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### Faster than light?

The latest news from Opera is that some of their neutrinos are breaking the speed limit. That’s not only a violation of road speed limits, it’s a violation of the universal speed limit, and the laws that it breaks are the fundamental laws of physics. So let’s spend a few minutes speculating wildly about what would happen if these neutrinos are actually going faster than the speed of light. (As Katherine said in her blog post, there will be a lot of serious discussion once we have more information.)

So, why can’t anything go faster than the speed of light? When a massive body approaches the speed of light, it’s effective mass increases. That means that every time we apply a force to the particle to accelerate it, the change in momentum is the same, but the change in speed decreases. The particle asymptotically approaches the speed of light, and never goes faster. This has been tested in the lab over and over again. Fortunately the electron is extremely light, so we can see it move at amazingly high speeds. (In fact the difference in speed between a photon and an electron accelerated at LEP is one part in 1011.)

What about a massless particle? Can that go faster than the speed of light? That’s actually a trickier question than it sounds. One approach would be to take a part with mass $$m$$, apply a force, $$F$$ to it for an infinitesimal amount of time and make the mass approach 0. When that happens you find that the particle should go at the speed of light. But as any mathematician will tell you there’s a difference between a value approaching a limit and a value being the same as another number. The equation also looks a little suspect:

$\lim_{m\to 0, \delta t\to 0}\left[ \frac{F}{m}\delta t = c \right]$

What does it really mean to say that the time interval $$\delta t$$ approaches zero in the context of quantum mechanics? If you want to make the time interval very small then the energy transfer must increase in accordance with the uncertainty principle.

You can work through the math of special relativity and come to the conclusion that going faster than the speed of light would violate causality (as well as common sense), but that’s circular logic, since one of the assumptions of special relativity is that the speed of light is a constant.

And again, quantum mechanics is at odds with special relativity, since Bell’s inequality shows that the universe “knows” what happens in regions of space that are not causally connected. The experiment goes like this: Two particles (usually photons) are created in an entangled state so that, for example, one has spin “up” and the other has spin “down”. If you measure the direction of spin of one particle with respect to a given axis you instantly know the direction of spin of the other with respect to the same axis. Now separate the spin measuring devices by a large distance and perform the experiment again. You’ll find that the directions of the spins agree (one “up” and one “down”), even though there has not been enough time for one particle to “tell” the other one what spin to have. So far it all seems fairly reasonable and even deterministic.

That is until the next step of the experiment is performed: the direction of the axis is not determined until the particles are halfway to the measuring devices. If we have two experimenters Alice and Bob, one at each measuring device, then it doesn’t matter what Alice chooses to be “up”, if Bob chooses the same axis then he’ll measure “down” whenever Alice measures “up” and vice versa. (If Bob picks an axis at 90 degrees to Alice’s then he’ll get garbage. Exactly half the time he’ll get the same as Alice by dumb luck.) By picking the “up” direction while the particles are in flight, we break any causal connection between the measuring devices. The conclusion is that Alice’s measurement determines Bob’s measurement, even though there’s no way to transfer that information without going faster than the speed of light. Quantum mechanics doesn’t respect locality, and details about a wavefunction can travel faster than the speed of light. (Quantum mechanics has an ironic sense of humor though, since this mechanism can’t be used to send real information.)

Why not look at some other massless particles to see how they behave? Unfortunately there aren’t any more that we can study, since gluons are always bound up in hadronic states, and they’re only theoretically massless. (We haven’t yet ruled out very small masses with experiment. Small masses would cause problems for QCD, but that’s a different story…) Until recently neutrinos were considered massless, but we now know that isn’t the case, and each year we get better knowledge about the mass differences of the various neutrinos.

Just to confuse things further, quantum mechanics allows particles to travel faster than the speed of light (and photons to travel slower than the speed of light) as long as the uncertainty principle isn’t violated. This is how an electron can emit a virtual photon and still catch up to it to reabsorb it before anyone notices. If you try to catch particles violating these laws by firing other particles at them, they suddenly conspire to come out with the correct masses and speeds after all.

