Fully general relativistic collapse simulations (i.e., without the conformally flat approximation) have
also been performed by Shibata [222]. He used an axisymmetric code that solves the Einstein
equations in Cartesian coordinates and the hydrodynamics equations in cylindrical coordinates.
The use of the Cartesian grid eliminates the presence of singularities and allows for stable,
long-duration axisymmetric simulations [4]. The focus of this work was the effect of rotation on the
criteria for prompt black hole formation. Shibata found that if the parameter
is less
than 0.5, black hole formation occurred for rest masses slightly greater than the maximum
mass of spherical stars. However, for
, the maximum stable rest mass is increased
by
70 - 80%. The results are only weakly dependent on the initial rotation profile. More
recent results suggest that this limit can be eased for differentially rotating massive stellar cores,
and systems with spin parameter
as high as 2.5 may collapse to form black holes [217].
Shibata did not compute the GW emission in his collapse simulations, but in a recent study
using axisymmetric calculations, GW signals have been calculataed focusing on this collapse
phase [267].
Duez et al. [62] found that if a black hole does form, but the disk is spinning rapidly, that the disk will
fragment and its subsequent accretion will be in spurts, causing a “splash” onto the black hole, producing
ringing and GW emission. Their result implies very strong gravitational wave amplitudes at
distances of
. Black hole ringing was also estimated by FHH, where they too assumed discrete
accretion events. They found that even with very optimistic accretion scenarios, that such radiation
will be of very low amplitude and beyond the upper frequency reach of LIGO-II (see [86
] for
details).
The new general relativistic hydrodynamics simulations of Zanotti, Rezzolla, and Font [269] suggest
that a torus of neutron star matter surrounding a black hole remnant may be a stronger source of GWs
than the collapse itself. They used a high resolution shock-capturing hydrodynamics method in conjunction
with a static (Schwarzschild) spacetime to follow the evolution of “toroidal neutron stars”. Their results
indicate that if a toroidal neutron star (with constant specific angular momentum) is perturbed, it could
undergo regular oscillations. They estimate that the resulting GW emission would have a characteristic
amplitude ranging from
, for ratios of torus mass to black hole mass in the
range 0.1 - 0.5. (These amplitude values are likely underestimated because the simulations of Zanotti et
al. are axisymmetric.) The corresponding frequency of emission is
. The values of
and
quoted here are for a source located at
. This emission would be just outside
the range of LIGO-II (see Figure 2
). Further numerical investigations, which study tori with
non-constant angular momenta and include the effects of self-gravity and black hole rotation, are
needed to confirm these predictions. Movies from the simulations of Zanotti et al. can be viewed
at [200].
Magnetized tori around rapidly spinning black holes (formed via either core collapse or neutron
star-black hole coalescence) have recently been examined in the theoretical study of van Putten and
Levinson [251]. They find that such a torus-black hole system can exist in a suspended state of accretion if
the ratio of poloidal magnetic field energy to kinetic energy
is less than 0.1. They
estimate that
10% of the spin energy of the black hole will be converted to gravitational
radiation energy through multipole mass moment instabilities that develop in the torus. If a
magnetized torus-black hole system located at
is observed for
rotation
periods, the characteristic amplitude of the GW emission is
. It is possible that this
emission could take place at several frequencies. Observations of x-ray lines from gamma-ray
bursts (which are possibly produced by these types of systems) could constrain these frequencies
by providing information regarding the angular velocities of the tori: preliminary estimates
from observations suggest
, placing the radiation into a range detectable by
LIGO-I [251
].
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