Why, when someone sees an equation, the first thing he/she thinks about is to run away? And why, when someone sees a t-shirt with a picture of an old man with white messy hair sticking out his tongue with an equation below the picture, the first thing he thinks is, cool! I want one of those too!? The old man is Albert Einstein and the equation is *E = mc ^{2}* which is the most famous equation in history, probably even more famous than the Pythagorean theorem

*(hypotenuse)*that, by the way, is related to the Einstein equation in way that I will explain later. The problem is that many of those who have, or want to have, such a t-shirt don’t know exactly what the meaning of the equation is and even less what its origin is. What we all probably know is that, in the equation

^{2}= (cathetus 1)^{2}+ (cathetus 2)^{2}*E*is the energy,

*m*the mass and

*c*the speed of light.

Another thing that very few people know is where the famous equation comes from and what Einstein wanted to explain when he worked it out, so lets do a bit of history about how Einstein came to it without using mathematical formulae (this is difficult, so if you are curious I recommend you to have a look at the original paper even if you don’t understand anything, it is so beautiful and simple that it is worthy to have a look at it)

The original paper where Einstein formulated the problem had as title ‘Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?’[1], what means, does the inertia of a body depend upon its energy-content? Einstein’s objective was to explain an aspect derived from his previous studies about electrodynamics where he used Maxwell’s equations, which explain why we see what we see and that are for me one of the most beautiful set of equations in history.

He also used the relativity principle as follows:

*‘The laws by which the states of physical systems alter are independent of the alternative, to which of two systems of coordinates, in uniform motion of parallel translation relatively to each other, these alterations of states are referred’ *

As you can see it does not say that everything is relative what it is used by many people when they want their opinion being in a better position than a different one.

From this point Einstein asked himself what happens when a stationary body, with an energy E_{1} in a system of coordinates, let us call it *C*, and *E _{2}* in another system of coordinates

*C*moving with a uniform parallel translation, emits energy in two directions. To understand it let us suppose that we are the body with energy

^{’}*E*, that we got it after a good breakfast and that we are stopped in train station’s platform. The train station is the system of coordinates

_{1}*C*. The system of coordinates

*C*would be a train moving in a railroad track parallel to the platform where we are. The emission of energy

^{’}*E*by the body means that we emit energy of a certain type. To be original, let us suppose that we have a pair of eyes capable of throwing X-rays then, when we throw X-rays, we emit energy in two directions (one per eye). If we take into account the energy we emitted and measure it with regard to the system of coordinates that is moving (the train) and to the stationary one, as well as we consider the principle of relativity and the energy conservation law (energy cannot be created or destroyed, but can change form, or in other words, the initial energy, the one gained after the breakfast, has to be the same that the one at the end, the energy emitted in the form of X-rays together with the one from the breakfast that we didn’t use), after subtracting the energies in both systems of coordinates we find the result that obtained Einstein which explains that if it is emitted energy in the form of radiation its mass diminishes, what is the same as ‘the mass of a body is a measure of its energy-content (I must admit that this is much more easier to explain with formulae). The

_{1}*c*appears because of measuring the energy of the body in the system of coordinates that is moving.

^{2}In summary, the energy and mass are equivalent through a constant which is the speed of light, because although the OPERA experiment says the opposite, wait… they don’t say it anymore because they forgot to correctly plug a wire, nothing can travel faster that the speed of light (in vacuum) which is constant with a numerical value of *300,000 km/s* approximately. Or what is the same *E = mc ^{2}*.

Although it may seem at useless result, it is not because the mass is continuously transforming into energy and the energy into mass. The former is easier to understand. In the interior of the sun there are permanent nuclear reactions transforming where two hydrogen atoms (this is quite more complex but enough to understand it) are fusing or ‘colliding’ together to produce a Helium atom plus an amount of energy that escapes from the reaction, which reaches us to heat and tan us. It is something similar to what happen to us when we are ‘big’ and run and make exercise to burn fat, the excess of mass disappear, doesn’t it? And, where does it go? It is the energy needed to run and make exercise!

