$E = mc^2$ 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 $E = mc^2$.  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 $E = mc^2$.  We will do so by re-considering one of the most basic ideas from high school physics: the center of mass.

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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 $\Delta x$ 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 $\Delta x \approx m L / M$, where L is the length of the boat.

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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 $t = 0$) the light bulb turns on, and it remains turned on for some time $\Delta t$:

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 $\Delta x = \frac{PAL}{Mc}$.  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: $I = \frac{E}{A \Delta t}$.  Put these together and you get our result:

$\Delta x = E L/(M c^2)$.

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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:

$\Delta x = \frac{m L}{M} = \frac{E L}{M c^2}$

$\Rightarrow E = m c^2$

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.

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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).

1. May 23, 2009 1:15 am

I have seen a blog today by Maien on velocity, mass,gravity. He writes that mass is form of energy but not energy itself. I am also writing same ideas during the last fortnight. I am convinced that mass is the highest level of condensed state of energy and energy is highest level of subtle state of mass.

It now appears that the discussion is becoming more interesting.

One question may be raised, what is the difference between mass and matter? Either mass just represent some quantity of matter or more?

2. May 23, 2009 3:29 am

According to Einstein Theory of Relativity, E=mc^2
1 kg mass of any matter is equivalent to 9 x 10^16 J of energy.

Mass of any matter is Condensed Form of Energy and Energy is Diffused Form of Mass of any matter ?

A question may now arise what existed before the creation of the Universe Energy or Mass or both?

May 23, 2009 11:14 am

Hi Anirudh,

You’re asking hard questions! Sometimes a question may have an answer but any proposed answer would be completely untestable. Like “what was there before the Big Bang?” A scientist is “not allowed” to ask that question, because it would be completely impossible to test. The Big Bang destroyed everything that came before it, and there’s no reason to think that any of the physical laws we have developed would hold. The pre-Big Bang universe could have been completely different from the one we observe now.

As far as the philosophy of mass and energy, it’s a little tricky. It’s clear that mass is a form of energy, but that not all energy can be called mass. Is mass the “most condensed” form of energy? Maybe. It depends on whether the things that we consider “point masses” actually have some size to them. But you’re right, mass energies tend to be enormously large. This is what makes nuclear technolgies so powerful.

Finally, “matter” isn’t a very scientific term. Usually matter means anything with both mass and volume. But your definition of volume changes depending on how closely you look at something. Your body, for example, seems to be solidly occupying a certain volume. But if you zoomed in to look at the scale of 10^-10 meters, you would find that you were made of tiny nuclei with tiny electrons orbiting them, and huge amounts of empty space between them.

• May 25, 2009 1:11 am

Thank you gravityandlevity for your quick reply. I have some more views which may follow in quich succession

E=mc^2 is called ‘Einstein’s energy-mass relation’. According to this relation, 1 kg mass of any matter is equivalent to 9×10^16J of energy. This is a huge amount of energy, equal to 2.5×10^10kWh. It is evident that the amount of energy is same irrespective of the matter taken, whether it is carbon, iron, copper or any other including radioactive elements. The amount of energy thus released does not depend on the atomic number, atomic weight, electronic configuration etc. It is the mass of the matter only based on which the amount of energy is calculated. It means that ‘mass’ is the connecting link between energy and matter.

May 25, 2009 5:38 pm

I think what you’re getting at is that this “mass energy” is a very different kind than what we are used to. It’s not like the energy you get from burning something (say, coal or gasoline) or from exploding something. It is both much, much larger in magnitude and it is completely independent of the properties of the material. A pound of water has as much mass energy as a pound of uranium.

That’s because most form of energies are in some way “chemical” energies. They come from rearranging the atoms within a material in such a way that their chemical potentials are released. It’s something like opening one of those jars that had a compressed spring snake in it. The energy came from having a compressed “spring” inside your material. You just have to agitate it enough for the energy to be released.

Mass energy, on the other hand, is released by destroying the component particles of which something is made. It is universal because all things in the world are made of atoms, which in turn are made of protons, neutrons, and electrons. If you look at this level, on the subatomic scale, there is really not much difference between water and uranium, or anything else for that matter.

