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Bearing in mind the application to the collapsar models of gamma-ray bursts(GRBs), we develop a numerical scheme and code for estimating the deposition ofenergy and momentum due to the neutrino pair annihilation ($\nu + {\bar \nu}\rightarrow e^{-} + e^{+}$) in the vicinity of accretion tori around a Kerrblack hole. Our code is designed to solve the general relativistic neutrinotransfer by a ray-tracing method. To solve the collisional Boltzmann equationin curved spacetime, we numerically integrate the so-called rendering equationalong the null geodesics. For the neutrino opacity, the charged-current$\beta$-processes are taken into account, which are dominant in the vicinity ofthe accretion tori. The numerical accuracy of the developed code iscertificated by several tests, in which we show comparisons with thecorresponding analytic solutions. Based on the hydrodynamical data in ourcollapsar simulation, we estimate the annihilation rates in a post-processingmanner. Increasing the Kerr parameter from 0 to 1, it is found that the generalrelativistic effect can increase the local energy deposition rate by about oneorder of magnitude, and the net energy deposition rate by several tens ofpercents. After the accretion disk settles into a stationary state (typicallylater than $\sim 9$ s from the onset of gravitational collapse), we point outthat the neutrino-heating timescale in the vicinity of the polar funnel regioncan be shorter than the dynamical timescale. Our results suggest the neutrinopair annihilation has a potential importance equal to the conventionalmagnetohydrodynamic mechanism for igniting the GRB fireballs.

General Relativistic Ray-Tracing Method for Estimating the Energy and Momentum Deposition by Neutrino Pair Annihilation in Collapsars