# 15.2. Equation of Motion in Gravitational Field¶

Cardall and Fuller derived the formalism for neutrino oscillations in gravitational field in [Cardall1996].

They derived the Schrodinger like equation of motion by generalizing the gravitational effect to some potential in Dirac equation,

$i \frac{d}{d\lambda} \chi = \left( \frac{M_f^2}{2} + \vec v \cdot \vec A_f\mathscr P_L \right) \chi,$

where $$\lambda$$ is an invariant parameter for world line of neutrinos, which is usually choose to be proper time $$\tau$$ ($$d\tau = \sqrt{ g_{\mu\nu} dx^\mu dx^\nu}$$), $$\chi$$ is the wave function which has two amplitudes

$\begin{split}\chi = \begin{pmatrix} \psi_e \\ \psi_x \end{pmatrix},\end{split}$

$$M_f^2$$ is the mass matrix in flavor basis, $$\mathscr P_L$$ is the left-handed projection operator.