Using E=m*c*c

The last time I gave you a first, brief glimpse of special relativity. Special relativity has one property on which all modern experiments at accelerators like the LHC are based on. It is encoded in Einstein's most famous equation E=m*c*c, where E stands for energy, m for mass, and c is the speed of light. But what does this equation, which is already part of pop culture, really mean?

Let us have a look at its part. The symbol c denotes the speed of light. As discussed last time, the speed of light is always and everywhere constant. It is thus a constant of proportionality, without any dynamical meaning. In fact, its value is no longer measured anymore. In the international most used system of physical units it is defined to have a certain value, roughly 300000 km/s. Since it is so devoid of meaning, most particle physicist have decided that you do not need it, really, and replaced it with one. Ok, this may sound pretty strange to you, since one is not even a speed, it is just a number. But all such things like physical units are man-made. Nature knows only what a distance is, but not what a kilometer is. Thus, you must be able to formulate all laws of nature without such man-made things like a second or a kilometer.

Indeed, this is possible. However, we are just human, and thus working only with numbers turned out to be inconvenient for the mind. Thus, we usually set only so much irrelevant constants to one until we are left with just one single physical unit. Depending on the circumstances, for a particle physicist this is either the so called femtometer (short fm) or fermi, 0.000000000000001 meter, what is roughly the size of a proton. Or we use energy, measured in giga-electron volt (short GeV), or 100000000 electron volt. An electron volt is the amount of kinetic energy an electron gains when it is accelerated by one volt of voltage. That is roughly the voltage of an ordinary battery. Both units are very convenient when you do particle physics. If you are an astrophysicist, this would not be the case. They measure distances, e.g., in megaparsec, which is roughly 3261567 light years.

Anyway, lets get back to the equation. If we set c to one, it reads E=m. Much simpler. The left hand side now denotes an energy E, and the right-hand side a mass m. This is actually not what you can read off a scale. This is called weight, and depends on the planet you are on. Mass is a unique property of a body, form which one can derive the weight, once you chose a planet.

Since the left-hand side is an energy, measured in GeV, so is the right-hand side. Thus, we measure mass not in kilogram, but in energy. A proton has then roughly the mass of 1 GeV, while an electron has a mass of about 0.005 GeV.

But this equation is not just about units. It has a much deeper meaning. As it stands, it says that mass is equal energy. What does this mean? You know that you have to invest energy to get an object moving, again the kinetic energy. But the right-hand side does not contain a speed, so the energy on the left-hand side seems not to be a kinetic energy. This is correct. The reason is that this formula is actually a special case of a more general one, which only applies if you consider something which does not move. It makes the explicit statement that a body with a mass m at rest has an energy E. Thus, the energy has nothing to do with moving, and is therefore called a rest energy. If the particle should start to move, this energy is increased by the kinetic energy, but never decreased. This means that every body has a minimum energy equal to its rest energy, which in turn is equal to its mass.

Why is this so? The first answer is that it necessarily comes out of the mathematics, once you set up special relativity. That is a bit unsatisfactory. In quantum field theory, mass comes out as an arbitrary label that every body has, and which can take on any value. Only by experiment we can decide what particles of which mass do exist. We cannot yet predict the mass a particle has. That is one of the unsolved mysteries of physics. Note that the Higgs effect or the strong force seem to create mass. Thus, it seems we can predict mass. But this is a bit imprecise. Both of them do not really create mass, but add more to Einstein's equation. This makes particles behave as if they have a certain mass. But it is not quite the same.

Let me get back to where I started. Why is this equation so important? Well, as I said, the energy gets only bigger by moving. Now, think of a single particle, which moves very fast. Thus, it has a lot of energy. At the LHC, the protons have currently 4000 times more energy than they have at rest. If you stop the proton by a hard wall, than most of this energy will go on and move the wall. But since a wall is usually pretty heavy, and even 4000 times the proton rest mass is not much on the scale of such walls, they do not move in a way that we would notice.

But now, let us collide this proton with another such proton. What will happen? We have a lot of energy and a head-on collision. One thing Einstein's equation permits, if you formulate it for more than one particle, is the following: You are permitted to convert all of this energy into new particles. At least, as long as the sum of kinetic and rest energy does not exceed the total energy of the two protons before-hand. By this, you can create new particles. And this is what makes this equation so important for modern experiments. You can create new particles, and observe them, if you just put enough energy into the system. And that is, why we use big accelerators like the LHC: To make new particles by converting the energy of the protons.

