Collapse of a rotating supermassive star to a supermassive black hole: Post-Newtonian simulations

We study the gravitational collapse of a rotating supermassive star by means of a (3+1) hydrodynamic simulation in a post-Newtonian approximation of general relativity. This problem is particularly challenging because of the vast dynamic range in space that must be covered in the course of collapse. We evolve a uniformly rotating supermassive star from the onset of radial instability at R-p/M=411, where R-p is the proper polar radius of the star and M is the total mass-energy, to the point at which the post-Newtonian approximation breaks down. We introduce a scale factor and a "comoving" coordinate to handle the large variation in radius during the collapse (8 less than or similar to R-p/M-0 less than or similar to 411, where M-0 is the rest mass) and focus on the central core of the supermassive star. Since T/W, the ratio of the rotational kinetic energy to the gravitational binding energy, is nearly proportional to 1/R-p for an n=3 polytropic star throughout the collapse, the imploding star may ultimately exceed the critical value of T/W for dynamic instability to bar-mode formation. Analytic estimates suggest that this should occur near R-p/M-0 similar to 12, at which point T/W similar to 0.27. For stars rotating uniformly at the onset of collapse, however, we do not find any unstable growth of bars prior to the termination of our simulation at R-p/M-0 similar to 8. We do find that the collapse is likely to form a supermassive black hole coherently, with almost all of the matter falling into the hole, leaving very little ejected matter to form a disk. In the absence of nonaxisymmetric bar formation, the collapse of a uniformly rotating supermassive star does not lead to appreciable quasi-periodic gravitational wave emission by the time our integrations terminate. The coherent nature of the implosion, however, suggests that rotating supermassive star collapse will be a promising source of gravitational wave bursts. We also expect that, following black hole formation, long-wavelength quasi-periodic waves will result from quasi-normal ringing. These waves may be detectable by the Laser Interferometer Space Antenna.