# That famous equation

By Sean Carroll | October 1, 2005 12:10 pm

Brian Greene has an article in the New York Times about Einstein’s famous equation E=mc2. The relation between mass and energy was really an afterthought, and isn’t as important to physics as what we now call “Einstein’s equation” — Rμν – (1/2)Rgμν = 8πGTμν, the relation between spacetime curvature and stress-energy. But it’s a good equation, and has certainly captured the popular imagination.

One way of reading E=mc2 is “what we call the `mass’ of an object is the value of its energy when it’s just sitting there motionless.” The factor of the speed of light squared is a reflection of the unification of space and time in relativity. What we think of as space and time are really two aspects of a single four-dimensional spacetime, but measuring intervals in spacetime requires different procedures depending on whether the interval is “mostly space” or “mostly time.” In the former case we use meter sticks, in the latter we use clocks. The speed of light is the conversion factor between the two types of measurement. (Of course professionals usually imagine clocks that tick off in years and measuring rods that are ruled in light-years, so that we have nice units where c=1.)

Greene makes the important point that E=mc2 isn’t just about nuclear energy; it’s about all sorts of energy, including when you burn gas in your car. At Crooked Timber, John Quiggin was wondering about that, since (like countless others) he was taught that only nuclear reactions are actually converting mass into energy; chemical reactions are a different kind of beast.

Greene is right, of course, but it does get taught badly all the time. The confusion stems from what you mean by “mass.” After Einstein’s insight, we understand that mass isn’t a once-and-for-all quantity that characterizes an object like an electron or an atom; the mass is simply the rest-energy of the body, and can be altered by changing the internal energies of the system. In other words, the mass is what you measure when you put the thing on a scale (given the gravitational field, so you can convert between mass and weight).

In particular, if you take some distinct particles with well-defined masses, and combine them together into a bound system, the mass of the resulting system will be the sums of the masses of the constituents plus the binding energy of the system (which is often negative, so the resulting mass is lower). This is exactly what is going on in nuclear reactions: in fission processes, you are taking a big nucleus and separating it into two smaller nuclei with a lower (more negative) binding energy, decreasing the total mass and releasing the extra energy as heat. Or, in fusion, taking two small nuclei and combining them into a larger nucleus with a lower binding energy. In either case, if you measured the masses of the individual particles before and after, it would have decreased by the amount of energy released (times c2).

But it is also precisely what happens in chemical reactions; you can, for example, take two hydrogen atoms and an oxygen atom and combine them into a water molecule, releasing some energy in the process. As commenter abb1 notes over at CT, this indeed means that the mass of a water molecule is less than the combined mass of two hydrogen atoms and an oxygen atom. The difference in mass is too tiny to typically measure, but it’s absolutely there. The lesson of relativity is that “mass” is one form energy can take, just like “binding energy” is, and we can convert between them no sweat.

So E=mc2 is indeed everywhere, running your computer and your car just as much as nuclear reactors. Of course, the first ancient tribe to harness fire didn’t need to know about E=mc2 in order to use this new technology to keep them warm; but the nice thing about the laws of physics is that they keep on working whether we understand them or not.

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• http://iso42.blogspot.com iso42

Peter Norvig wrote a performance review for Albert for 1905. See
http://iso42.blogspot.com/2005/09/performance-review.html
for the link.He suggests that the PR department changed the original L/c^2
equation into the more marketable E=mc^2

• http://pantheon.yale.edu/~eal48 Eugene

That’s a very nice article. And he managed it without mentioning “string theory”!

• http://iso42.blogspot.com iso42

By the way, Norvig’s comment is not only funny but correct. Albert’s paper does NOT contain the famous E=mc^2. The closests is a remark about L/V^2, where L is the energy and V the speed of light: “Gibt ein Koerper die Energie L in Form von Strahlung ab, so verkleinert sich seine Masse um L/V^2.”
I doubt an equation involving a fraction would have been a gret success (in the US).

• http://www.jessemazer.com Jesse M.

In particular, if you take some distinct particles with well-defined masses, and combine them together into a bound system, the mass of the resulting system will be the sums of the masses of the constituents plus the binding energy of the system (which is often negative, so the resulting mass is lower).

Would “the sums of the masses of the constituents plus the binding energy” be equal to the sums of the masses of the constituents plus the kinetic energy of the constituents plus the potential energy of the constituents? In other words, is the “binding energy” of a bound system the same as kinetic energy + potential energy for that system? This would sort of make sense, because I think the potential energy of a bound system would be negative, so this would be the difference between the depth of the potential well and the system’s total kinetic energy, and if you added enough kinetic energy so the two were equal, that would be the point where the system would tear itself apart (come unbound). Am I getting this right?

If so, then for me it’s a little easier to think of the implications of E=mc^2 as “both kinetic energy and potential energy contribute to the inertia of a bound system”–a hot brick is a little harder to accelerate than a cold one, and a compressed spring is harder to accelerate than a relaxed one. And because of the equivalence principle, this also means the hot brick and the compressed spring would weigh a little more too.

• Annonymous

Sean Wrote:
One way of reading E=mc2 is “what we call the `mass’ of an object is the value of its energy when it’s just sitting there motionless.” The factor of the speed of light squared is a reflection of the unification of space and time in relativity. What we think of as space and time are really two aspects of a single four-dimensional spacetime, but measuring intervals in spacetime requires different procedures depending on whether the interval is “mostly space” or “mostly time.” In the former case we use meter sticks, in the latter we use clocks. The speed of light is the conversion factor between the two types of measurement…
Greene makes the important point that E=mc2 isn’t just about nuclear energy; it’s about all sorts of energy

Interesting. Has anybody ever figured out why the rest mass is related to energy by velocity? We know that the space/time (note: not spacetime) dimensions of velocity squared are (cm/sec)^2, but if you try to find the dimensions of mass, or the dimensions of energy, on Google, you get something like this:

“The physical dimension of energy is defined as mass times length squared over time squared.”

