How fast are neutrinos?

I realise that I have not been making any effort to write at a level understandable to non-physicists. I don't apologise for this, but I thought it was time for another post aimed at a broader audience. There will be some equations, because I want to be quantitative, but nothing too complicated!

Neutrinos are the most elusive of sub-atomic particles, interacting only very weakly with other matter. In fact, according to Wikipedia, the total flux of solar neutrinos (neutrinos produced by the Sun) at the distance of the Earth is about 65 billion neutrinos per square centimetre per second(!), and these pass straight through without us noticing them. Nevertheless, they do very occasionally 'bounce off' an atom, and this allows us to detect them, and do experiments with them. Indeed, neutrinos made headlines last year, when the Opera experiment claimed to measure them moving faster than light. This turned out (as most of us expected all along) to be due to an error in the equipment, rather than a genuine physical effect, which would have implied a breakdown of special relativity.

Judging by various comments on blogs and the like, the whole 'superluminal neutrino' affair did raise one puzzling point for some people. Since the 1990s, we have known that neutrinos have mass, albeit very small, and special relativity tells us (as we will see below), that no particle with mass can travel at the speed of light. Yet all the news stories reported that the neutrinos were expected to travel at the speed of light! The point is that they were expected to travel so close to the speed of light that no difference could be measured, and in this post, I want to explain why.

In relativity, we define the 'gamma factor' for a massive particle to be $\gamma = \frac{E}{m\,c^2}$, where $E$ is the energy of a particle, and $m$ is its mass. This quantity is related to the speed, $u$, at which the particle is travelling, by $$ \gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}} $$ Now we see why massive particles can't travel at the speed of light: if we set $u = c$, we get $\gamma = \infty$, which corresponds to infinite energy!

If we know the mass of a particle, and can measure its energy, then we immediately know the value of $\gamma$, and to find the particle's speed, we need only solve the equation above for $u$. This is a simple exercise in high-school algebra, and we get $$ u = c\sqrt{1 - \frac{1}{\gamma^2}} $$ This looks a bit complicated, but if $\gamma$ is large, we can get a good approximation by $$ u \approx c\left(1 - \frac{1}{2\gamma^2}\right) ~. $$ (This is just a first-order Taylor expansion in $\frac{1}{\gamma^2}$; if you don't know what this means, please just believe that the above is a very good approximation for large $\gamma$ — say, bigger than $10$.) With a little bit of re-arranging, we can write this as $$ \frac{c - u}{c} = \frac{1}{2\gamma^2} ~, $$ which will be convenient for speeds close to $c$.

Using the expression above, we can ask about the typical speeds of neutrinos. First of all, how large are the neutrino masses? Well, we don't actually know, but roughly speaking, $m\,c^2$ can't be bigger than about 1 eV, where eV stands for 'electron Volt', a funny unit of energy which is convenient for high energy physics. The energy of the neutrinos measured at Opera is tens of GeV, or 'Giga-electron Volts', i.e., tens of billions of electron Volts! If we plug in $m\, c^2 = 1\,\mathrm{eV}$ and $E = 10\,\mathrm{GeV}$, we find $\gamma = 10^{10}$, which is quite a large number. We can therefore use our approximation above to get the speed of the neutrinos: $$ \frac{c - u}{c} \approx 5 \times 10^{-21} ~. $$ So the speed of these neutrinos differs from that of light by only 5 parts in $10^{21}$ — this can't possibly be measured!

Before I sign off, I want to mention another interesting source of neutrinos, which is supernovae — the huge explosions which occur when a very massive star collapses in on itself. A famous, and scientifically important, supernova goes by the glamourous name of supernova 1987A, which occurred approximately 160,000 light years away, and was observed on Earth in 1987 (hence the name). Neutrinos from this event were detected with energies of $20 - 40\,\mathrm{MeV}$, where MeV is 'Mega-electron Volt', or one million electron volts. Doing the same calculation as above with $E = 20\,\mathrm{MeV}$, we find $$ \frac{c - u}{c} \approx 1.2 \times 10^{-15} ~. $$ Over a distance of 160,000 light years, how far behind light would these neutrinos lag? This is easy to work out: they will be about $2 \times 10^{-10}$ years later than light emitted at the same time. If I've done my maths right, this is about 6 milliseconds! So even over such a vast distance, we can safely assume that neutrinos travel at the speed of light, without introducing any significant errors into our calculation.

(I should add that the neutrinos from supernova 1987A were actually detected before the light, but this is because the light takes a while to escape from the hot dense plasma of the supernova, whereas the neutrinos basically stream straight through.)