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The conventional way physicists describe neutrinos is that they have a very small amount of mass which entails they are traveling close to the speed of light. Here's a Wikipedia quote which is also reflected in many textbooks:

It was assumed for a long time in the framework of the standard model of particle physics, that neutrinos are massless. Thus they should travel at exactly the speed of light according to special relativity. However, since the discovery of neutrino oscillations it is assumed that they possess some small amount of mass.1 Thus they should travel slightly slower than the speed of light... -- Wikipedia (Measurements of Neutrino Speed)

Taken at face value, this language is very misleading. If a particle has mass (no matter how small), its speed is completely relative, and to say that neutrinos travel close to the speed of light, without qualification, is just as incorrect as saying electrons or billiard balls travel close to the speed of light.

So what is the reason everyone repeats this description? Is it because all the neutrinos we detect in practice travel close to the speed of light? If so, then I have this question:

Neutrinos come at us from all directions and from all sorts of sources (stars, nuclear reactors, particle accelerators, etc.), and since they have mass, just like electrons, I would have thought we should see them traveling at all sorts of speeds. (Surely some cosmic neutrino sources are traveling away from the earth at very high speeds, for example. Or what about neutrinos emitted from particles in accelerators?)

So like I said at the start: Where are all the slow neutrinos? And why do we perpetuate the misleading phrase: 'close to the speed of light' (i.e. without contextual qualification)?

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    $\begingroup$ The contribution of nonrelativistic neutrinos in beta decay can in principle be detected, it can be used to measure the neutrino mass: katrin.kit.edu $\endgroup$ – Count Iblis Jul 8 '16 at 20:36
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    $\begingroup$ @CountIblis: They aren't detecting the neutrinos, though, just the missing energy momentum. Having said that, it's a beautiful precision experiment. $\endgroup$ – CuriousOne Jul 8 '16 at 20:50
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    $\begingroup$ Does anyone know why part of the comment trail was deleted? It was a useful and relevant dialog. $\endgroup$ – Physics Footnotes Jul 8 '16 at 22:11
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    $\begingroup$ If neutrinos have very low mass then if they're not moving fast relative to the detection apparatus then, relative to it, they have very tiny energy. Does this make such neutrinos very hard to detect, thus meaning we only ever see rather fast neutrinos? This isn't a rhetorical comment: I'm not a particle person so I don't know, but it would explain the lack of detection of slow neutrinos nicely. $\endgroup$ – tfb Jul 8 '16 at 22:46
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    $\begingroup$ @PhysicsFootnotes Much better. Also, I deleted some of the comments because they should have been posted as an answer. The comments are meant for suggesting improvements, requesting clarifications, and to some extent, linking to related resources - that's all. Comments are always transient and may be deleted once their purpose has been served. Any useful information from the comments should be incorporated into the question or an answer. $\endgroup$ – David Z Jul 9 '16 at 6:13
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Strictly speaking, it is indeed incorrect that neutrinos travel at "close to the speed of light". As you said, since they have mass they can be treated just like any other massive object, like billiard balls. And as such they are only traveling at nearly the speed of light relative to something. Relative to another co-moving neutrino it would be at rest.

However, the statement is still true for almost all practical purposes. And it doesn't even matter in which reference frame you look at a neutrino. The reason is that a non-relativistic neutrino doesn't interact with anything. Or in other words: all the neutrinos you can detect necessarily have to have relativistic speeds.

Let me elaborate. Since neutrinos only interact weakly they are already extremely hard to detect, even if they have high energies (> GeV). If you go to ever lower energies the interaction cross-section also decreases more and more. But there is another important point. Most neutrino interaction processes have an energy threshold to occur. For example, the inverse beta decay

$$ \bar\nu_e + p^+ \rightarrow n + e^+$$

in which an antineutrino converts a proton into a neutron and a positron, and which is often used as a detection process for neutrinos, has a threshold of 1.8 MeV antineutrino energy. The neutron and the positron are more massive than the antineutrino and the proton, so the antinneutrino must have enough energy to produce the excess mass of the final state (1.8 MeV). Below that energy the (anti)neutrino cannot undergo this reaction any more.

A reaction with a particularly low threshold is the elastic scattering off an electron in an atom. This only requires a threshold energy of the order of eV (which is needed to put the electron into a higher atomic energy level). But a neutrino with eV energies would still be relativistic!

