Dynamic black holes through gravitational collapse: Analysis of the multipole moment of the curvatures on the horizon

We have investigated several properties of rapidly rotating dynamic black holes generated by the 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) final approximate to 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) approximate to 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. First, 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. Second, 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 quasiperiodic waves in the gravitational waveform after the ringdown in terms of the multipole moment of the dynamic hole.