  The neutrino-antineutrino annihilation into electron-positron pairs near the
surface of compact general relativistic stars could play an important role in
supernova explosions, neutron star collapse, or for close neutron star binaries
near their last stable orbit. General relativistic effects increase the energy
deposition rates due to the annihilation process. We investigate the deposition
of energy and momentum due to the annihilations of neutrinos and antineutrinos
in the equatorial plane of the rapidly rotating neutron and quark stars,
respectively. We analyze the influence of general relativistic effects, and we
obtain the general relativistic corrections to the energy and momentum
deposition rates for arbitrary stationary and axisymmetric space-times. We
obtain the energy and momentum deposition rates for several classes of rapidly
rotating neutron stars, described by different equations of state of the
neutron matter, and for quark stars, described by the MIT bag model equation of
state and in the CFL (Color-Flavor-Locked) phase, respectively. Compared to the
Newtonian calculations, rotation and general relativistic effects increase the
total annihilation rate measured by an observer at infinity. The differences in
the equations of state for neutron and quark matter also have important effects
on the spatial distribution of the energy deposition rate by
neutrino-antineutrino annihilation.
