  The search for an understanding of an energy source great enough to explain the gamma-ray burst (GRB) phenomena has attracted much attention from the astrophysical community since its discovery. In this paper we extend the work of K. Asano and T. Fukuyama, and J. D. Salmonson and J. R. Wilson, and analyze the off-axis contributions to the energy-momentum deposition rate (MDR) from the neutrino anti-neutrino collisions above a rotating black hole/thin accretion disk system. Our calculations are performed by imaging the accretion disk at a specified observer using the full geodesic equations, and calculating the cumulative MDR from the scattering of all pairs of neutrinos and anti-neutrinos arriving at the observer. Our results shed light on the beaming efficiency of GRB models of this kind. Although we confirm Asano and Fukuyama's conjecture as to the constancy of the beaming for small angles away from the axis; nevertheless, we find the dominant contribution to the MDR comes from near the surface of the disk with a tilt of approximately \pi/4 in the direction of the disk's rotation. We find that the MDR at large radii is directed outward in a conic section centered around the symmetry axis and is larger, by a factor of 10 to 20, than the on-axis values. By including this off-axis disk source, we find a linear dependence of the MDR on the black hole angular momentum (a). In addition, we find that scattering is directed back onto the black hole in regions just above the horizon of the black hole. This gravitational ``in scatter'' may provide an observable high energy signature of the central engine, or at least another channel for accretion. 
Off-Axis Neutrino Scattering in GRB Central Engines