# Faster than the Speed of Light

By | 2011-11-10

On the potential discovery of faster than light neutrinos.

Einsteinâ€™s seminal work was â€œgeneral relativityâ€, which is about gravity, space and time. General relativity is utterly beyond me; but special relativity (broadly) is taught to A-Level students without much difficulty.

Firstly: itâ€™s not really about â€œphotonsâ€ having a particular speed; and us suddenly discovering some other particle that can go faster. Photons arenâ€™t Corsas and weâ€™ve suddenly discovered BMW-shaped neutrinos.

The speed of light (or radio waves, or whatever else you want to call it, but â€œlightâ€ is probably best) is not just a number, rather it is an inherent property of the universe. Similar to the universal gravitational constant, the charge on an electron, or the number of protons in a sodium nucleus. The speed of light is more than itâ€™s mere numerical value though, in fact the actual number is not as important as the ways in which it affects the universe. I think the problem is that many reporters are confusing these two parts: concept and number.

The relevant equation is not the popular

$E=m{c}^{2}$

. Rather it is the particleâ€™s â€œrelativistic massâ€,

$m\left(v\right)=\gamma \left(v\right)×{m}_{0}$

$m\left(v\right)$

is the observed mass of an object moving at relative velocity

$v$

which has a rest mass of

${m}_{0}$

.

$\gamma$

is the Lorentz factor, and is dependent on the relative velocity between the observer and the objectâ€¦

$\gamma \left(v\right)=1/\sqrt{\}left\left(1–\left(\frac{{v}^{2}}{{c}^{2}}\right)\right)$

Nasty looking; but forget most of it and concentrate on

${v}^{2}/{c}^{2}$

.

• As
$v$

approaches

$c$

,

${v}^{2}/{c}^{2}$

gets closer and closer to 1

• Therefore
$1–{v}^{2}/{c}^{2}$

gets closer and closer to 0

• Therefore the square root gets closer and closer to zero
• Therefore gamma gets closer and closer to infinity
• Therefore the (observed) mass of a moving object approaches infinity
• Energy needed for acceleration is proportional to mass and therefore, the energy needed to accelerate an object approaching infinite mass approaches infinity.

There is one caveat. The mass approaches infinity provided its rest mass (

${m}_{0}$

) is greater than zero. If its rest mass is zero, then

$m(v) = \infinity \times 0$

Which has no defined static answer; but can be finite. In the case of photons/neutrinos it certainly is finite. Yep: photons (and neutrinos) have mass because they are travelling at the speed of light (and no other speed will do). That mass is what powers, for example, solar sails. (actually itâ€™s more accurate to say that they have momentum rather than mass)

Soâ€¦ given current understanding (i.e.Â that set of equations), itâ€™s very hard to understand how anything travels faster than

$c$

. That is what all the excitement is about. If something did travel faster than the speed of light, that would lead to this:

$\gamma =1/\sqrt{\left(}1–\left(biggerthan1\right)$

$\gamma =1/\sqrt{\left(}negativenumber\right)$

Weâ€™ve got the maths to deal with square roots of negative numbers (imaginary numbers); just not the universe. What exactly would an imaginary mass look like? Itâ€™s not ruled out by relativity at all (you might have heard of the theoretical group of particles called tachyons); itâ€™s just we have no idea what the physical meaning of such results would be. Hence more excitement.

Einstein didnâ€™t just pluck

$c$

out of the air and say â€œwell thatâ€™s a good fast numberâ€, as the BBC news seem to think. He came to it in one of the most beautiful thought experiments ever done. It is a derivation that is covered at A-Level physics level it is so simple and elegant. The number for

$c$

is barely relevant, the relevancy is that Einsteinâ€™s theory shows us that whatever

$c$

is, it acts as a speed limit for the universe. The maths breaks if you try to pass through it (from either direction, tachyons, if they exist would be impossible to slow down to less than the speed of light by exactly the same logic â€” infinite energy would be required).

Particles with zero rest mass donâ€™t exist. Unless of course, they are travelling at the speed of lightâ€¦ it is at that speed only that they have mass (and hence existence); any lower and gamma is not infinite, and hence the zero rest mass makes their relativistic mass zero. Particles with zero rest mass must travel at the speed of light. Neutrinos have zero rest mass so must travel at the speed of light. There is no mass smaller than zero, so there can be no travelling faster than the speed of light.

