It is now said that neutrinos have mass. If an object has mass then it also emits a gravitational field. I appreciate the neutrinos mass is predicted to be small, but as there are so many produced by our sun and gravity works collectivity should we not be able to detect neutrinos through gravitational differences?
An extract from Wikipedia states:-
"Most neutrinos passing through the Earth emanate from the Sun. About 65 billion ($6.5 \times 10^{10}$) solar neutrinos per second pass through every square centimeter perpendicular to the direction of the Sun in the region of the Earth."
That is a great deal of neutrinos. Would that many not produce a noticeable gravitational signature that we could detect? If we can't detect it does that mean that neutrinos are massless?
Answer
The gravitational field of a fast moving particle is from its energy, not its rest-mass. The source of the gravitational field is the energy divided by c2 if you are using unnatural units, or what used to be called "relativistic mass" before that term fell out of favor.
The neutrinos we observe are moving at essentially the speed of light, so we cannot distinguish their gravitational signature from that of a massless particle. Both a neutrino at 1KeV and an exactly massless fermion at 1KeV have pretty much the same gravitational field. The difference is suppressed by the ratio of the mass to the energy, and it is as hard to detect as the difference in the neutrino speed from the speed of light (which is a more direct way to measure the mass, but also impractical). There are not enough neutrinos which are moving slowly compared to the order .01 eV masses, so that the number of nonrelativistic neutrinos is too small to allow their mass to be measured gravitationally.
In short, the answer is just no.
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