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Friday, December 26, 2014

Was Einstein Wrong? A Faster-than-Light Neutrino Could Be Saying Yes


Albert The Greatest Scientist Ever
Physicists have a stock phrase they trot out whenever someone claims to have made an astounding new discovery about the universe. "Important," they say, "if true."

It's a tactful way of saying "Don't bet on it," and they've been saying it a lot over the past day or so. The reason: a team of European scientists has reportedly clocked a flock of subatomic particles called neutrinos moving at just a shade over the speed of light. According to Albert Einstein's special theory of relativity, that can't be, since light, which cruises along at about 186,000 miles per second (299,000 km/sec.), is the only thing that can go that fast.

If the Europeans are right, Einstein was not just wrong but almost clueless. The implications could be huge. Particles that move faster than light are essentially moving backwards in time, which could make the phrase cause and effect obsolete.

"Think of it as being shot before the trigger is pulled," wrote University of Rochester astrophysicist Adam Frank on his NPR blog. Or, as Czech physicist Lubos Motl put it on his blog, "You could kill your grandfather before he had his first sex with your grandmother, thus rendering your own existence needed for the homicide inconsistent with the result of the homicide."

The evidence for this complete upending of modern physics and cosmic decorum comes from an experiment involving two top-notch physics installations. The first is CERN, the European Center for Particle Physics, near Geneva, where a particle accelerator created the swarm of neutrinos in the first place. These bits of matter are bizarre no matter how you look at them: they're so elusive that one of them could pass through a chunk of lead a trillion miles thick without a bump.

It's no surprise, then, that the swarm created at CERN could fly out of the accelerator, zip right through the Alps and appear in the Gran Sasso Observatory, located in a tunnel deep beneath Italy's Apennine Mountains. Most of the neutrinos kept on going, but just a few, by pure chance, were intercepted by one of the observatory's neutrino detectors. And when the two labs synchronized their watches, it appeared that the particles had made the 450-mi. (724 km) journey 0.0025% faster than a beam of light would have (if light could travel through mountains, that is).

That splinter of a second isn't much, but it's enough to overturn a century of firmly established physics, rewrite the textbooks and throw the faculties at major universities around the world into a collective tizzy. In short, it's really important.

If true.

No one is tearing up the Einsteinian rule book just yet. As physicists well know, astonishing results like this often turn out to be wrong, especially when they haven't been double-checked. Sometimes that means the group announcing the big news has done shoddy work, like the Utah chemists who announced to great fanfare back in 1989 that they'd achieved controlled nuclear fusion on a tabletop — the cold-fusion kerfuffle — trumping the physicists who'd been struggling for years to do the same thing with billion-dollar machines. Sometimes it just means the researchers have overinterpreted what they're seeing, as when NASA scientists said they'd found evidence of life in a rock from Mars.

And sometimes, the researchers have gone about things the right way, carefully checking their equipment and their calculations to make sure they aren't being fooled by some mundane, potentially embarrassing glitch. The Grand Sasso scientists have done just that kind of due diligence here, and you know what? They still can't find any evidence that they've missed anything.

But that doesn't mean they haven't. It's always possible that their instruments are misbehaving in too subtle a way for anyone to detect at this point. Given the stakes if the equipment is right — if neutrinos really can move faster than light — nobody's buying the shocking result until another set of researchers, using another set of instruments, gets the same answer. Indeed, that's exactly what Antonio Ereditato, of the University of Bern, leader of the Gran Sasso end of the experiment, is hoping for. He told the BBC: "My dream would be that another, independent experiment finds the same thing. Then I would be relieved." This very willingness to be double-checked — and proved wrong — gives the scientists greater credibility, even if the jury is still out on their findings.

A second opinion may be coming soon. A group at the Fermilab accelerator complex, near Chicago, says it's preparing to do just the follow-up round of studies Ereditato welcomes. As it happens, Fermilab physicists made their own faster-than-light neutrinos claim back in 2007. It too would have been important if true, but on closer analysis, the evidence went away. The Fermilab scientists immediately accepted the verdict that time, just as the Europeans undoubtedly will if this new "discovery" goes up in smoke, as physicists everywhere are betting it will.