Hmmm, that’s not very interesting. Try again. Whoa, now we find something different, “The unit of energy is the joule, with dimensions kgÂ·m^2/s^2.” Grrrrr! Let’s try for the dimensions of mass… Turns out that we run into the same tautology, but if we dig hard enough we can find that the dimensions of mass can be expressed in terms of geometric units, which are meters per kilogram. Well at least we see something in common with velocity, but then this space unit is not much help really, because it’s just a unit of time expressed as length and mass expressed as the magnitude of four-momentum vector.

Geezzz, guys, can’t we do better than that? What if the space/time dimensions of energy were the inverse of the space/time dimensions of velocity, with dimensions s/l? Now that would be interesting! And it’s not all that unreasonable given the emphasis on symmetry these days, huh? I mean come on, what’s a little spontaneous symmetry swapping among friends?

Lets see now. With that little discovery, and Einstein’s un-American equation of mass,

M = L/V^2,

we get

m = t/s / (s/t)^2 = (t/s)^3, [1]

where s and t are units of space (length) and time (seconds). Now we’re getting somewhere, at last! Can we go further? What about momentum, p?

p = mv (we’ll use the more modern lower case letters), so from [1]
we get

p = (t/s)^3 * s/t = (t/s)^2.

Hmmm, what can we do with that? Well, we see that these dimensions are really similar to the dimensions of that enigmatic concept of action, or momentum times space, the foundation of the uncertainty principle. That’s interesting, because the energy of radiation is related to h by another form of velocity, a frequency, 1/t. Let’s see:

E = hv,

or with our new dimensions of energy,

h = E/v = (t/s)/(1/t) = t/s * t = t^2/s,

but, if frequency really is a velocity, the space unit is still there, it’s just used over and over again in a rotation, right? So, then, putting that ole sneaky space unit back where it belongs, let’s do that equation again,

h = E/v = (t/s)/(s/t) = t/s * t/s = (t/s)^2,

voila! How interesting is that? Momentum with the same dimensions as h, and a symmetry between the way mass and momentum relate to energy through velocity. Too bad we can’t come up with something like that, instead of that horrible four-momentum vector, Sean.

• John Quiggin

That all helps a lot, thanks.

• amanda

EEk! My teacher used to rage about this! Mass is a scalar, energy a component of a vector! The mass of an object does NOT change when it moves! Expecting the masses of different objects to add up makes precisely as much sense as expecting the lengths of non-parallel vectors to add up!
In short, I cringed all the way through reading Greene’s article. And one more thing: there is no “mass being converted to energy” in a nuclear reaction, any more than there is in a chemical reaction. In both cases, one form of energy is being released and converted to other forms of energy.

• Thomas Palm

The myth about E=mc2 has another half: that it is important for designing nuclear bombs, but Sean’s final paragraph is just as valid for primitieve tribes harnessing nuclear power.

It’s more or less a coincidence that the theory of relativity and nuclear physics were developed at the same time. One might easily imagine a world where it was measurements of nuclear reactions and their mass loss that gave inspiration for the theory of relativity.

• http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

Jesse M– that’s basically right. Really “binding energy” is pretty much the same as “potential energy,” and I was simply neglecting the kinetic energies of the constituents.

• Doug

John wrote:

That all helps a lot, thanks.

I can’t tell if that quip is missing a smilie or not. When you were discussing this on CT, almostfamous wrote:

re: the units. doesn’t dimensional analysis not make sense if you could just equate mass to energy? that would be M=ML2T-2

i realize we aren’t talking about billiard balls anymore, but i thought dimensional analysis held up in most cases. unless, of course something has changed in the way physicists think about these things since early 2002, the last date at which i took a class in physics.

He makes an important point. Mass and energy are not the same, but physicists have to assume that they are in order to be able to explain what’s happening. The fact is, though, if mass is the equivalent of energy, then the dimensional inconsistency that almostfamous points out has to be swept under the rug and ignored. On the other hand, if they are not the same, which commonsense seems to insist, then the fact that they each can be changed into something else reveals that they are not fundamental entities, because that which can be changed into something else is obviously not fundamental. This indicates that the more fundamental entity is common to both mass and energy, and since both matter and radiation are related to energy by velocity, and velocity is space and time, then we ought to take the hint, and look to motion, s/t, as the fundamental constituent of the universe.

If a universal motion exists, it consists of a universal increase of space and time, delta s/delta t. We know that time progresses universally, because we observe it. However, we haven’t considered an analogous progression of space, because we haven’t observed it, except for the last 75 years or so (these things take a long time to sink in). Nevertheless, it’s really only a matter of a slight interpolation to assume that space has the properties of time, i.e. space, like time, progresses. In fact, it’s a sound scientific procedure to interpolate like this, much more justified scientifically than the wild, ad hoc, theoretical inventions so common today.

Of course, a hypothesis is justified by its results, and the above results are very encouraging (I don’t know if you think so. I can’t read the intent of your one-liner.) At any rate, this assumption opens up a whole new basis for physics in which the speed of light is viewed as the unit ratio of a universal progression of space and time, which redefines space and time as the reciprocal aspects of a universal motion that forms the reference system of all physical phenomena.

It’s just this type of fundamental change in the nature of space and time that people like Gross and Smolin are expecting, as I have been explaining in detail in a thread on the Bad Astronomy Universe Today discussion forum (see: The Reciprocal System of Physical Theory thread). If you want to know more, I’ll be happy to discuss it with you there.

Regards,

Doug

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