A Possible Energy Mechanism for Cosmological Gamma-ray Bursts

We suggest that an extreme Kerr black hole with a mass $\sim {10}^6{M}_{\odot }$, a dimensionless angular momentum $A\sim 1$ and a marginal stable orbital radius $r_{ms}\sim 3r_s\sim {10}^{12}{M}_{6}cm$ located in a normal galaxy, may produced a Gamma-ray Burst by capturing and disrupting a star. During this period, a transient accretion disk is formed and a strong transient magnetic field $\sim 2.4\times 10^9M_6^{-1/2}$Gauss,lastingfor$r_{ms}/c\sim 30 M_6 s$, may be produced in the inner boundary of the accretion disk. A large amount of rotational energy of the black hole is extracted and released in the ultra relativistic jet with a bulk Lorentz factor $\mathrm{\Gamma }$ larger than ${10}^3$ via Blandford-Znajek process. The relativistic jet energy can be converted into $\gamma$-ray radiation via internal shock mechanism. The gamma-ray burst (GRB) duration should be the same as that of the life time of the strong transient magnetic field. The maximum number of sub-bursts is estimated to be $r_{ms}\mathrm{/}h\sim (10-{10}^2)$ because the disk material is likely broken into pieces with the size about the thickness of the disk $h$ at the cusp ($2r_s\le r\le 3r_s$). The shortest rising time of the burst estimated from this model is $\sim h/\Gamma c\sim 3\times 10^{-4}\Gamma^{-1}3(h/r){-2}M_6$ s. The model gamma-ray burst density rate is also estimated.