We’re all taught in school that the Earth goes round the Sun. But if you look around on the internet, you will find websites that say something quite different. There you will find the argument that Einstein’s great insights imply otherwise — that in fact the statements “The Earth goes round the Sun” and “The Sun goes round the Earth” are equally true, or equally false, or equally meaningless.

Here, for example, is this statement as written in *Forbes* by professor Richard Muller at the University of California, Berkeley. It opens as follows: “**According to the general theory of relativity,*** the Sun does orbit the Earth*. And the Earth orbits the Sun.” I invite you to read the rest of it; it’s not long.

What’s his point? In Einstein’s theory of gravity (“general relativity”), time and three-dimensional space combine together to form a four-dimensional shape, called “space-time”, which is complex and curved. And in general relativity, **you can choose whatever coordinates you want on this space-time.**

So you are perfectly free to choose a set of coordinates, according to this point of view, in which the Earth is at the center of the solar system. In these coordinates, the Earth does not move, and the Sun goes round the Earth. The heliocentric picture of the planets and the Sun merely represents the *simplest* choice of coordinates; but there’s nothing wrong with choosing something else, as you like.

This is very much like saying that to use latitude and longitude on the Earth is just a choice. I could use whatever coordinates I want. The equator is special in the latitude-longitude system, since it lies at latitude=0; the poles are special too, at latitude +90 degrees and -90 degrees. But I could just as well choose a coordinate system in which the equator and poles don’t look special at all.

And so, after Einstein, the whole Copernican question — “is the solar system geocentric or heliocentric?” — is a complete red herring… much ado about nothing. As Muller argues in his article, “**the revolution of Copernicus was actually a revolution in finding a simpler way to depict the motion, not a more correct way.**“

Well? Is this true? If not, why not? Comments are open.

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My two cents: I agree that “what goes around what” is subjective (and is even with Galilean relativity, I would have thought). What you *can* say is that the curvature of space in the Earth-Sun system (as measured by a deflection of a test particle flung from far away, say) is mostly due to the mass of the Sun, not the mass of the Earth.

These are some good ingredients for an answer to the question, yes. We need some more.

When the Earth orbits the Sun, then the point on the sky where we see the Sun is also in the direction where the sun is at that moment. The Sun always emits light and no matter where the Earth is on its orbit, sunlight will reach the Earth and it originates from the same point. If the Sun orbited the Earth, the Sun wouldn’t be where we see it. As the light takes 8 minutes to reach us, the Sun would actually be 2 degrees away from where it appears to be.

Not 2 degrees; only about 8.5 minutes/1 year * 360 degrees, which is .006 degrees, which in turn is only 1.2 percent of the angular diameter of the Sun on the sky, (about half a degree.)

But the real problem with your suggestion is that when you say “the Sun wouldn’t be where we see it”, how would we know that? How can we tell where the sunlight originates other than by looking at the Sun? There’s no monitor there to tell us “the light originated from here…” So I’m afraid this argument holds no water. In fact, if it were correct, then we ought to be able to use it to tell that the Sun is moving at 130 miles per second around the galaxy (much faster than Earth’s motion around the Sun, which is only 20 miles per second). But we cannot.

Gravity: Imagine a sheet called space moving through a 4th dimension where the 4th dimension was called time. The fabric’s direction of travel is from a place we call the past to a place we call the future. The surface of the sheet is what we will call the present. This 4th dimension is not the same as our ordinary experience of time but is related to it. If you put a lead ball on the moving fabric the ball would resist the movement so as to cause the fabric to deform. The ball and the fabric pushing the ball would lag behind the rest of the fabric. According to our definitions, the ball could be said to be in the past. If another ball were put on the fabric it would deform the fabric too. If the two balls were close together they would approach. The balls would seem to be attracted to one another but in reality, they would be both moving toward the same place.

Your images bear some similarity to the way general relativity works, but not enough to actually capture what the math actually says. There are some very important differences. So I’m afraid this imagistic approach to the problem won’t help us answer these questions.

Maybe one way forward is to calculate the motion of the two parts relative to the centre of mass of the system?

Muller’s argument implicitly claims that the center of mass is irrelevant to the facts in general relativity; putting the center of mass at the center of your coordinates is just a choice. That’s one of the questions we have to think about, to decide if (and to what extent) he is right about that.

This was an interesting question. In special relativity, there is no difference between moving & not moving, in that being at rest, is exactly the same as being in uniform motion. Whether you are moving or not depends on the observer. Actually, this is the basis of regular, as per Galilean relativity also. However, in special relativity, there is a difference between rotating & not rotating. Two objects rotating relative to each other, will be agreed upon by all observers.

If we just consider the sun & earth, ignoring all the other planets, the earth & sun are orbiting the center of mass of the earth /sun system (barycenter)

If we were to take the earth as stationary, and the sun orbiting around it…the distant stars would also need to rotate relative to the earth, and how would we explain the centripetal force on the earth from the sun?

