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# Talking of Faster Than Light Neutrinos

A heuristic approach suggests that the distance through the rock travelled by the OPERA experiment neutrinos might be slightly less than the ideal value.

In late September 2011, the science community was presented with experimental evidence which seems to challenge the understanding that nothing can travel faster than the speed of light in vacuum. Three years of data-gathering for muon neutrino “ghost particles” travelling 730 kilometres (km) through the Earth from a particle accelerator run by CERN at Geneva to a detector at Gran Sasso in Italy showed to a high statistical significance that their time of flight was less than expected. The neutrinos seemed to be arriving on average about 60 nanoseconds (nS) earlier than if they had travelled the same distance at the speed of light through empty space.

The equivalent distance discrepancy was small, just 18 metres at light-speed, but still much larger than the estimated maximum 20 centimetres error in positioning, carefully measured using satellite technology. Since all the possible timing delays and timing errors had been taken into account to a statistical accuracy of less than 10 nS and the clocks between starting and finishing locations had been synchronised to 2 nS, the experimental physicists were faced with a three-way choice of possible explanations. Either the neutrinos were truly faster-than-light, or there had been an unknown source of error in the measurements, or the physics of neutrinos travelling inside Earth was slightly different due to some other reason.

Not surprisingly, the first option is now generating the most heat in the world outside CERN and Gran Sasso. Einstein's theory of special relativity, which says that nothing with any mass can travel faster than the speed of light, came out in 1905 and has been the gold standard ever since. Nobody yet knows for sure whether neutrinos, which come in three types or “flavours”, have a tiny mass or not. If so, it is less than a millionth of the mass of an electron. So they should be able to travel at speeds of very close to that of light, but not above it.

If you were older than six at the time, you might recall that in 1987 this theoretical speed-of-light prediction was put to the test when a star in a nearby dwarf galaxy called the Large Magellanic Cloud (LMC) exploded into a supernova. Neutrinos were emitted instantaneously during the collapse of the star's core, while light fought to get out and was emitted about three hours after the neutrinos. As the distance travelled was a cool 168,000 light-years, the neutrinos and the following light photons must have been travelling at exactly the same speed as each other to arrive here, as they did, with that time gap still intact.

Outer space is not inner Earth, however, which brings us to the (maybe combined) second and third possibilities. The distance measurement, though carried out very precisely, only gives a theoretical solution to the shortest path through the rock at depths of up to 10 km below Earth's nominal surface between CERN and Gran Sasso. It does this using the time-honoured technique of triangulation using an advanced form of the GPS satellite navigation system. What if spacetime consists of something which inside Earth happens to be measurably different to the spacetime environment outside Earth?

One thing we can't do is shine a laser light to find out, as there's the small matter of 730 km of solid rock in the way. Neutrinos can penetrate rock unhindered to a phenomenal extent, but we don't know how they propagate through spacetime which is cluttered up with all the electrons, protons and neutrons of familiar molecular compounds. Do they “fail to see” point particles by subtracting their tiny background presence from the relatively enormous volume of spacetime surrounding them? Or could there be a more esoteric connection involving a kind of quantum reality, which at increasing depths into the Earth enables them to tunnel through the light-speed limit when there are no speed cameras watching?

We don't yet know. Speculative physics makes pleasant reading, but provides no definite answers. However, what we can do is use a heuristic approach: assuming that the GPS triangulation is giving the correct theoretical solution on the outside, what part of the geometry inside Earth would we need to change to produce the same result?

The diagram above shows the details of the calculation involved, which is fairly simple trigonometry. If R = 6370 km is Earth's average radius, and *a* = 730/2 = 365 km is the half-distance from the CERN source to the Gran Sasso detector, the ideal depth *b* (also in km) below Earth's surface at that midpoint is given by the Pythagorean relation and a little thought as:

(1) 1 – *b*/R = √ ( 1 – *a*^{2}/R^{2} ) : from which *b* = 10.466 km

For small *b*/R, as in this case, we can approximate the square root term by 1 – *a*^{2}/2R^{2}, so that (1) becomes a much simpler:

(2) *b*/R = *a*^{2}/2R^{2} : from which* b* = 10.457 km

A more accurate calculation gives the difference between exact and approximate *b*/R values as 8.60 metres. This is an interesting coincidence, as it closely represents half of the distance measurement deficit, say ε, of about 18 metres in the 730 km travelled by the neutrinos – that is, the midpoint value of the deficit assuming it to be equal on both halves of the journey. So it looks as if the distance deficit could be a simple matter of trigonometry within Earth not working as expected.

The above two equations can be matched up nicely by squaring the first one, then using a little algebra to expand the ( 1 – *b*/R)^{2} term on the left-hand side, remove the 1s on both sides and change the signs to give:

(3) 2*b*/R – *b*^{2}/R^{2} = *a*^{2}/R^{2}

Conveniently, this comes out at twice the value of equation (2), so the distance deficit ε/R over the entire path travelled by the neutrinos can be identified with *b*^{2}/R^{2}, the exact difference between twice (2) and (3). Then from (2) we can substitute for *b*/R to give:

(4) ε/R = *b*^{2}/R^{2} = *a*^{4}/4R^{4} : from which ε = 17.17 metres

This is well within the large margin of error in the original time deficit and equivalent distance seen by the OPERA experiment result of 60 nS/ 18 metres +/- 10 nS/ 3 metres. Assuming then that (4) turns out to be true, special relativity and the light-speed limit are both safe, as it is Euclidean space which is somehow being violated.

A final step in the procedure replaces *a*/R in (4) with its exact equivalent of sin θ, where θ is the half-angle between the neutrino source and detector referred to the centre of Earth, as shown in the diagram. The distance deficit ε/R is then given by a rather more aesthetic:

(5) ε/R = sin^{4} θ/4

Of course, so far this only works for one value each of ε and θ. At present it isn't a general law, just a nice simple maths fit, with no theoretical physics behind it as yet. So it would need to be experimentally tested for several values before anyone could be sure of its validity or otherwise. Are we in a position to be able to do this?

Since 2005, a similar experiment to OPERA called MINOS, located in the U.S., has been sending muon neutrinos about 735 km from the particle accelerator at Fermilab near Chicago to a detector inside the Soudan mine in Minnesota. They had previously spotted a possible speed anomaly in their data, but with too high an uncertainty to be sure it was a real effect. With a timing upgrade to their equipment, they expect to be able to review their existing data and come up with a more decisive answer in four to six months. If the above analysis is correct, they should obtain the same result as OPERA of 60 nS/ 18 metres time and distance deficit.

Across the other side of the world, there is the Tokai to Kamiokande neutrino link T2K in Japan, but as this is over a much smaller distance of about 295 km the highly nonlinear equation (5) predicts an anomaly of only 45 centimetres, just over 1 nS at light-speed and essentially undetectable.

So the answer is yes and no. In the meantime, a variety of seriously exotic theoretical approaches to OPERA's faster-than-light paradox which involve extra dimensions, tachyons, string theory or whatever else comes to mind have been given a boost following the excitement generated by their results. More prosaically, on the Fermilab website there is a section entitled: MINOS for neighbors. Somehow I think this controversy is unlikely to be resolved by a cosy chat across a garden fence, but you never know.

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