Are neutrinos affected by gravity?
If not, could that be a plausible reason for a neutrino taking a shorter path than light, since light is affected by gravity?
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Everything is affected by gravity. Gravity warps of space-time according to the Einstein Field Equations, and traveling on "geodesics" (shortest path curves) on that curved surface is how gravity is manifested.
Thinking as though there is some sort of euclidean space underneath the non-euclidean space-time in which neutrinos can take a more direct "straight line" between the points is not at all supported. Everything we know of travels on this curved spacetime.
Also, speaking more Newtonianly, neutrinos do seem to have mass in the more classic sense.
Both neutrinos and anti-neutrinos are affected by gravity (same magnitudes and direction).
The 1987 Supernova event was the first instance of neutrino and anti-neutrino detection of a source outside of our solar system. Not only were they detected, but the neutrino event recorded occurred a short interval of time AFTER the visible light and gamma ray burst arrived from the same supernova. This was the first hint that neutrinos might actually have mass. Formerly, they had been considered to be massless particles.
The arrival times of the anti-neutrinos for this event were close enough to the time for the neutrinos that any curvature of the path lengths by gravity had to be the of the same magnitudes and directions. This has been cited as proof of the idea that particles of antimatter are affected by ordinary gravitation in the same manner and direction as matter.