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 ≲ R_p/M₀ ≲ 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 ∼ 12, at which point T/W ∼ 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 ∼ 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.