Large Changes of Atmospheric Mixing Angle $\theta_{23}$ with Four Flavor Mixing from Planck Scale Effects

We consider the effects of Planck scale on four flavour neutrino mixings. The gravational interaction at $M_{x}=M_{\rm planck}$, we find that for degenerate neurino mass order, the Planck scale effects changes the mixing angle $\theta'_{23}$, $\theta'_{12}$ values and $\theta'_{13}$, $\theta'_{14}$, $\theta'_{34}$, $\theta'_{24}$ are unchanged above the GUT scale. In this paper, we study neutrino mixing in four flavor above the GUT scale.