Unfortunately, we cannot predict to what the energy is converted, as already noted earlier. But well, at least we can create particles.

Oh, and there is a subtlety with the wall: If we are good, and hit a single particle inside the wall, then the same happens as when we collide just two protons. But in most cases, we do not hit a single particle, but it is more like the first shot of billiard, giving just a bit of energy to every particle in the wall. And then the wall as a whole is affected, and not a single particle. Just when you wonder if you ever hear of fixed-target experiment, instead of a collider. This it, how this works: Shot at a wall, and hope to hit just a single particle.

The speed of light - and its consquences

So far, I did not say anything about gravity. This will remain so. However, I will have to say something about special relativity. Somehow, special relativity is often associated with gravity. This is actual not the case. Einstein's theory of special relativity does not make any reference to gravity. Only the theory of general relativity, of which special relativity is just a small subset, does so.

If special relativity is not about gravity, what is it about? Well, it is about the fact that our universe is a bit more weird than one expects.

What do you expect of a law of nature? One property is likely that it is always valid. This simple requirement has quite profound consequences. Assume for a second that you and your experiment are alone in the universe. This means ta you have no point of reference. If now the experiment moves, you could not say whether it is moving, or you. Still, you would expect that it gives the same results, irrespective of whether it or you are moving. We made experiments to test this idea, and it was confirmed beautifully. This fact that experiments are independent of relative motion, is one of the basic observations leading to special relativity.

The next ingredient is much more harder to believe. Take a light beam. What speed does it have? Well, the speed of light, of course. Now, if you move the thing creating the light at a fixed speed, how fast should the light move? Naively, one would expect that the light would now move faster. Unfortunately, our universe does not tick that way: The light still moves with the same speed. Actually, the light is actually made up out of massless photons, which travel at the speed of light. And this observation is the same for anything which is massless: All massless particles move at the speed of light. And the speed of light is always the same, no matter how fast the light source moves.

This is nothing we can really explain. It is an experimental fact. Our universe is like this. But this observation is the second basic fact underlying special relativity.

If we cannot explain why this second fact comes about, can we at least describe it? We can, and that is what leads to special relativity.

Now, how do we describe this? Well, this is a bit more involved. Take the universe. Then at each instance you have three directions in space. Distances you measure do not depend on the direction in space. You can also measure time elapsing. Works out also nicely. But now, try to measure a space-time distance. You do this by measuring a distance in the space direction. Then you take the elapsed time, and calculate what distance a light ray would have moved during this time. By this, you can talk about a distance in time direction.

The speed of an object is given (if the speed does not change) by the ratio of distance over time required to move over this distance. If you now want that the speed of light is independent of whether the light source moves or not, something peculiar is found: To get this, a distance in space-time direction is obtained by subtracting the distance in the time direction and in the space direction, when you do your Phytagorean geometry. That is completely different than what you have in the three time directions, but the only way to get the light speed to agree with experiment. The geometry of space-time is hence quite different from the one we know just from space.

As a theoretician, I say that the way we measure the distances is not like the ordinary one in three space directions (a so-called Euclidean measure), but we have rather a Lorentz measure. This is in honor to the first one who has described it. Again, we cannot yet explain why this is the case, it just is an experimental fact.

You may wonder why you never noticed this in real lifer. The answer is that when the light source moves very slowly compared to the light ray, the effect is negligible. This becomes only relevant, if you move at a considerable fraction of the speed of light. But then all the nice effects result of which you may have heard in Science Fiction movies or novels: Things like the twin paradox, time dilatation, length contraction, and so on. All of these result from these two basic observations. And are described by the theory of special relativity.

This all sound pretty weird, and it is. It is nothing we have a real handle on with our everyday experience. Just like the quantum effects. The universe is just this way.

As you see, gravity enters here nowhere. And also no quantum stuff. If you add gravity, you get to the theory of general relativity. If you add quantum stuff, you end up with quantum field theory. The standard model is of the latter kind. And this combination leads to very interesting effects, which I will discuss in more detail next time.