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Hyperlatives

No little thing is to small for grandiose words chiseled by some marketing war machine.

Seen on a Lampe Berger anti-mosquito product this morning:

Parfum “Absolu de vanille”

Vanilla Gourmet Scent

Not only is this ridiculously hyperlative, but they also have a different “tint” for the Engish and French version. English reader will notice that the French version sounds more like “Absolute Vanilla”, because that’s basically what it means. Who on Earth paid people to tell their customers that their anti-mosquito drug had a “Vanilla Gourmet scent?”

Let’s not get used to this kind of marketing hyperbole…

Hyperbole in science

In despair, I turned to a slightly more serious text, the first page of this month’s issue of Science et Vie. And here is what I read there about faster than light neutrinos:

Incroyable? Alors là oui, totalement! Et même pis. Que la vitesse de la lumière puisse être dépassée, ne serait-ce que de très peu, n’est pas seulement incroyable, mais totalement impensable. Absolument inconcevable. [...] c’en serait fini d’un siècle de physique. Mais, et ce serait infiniment plus grave, c’en serait aussi fini avec l’idée selon laquelle la matière qui compose notre univers possède des propriétés, obéit à des lois. Autant dire que la quête de connaissance de notre monde deviendrait totalement vaine.

Incredible? Absolutely! And even worse. That the speed of light can be exceeded, even a little, is not only unbelievable, but totally unthinkable. Absolutely inconceivable. [...] This would end a century of physics. Even more serious, we would be done with the the idea that matter making up our universe has properties, obeys laws. This would mean that the quest for knowledge in our world would become totally hopeless.

Whaaaaat? I really don’t like this kind of pseudo-science wrapped in dogma so pungent to be the envy of the most religious zealots. How can anybody who understood anything about Einstein’s work write something like that? Let’s backpedal a little bit and remember where the speed of light limit comes from.

Where does the speed of light limit come from?

At the beginning was Maxwell’s work on the propagation of electromagnetic waves, light being such a wave. These equations predicted a propagation of light at a constant speed, c, that could be computed from other values that were believed at the time to be physical constants (the “epsilon-0″ and “mu-0″ values in the equations). The problems is that we had a physical speed constant, in other words a speed that did not obey the usual law of speed composition. If you walk at 5 km/h in a train that runs at 200 km/h, your speed relative to the ground is 205 km/h or 195 km/h depending on whether you walk in the same direction as the train or in the opposite direction. We talk about an additive composition rule for speed. That doesn’t work with a constant speed: if I measure the speed of light from my train, I won’t see c-200 km/h, since c is constant. The Michelson-Morley experiment proved that this was indeed the case. Uh oh, trouble.

For one particular speed to be constant, we need to change the law of composition. Instead of adding speeds, we need a composition law that preserves the value of c. It’s the Lorentz transformation. What Einstein acknowledged with his special relativity theory is that this also implied a change in how we consider space and time. Basically, Lorentz transformation can be understood as a rotation between space and time. And in this kind of rotation, the speed of light becomes a limit in a way similar to 90 degrees being the “most perpendicular direction you can take”. Nothing more, nothing less. Of note, that “c” value can also be interpreted as the speed at which we travel along time when we don’t move along any spatial dimension.

There are limits to limits

Once you understand that, you realize how hyperbolic what Science et Vie wrote is.

First, the value of c was computed as a speed of light, for equations designed for electromagnetism. It was never intended to say anything about neutrinos. We don’t know how to measure space and time without electromagnetic interactions somewhere. So the speed of light limit is a bit like the speed of sound limit for bats who would measure their world using only echo-location. It doesn’t necessarily mean nothing can travel faster than light, it only means that no measurement or interaction based on electro-magnetic interactions can ever measure it. I have tried to elaborate a bit on this in the past.

Second, Einstein revised his initial view to include gravity, and this made the world much more complex. Now space-time could be seen as modified locally by gravity. Now imagine how solid your “90 degrees is the most perpendicular direction” argument is if you look at a crumpled sheet of paper. The reasoning doesn’t mean much beyond very small surfaces. Remember that in the neutrinos experiments, we are in a very complex gravitational environment (mountains, …) and you’ll see that this “crumpled sheet of paper” analogy may not be so far off.

