This week the OPERA experiment released a statement about their famous “faster than light” neutrino measurement. In September scientists announced that they had measured the speed of neutrinos traveling from CERN to Gran Sasso and they found that they arrived slightly sooner than they should do according to special relativity. There was a plethora of scientific papers, all kinds of rumors and speculation, and most physicists simply refused to believe that anything had traveled faster than light. After months of diligent study, OPERA announced that they may have tracked down two sources of experimental error, and they are doing their best to investigate the situation.

But until we get the results of OPERA’s proposed studies we can’t say for sure that their measurement is right or wrong. Suppose that they reduce the lead time of the neutrinos from 60ns to 40ns. That would still be a problem for special relativity! So let’s investigate how we can get faster than light neutrinos in special relativity, before we no longer have the luxury of an exciting result to play with.

Special relativity was developed over a hundred years ago to describe how electromagnetic objects act. The electromagnetic interaction is transferred with electromagnetic waves and these waves were known to travel extremely quickly, and they seemed to travel at the same speed with respect to all objects, no matter how those objects were moving. What Einstein did was to say that the constancy of the speed of light was a fundamental law of nature. Taking this to its logical conclusion meant that the fastest speed possible was the speed of light. We can call the fastest possible speed \(s\) and the speed of light \(c\). Einstein then says \(c=s\). And that’s how things stood for over a century. But since 1905 we’ve discovered a whole range of new particles that could cast doubt on this conclusion.

When we introduce quantum mechanics to our model of the universe we have to take interference of different states into account. This means that if more than one interaction can explain a phenomenon then we need to sum the probabilities for all these interactions, and this means we can expect some strange effects. A famous example of this is the neutral kaon system. There two lightest neutral kaons are called \(K^0\) and \(\bar{K}^0\) and the quark contents of these mesons are \(d\bar{s}\) and \(s\bar{d}\) respectively. Now from the “outside” these mesons look the same as each other. They’ve got the same mass, they decay to the same particles and they’re made in equal numbers in high energy processes. Since they look identical they interfere with each other, and this gives us clues about why we have more matter than antimatter in the universe.

Since we see interference all over the place in the Standard Model it makes sense to ask if we see interference with a photon. It turns out that that we do! The shape of the Z mass peak is slightly asymmetric because of interference between virtual Z bosons and virtual photons. There are plenty of other particles that the photon can interfere with, including the \(J/\psi\) meson, and the \(\rho\) meson. In fact, any neutral vector meson with no net flavor will do. Einstein didn’t know about any of these particles, and even if he did he never really accepted the conclusions of quantum mechanics, so it’s no surprise that his theory would require that the speed of light is the fastest speed (that is, \(c=s\).) But if the photon interferes with other particles then it’s possible that the speed of light is slightly lower than the fastest possible speed (\(c<s\)). Admittedly, the difference in speed would have to be very small!

In terms of quantum mechanics we would have something like this:

\[

|light>_{Einstein} = |\gamma>

\]

\[

|light>_{reality} = a_\gamma |\gamma> + a_{J/\psi} |J/\psi> + a_Z |Z> + \ldots

\]

As you can see there are a lot of terms in this second equation! The contributions would be tiny because of the large difference in mass between the massive particles and the photon. Even so, it could be enough to make sure that the speed of light is ever so slightly slower than the fastest possible speed.

At this point we need to make a few remarks about what this small change in speed would mean for experiments. It would not change our measurements of the speed of light, since the speed of light is still extremely fast and no experiment has ever showed a deviation from this extremely fast speed. Unless somebody comes up with an ingenious experiment to show that the difference between the speed of light and the fastest possible speed is non-zero we would probably never notice any variation in the speed of light. It’s a bit unfortunate that since 1983 it’s been technically impossible to measure the speed of light, since it is used in the definition of our unit of length.

Now we know that photons can interfere with other particles it makes sense to ask the same question about neutrinos. Do they interfere with anything? Yes, they can interfere, so of course they do! They mix with neutrinos of other flavors, but beyond that there are not many options. They can interfere with a W boson and a lepton, but there is a huge penalty to pay in the mass difference. The wavefunction looks something like this:

\[

|\nu_e>(t) = a(t)_{\nu_e}|\nu_e> + a(t)_{\nu_{\mu}}|\nu_\mu> + a(t)_{\nu_{\tau}}|\nu_\tau> + a(t)_{We}|We>

\]

(I’ve had to add a time dependence due to neutrino mixing, but it’s essentially no more complicated than what we had for the photon.)

