The question addressed here is the following: if a single weak-type neutrino with a definite energy E is a mixture of three mass-type neutrinos (so it does not have a definite mass), why is it that the three mass-type neutrinos travel at different speeds?
The speed v of a particle in Einstein’s relativity can be written in terms of the particle’s mass m and energy E [that’s total energy, i.e. motion energy plus mass-energy E=mc2] and the speed of light c as
- v = c ( 1- [m c2/E]2 )1/2
Recall the raised 1/2 means “take-the-square-root”. If the particle has very high velocity and its total energy E is much, much larger than its mass-energy mc2, then
- v = c (1 – [m c2/ E]2/2 + …)
where the dots mean that this formula isn’t exact but is an extremely good approximation for E very large. [In other words, the velocity of a particle traveling near the speed of light differs from the speed of light by a shift equal to half the square of the ratio of the particle’s mass-energy to its total energy.] So, from this formula you can see if two neutrinos have different masses m1 and m2 but the same very large energy E, then their velocities will differ by a tiny amount.
Let’s see what this means. All the measured neutrinos from the 1987 supernova arrived on earth within about 10 seconds of one another. Think about an electron neutrino emitted from the supernova with an energy of 10 MeV (an MeV is a million eV [electron-volts], or 1/1000 of a GeV; read here for the definition of these terms). Well, that electron neutrino was a mixture of neutrino-1, neutrino-2 and neutrino-3, each of which traveled with a slightly different speed! Is this something we would have noticed? We don’t precisely know the masses of the neutrinos, but suppose that neutrino-2 has a mass-energy of 0.01 eV (electron-volts; see this article for the definition) and neutrino-1 has a mass-energy of 0.001 eV. Then their two velocities, remembering that they have the same energy, would differ from the speed of light and from each other by less than a part in a hundred thousand trillion
- v1 – v2= c [ (m22 – m12) c4/ 2 E2 + … ] = 0.0000000000000000005 c
(all equations accurate and precise to a one percent or better.) That velocity difference would mean the neutrino-2 part and neutrino-1 part of the original electron-neutrino would both arrive at the earth within a millisecond of each other — a undetectable difference for a variety of technical reasons. (Keep in mind that OPERA claims a difference of neutrino speeds from the speed of light of one part in 100,000, a much, much larger effect, though the measurement involves neutrinos at an energy a few hundred times larger than those from the supernova.)