Effects of gravitational lensing on neutrino oscillation in $ \gamma $-spacetime

We study the effects of gravitational lensing on neutrino oscillations in the $\gamma$-spacetime which describes a static, axially-symmetric and asymptotically flat solution of the Einstein's field equations in vacuum. Using the quantum-mechanical treatment for relativistic neutrinos, we calculate the phase of neutrino oscillations in this spacetime by considering both radial and non-radial propagation. We show the dependence of the oscillation probability on the absolute neutrino masses, which in the two-flavor case also depends upon the sign of mass squared difference, in sharp contrast with the well-known results of vacuum oscillation in flat spacetime. We also show the effects of the deformation parameter $\gamma$ on neutrino oscillations and reproduce previously known results for the Schwarzschild metric. We then extend these to a more realistic three flavors neutrino scenario and study the effects of the parameter $\gamma$ and the lightest neutrino mass while using best fit values of neutrino oscillation parameters.