So what can we do if these neutrinos are going faster than the speed of light? Throw out special relativity? Adjust it somehow? Appeal to quantum mechanics? We have an immense mountain of evidence in favor of special relativity, and it’s passed about as many experimental tests as any theory in the history of science. (Okay, technically we have no evidence in favor of special relativity, but we have an immense amount of evidence which has failed to falsify it!) Abandoning special relativity would also upheave huge areas of physics, notably electromagnetism, which has been through all kinds of stringent testing.

Let’s leave special relativity alone for now and see if we can extend it somehow. It’s possible that there extra dimensions curled up that we can’t usually see. If that’s the case then there may be a shorter path between Gran Sasso and CERN that isn’t available to most particles, but which can be used by neutrinos. That’s an interesting proposition as it could explain why the neutrinos couple so weakly- they could spend most of their time interacting in the other dimensions and this could “dilute” the coupling in our 3+1 dimensions.

In the distant past, after the big bang, the universe was full of neutrinos (and lots of other particles!) and these neutrinos lingered until today and they’re known as relic neutrinos. These relic neutrinos form a field that occupies all the space around us (in the same way as the cosmic microwave background radiation does.) It’s possible that neutrinos traveling through “empty” space could interact with relic neutrinos, shaking up their wavefunctions. According to special relativity these neutrinos couldn’t transmit any information faster than the speed of light, but as we already know, quantum mechanics isn’t local. Could a field of relic neutrinos account for faster than light travel? Maybe, but I doubt it, as we’d have probably seen other effects elsewhere before now.

Special relativity says that a massive particle cannot travel faster than the speed of light, and that massless particles must travel at the speed of light, but it doesn’t say anything about particles with imaginary mass. Take one of these exotic particles, known as tachyons, and apply special relativity and you’ll find that it goes faster than the speed of light! Could the neutrinos be communicating with these particles? Perhaps…

Maybe we can turn to general relativity for help. The huge gravitational field of the Earth should not be neglected when calculating the length of the path from CERN to Gran Sasso. The distance may be smaller than we think. It’s something that would be hard to verify directly, as the matter between the two points is not completely transparent to light. Calculating the true interval would require knowledge of the gross structure of the Earth and the path taken by the neutrinos, which isn’t insurmountable, but the calculation isn’t trivial. Without seeing any papers on the subject it’s hard to make a statement about it, but if the true interval is smaller than we expect by one part in 50,000 this could explain the result, and the speed of light would live to fight another day.

A lot of people have commented on the timing measurement itself. It’s calibrated with GPS, which is precise to within about 10 nanoseconds. But we need to be careful when dealing with differences in height. Once again, general relativity comes in and messes things up, and 1 second at sea level is not the same as 1 second in space or 1 second deep underground. This means that the accuracy might not be spot on, unless we take special precautions. (In fact satellites are continually updated to make sure they don’t go too far afield. Blindly trusting their on board clock isn’t enough when gravity plays around with the local time.) Synchronizing two distant clocks is a long standing problem for physicists, even in hypothetical scenarios and thought experiments. How could we guarantee that two clocks agree that lunchtime at CERN is the same as lunchtime at Gran Sasso? It’s easy to imagine sending pulses of light to a satellite and counting how many arrive between each measurement, but that only tells us about the time differences at a given site, and not about the time differences between the sites, which is what we want. I suppose the way to measure synchronicity is to perform the same experiment in reverse, making a different beam of neutrinos travel the same journey in reverse. If they arrive 60ns later than expected then it shows an asynchronicity between the clocks. But that’s hardly practical!

So the conclusion? There is none really. This is just a fun post of wild speculation, not to be taken too seriously. So let’s wait for more information and in the meantime we can finish on a joke:

So the neutrino leaves.
The barista replies “We don’t serve faster than light particles here at Gran Sasso”
A neutrino passes through a coffeeshop and asks the barista for “An espresso, and make it quick!”