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 hol
e 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.
We investigate the collapse of differentially rotating supermassive stars (SMSs) by means of 3+1 hydrodynamic simulations in general relativity. We particularly focus on the onset of collapse to understand the final outcome of collapsing SMSs. We fin
d that the estimated ratio of the mass between the black hole (BH) and the surrounding disk from the equilibrium star is roughly the same as the results from numerical simulation. This suggests that the picture of axisymmetric collapse is adequate, in the absence of nonaxisymmetric instabilities, to illustrate the final state of the collapse. We also find that quasi-periodic gravitational waves continue to be emitted after the quasinormal mode frequency has decayed. We furthermore have found that when the newly formed BH is almost extreme Kerr, the amplitude of the quasi-periodic oscillation is enhanced during the late stages of the evolution. Geometrical features, shock waves, and instabilities of the fluid are suggested as a cause of this amplification behaviour. This alternative scenario for the collapse of differentially rotating SMSs might be observable by LISA.
Nonlinear growth of the bar-mode deformation is studied for a differentially rotating star with supercritical rotational energy. In particular, the growth mechanism of some azimuthal modes with odd wave numbers is examined by comparing a simplified m
athematical model with a realistic simulation. Mode coupling to even modes, i.e., the bar mode and higher harmonics, significantly enhances the amplitudes of odd modes, unless they are exactly zero initially. Therefore, other modes which are not axially symmetric cannot be neglected at late times in the growth of the unstable bar-mode even when starting from an almost axially symmetric state.
We investigate the nonlinear behaviour of the dynamically unstable rotating star for the bar mode by three-dimensional hydrodynamics in Newtonian gravity. We find that an oscillation along the rotation axis is induced throughout the growth of the uns
table bar mode, and that its characteristic frequency is twice as that of the bar mode, which oscillates mainly along the equatorial plane. A possibility to observe Faraday resonance in gravitational waves is demonstrated and discussed.
