Invariance Violation Extends the Cosmic-Ray Horizon?

We postulate that the energy-momentum relation is modified for very high energy particles to violate Lorentz invariance and the speed of photon is changed from the light velocity c. The violation effect is amplified in a sensitive way to detection through the modified kinematical constraints on the conservation of energy and momentum, in the absorption process of γ-rays colliding against photons of longer wavelengths and converting into an electron-positron pair. For γ-rays of energies higher than 1013 eV, the minimum energy of the soft photons for the reaction and then the absorption mean free path of γ-rays are altered by orders of magnitude from those conventionally estimated. Consideration is similarly applied to high-energy cosmic-ray protons. The consequences may require the standard assumptions on the maximum distance from which very high energy radiation can travel to be revised.