Physics Question #1926

Donald, a 22 year old male from Halifax asks on February 23, 2004,

If neutrinos have mass, as now seems to be the case, they cannot travel at the speed of light … explain why not?

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The answer

Doug Hube answered on March 2, 2004

Short answer: they do have a tiny bit of mass, and hence, yes they cannot quite go at the speed of light, but they get pretty close.

The details:Until a few years ago it was generally believed that neutrinos had zero rest mass. Or, at least, there was no experimental evidence nor was there any convincing theoretical argument that neutrinos have a finite mass. That has changed as a result of several experiments including, in particular, observations made at the Kamiokande laboratory in Japan and others made at the Sudbury Neutrino Observatory. It is now claimed that there is a 99.995 percent certainty that neutrinos have small but finite masses.

Each of the three known leptons -- electron, muon and tau particle -- has an associated neutrino, as does each of the antiparticles of those leptons. Hence, there are three types of neutrinos plus three antineutrinos. Each of those neutrino/antineutrino pairs has a mass that differs from the other two. The masses are still very uncertain and, in general, only an upper limit is given for each.

Leaving out many of the details ... The most common neutrino is the one associated with the electron. It has a mass which is much smaller than that of the electron. The energy equivalent of the electron mass is 0.511 MeV that is, 511000 electron volts. The mass of the electron neutrino is much smaller, probably less than 1 electron volt, and one group of researchers suggests that it is less than 0.24 electron volts: Almost zero, but not quite, and too small to account for the hidden/dark matter in the Universe.

A photon, the particle of light, that really does have zero rest mass does travel precisely at the speed of light. If a neutrino really had zero mass it, too, would travel precisely at the speed of light. However, the neutrino has a small, non-zero rest mass and, therefore, travels at less than the speed of light. When discussing motion of matter near the speed of light one must use the principles of Special Relativity. There are several mathematical relationships that are appropriate in Special Relativity and that we could employ here. Hence, there are several routes that one could follow from this point.

Consider a neutrino in motion, a neutrino that has been released, say, during the sequence of nuclear fusion reactions that power the Sun. That neutrino has some total energy, E, or an effective/relativistic mass, m, where E = mc**2. The neutrino also has a rest mass, m', and a corresponding rest energy, E', where, again, E' = m"c**2.

The ratio E'/E = m'/m = SQRT[1 - (v/c)**2]

Consider the following important-for-us situation. Approximately 3-percent of the power output from the Sun, 4x10**26 watts is carried away by neutrinos, i.e. 12x10**24 watts carried by neutrinos. Solar models predict that (all these numbers are approximate) 10x10**37 neutrinos are emitted by the Sun each second. Hence, each neutrino has an energy of E = 1.2x10**-13 joule.

If the rest energy is, say, 1 electron volt, that is E' = 1.6x10**-19 joule. Therefore E'/E = 10**-6 (approximately) ... a very small number. Hence, SQRT[1 - (v/c)**2] = 10**-6 and solving for the speed, v, demonstrates that v is only marginally different from c, the speed of light.

In effect, the energy of the neutrino, really its kinetic energy has a finite value because of its very high speed which compensates for the very low mass of the particle. In an analogous classical situation where kinetic energy equals 1/2(m)(v**2) a very tiny, low mass particle would have a significant kinetic energy only if traveling at a very high speed. A meteoriod with a mass of 1 gram but traveling at 30 km per second, has the energy of a small car traveling at normal highway speeds. At that speed the 1 gram particle has great energy, whereas it would have negligible energy if thrown by hand. The equivalent statement applies to the neutrino: very small mass combined with very high speed results in 'significant' energy.

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