# That famous equation

Brian Greene has an article in the New York Times about Einstein’s famous equation E=mc^{2}. 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” — R_{μν} – (1/2)Rg_{μν} = 8πGT_{μν}, 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=mc^{2} 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=mc^{2} 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 c^{2}).

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=mc^{2} 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=mc^{2} 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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