So, I think you are mixing up a couple of things here. First, there’s no need to have the distant stars rotate; that can be soaked up by the daily spin of the Earth, which is irrelevant to the question of whether the Earth goes around the Sun. You might want to read this earlier blog post: https://profmattstrassler.com/2022/02/07/the-best-proof-that-the-earth-spins/ What this shows is that we can prove that the Earth spins *and* the Earth-Sun line rotates, relative to the fixed stars. But we can’t use that to determine who goes around who.

Muller’s claim is that in general relativity even your statement that ” the earth & sun are orbiting the center of mass of the earth /sun system (barycenter)” is an arbitrary coordinate choice. It would be just as accurate, he says, to say the Earth orbits the Sun or the Sun orbits the Earth or all of us are orbiting Venus, or a point midway between Jupiter and Saturn. We need to address his claim somehow.

When I said the stars would need to rotate, I was not speaking of the rotation observed nightly by the spin of the earth on its axis. I was actually referring to the fact that different constellations are only observable at certain times of the year depending upon where there earth is on its orbit around the sun.

However, I see your point.

I’m afraid you’re making a geometrical error. “Night” is determined by where the Earth-Sun line is. If the Sun orbits the Earth, the constellations visible at certain times of year will be exactly the same as if the Earth orbits the Sun. The stars will be fixed in either case, only the plane dividing day from night has to rotate.

Yes, I realized that, that’s why I said I see your point.

Completely irrespective of GR vs Newtonian mechanics, it has always been possible to choose whatever frame of reference you desire. But inertial frames are special in a principled way, and the fixed-earth frame of reference is very much not an inertial frame. I think this idea (for instance as you quoted from Muller) comes from an overcommitment to the “relative” part of relativity.

This gets into Mach’s principle, but my understanding is that that’s not really a live physical principle. If you have a hydrostatic planet in intergalactic space that’s ellipsoidal, I don’t think it really works to say it’s not rotating but somehow the distant galaxies rotating around it are creating a field that gives it an equatorial bulge. But my understanding is imperfect here.

There’s a lot of merit in what you say, but Muller is making the claim that the fact that general relativity’s equations are covariant, and completely the same in all coordinate systems, proves that one should *indeed* view things just as “relatively” as he does in his article. So he’d disagree with you.

That said, I don’t think Mach’s principle (can one fix a frame by insisting the distant stars appear fixed?) is an issue here. The coordinate system in question need not rotate; it’s merely the Sun or Earth’s position which travels in a circle as specified by those coordinates, but the coordinates may leave the distant stars roughly fixed.

The issue with Mach’s principle would only arise if Muller attempted to deny that it’s clear that the Earth rotates. (Should he, for logical consistency? this is actually a subtle point, because it has to do with asymptotic boundaries.)

A couple thoughts on this:

(1) The proper acceleration of the Earth is a lot larger than the Sun’s. That’s a coordinate-independent statement, and suggests that it’s more convenient to use Sun-centered coordinates rather than Earth-centered ones (in Newtonian mechanics the latter would give you larger fictitious forces, in GR it would give you larger obnoxious off-diagonal terms in the metric). Also, the problem is not about spacetime curvature, since the amount of curvature (as measured by various scalars) is coordinate invariant.

(2) Astronomers are often concerned with observing Lorentz invariance violating backgrounds, such as the CMB, cosmic rays, the other stars in our galaxy, or other galaxies. These all define preferred frames, and you would want to perform the data analysis in a frame that isn’t accelerating with respect to them. So people subtract out the Earth’s orbital motion to effectively get Sun-centered coordinates, or even additionally subtract out the Sun’s motion relative to those frames.

(3) If neither of the above two factors are important (i.e. if you’re not sensitive to the Earth’s proper acceleration, and you’re not looking at things outside the Earth-Sun system) then the two descriptions are equally good.

On (2) and (3): I’m not sure one needs to look at “things”. That is, I’m not sure you need a Lorentz-invariance violating background of actual physical material. You may only need an asymptotically flat external region and an inertial observer in that region in order to say which description is better, and an isolated two-body system, observed at a distance, provides that. It’s an interesting and debatable point to which I’ll return in later posts; we can discuss it more.

On (1), a technical point: remind us why proper acceleration is coordinate-invariant? Locally, of course, we can replace acceleration by gravitation by changing coordinates, so what exactly is the invariant quantity? (I agree about the various nonvanishing curvature scalars, of course.)

The classical argument is simply that two or more bodies orbit around a common Barycentre. While any coordinate system can be picked, the underlying mathematical representation will always contain that reality – it’s an orbit description, plus an observer’s perspective transformation. The location of the barycentre is dependent on the relative masses of the bodies involved

And a relativistic representation adds things like frame dragging to that, adding massive spin to correctly predict things. My simplistic understanding is the drag is about the spin axis, not the barycentre. For the Sun/Mercury, though, those are almost idenical.

You’re not really wrong on the main points. But if it’s that simple, mustn’t we conclude that Professor Muller is simply an idiot? That seems unlikely. The problem with your argument is that it is, indeed, Newtonian. So if we are to refute Muller’s argument, we need to speak his language, not Newton’s. How, from Einstein and/or Muller’s point of view, could you define the “barycenter?” And what would stop Muller from claiming that coordinates that put the barycenter at the center of the two-body system are no better than coordinates that put the Sun or the Earth at the center of the two-body system?