3. May 26, 2009 1:29 am

It is written in the Text-Books of Physics that if we give ∆E energy to some matter, then according to E=mc^2, its mass will increase by ∆m, where

∆m=∆E/c^2

Since the value of c is very high, the increase in mass ∆m is very small. For example, if we heat a substance, then the heat-energy given to this substance will increase its mass. But this increase in mass is so small that we cannot measure it even by the most sensitive balance. Similarly, if we compress a spring, its mass will increase, but we cannot confirm this mass-increase by any experiment.

Now the question is whether the change in mass as quoted in these two examples is reversible i.e. when the same substance of example one is cooled down, energy is produced equal to ∆m x c^2 (∆E=∆m x c^2) and in second example when we release the spring , energy is produced equal to ∆m x c^2 and initial mass is retained in both the cases ? Or the above changes are irreversible ?

May 26, 2009 9:18 am

The changes are certainly reversible. A compressed spring is actually more massive than an uncompressed spring, usually by an imperceptible amount. Allowing the spring to expand (and give its energy to some other object) decreases the spring’s mass.

This topic of energetic objects being more massive is a little tricky, and will be the subject of another post in the near future.

4. May 28, 2009 1:46 am

Incidentally, you’re not quoting the full energy-mass relationship. In full, it is:
E^2 = m^2c^4 + p^2c^2
Which you can see reduces to E = mc^2 when p = 0, p being the momentum, which is also a quantity associated with movement.
(aiusepsi)

The formula describes an equivalence of mass and energy, rather than a convertibility between mass and energy.
Note though that the formula is only valid for stationary objects. For moving objects, which have non-zero momentum p, the correct formula is E^2-p^2c^2=m^2c^4, thus a moving object would have more energy than just mc^2. (terence tao)

Dear Mr. gravityandlevity , do you agree with the above mentioned views of two bloggers ?

If yes, kindly let me know what is difference between weight and momentum ? Kindly also let me know if there is some commonality between weight and momentum ?

Warm regards

Anirudh

May 28, 2009 10:29 am

E = mc^2 describes only one particular kind of energy: the MASS energy. There are plenty of other kinds of energy out there, including kinetic energy and all kinds of potential energy. When people write E = mc^2, they don’t mean that the total energy of an object is equal to mc^2, just that its mass energy is mc^2.

The formula you posted from other blogs is completely correct. It calculated the Total energy E when there are both mass energy and kinetic energy present.

Weight and momentum are very different concepts. Weight is the force of gravity (from the Earth, usually) pulling on object. There is nothing particularly special about it. An object with a lot of weight on Earth can be taken into outer space and become weightless. Momentum is a conserved quantity which can generally be defined as an object’s mass times its velocity. See the wikipedia article for a more thorough explanation.

5. May 29, 2009 3:35 am

Thank you dear gravityandlevity for your response which may be O.K. at Earth. But the same may not be applicable at Sun where the situation is altogether different. The concepts ‘weight’ and ‘momentum’ should be defined differently for Sun.

6. May 30, 2009 1:20 am

“Einstein came out with the general theory of relativity (GRT)in its final form in 1915. By that time he had proposed his three famous (“critical”) effects to be used for verification of the theory: gravitational displacement of spectral lines, light deflection in the gravitational field of the Sun and displacement of prehelion of Mercury. More than half a century has passed but the problem of the experimental verification of GRT is still as urgent as ever.

All the Einstein’s effects have been observed but the experimental accuracy is still low. The error of measurement of the gravitational displacement of spectral lines is about 1%. The deflection of light rays in the field of the Sun has been measured to an accuracy of about 10\$ and so on. (From Key Problems of Physics and Astrophysics by V.L.Ginzburg, Published by Mir Publishers, Moscow, 1976)”

I don’t know what is the current status of experimental verification now ? Would you like to enlighten me on this topic?

• August 21, 2009 1:43 pm

In case anyone is still reading this: The HIPPARCOS satellite, launched by the European Space Agency, I believe, measured stellar displacement due to the gravitational effect of the Sun. Because of its (then) unprecedented precision, it was able to use the displacement of stars a great angular distance away from the Sun (unlike solar eclipse measurements by hand). I believe Einstein’s particular brand of GR (as opposed to other more complex brands) has been verified to a high level of certainty in this arena.

Incidentally, it may interest some folks to know that Newton’s laws also predict a deflection (by treating the effect of gravity on light particles as their mass approaches zero in the limit); this deflection is just half of what is predicted by GR. The Eddington experiment of 1919 came out on the side of GR, but only just barely; today, it is not considered a definitive demonstration of GR.