# Does the Earth move around the Sun?

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.”

This entry was posted in Science. Bookmark the permalink.

### 19 Responses to Does the Earth move around the Sun?

1. Alex R says:

Quibble. You write: 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.

As you should know, the Coriolis forces that we usually measure, such as with Foucault’s Pendulum, are due to the once-per-sidereal-day *rotation* of the reference frame that is fixed with respect to the Earth’s surface, not (to first approximation) the once-per-year *orbit* of the Earth around the sun.

If we were to do experiments in a non-rotating frame which was fixed with respect to the center of mass of the Earth, we could presumably also measure its departures from being inertial, but they would be substantially smaller than the effect of rotation with respect to the fixed stars seen by Foucault’s pendulum.

2. Sean says:

Yes, but by “Earth stationary at the center” I really meant it. I.e. a coordinate system which rotates with the Earth. Since, after all, if the Sun moves around the Earth, it does so once per (solar) day, not once per year.

3. Thanks, Sean, for explaining my point better than I could, and bailing out my sad attempt at humor.

4. Pingback: thomasmarquart.net

5. Arun says:

Since we have to set up a reference frame w.r.t. something, setting up the frame w.r.t. far away things wins out. While GR says there is no “more natural global coordinate system – inertial or otherwise”, in practice, each time we discovered one more term in earth’s wobbly rotation, we’d have to revise all our calculations. It is simpler to fix a coordinate frame w.r.t things that are distant and virtually unmoving. (That is why theory remains theory, and experimental or observational usage is what makes physics into a science.)

In a reference frame with simpler motion w.r.t. to distant stars or to the CMB, the Sun’s reference frame wins out over the Earth’s reference frame.

6. Arun: if you do that, you can sometimes obscure physics of nearby objects. Take the Schwarzschild metric, given in its standard form. This metric is easily derived from the Einstein Equations by requiring vacuum and “stationarity with respect to distant objects (i.e., that the metric be asymptotically Minkowski).

This metric, however, looks very awful in the neighborhood of the black hole horizon. For a sufficently large mass black hole, however, you can show that a local observer doesn’t really observe any significant effects as (s)he crosses the r = 2M threshold.

In the Schwarzchild coordinates, this seems extremely puzzling, but switching to a coordinate system in which the time coordinate is (roughly) given by the proper time elapsed by a family of geodesically infalling observers (or null light rays), it can be shown that the singular behavior of the metric at the horizon completely evaporates. The price? The loss of the convenient interpretation of the metric components near infinity.

So, my point is, what coordinate system is best depends on what type of calculation you are doing…

7. Arun says:

bittergradstudent, yes, I was thinking just of the enormous and unglamorous work that goes into establishing a good coordinate system for practical astronomy, solar system navigation, etc., etc..

8. Jack says:

OK, I concede that the question depends on your interpretation of “goes around”. My definition has nothing to do with coordinate systems — just look at the worldlines and you will see by inspection which one is “going around” the other. One point I do want to insist on though: it makes no difference whether we are talking about SR or GR here, because you are just as free to make arbitrary choices of coordinates in SR as in GR.
And yes, I did realize that bittergradstudent was joking 🙂

9. spyder says:

Not that it matters so much now that we use GPS systems with which to navigate, but having had to learn celestial navigation in them “old” days, i always wondered if such knowlege would have been fully realized in the “euro-west” had we remained obstinately stubborn about terra-centrism????

10. Phil Plait says:

As it happens, I get real live geocentrists trying to use GR to claim that geocentrism is just as valid as heliocentrism. I find that amusing, because what they are really trying to show is that geocentrism is really the One True Way. It got so bad on my old bulletin board I wrote a FAQ for it:

Geocentrism, as advocated by creationists or other religiously fundamental people, is certainly wrong.

How, you may ask? What is going on is that you can do a change of reference frame to a geocentric one, and by relativity the math must still work out. I readily admit that. I do not understand all the math involved, but I will take it for granted that it works out, and that physically, geocentrism is just as valid as, say, heliocentrism.

But note the words “just as valid”. Also, by relativity, it cannot be any more valid; geocentrism is just another change of frame (although to a non-inertial one).

What geocentrists are saying is that geocentrism is the one, true frame. Creationists must say that because that is what is says in the bible. Now pay attention here, because this is the important bit: to say geocentrism isn’t wrong, you have to accept the premise that any frame of reference is just as valid as any other. But to claim that geocentrism is correct, you have to ignore that very same premise.

Geocentrism as the One True Way is therefore self-contradictory. It doesn’t work.

11. agm says:

In the right reference frame, both the earth and the sun revolve around the server hosting this blog.

12. Jack says:

Oh yes, I meant to ask Sean: What is a “dark energy task force?” Sounds scary.

13. Belizean says:

This is related to the old issue of Mach’s Principle in GR.

Even though things LOOK the same, you CAN tell the difference between 1) the stars spinning about you, and 2) your body spinning at the same angular rate.

So Mach’s Principle is not part of GR (even though it motivated Einstein).

And inertial reference frames have a preferred place in the theory in the sense that they are all connected by the modern version of the Equivalence Principle in a way that accelerated frames are not.

Hence, the Earth revolves around the Sun, because the former’s acceleration well exceeds the latter’s.

14. Belizean:

Mach’s principle makes explicit reference to “distant stars.”–i.e., objects infinitely far away. It then picks out a preferred reference frame by demanding stationarity with respect to these.

A priori, there is no reason to postulate the existence of such objects (in fact, until recently, it was reasonable to believe that the universe might have a finite volume, and therefore, a finite “maximal” diameter).

The equivalence principle is essentially a statemetn that there is no such thing as absolute acceleration–acceleration and four dimensional curvature are interchangeable concepts (speaking very loosely, find a coordinate system where F/m is the same as the Christoffel symbols). Hence, in GR, there are only equivalence classes of inertial reference frames (i.e., sets of frames which do not accelerate with respect to each other), rather than an absolute notion of inertial/non-inertial. That is, unless you invoke something like Mach’s principle, or a Geocentric principle, or a Heliocentric principle.

15. Mark Wyatt says:

I have put together a three part series on geocentrism. The first two parts are dealing with science, and the third deals with the Church position. Take a look.

Geocentricity 101: A beginner’s Course
Geocentricity 101, Part I: Basic Principles
Geocentricity 101, Part II: Basic Physics
Geocentricity 101, Part III: Scriptural and Church Position

Mark Wyatt

16. Amanda says:

How many miles is the atmosphere from the earth’s surface?

17. rosey says: