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### E = mc^2

Last month when I gave a talk about Angels & Demons to a group of high school teachers one of the big discussion topics was the nature of Einstein’s famous equation. Since E=mc2 is at the heart of the entire program of collider physics, I thought it’d be a good thing to go over with everyone.

E=mc2 explained in one sentence

In one line, E=mc2 is the statement that energy E and mass m are somehow the same thing where c is the speed of light, which is a fundamental constant and allows us to convert units of mass into units of energy.

How do I use it?

In the context of the LHC, this equation tells us roughly how much energy is needed to create a particle of a certain mass. In the same way, it also tells us how much energy is contained in some lump of matter. For example, if we annihilated a lump matter with mass m with the exact same amount of antimatter, we would expect to cause an explosion of photons with energy E=2mc2 .

Okay. Is that really all there is to it?

Actually, the common form E=mc2 isn’t the whole story. The famous version of Einstein’s relation is actually just an approximation for the full expression, which is: The new letter p is momentum. These are all familiar concepts from high school physics: energy is the ability to do ‘work’ (e.g. move stuff around), mass is some concept of how heavy something is, and momentum characterizes an object’s motion. This equation is telling us that these are all somehow the ‘same’ thing, up to factors of the speed of light.

The first thing you should do is check that this reproduces E=mc2 . Certainly if p=0 we get the old relation. More generally, if the p term is much smaller than the m term then we’re valid in using the old equation as an approximation. So this equation is at least consistent with what all the popular science books tell you.

Potential and Kinetic Energy

The reason why I wanted to write this out is that this explicitly separates energy into kinetic and potential parts, just as we’re used to from basic science. Before I explain this, you should be a little surprised: there is no gravitational or electrical background causing a potential, how do we get potential energy for a free particle drifting through empty space?

It’s all in the mass! The m2c4 term is a kind of potential energy for the particle: it’s the amount of energy borrowed from the universe that is bundled up and stored in the particle. When we annihilate matter and antimatter, we are really releasing this mass potential energy back into the universe.

But wait! Then the p2c2 term is supposed to be some kind of kinetic energy. But you already know what kinetic energy looks like for a particle; it’s 1/2mv2. I know that p=mv, and if I plug this in it doesn’t look right at all!

Lies they taught you at school…

Good. Now we can discuss another ‘lie’ they taught you at school: the usual expression for kinetic energy is also just an approximation! Look back at our main equation: the mass term is multiplied by c4, while the momentum term is only multiplied by c2. Since the speed of light is a big number compared to the usual velocities that we’re used to, we can see that the mass term is much, much bigger than the kinetic energy term.

The reason why we never talk about the mass potential energy in high school physics is that usually it’s not possible to convert mass energy into energy useful for work; a particle’s mass doesn’t change. The first time we use such a conversion is in chemistry when we look at nuclear beta decay. (So E=mc2 is part of the explanation for “why does the sun shine?”)

In order to recover the usual form of the kinetic energy, we can make an approximation. Mathematically this means we do a Taylor expansion. (For those unfamiliar with calculus: this is just a natural way of expanding a function in terms of smaller and smaller corrections.) A good chunk of physics has to do with making clever Taylor expansions. 🙂 In order to do an expansion we need an expansion parameter which is small and dimensionless (it doesn’t make sense to call a dimensionful quantity ‘small’ without a reference point). In this problem we are saying p2 is much smaller than m2c2, so we can write the expression for the energy as: Voila! We’ve explicitly written out the energy as a mass potential term plus the usual kinetic energy form. Here the dots mean terms which are smaller by factors of (p2/m2c2), which is indeed a very small number for everyday velocities much smaller than the speed of light.

Another short summary

Our conclusion is that the Einstein relation tell us that a particle’s energy is given by a [quadratic] sum of its mass and kinetic energy. Momentum, energy, and mass are all the same thing in different forms. A particle’s mass is energy that stored up in making that particle heavy while a particle’s momentum is energy that is used to make that particle move.

A hint of more advanced stuff

That’s it for the main idea of this post. While we’ve done some work, however, I wanted to share something to entice any future physicists (or recreational physicists) out there. We can compare our ‘complete’ energy-mass-momentum with the equation for a circle from high school algebra to motivate a mathematical understanding of Einstein’s so-called ‘special relativity’. In the first line is the equation for a circle of radius r. In the second line we’ve rewritten our energy-mass-momentum relation in a suggestive way. The left-hand sides of both equations are constants.

The first equation tells us that a point (x,y) is part of a circle of radius r if the sum of the squares of its coordinates is equal to r2. The actual point (x,y) can change, but in order for the point to stay on the circle it has to change in such a way that the relation is maintained. If x increases, y has to decreases; and neither x nor y can increase/decrease too much or else it’s impossible to satisfy the equation. (e.g. the point (2,y) doesn’t live on the circle of radius 1 for any y.)

Let’s understand the second equation in the same way. Now I’m telling you that a particle’s mass is constant. It’s a fundamental property of the particle. (There’s an old notion of ‘relativistic mass’ which has been discarded in the modern way of looking at this.) The particle’s energy and momentum can change (e.g. through elastic collisions), but they must change in such a way that the above relation is satisfied. If the momentum increases, then the energy increases. Well that makes sense from our intuitive understanding of momentum. This also tells us that there is a minimum energy given by p2=0, E2=m2c4. I.e. when the particle is at rest the energy is just the mass potential energy.

Great. This all seems like I’m stating the obvious in an overly complicated way. The point is this: the equation for a circle is also a way of defining length. The distance from the point (0,0) to (x,y) is given by r, the length of the ‘radius’ to the point. A definition of length (called a ‘metric’) defines a particular kind of geometry. The symmetries of the metric are symmetries of the geometry: for example, the rotational symmetry of the circle manifests itself in the rotational symmetry of the two-dimensional Euclidean plane.

In the same way the rewritten mass-energy-momentum equation is also a definition of ‘length’ in this energy-momentum space. It has a funny minus sign. The relation can be written in terms of space-time (the combined coordinates of space and time) as where t is time, x is distance in space, and s is some constant (like r or m2c4) called the proper distance.. In special relativity the trajectory of a free particle must obey this equation of constraint. The symmetries of this ‘metric’ (this thing which defines a preserved length) are called Lorentz transformations. The space defined by these symmetries is called Minkowski space (versus Euclidean space that we’re used to). Just as the rotations caused a point to move around in a circle of constant radius, Lorentz transformations are a rotation in spacetime that preserve the proper distance, s.

In particular, what that means is this remarkable fact:

Space and time are in some sense the same thing.

Of course this statement needs to be understood in a mathematical context. Of course space and time are different: we can move back and forth in space but only in one direction in time, etc. But mathematically one can do rotations between space and time. This is precisely the origin of the magnificent results of special relativity: length contraction and time dilation!

It turns out that the analog of a rotation in Euclidean space by some angle is a boost in Minkowski space. The name is chosen specifically to make clear the relation to picking a reference frame.

Anyway, this opens up the rabbit hole to the fantastic story of special relativity which one can find in any number of excellent books or online references. For those who really want to pursue the mathematical story at a basic level, I cannot recommend enough Sander Bais’ book Very Special Relativity. Those with a high-school physics background can read the relevant chapter of The Feynman Lectures on Physics.

Flip