# Why Does E=mc2?

Once again this semester, I’m teaching a ridiculously fun course – Physics 312: Relativity and Cosmology; Einstein and Beyond. As I’ve mentioned before, this course is so enjoyable because one gets to expose undergraduates with not much physics background to some of the mind-bending results of relativity, and watch them struggle with it and, usually, finally come to understand it. Great stuff.

While I have a blast with the later parts of the course – general relativity and cosmology – I have a particular soft spot for something rather close to the beginning, in the special relativity portion – the most famous equation in physics – E=mc^{2}.

So how does one go about motivating this equation for a class of students with only a little physics background, but who know some calculus? Well, perhaps the first thing to say is that, for the purposes of this course, it is much more important that they understand why an equation relating mass and energy is required, and how one might derive its form, than that they actually be able to do the detailed derivation themselves. So that is the tack that I take.

Early in the course, we review the idea that light is an electromagnetic wave. We do this by starting with Maxwell’s equations, which describe how moving and spatially varying electric and magnetic fields are related, and using them to show that, even in vacuum, if one tweaks the electric or magnetic field, then that disturbance propagates as a wave, with a given speed. We then see that the speed that arises is empirically equal to the speed of light, and hence we identify light itself with electromagnetic waves. This is a powerful idea, because students have a great deal of intuition about waves. In particular, they know that waves carry energy and momentum. So, at this point, students are pretty comfortable with the idea of light as a wave, and that light therefore carries energy and momentum.

Now, this is a great point for one of those staples of relativistic reasoning – the thought experiment. We start with one that doesn’t involve any of those worrisome relativity ideas. Think of a physicist, standing at one side of a large box, which itself is sitting on a perfectly frictionless surface (think of ice if you like). The physicist possesses a large cannon, which she is using to hurl heavy cannonballs across the box. What happens to the whole system?

Well, the box, physicist, cannon and cannonballs are a closed system, with no external forces acting on it. So one thing we know is that the center of mass of the system won’t move. Of course, that doesn’t mean that nothing will happen. As a cannonball is launched, it acquires a certain momentum, and conservation of momentum means that the box acquires the equal and opposite momentum, and sets off sliding backwards on the ice.

The next important event is that the cannonball collides with the opposite wall of the box, imparts it’s momentum to the box, and both cannonball and box come to a halt. At this point, the distribution of mass in the box is different from at the beginning (a cannonball has been transferred from one side to the other), and the position of the box has shifted. These two differences conspire in such a way that the center of mass of the system as a whole remains in the same place. All is right with the world.

Now let’s think about a second thought experiment, which is closely related to the first. All I want to make different is to replace the cannon by a powerful laser. Instead of a cannonball being propelled across the box, we’ll now think about the laser firing a pulse of light. Now, the light carries momentum, and so when the laser fires and the pulse sets off, the box will once again begin a backwards slide in order that momentum be conserved. Also once again, when the light reaches the other side and is absorbed by the opposite wall, the momentum will be transferred back to the box, which will then come to a halt. But now you see the problem. The distribution of mass in the box is the same as it was at the beginning, and no external forces have acted on the system, and yet because the box has slid backwards and no mass has been moved, the center of mass of the entire system has moved! All no longer seems right with the world.

This kind of thought experiment is what forces one to the conclusion that the idea of “center of mass” needs to be replaced by a more general concept – that of a “center of energy”. Obviously, this means that one must take into account how the distribution of energy in the system has changed, as well as the mass, when figuring out how a system should behave under no external forces. Another way to say this is that moving energy to one side of the box to the other is equivalent to moving some mass across the box – *mass-energy equivalence*!

This is the punch line, but one can do a little better. One can, of course, ask, when I’ve fired my laser pulse and had it absorbed on the other side, how far has the box moved? One can then ask, how much mass would I have to have moved from one side to the other in order that this movement of the box, combined with the mass movement, leave the center of mass unchanged. Equating the answer, m, to this question, with the energy, E, of the pulse, moving at the speed of light, c, yields: **E=mc ^{2}**.

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