October 2005

In the comments to Mark’s post about the embarassment being caused to the U.S. by the creationism trial in Dover, a scuffle has broken out over another deep question: does the Earth go around the Sun? See here and here and here.

It’s actually a more subtle question than you might think. The question is not “Was Ptolemy right after all?”, but rather “in the context of modern theories of spacetime, is it even sensible to say `X goes around Y,’ or is that kind of statement necessarily dependent on an (ultimately arbitrary) choice of coordinate system?”

You’ve come to the right place for this one; biologists can have their fun demolishing creationism, but we’re the experts on the whole geocentrism/heliocentrism thing. The answer, of course, does indeed depend on what one means by “move around,” and in particular the comments refer to the notion of a “reference frame.” I can think of at least three different things one might mean by that phrase. First there is the idea of a “global reference frame.” By this we mean, set up some perpendicular axes (some choice of coordinates x, y, and z) locally, right there in the room where you are sitting. Now extend these coordinates globally throughout space, by following straight lines and keeping everything appropriately perpendicular. That would be a global reference frame. (I am implicitly assuming that the coordinates are “Cartesian,” rather than using polar coordinates or some such thing — no reason to contemplate that particular complication.)

The second notion is that of an “inertial reference frame.” Inertial frames are actually a subset of all possible global frames; in particular, they are the global frames in which free (unaccelerated) particles appear to move on straight lines. Basically, this simply means that we allow the coordinate axes to float freely, as would gyroscopes in free-fall, rather than rotating them around. Newton figured out long ago that we could decide whether we were in an inertial frame or not by examining whether the water in a bucket that was stationary with respect to our frame began to creep up the sides (as it would if our bucket were rotating with respect to a really inertial frame).

Finally, we have the more flexible notion of a “coordinate system.” Unlike a global frame or the even-more-restrictive inertial frame, a coordinate system can be set down throughout space in any old way, so long as it assigns unique coordinates to each point. No mention is made of extending things along straight lines or keeping angles perpendicular; just put down your coordinates like a drunken sailor and be done with it.

Now what does all this pedantic geometry have to do with the Earth going around the Sun? Well, what Copernicus was really saying was that there is no inertial reference frame in which the Earth is stationary at the center and the Sun moves in a circle around it. Of course we could still imagine some global frame with the Earth stationary at the center; in fact, such geocentric reference frames are often quite useful. But it wouldn’t be inertial, as we could easily tell by the existence of Coriolis forces (as measured for example by Foucault’s pendulum). That is the sense in which it’s “really” the Earth that goes around the Sun, not vice-versa.

But now comes along Einstein and general relativity (GR). What’s the situation there? It actually cuts both ways. Most importantly, in GR the concept of a global reference frame and the more restrictive concept of an inertial frame simply do not exist. You cannot take your locally-defined axes and stretch them uniquely throughout space, there’s just no way to do it. (In particular, if you tried, you would find that the coordinates defined by traveling along two different paths gave you two different values for the same point in space.) Instead, all we have are coordinate systems of various types. Even in Newtonian absolute space (or for that matter in special relativity, which in this matter is just the same as Newtonian mechanics) we always have the freedom to choose elaborate coordinate systems, but in GR that’s all we have. And if we can choose all sorts of different coordinates, there is nothing to stop us from choosing one with the Earth at the center and the Sun moving around in circles (or ellipses) around it. It would be kind of perverse, but it is no less “natural” than anything else, since there is no notion of a globally inertial coordinate system that is somehow more natural. That is the sense in which, in GR, it is equally true to say that the Sun moves around the Earth as vice-versa.

On the other hand, sometimes one is able to make useful approximations, and there’s no reason to forget that. In particular, gravity in the Solar System is extremely well described as “flat spacetime (as in special relativity) plus a small perturbation.” From this perspective, we can very well define inertial frames in the flat background spacetime on top of which gravity is a tiny perturbation. And in those frames, it’s the Sun that is basically stationary and the Earth that is truly moving. So even the most highly sensitive general-relativists would not complain if you said that the Earth moved around the Sun, unless they hadn’t yet had their coffee that morning and were feeling especially confrontational.

Tune in tomorrow for a detailed examination of “what goes up, must come down.”

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” — 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=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.