That means that the photon could get slowed down slightly by the interference with other particles (including particles in the vacuum) and that neutrinos could get slowed down more slightly by their interference terms with other particles. And that way we could get neutrinos traveling faster than the speed of light and special relativity could remain intact. (In this description of the universe we can do what used to seem impossible, we can boost into the rest frame of a photon. What would it mean to do that? Well I suppose it would mean that in this frame the photon would have to be an off-shell massive particle at rest.)

Now I’ll sit back and see people smarter than I am pick holes in the argument. That’s okay, this isn’t intended to be a serious post, just a bit of fun! There are probably predictions of all kinds of weird effects such as shock waves and time travel that have never been observed. And there are plenty of bits I’ve missed out such as the muon neutrinos traveling faster than electron neutrinos. It’s not often we get an excuse to exercise our analytic muscles on ideas like this though, so I think we should make the most of it and enjoy playing about with relativity.

Tags: faster than light, neutrinos, OPERA, relativity

It is possible that the particles can travel fastest than light, But the question arises what are the consequences of this. As einstien says that no matter could be stable when it goes to travel faster than light. We have observed only the light photons which can travel at such speed but how about matter?

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That means that the photon could get slowed down slightly by the interference with other particles (including particles in the vacuum)”Massless boson photons detect zero vacuum anisotropy, refraction, dispersion, dichroism, or gyrotropy to 14+ signficant figures, arxiv:0912.5057, 0905.1929, 0706.2031, 1106.1068. Empirically falsifying relativity requires attacking a founding postulate. Nothing constrains massed fermions to wholly obey massless boson observations. Theory already knows where the leak occurs.

The vacuum is observed to be chiral toward mass all the way down (strong interactions are racemic). Hierarchies of symmetry breakings must be manually inserted whenever physical theory postulates vacuum mirror symmetry toward mass. String/M-theory and quantum gravitation have no empirical validation. SUSY; superpartners, solar axions, and proton decay have no empirical validation. The testable failure of theory is physical chirality.

GR postulates the Equivalence Principle, then racemic spacetime curvature (Ashtekar variables split General Relativity into chiral halves; Chern-Simons correction of Einstein-Hilbert action). Einstein-Cartan-Kibble-Sciama gravitation ignores the EP, then chiral spacetime torsion. ECKS + EP = true is GR. ECKS + EP = false is bench top testable. If the vacuum is trace chiral toward femionic matter, enantiomorphic atomic mass distributions vacuum free falling along local minimum action trajectories violate the EP. Test mass opposite shoes fit upon a given vacuum foot with different energies. Crystallography defines maximally opposite shoes: single crystals in one of the 11 enantiomorpic space group pairs of 230 total space groups.

Crystals in paired enantiomorphic space groups P3(1) | P3(2) and P3(1)21 | P3(2)21 are commercially available. They do not contain opposing or racemic screw axes within a given space group. 3(1) is a right-handed threefold helix, 3(2) is a threefold left-handed helix.

http://www.mazepath.com/uncleal/erotor1.jpg

Two geometric Eotvos experiments. Observe the empirical answer.

No composition, dynamic observable, or field contrast violates the EP – Eotvos experiments, Nordtvedt effect; pulsar binaries with solar stars or white dwarfs (orbit, periastron precession, gravitation radiation orbital decay). Physics actively excludes physical chirality from fundamental observables: It is emergent, extrinsic, external; it can be observed and calculated but not measured (how different are your shoes?) If you were the universe, where would you hide the answer?

I very much doubt I’m smarter than you, but…

The neutrinos were clocked at 1.000025c, where c is the known speed of light in a vacuum. The protons in the LHC are accelerated to at least 0.999998c. As more energy is added to those protons, they must approach the true relativistic speed limit asymptotically, right? So then shouldn’t the protons actually be traveling closer to 1.000025c? That seems like it would be noticeable seeing as how the LHC has to time everything precisely… or at least I thought it did.

Oh, and also: cosmic rays must be something like 0.999999999(more nines)x, where x is the true speed limit. Since x >= 1.000025c, we can see that clearly cosmic rays must go faster than the speed of light, and therefore must emit cherenkov radiation.

*walks outside*

*looks up*

I don’t see any.

