  We study neutrino oscillations in a rotating spacetime under the weak gravity
limit for the trajectories of neutrinos which are constrained in the equatorial
plane. Using the asymptotic form of the Kerr metric, we show that the rotation
of the gravitational source non-trivially modifies the neutrino phase. We find
that the oscillation probabilities deviate significantly from the corresponding
results in the Schwarzschild spacetime when neutrinos are produced near the
black hole (still in the weak-gravity limit) with non-zero angular momentum and
detected on the same side, i.e., the non-lensed neutrino. Moreover, for a given
gravitational body and geometric parameters, there exists a distance scale for
every energy scale (and vice versa), after which the rotational contribution in
the neutrino phase becomes significant. Using the sun-sized gravitational body
in the numerical analysis of the one-sided neutrino propagation, we show that
even a small rotation of the gravitational object can significantly change the
survival or appearance events of a neutrino flavor registered by the detector,
which is located on the earth. These effects are expected to be prominent for
cosmological/astrophysical scenarios where neutrinos travel past by many
(rotating) gravitational bodies and for large distances. Thus rotational
effects of all such bodies must be incorporated in analyzing oscillations data.
