Slow photon delay and the neutrino velocity

Abstract

Starting from the coordinate system used by Einstein to find the bending of light rays by gravitational fields we calculate the effect of the Earth gravitational energy along a hypothetical photon null path on the geoid non inertial system. There is an energy term, relative to an inertial system, which may be interpreted as a small time-relative "dressed" physical rest mass correction to the photon null mass. This relative gravitational potential energy determines a proper time delay proportional to the laboratory non inertial flying time interval along the trajectory with a small factor 5.276x10-5. We also use a geometric technique based on Ehresmann connections to compare the parallel transport of the photon vector and the neutrino spinor on null trajectories. The result is that there is a theoretical neutrino time delay from its inertial path which is half the delay of the photon. Applying these delays to a hypothetical particle trajectory from the CERN SPS/CNGS target to the LNGS OPERA detector we find that the difference between the experimental and theoretical results falls within the reported experimental errors. The delayed neutrino is faster than the delayed photon but its speed is smaller than the fundamental constant c. The results also indicate that neutrino trajectories, as photon trajectories, are affected by gravitational fields but with smaller effects.