Neutrino radiation fields from stationary, axisymmetric, non-selfgravi
tating tori around stellar mass Kerr black holes are calculated. Such
objects may form as a result of a merger between two neutron stars, a
neutron star and a stellar mass black hole, or a ''failed supernova''
collapse of a single rapidly rotating star. The neutrino annihilation
rates near the axis of rotation are found with fully general relativis
tic methods. Our calculations show that about 80% of the energy deposi
ted by annihilating neutrinos is swallowed by the black hole, lowering
the energy which can be transported to infinity. The importance of th
is effect for the modeling of the gamma ray burst sources is discussed
.