The outcome of SMS collapse can be determined only with numerical, relativistic 3D hydrodynamics simulations. Until recently, such simulations had been published only for nearly spherical collapse. The spherical simulations of Shapiro and Teukolsky [275] produced collapse evolutions that were nearly homologous. In this case, the collapse time is roughly the free-fall time at the horizon

The peak GW frequency is then 10The amplitude of this burst signal can be roughly estimated in terms of the star’s quadrupole moment

Here is the distance to the star and is a measure of the star’s deviation from spherical symmetry. In this case, will be much less than one near the horizon, since the collapse is nearly spherical.There are two possible aspherical collapse outcomes that have been studied. The first outcome is direct collapse to a SMBH. In this case, will be on the order of one near the horizon. Thus, according to Equation (31), the peak amplitude of the GW burst signal will be

Alternatively, the star may encounter the dynamical bar mode instability prior to complete collapse. Baumgarte and Shapiro [12] have estimated that a uniformly-rotating SMS will reach when . The frequency of the quasiperiodic gravitational radiation emitted by the bar can be estimated in terms of its rotation frequency to be

when . The corresponding , again estimated in terms of the star’s quadrupole moment, is The LISA sensitivity curve is shown in Figure 30 (see [149] for details on the computation of this curve; a mission time of three years has been assumed). The GW signal from this dynamical bar mode could be detected with LISA.Shibata and Shapiro [284] have published a fully general-relativistic, axisymmetric simulation of the collapse of a rapidly, rigidly-rotating SMS. They found that the collapse remained homologous during the early part of the evolution. An apparent horizon does appear in their simulation, indicating the formation of a black hole. Because of the symmetry condition used in their run, non-axisymmetric instabilities did not develop.

The collapse of a uniformly-rotating SMS has been investigated with post-Newtonian hydrodynamics, in 3+1 dimensions, by Saijo, Baumgarte, Shapiro, and Shibata [259]. Their numerical scheme used a post-Newtonian approximation to the Einstein equations, but solved the fully relativistic hydrodynamics equations. Their initial model was an polytrope.

The results of Saijo et al. (confirmed in conformally-flat simulations [256]) indicate that the collapse of a uniformly-rotating SMS is coherent (i.e., no fragmentation instability develops). The collapse evolution of density contours from their model is shown in Figure 31. Although the work of Baumgarte and Shapiro [12] suggests that a bar instability should develop prior to black-hole formation, no bar development was observed by Saijo et al. They use the quadrupole approximation to estimate a mean GW amplitude from the collapse itself: , for a star located at a distance of 50 Gpc. Their estimate for at the time of black-hole formation is . This signal would be detectable with LISA (see Figure 30).

Saijo et al. also consider the GW emission from the ringdown of the black-hole remnant. For the quasi-normal mode of a Kerr black hole with , they estimate the characteristic amplitude of emission to be at for an source located at a luminosity distance of 50 Gpc (see [174, 307, 285] for details). Here, is the radiated energy efficiency and may be [291]. This GW signal is within LISA’s range of sensitivity (see Figure 30).

Living Rev. Relativity 14, (2011), 1
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