What makes

$c$

even more significant is that it turns up all over the place. Not least of which (in my field) is the strangely attractive fact that

$c=\frac{1}{\left(permittivityoffreespace×permeabilityoffreespace\right)}$

â€œFree spaceâ€ being a vacuum. These two numbers tell us about the strength of electric fields in a vacuum; and the strength of magnetic fields in a vacuum and can be experimentally measured on your desk if you wanted. The fact that they can be combined to form

$c$

is another confirmation of the â€œrightnessâ€ of

$c$

. This property comes from the fact that light is a self-propagating combination of an alternating electric field, which induces an alternating magnetic fieldâ€¦ hence electromagnetic waves.

So had Einstein and all his contemporaries known about neutrinos, they might well have decided that the things which travel fastest is neutrinos – because gravity cannot slow them down, heck, not even solid rock can slow them down.

Gulp. Hopefully Iâ€™ve shown you that it is not the fact that neutrinos werenâ€™t known about (and in fact, it was one of Einsteinâ€™s chums, Pauli, who first postulated their existence) that make

$c$

the fastest number in the universe. If you do the analysis with neutrinos then youâ€™d come up with exactly the same

$c$

as you do for photons, or any other zero rest mass particle. Some quick points about the above paragraph:

• Gravity doesnâ€™t slow down neutrinos or photons; as Einstein showed, the speed of light is constant.
• Thanks to badly written sci-fi, the world is under the impression that a black hole has such massive gravity that not even light can escape it. That leaves you with the impression that light is pulled in. No. Light travels in straight lines always; black holes are so massive that they bend space, so that a â€œstraight lineâ€ from the point of view of the light is bent from a point of view outside the black hole..
• In fact, black holes bend space to breaking point, so much so that lightâ€™s â€œstraight lineâ€ path is so bent that it has no way back to the universe we can see.
• Light around a black hole is red shifted, not changed in speed. Light that is inside the event horizon is infinitely red shifted; making it disappear â€” not slow down.
• Neutrinos are no more immune to a black hole than anything else in the universe: it doesnâ€™t matter that they are weakly interacting, it matters that the black hole has distorted space â€” everything that travels in that space is affected.
• There is a small detail that itâ€™s likely that black holes â€œemitâ€ neutrinos. They donâ€™t really emit them, they appear in the same way that Hawking radiation is created (thatâ€™s a whole new story).
• Gravity is nothing to do with electromagnetism; and there is certainly nothing that guarantees that gravity propagates at the speed of light (although Einstein assumes it in his general theory). In fact, thatâ€™s still a difficult question for physicists (and would be helped enormously if they could find the Higgs Boson, the supposed carrier particle of the gravitational force); Markâ€™s New Scientist link is (I believe) actually talking about the measurement of the propagation speed of gravitational waves, which are different. Gravity is so completely not understood that it would be difficult to say anything definitive about how it travels. Should there not be a Higgs Boson, then gravity will turn out to be not like any of the other three fundamental forces in the universe; and therefore not necessarily subject to the same rules as them.

Personally, I donâ€™t believe the results are valid. I suspect that something will turn out to be the explanation for the faster-than-light measurement. The best one Iâ€™ve heard so far is that theyâ€™ve assumed the shape of the reception signal, which is created by oversampling many repetitions of the experiment, is the same as the shape of the transmission signal. The problem is that the transmission signal is a measurement of the generating protons, not neutrinos. There is absolutely no way to be confident about the correlation of these two shapes. If the neutrino signal is spread out a little relative to the proton signal, then the 0.00000000001 of a second â€œfasterâ€ could easily vanish.

However, despite the above, the result is certainly an exciting one â€” itâ€™s not just a matter of a new value for a known physical constant; it is more that, if true, it would require that a great deal of our understanding of physics get chucked in the bin.

Update: I forgot to mention my crazy thought on this subject. Tachyons are weird, right? We have no idea what a real tachyon would look like in our universe nor how their effects would be felt. Imagine, a particle with a negative or imaginary mass; that travelled backwards through time and faster than the speed of light; in fact they would probably travel towards their source rather than away from it. Whoah. It would be something that behaved completely differently to anything else we have any experience of.

Something that behaves differently from everything elseâ€¦ strange interactions, with no explanationâ€¦ sounds a bit like gravity.

(Note: I accept the craziness of this idea; Iâ€™m not actually that serious about it).

Update: Itâ€™s nice when my lowly engineerâ€™s brain gets it right (I wasnâ€™t alone of course, plenty of physicists had the same concerns). â€œI donâ€™t believe the results are validâ€, I said.

Now another experiment located just a few metres from OPERA has clocked neutrinos travelling at roughly the speed of light, and no faster. Known as ICARUS, the rival monitored a beam of neutrinos sent from CERN in late October and early November of last year. The neutrinos were packed into pulses just 3 nanoseconds long. That meant that the timing could be measured far more accurately than the original OPERA measurement, which used 10-microsecond pulses.