Hi Xerlec! Your point about accelerators is well taken (I’ve had LEP in the back of my mind, since those particles were even faster than the LHC’s electrons, and more closely monitored.) As particles exceed the speed of light in an accelerator they would give off Cherenkov radiation, and this would presumably have been seen already. On the other hand, suppose a particle exceeding the speed of light gives of Cherenkov radiation faster than energy can be absorbed by the particle. In that case the particle would approach the speed of light, and for an extremely short time it would give up it’s “excess” energy and become subluminal again. Has anyone ever seen such radiation? I don’t think so. Although if it was seen I imagine some engineers would grumble about gas leaking into the vacuum in the LHC, and there would be years of debate and experiment to work out exactly what it was!

Your point about cosmic rays is a bit more subtle. The most likely candidates for cosmic ray sources are things like gamma ray bursts where protons are accelerated from behind by the photons which are being emitted by the burst. Once these protons travel at the speed of light they can no longer be accelerated because the photons cannot catch up to them. As a result these protons would never exceed the speed of light.

Perhaps particles with a non-zero rest mass can appear to travel faster than light, but only because they are taking short cuts through other dimensions of spaces. M-Theory says there are several rolled up space dimensions. So locally, the particles are always travelling slower than light.

Take a sheet of paper, draw dots on the diagonally opposite corners, now bend the paper – the dots could be closer, travelling direct through the 3rd dimension would be faster than following the paper’s surface.

To make use of these short cuts, particles may need special properties – possible the extreme ‘smallness’ & ultra low mass of neutrinos might qualify them?

So an apparent FTL particle may not actually be breaking the light barrier – it just appears to, as we measure the distance it actually travels incorrectly

Hi Gavin, thanks for your comment! The ideas of M theory are very interesting and M theory has a neat trick up its sleeve. Suppose that neutrinos spend much of their time in the other space, then it seems reasonable that most of their interactions would reside in this space as well. That would mean that a neutrino would seem to react very rarely to us when compared to other particles, which is exactly what we see. If the mass of a particle is correlated to its interactions then we would also see neutrinos as being very light. It’s a neat idea, and it would be great if it turned out to be true. Let’s hope we get some more exciting results in the future!

May I point out that the LHC was not part of OPERA

OPERA sent the high-energy beam of muon neutrinos produced at the CERN SPS (Super Proton Synchrotron) in Geneva on a 730 km ride to the LNGS underground laboratory at Gran Sasso.

These muon-neutrinos took their 3 millisecond trip to the LNGS underground laboratory to detect for the first time the appearance of tau-neutrinos from the transmutation or oscillation of muon-neutrinos.

Then observed in bricks of photographic emulsion films interleaved with lead plates.

Hi Samson, thanks for the extra info! It’s important to keep all these distinctions clear, and it’s easy for people to mix up the names/experiments/facilities if we don’t make the effort to explain which parts do and do not belong together.

OK, let me try again.

The amount of mass lost when energy is released in nuclear reactions has been pretty accurately measured as the energy divided by the measured speed of light squared, has it not? And that proportionality constant really ought to be the square of the maximum possible speed. Therefore, the maximum possible speed is the measured speed of light.

Ooh! And what about Hawking radiation? Could it still escape the event horizon of a black hole if it were traveling slower than the maximum possible speed?

Hi Xezlec, I think that’s a great argument! If any nuclear process has had the masses measured to a precision of 1 part in 100,000 without using the speed of light in any of the calculations then this would be excellent evidence that the speed of light is the fastest speed possible. I’m pretty sure these kinds of measurement have been performed so it looks like relativity is probably safe after all.

I am not a physicist, but is if possible that Einstein’s famous equation is wrongly stated, and that c is not the speed of light but the speed of gravity.

Hi John. Einstein’s general theory of relativity does a very good job of explaining how gravity works and he made the theory in such a way that locally, the effects of gravity always propagate at the speed of light. That’s one of the assumptions built into the theory. If the force of gravity is transmitted via a spin-2 gauge boson, then it could be the case that the speed of the graviton is closer to the speed limit, and it might actually be the speed limit!

Would the photon interfering (very slightly) with the Z boson etc. be equivalent to saying the photon has a very small mass?

Hi Stephen, that’s a good question! It wouldn’t mean that the photon had mass, because Einstein’s photon state would still be massless (as required by gauge invariance.) It would mean that if we could take a snapshot of the real photon at different points in time we’d get different values for its mass as it went into states of off-shell Z, off-shell J/Psi etc. This is one of the weirder aspects of quantum mechanics that causes a lot of conceptual problems. The superposition of states is not the same as being “between states” somehow. It’s all possible states at the same time until something comes along to collapse the wavefunction.