In short, it we find conditions where something appears to travel faster than light, it is exciting, it is interesting, it is worth investigating, but it’s certainly not the End of Science as Science et Vie claimed. Let’s not get used to this kind of crap.

How To Teach Special Relativity

How to Teach Special Relativity is a famous article by John Bell where he advocates that the way we teach relativity does not give good results. He describes an experiment now known as Bell’s spaceship paradox (even if Bell did not invent it):




In Bell’s version of the thought experiment, two spaceships, which are initially at rest in some common inertial reference frame are connected by a taut string. At time zero in the common inertial frame, both spaceships start to accelerate, with a constant proper acceleration g as measured by an on-board accelerometer. Question: does the string break – i.e. does the distance between the two spaceships increase?

Considered a difficult problem

The correct answer is that the string does break, even if the spaceships appear to always be at the same distance from one another as seen from an observer who did not accelerate with the spaceships. Yet, according to Wikipedia:

Bell reported that he encountered much skepticism from “a distinguished experimentalist” when he presented the paradox. To attempt to resolve the dispute, an informal and non-systematic canvas was made of the CERN theory division. According to Bell, a “clear consensus” of the CERN theory division arrived at the answer that the string would not break.

In other words, this problem was considered hard by a majority of serious physicists at the time Bell raised the question in 1976. I would venture to say that this remains the case today, except that this particular paradox is probably well known now. But the teaching of special relativity has not changed. This is how we explain the paradox today. I have a lot of admiration for John Baez in general, his “blog” is even in the sidebar of this one. But with all due respect, the explanation of the paradox posted on his web site is utterly complicated (I know he gives credit to someone else for it, but by hosting it on his web site, I would say that he condones it).

It should be easy

This particular formulation of the paradox was not known to me until someone recently asked me if the string would break. Using my little technique, it took me less than one minute to have the correct answer, without looking it up, obviously, but also without any computation or complicated diagram. Here is the mental diagram I used (click to see it in high resolution):

On this diagram, time is represented horizontally, and the two space ships are represented by the green and red curves, which are identical but separated by a distance along the vertical “spatial” axis. The distance at rest is represented by the blue arrow. The distance as measured between the two ships after they started moving is measured by the green and red arrows. The distance as measured “from the ground” is along the vertical axis, and remains constant.

Remember the only trick is that a “cosine” contraction on this diagram corresponds to a dilatation in relativity and conversely. On the diagram, the red and green arrows are obviously shorter than the blue arrow. The contraction factor is the cosine of the angle between these arrows and the vertical (space) axis, which is the same as the angle between the red or green curve and the horizontal axis. Therefore, relativity predicts that the distance between the two ships, as seen from the ship, will increase. Specifically, it increases by a factor usually denoted “gamma” (but which I prefer writing as the cosine of an angle myself), which can also be seen as a hyperbolic cosine, and which plays in Minkowski geometry the exact same role as the cosine in the Euclidean diagram above. You can find the precise mathematical relationship here.

Consequently, the string will break.

Accelerated solids in relativity

Another interesting observation one can make from the diagram is that you cannot draw a straight line that is perpendicular to both curves. What is “space” for one ship is not just “space” for the other. You need to draw a curved line between the two rockets if you want to always be perpendicular to the local “time” direction. In other words, the “time” direction for the string is not constant along the way, so all parts are not moving at the same speed. Someone sitting anywhere on the string will see other parts of the string move relative to him. That’s another way to explain why the string will break.

You can easily verify that this problem exists for any kind of accelerated solid. All parts of an accelerated solid in special relativity do not move at the same speed.

My own puzzle

Here is the interesting other thing that I realized within this short moment of reflection: there is a way for the two ships to accelerate “identically” (for a suitable definition of identically which remains to be given) so that the string will not break. Can you find it?

C’est tout de même dur à avaler.
Comment voulez vous qu’on enseigne cela à des élèves de Terminale?

Jean-Claude Carrière, in Entretiens sur la multitude du monde with Thibault Damour

Categories: Physics, Relativity, Teaching
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