Assuming that a neutrino has a mass of around 0.1 eV, this would still mean a gamma factor of $\gamma\approx 10$. For a neutrino to be non-relativistic it would have to have a kinetic energy in the milli-eV range and below. This is the expected energy range of Cosmic Background Neutrinos, relics from the earliest times of the universe. They are so to say the neutrino version of the Cosmic Microwave Background. So not only do non-relativistic neutrinos exist (according to mainstream cosmological models), they are also all around us. In fact, their density at Earth is $\approx$50 times larger than neutrinos from the Sun!

There is a big debate if they can ever be detected experimentally. There are a few suggestions (and even one prototype experiment), but there are differing opinions about the practical feasibility of such attempts. The only process left for neutrinos at such small energies is neutrino-induced decay of unstable nuclei. If you have an already radioactive isotope, it's like the neutrino would give it a little "push over the edge". The $\beta$-electron released in the induced decay would then receive a slightly larger energy than the Q-value of the spontaneous decay and the experimental signature would be a tiny peak to the right of the normal $\beta$-spectrum. This will still be an extremely rare process and the big problem is to build an apparatus with a good enough energy resolution so that the peak can be distinguished from the spectrum of normal spontaneous nuclear decay (amidst all the background). The Katrin experiment is trying to measure the endpoint of $\beta$-spectrum of Tritium in order to determine the neutrino mass. But under very favorable circumstances they even have some chance to detect such a signature of cosmic background neutrinos.

TL;DR: In fact there are non-relativistic neutrinos all over the place, but they they interact so tremendously little that they seem to not exist at all.

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    $\begingroup$ re; "and the big problem is to distinguish it from normal spontaneous nuclear decay." Posit for a moment then that there is no such thing as normal spontaneous nuclear decay;, but rather that all events appearing to be such are actually triggered by slow neutrinos. What would the consequences to nuclear physics be? Would such a possibility be consistent? Is there an energy density of slow neutrinos calculable based on that assumption? $\endgroup$ – Pieter Geerkens Jul 9 '16 at 20:56
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    $\begingroup$ If there are so many slow neutrinos, any chance they contribute significantly to dark matter despite being so tiny? $\endgroup$ – John Dvorak Jul 10 '16 at 8:25
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    $\begingroup$ @PieterGeerkens: sorry, my wording was incorrect and I changed the paragraph in my answer. Spontaneous and neutrino-induced $\beta$-decay are two distinct processes (the latter being a two-body reaction in which the $\beta$-electron always gets a fixed amount of energy, while the former is a three-body decay which produces a continuous spectrum of $\beta$-energies). I meant the difficulty to distinguish this process experimentally, which is extremely challenging because the structures you are looking for are smaller than the resolution of your apparatus. $\endgroup$ – Sentry Jul 10 '16 at 16:24
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    $\begingroup$ @JanDvorak: on a cosmological scale the C$\nu$B neutrinos do play a certain role, but they are not the kind of dark matter everyone is looking for. Even if C$\nu$B neutrinos are non-relativistic, they are still extremely fast (several hundred km/s) and do not cluster significantly under gravity. So they cannot form halos in the way one would expect of dark matter. $\endgroup$ – Sentry Jul 10 '16 at 16:34
  • $\begingroup$ @Sentry Thanks very much for this excellent detailed answer. I'd also like to hear your opinion about the following paper which I have found very helpful. In particular, does it look reliable to you, and/or would you recommend something else along these lines? J.A. Formaggio, G.P. Zeller "From eV to EeV: Neutrino Cross Sections Across Energy Scales" arxiv.org/pdf/1305.7513v1.pdf $\endgroup$ – Physics Footnotes Jul 15 '16 at 3:04
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The experimental detection of slow neutrinos is indeed a big problem, but one that is very important.

The cosmic neutrino background is at a temperature of around 2K and likely to consist of non-relativistic neutrinos for plausible neutrino rest masses - with a density of around 340 cm$^{-3}$ (all flavours). It is at this low temperature for precisely the reason you suggest - it was emitted at a redshift of around $10^{10}$.

There is of course indirect evidence for these neutrinos from the cosmic microwave background (Follin et al. 2015), but efforts to directly detect these neutrinos is underway - see Faessler et al. (2016) and KATRIN.

An interesting thought, is that if you could somehow get your apparatus onto a moving platform, then there would be an appreciable change to the C$\nu$B detection efficiency in the "foward" facing direction if you could accelerate to relativistic velocities. I suppose this is the opposite scenario to your question - you would be making the slow neutrinos relativistic by your relative motion.

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