We study gravitational collapse of rapidly rotating relativistic polytropes of the adiabatic index \Gamma = 1.5 and 2 , in which the spin parameter q \equiv J / M ^ { 2 } > 1 where J and M are total angular momentum and gravitational mass , in full general relativity .
First , analyzing initial distributions of the mass and the spin parameter inside stars , we predict the final outcome after the collapse .
Then , we perform fully general relativistic simulations on assumption of axial and equatorial symmetries and confirm our predictions .
As a result of simulations , we find that in contrast with the previous belief , even for stars with q > 1 , the collapse proceeds to form a seed black hole at central region , and the seed black hole subsequently grows as the ambient fluids accrete onto it .
We also find that growth of angular momentum and mass of the seed black hole can be approximately determined from the initial profiles of the density and the specific angular momentum .
We define an effective spin parameter at the central region of the stars , q _ { c } , and propose a new criterion for black hole formation as q _ { c } \lesssim 1 .
Plausible reasons for the discrepancy between our and previous results are clarified .