That the energy transforms into mass is more difficult to understand, but it is, for example the fundamental principle that particle accelerators are based on. When in the CERN’s Large Hadron Collider (LHC) two protons travelling at almost the speed of light collide, protons (in fact the quarks and gluons that make up protons) transform into energy for a tiny period of time. Until now, mass transformation into energy again. If this was the only thing that happens it would be useless to spend so much money in a particle accelerator, but what happens next is the important thing, energy transforms into mass again! However instead of producing protons again, the energy transforms in a particle of the immense particle zoo that has been discovered and in particles that are waiting to be discovered. This is how, for example, the famous Higgs boson was found.

This is interesting, but… is *E = mc ^{2}* everything? The answer is a big NO. If you remember, in the example of the protons, I said that they are travelling at the speed of light, but in addition they have mass on their own, known as energy at rest. This mass is the one that appears in the Einstein equation. Where is then the speed that protons have? Einstein equation does not say anything about the bodies that, apart from the mass they have, are moving. In addition, it does not say anything either about the bodies that does not have mass, as it is the case of photons (light) because if there no mass, then energy would be zero, what is not possible because, for example the photons (light in the range of infrared frequency, which is made of photons too) heat us, which is the same that saying that they transmit energy to us. This problem is fixed by including the momentum of the particle in the equation. Momentum is, broadly speaking, a measure of the speed of the particle. In the case of photons, they don’t have mass but momentum, in other words they are moving. If we introduce the momentum in the Einstein equation, we have the extended form of the equation, which is as follows:

E^{2 }= (mc^{2})^{2} + (pc)^{2}

Where p is the momentum previously mentioned. Now we have a complete formula for the energy because if the particle does not have mass it can have energy (E = pc) and if it has not momentum (speed), it is at rest, it also has energy given by the famous Einstein equation. The fact that E is squared means that we have to take the square root to find the solution. But wait a moment, maths tells us that when we take the square root of a number, we always have two solutions, a positive and a negative one. For example has two solutions, 2 and -2, this is because 2^{2} = 4 and (-2)^{2} = 4, too! Does it mean that we can have negative energies? Well, the answer is not so easy. History attributes Paul Dirac in 1931 [2] the interpretation of negative energies as antiparticles. In this way, all (charged) particles have associated an antiparticle, the proton has its antiproton, the electron its antielectron or positron, etc.

Coming back to the extended form of the Einstein equation, if we have a close look at it, it has a similar form to the one of the Pythagorean theorem as I said before. How do we interpret this from a Physics point of view? Lets draw each element of the equation as part of a right-angled triangle to represent it as in the Pythagorean theorem [3]

This means that if a particle has mass, we could only give it energy to a certain limit; we could never give it an infinite energy. The reason can be seen in the figure. If we raise energy, then the hypotenuse *E* will be longer. When we raise energy we make that the particle increases its momentum (it increases its speed to keep it simple) and therefore the cathetus *pc* would be longer. To keep the form of a right-angled triangle the cathetus *mc ^{2}* should be longer which is the same as saying that its mass should increase. Thus, if we keep on giving energy the mass increases continuously giving as result that we would need more and more energy to make the particle keep moving. This is not efficient or worthwhile. It is not even useful! The limit on the speed is imposed by the speed of light; therefore if we increase the energy until the speed of the particle reaches the speed of light, we would need more and more energy to keep it moving!

I admit that this is difficult to ‘visualize’ it with the bunch of text I wrote, thus I leave you with a video of MinutePhysics (@minutephysics) where he explains it, in a wonderful way, in a bit more than two minutes.

http://www.youtube.com/watch?v=NnMIhxWRGNw

*References*

[1] Einstein, A. Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?, Annalen der Physik. 18:639, 1905 (versión en inglés: Does the Inertia of a Body Depend upon its Energy-Content)

[2]http://francisthemulenews.wordpress.com/2012/06/24/paul-a-m-dirac-y-el-descubrimiento-del-positron/ (in Spanish)