  We introduce a new effect in the neutrino oscillation phase which shows the
neutrino-antineutrino oscillation is possible under gravity even if the rest
masses of the corresponding eigenstates are same. This is due to CPT violation
and possible to demonstrate if the neutrino mass eigenstates are expressed as a
combination of neutrino and antineutrino eigenstates, as of the neutral kaon
system, with the plausible breaking of lepton number conservation. For Majorana
neutrinos, this oscillation is expected to affect significantly the inner edge
of neutrino dominated accretion disks around a compact object by influencing
the neutrino sphere which controls the accretion dynamics, and then the related
type-II supernova evolution and the r-process nucleosynthesis. On the other
hand, in early universe, in presence of various lepton number violating
processes, this oscillation, we argue, might lead to neutrino asymmetry which
resulted baryogenesis from the B-L symmetry by electro-weak sphaleron
processes.
