated several properties of rapidly rotating dynamic black
holes generated by gravitational collapse of rotating relativistic stars. At
present, numerical simulations of the binary black hole merger are able to
produce a Kerr black hole of J_final / M_final^2 up to = 0.91, of gravitational
collapse from uniformly rotating stars up to J_final / M_final^2 ~ 0.75, where
J_final is the total angular momentum and M_final the total gravitational mass
of the hole. We have succeeded in producing a dynamic black hole of spin
J_final / M_final^2 ~ 0.95 through the collapse of differentially rotating
relativistic stars. We have investigated those dynamic properties through
diagnosing multipole moment of the horizon, and found the following two
features. Firstly, two different definitions of the angular momentum of the
hole, the approximated Killing vector approach and dipole moment of the current
multipole approach, make no significant difference to our computational
results. Secondly, dynamic hole approaches a Kerr by gravitational radiation
within the order of a rotational period of an equilibrium star, although the
dynamic hole at the very forming stage deviates quite far from a Kerr. We have
also discussed a new phase of quasi-periodic waves in the gravitational
waveform after the ringdown in terms of multipole moment of the dynamic hole.
