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We explore the formation process of a black hole (BH) through thepair-instability collapse of a rotating Population III very massive star inaxisymmetric numerical relativity. As the initial condition, we employ aprogenitor star which is obtained by evolving a rapidly rotating zero-age mainsequence (ZAMS) star with mass $320M_\odot$ until it reaches a pair instabilityregion. We find that for such rapidly rotating model, a fraction of the mass,$\sim 10M_\odot$, forms a torus surrounding the remnant BH of mass $\sim130M_\odot$ and an outflow is driven by a hydrodynamical effect. We alsoperform simulations, artificially reducing the initial angular velocity of theprogenitor star, and find that only a small or no torus is formed and nooutflow is driven. We discuss the possible evolution scenario of the remnanttorus for the rapidly rotating model by considering the viscous andrecombination effects and show that if the energy of $\sim 10^{52}$ erg isinjected from the torus to the envelope, the luminosity and timescale of theexplosion could be of the orders of $10^{43}$ erg/s and yrs, respectively. Wealso point out the possibility for observing gravitational waves associatedwith the BH formation for the rapidly rotating model by ground-basedgravitational-wave detectors.

Black Hole Formation and Explosion from Rapidly Rotating Very Massive Stars