Trajectories of astroparticles in pseudo-Finsler spacetime with the most general modified dispersion

Abstract

Finsler geometry is a natural and fundamental generalization of Riemann geometry, and is a tool to research Lorentz invariance violation. We find the connection between the most general modified dispersion relation and a pseudo-Finsler structure, and then we calculate the arrival time delay of astroparticles with different modified dispersion relations in the framework of Finsler geometry. The result suggests that the time delay is irrelevant with the exact form of the modified dispersion relation. If the modified term becomes 0 when $E=p$, there is no arrival time difference, otherwise the time delays only depend on the Lorentz violation scale and the order at which the Lorentz invariance breaks.