Or maybe it won't: the history of science may be littered with claims that were ultimately proved false, but some outrageous ideas turn out to be true in the end. Take dark matter, the mysterious, invisible stuff that outweighs the visible stars and galaxies by a factor of 10 to 1. When it was first proposed in the 1930s, nobody believed it. When it reappeared in the 1960s, everyone laughed. Now it's firmly accepted as a fundamental part of the universe.

That kind of thing just might happen again. "Based on past experience, these results are probably wrong," writes Adam Frank at NPR.org, "but it sure would be a wild ride if they prove correct."

Monday, November 10, 2014

The Derivation of E=mc2

 
God's Equation

The Derivation of E=mc2

Perhaps the most famous equation of all time is E = mc2. The equation is a direct result of the theory of special relativity, but what does it mean and how did Einstein find it? In short, the equation describes how energy and mass are related. Einstein used a brilliant thought experiment to arrive at this equation, which we will briefly review here.

First of all, let us consider a particle of light, also known as a photon. One of the interesting properties of photons is that they have momentum and yet have no mass. This was established in the 1850s by James Clerk Maxwell. However, if we recall our basic physics, we know that momentum is made up of two components: mass and velocity. How can a photon have momentum and yet not have a mass? Einstein’s great insight was that the energy of a photon must be equivalent to a quantity of mass  and hence could be related to the momentum.

Einstein’s thought experiment runs as follows. First, imagine a stationary box floating in deep space. Inside the box, a photon is emitted and travels from the left towards the right. Since the momentum of the system must be conserved, the box must recoils to the left as the photon is emitted. At some later time, the photon collides with the other side of the box, transferring all of its momentum to the box. The total momentum of the system is conserved, so the impact causes the box to stop moving.

Unfortunately, there is a problem. Since no external forces are acting on this system, the centre of mass must stay in the same location. However, the box has moved. How can the movement of the box be reconciled with the centre of mass of the system remaining fixed?

Einstein resolved this apparent contradiction by proposing that there must be a ‘mass equivalent’ to the energy of the photon. In other words, the energy of the photon must be equivalent to a mass moving from left to right in the box. Furthermore, the mass must be large enough so that the system centre of mass remains stationary.

Let us try and think about this experiment mathematically. For the momentum of our photon, we will use Maxwell’s expression for the momentum of an electromagnetic wave having a given energy. If the energy of the photon is E and the speed of light is c, then the momentum of the photon is given by:

                              

 (1.1)
The box, of mass M, will recoil slowly in the opposite direction to the photon with speed v. The momentum of the box is:
                              

 (1.2)
The photon will take a short time, Δt, to reach the other side of the box. In this time, the box will have moved a small distance, Δx. The speed of the box is therefore given by
                              

 (1.3)
                              

 (1.4)
If the box is of length L, then the time it takes for the photon to reach the other side of the box is given by:
                              

 (1.5)
Substituting into the conservation of momentum equation (1.4) and rearranging:
                              

 (1.6)
Now suppose for the time being that the photon has some mass, which we denote by m. In this case the centre of mass of the whole system can be calculated. If the box has position x1 and the photon has position x2, then the centre of mass for the whole system is:

                              

 (1.7)
We require that the centre of mass of the whole system does not change. Therefore, the centre of mass at the start of the experiment must be the same as the end of the experiment. Mathematically:
                              

 (1.8)

The photon starts at the left of the box, i.e. x2 = 0. So, by rearranging and simplifying the above equation, we get:

                              
Substituting (1.4) into (1.9) gives:
                              

 (1.10)
Rearranging gives the final equation:
                              

So, let’s think about what this equation means. The equation suggests that a given mass can be converted into energy. But how much energy? Well, suppose we have a kilo of mass. Conversion of this mass into pure energy would result in (1kg * c2) joules of energy. Now note that c2 = 8.99 * 1016 m2s-2  so that's a WHOLE lot of energy - equivalent to 21.48 megatons of TNT!

In practice, it is not possible to convert all of the mass into energy. However, this equation led directly to the development of nuclear energy and the nuclear bomb - probably the most tangible results of special relativity.