The equivalence of mass and energy: the “center of energy”
is the world’s most famous equation. It is a remarkably simple one, but its meaning is quite difficult to grasp and is frequently misinterpreted. It says that mass is a form of energy, just like kinetic energy or spring energy or any other energy. Somehow, physical matter is a form of “locked up” and intensely concentrated energy. This is the driving principle behind nuclear power and nuclear weapons: by converting “locked up” mass energy into “useful” energy, we can extract an immense amount from a small amount of mass.
But how is mass a form of energy? How do you think about “mass energy”? That’s a difficult question, and any answer I can come up with feels incomplete. But I do know a couple of interesting thought experiments that are worth discussing, and which shed some light on Einstein’s mysterious . The arguments in this post are stolen once again from The Einstein Paradox: And Other Mysteries Solved By Sherlock Holmes (as I did with the double slit experiment post). There is something a little questionable about what I’m doing here, I must admit, and I highly encourage skeptics to leave their comments.
The end goal this post is to derive . We will do so by re-considering one of the most basic ideas from high school physics: the center of mass.
For a given group of objects, the “center of mass” can be found by averaging their position, giving more weight to the objects with larger mass. It is the mass-averaged position of the group. If there are no outside forces acting on the group, then the center of mass does not move, no matter how the individual objects are pulling on each other. For example, when the moon orbits the earth it causes both the moon and the earth to “wobble” . But they do so in such a way that their common center (the center of mass) does not change.
Here’s a more common example, which appears in every introductory physics course, and whose result we will need later. Imagine that you are standing up in a large boat on the (frictionless) water, and you walk from one end of the boat to the other. Like this:
As you walk to the right, you push off the boat in order to move forward. As a result, the boat moves to the left. If the boat is much heavier than you, it will only move a little bit. If the boat is much lighter than you, then you will mostly just stay in place and push the boat underneath you. Anyone who (like me) has ever tried to jump out of a canoe onto a dock is aware of this! Either way, the end result is that the center of mass of (you + boat) is unchanged. You can use this fact to find the distance that the boat shifts as you walk from one end to the other. I will quote the result here because we will need it a little later on. If the boat’s mass M is much greater than your mass m, then the distance the boat shifts is , where L is the length of the boat.
What I’m going to show you now is a violation of our “center of mass doesn’t move” doctrine. We’re going to see a simple collection of objects whose center of mass can apparently shift from one place to another, even though no outside forces are acting on it.
Imagine a long tube with a light bulb inside it. You can consider it to be sitting on a frictionless surface like our boat if you want, but we might as well imagine it to be in outer space, with absolutely nothing acting on it. At either end of the tube is a surface of area A which can absorb light. At some instant in time (which we’ll call ) the light bulb turns on, and it remains turned on for some time :
You can ask what’s powering the light bulb (which is a good question, because this might be a subtle point!), but for now let’s assume it has some small battery built into its base.
What effect does the light bulb have on the tube? Well, to figure that out you have to know that light can apply pressure. It doesn’t require any quantum mechanics or “light as a particle” thinking to understand this. You can just think that light is an electromagnetic wave, with both electric and magnetic fields. When the light hits one of the absorbing surfaces, its electric field accelerates the charges that comprise the surface. The moving charges then get pushed outward by the light’s magnetic field. This is called “radiation pressure”, and its strength is related to the intensity of the light (power/area = I) by P = I/c.
So why does the pressure cause the tube to shift? The answer is that the light travels outward from the lightbulb at a finite speed c, so it hits the left side first. The pressure on the left side of the tube begins almost instantaneously, whereas there is no pressure on the right side for a time L/c, when the light has propagated to the far end of the tube. This gives the tube some leftward momentum, which it keeps until the light shuts off. Right after the shutoff there is a short period where there is pressure to the right but not to the left. This brings the tube to a stop. But the net effect is to have shifted the tube to the left. Graphically, you can represent the process like this:
From this sloppy graph it’s fairly easy to figure out how far the tube shifts. If the tube has a mass M, then the total shift is . The pressure P is related to the intensity as P = I/c, and I can be written as the total energy transferred by the light bulb divided by the area and the time over which it was transferred: . Put these together and you get our result:
So there we have it: an absolute violation of the doctrine that we all learned in introductory physics. The center of mass has moved with no outside forces involved. Maybe, then, we need to re-examine our doctrine. Maybe the thing that really remains unchanged is not the center of mass, but the center of energy. After all, we didn’t move any mass around by turning on the light bulb, and the center of mass still changed. We did, however, move energy. Some amount of energy E that was stored in the light bulb on the left side of the tube got moved to the right side. As a result the tube shifted to the left. This is exactly the same as our canoe example: walking to the right makes the canoe move to the left. But this time we didn’t move any physical matter, we just moved energy.
If moving energy from the left to the right produces the same effect as moving mass from the left to the right, then what is their conversion? In other words, moving energy E from left to right must be the same as moving some amount of mass m from left to right. What is the relation between E and m? Well, here we can use our result from the canoe problem:
This is a fairly profound result, and it gives new insight into the way we think about the center of mass. When I walk from one end of a boat to the other, I am moving my considerable mass energy to one side. The system must retain a fixed center of energy, though, so the boat shifts its considerable mass energy in the opposite direction. For most problems that we think about, the mass energies are so enormous compared to other forms of energy in the problem that we might as well think that the “center of mass” is conserved. In reality, however, the idea of a “constant center of mass” is just an approximate consequence of a much bigger idea: the center of energy.
Update: Apparently this thought experiment is called “Einstein’s box”: http://galileo.phys.virginia.edu/classes/252/mass_and_energy.html. And there is a legitimate explanation for anyone who realized that the tube cannot be perfectly rigid (the momentum of the left side of the tube is not communicated to the right side instantaneously).