  Kinematics in Finsler space is used to study the propagation of ultra high
energy cosmic rays particles through the cosmic microwave background radiation.
We find that the GZK threshold is lifted dramatically in Randers-Finsler space.
A tiny deformation of spacetime from Minkowskian to Finslerian allows more
ultra-high energy cosmic rays particles arrive at the earth. It is suggested
that the lower bound of particle mass is related with the negative second
invariant speed in Randers-Finsler space.
