Gravitational Effects on Quantum Coherence in Neutrino Oscillation

In this paper, we investigate the quantum coherence for two flavor neutrinos propagating in a Schwarzschild metric. In fact, this issue is explored both qualitatively via calculating the parameter $K_{3}$ in Leggett-Garg inequality (LGI) and also quantitatively by evaluating the $l_{1}$-norm, ${\cal C}(\rho)$. Using the weak field approximations, we show that the gravitational effects decrease the maximum value of $K_{3}$ for some intervals of energy such a way that there is no violation, while it leaves the maximum amount of the quantum coherence, ${\cal C}(\rho)$ unchanged.