Even if we do not consider general relativity, we teach in freshman physics that dynamics can be done in rotating frame like merry-go-round or a car rounding a curve if you introduce a fictitious force, centrifugal force. This has a measurable effect on measured value of g at the equator. Similarly Coriolis force is important for winds on rotating earth. So what is wrong in the simplicity argument, doing dynamics in the good old inertial frame of Newton?

Ah, now you’ve hit on a very interesting point. Yes, of course, when we choose a rotating (or otherwise accelerating) set of coordinates, there’s a fictitious force to be accounted for; that’s why the Foucault pendulum, or a gyroscope, can prove to us that the Earth rotates. Can’t we just build a Foucault-like pendulum device that that would demonstrate that the Earth goes around the Sun? (Maybe it’s not practically possible with today’s technology, but couldn’t we invent one as a matter or principle?) Wouldn’t that determine who goes around who?

If not, why not?

Well, I’m having a hard time not seeing the barycenter argument, as a legitimate answer as to who is orbiting who. The barycenter is the center of mass of the system. Relative to the “fixed stars” the center of mass can be moving, just not accelerating, unless it is exchanging momentum with something external to the system. If I define my system as the earth and sun, how could the sun orbit the earth without accelerating the center of mass of my system?

I realize this is a Newtonian argument, but is not conservation of momentum upheld even in relativity?

Of course, I could argue that the center of mass of my earth sun system is accelerating if I start including stuff external to my system, like the galactic center which everything is orbiting around. In principle, I could keep defining more & more centers of mass by including more & more stuff.

If I include all the planets, and everything else in the solar system, I could argue that everything is orbiting the center of mass of the entire solar system.

However, I believe we are restricting this to just the sun & earth.

I’m not faulting your physical intuition; your understanding of the Newtonian viewpoint is clear.

But professor Muller is using the fact that general relativity is just what it says: a generalization of the concept of relativity, in which it can be difficult to say what is acceleration and what is gravity, and in which gravity emerges from the geometry of spacetime and is not merely a force like all the others. And in this context, yes, momentum is conserved, but in a much more subtle way that makes it legitimately conserved in any coordinate system you choose. It’s no longer true that certain frames of reference are special.

So the problem here is this: how do we take our Newtonian intuition for the existence of a barycenter and find meaning for it in Einsteinian, general relativistic language? Is Muller claiming this is impossible, and if so, is he correct? Or is Muller claiming this is irrelevant, and that even if you can determine a barycenter, that does not mean that the two objects orbit it in any unique meaningful sense?

I realize that in relativity, gravity emerges as the geometry of spacetime, and you often hear statements about it “not being a real force”. However, if I define a “real” force as one that obeys Newton’s third law, would not gravity meet that definition as “real”? the earth pulls on the moon as much as the moon pulls on the earth….thru gravity….obeying Newton’s third.

I have read Feynman’s lectures, and he asks the question…Is gravity a real force? but never answers the question.

So, with that said, if a real force is one that obeys Newton’s third law, which is tantamount to conservation of momentum….isn’t gravity real?

This is just not the way that general relativity works. Newton’s third law never appears. It only emerges in simplified situations, and is not generally true. The question is not really whether gravity is a force — it is “when *can* you view gravity as a force?” And a question you never ask is “when *must* you view gravity as a force?” because the answer is “never”. So you see, you’re never required to use Newton’s third law and apply it to gravitational forces, and so you have to ask whether, when you take advantage of this new freedom, you are justified in saying “the Sun orbits the Earth” from some point of view that’s just as good as the Newtonian one. Muller would say that you are.

I understand your point, I was just saying that when “corrections” are applied, it appears that Newton’s third law is still upheld. For instance you can argue that F=MA does not apply at very high speeds, however when you apply the “gamma” correction, it appears that it does. Maybe not in the way Newton originally stated it, as the gamma correction is basically “1” at normal everyday speeds. The true quantity of momentum M*V*gamma. Anyway, I enjoy your website, really good thought provoking articles.

Again, this is simply not the way general relativity works. The true quantity of momentum being m*v*gamma, and force being simply the derivative of momentum with time, involves approximations that work in freshman level special relativity (without electromagnetism) but are drastically more subtle by the time you get to general relativity. We can’t make an argument that Muller’s wrong by appealing to approximate facts that can only be true in a small fraction of the huge range of allowed coordinate systems — the range that he’s trying to bring to our attention. In my opinion, you’re essentially assuming what in fact you have to prove. That won’t buy you anything.

Ok, I understand your point. I realize there are other corrections, especially concerning electromagnetism, such as when two charged particles, say protons, cross each other’s paths, say 90 degrees to each other. one particle will only feel an electric force, and the other particle will feel both an electric force & magnetic force. And you have to account for the momentum in the field. Thanks, I enjoy your site.

The article’s statements are technically correct, and the author tries to justify the choice of one method or another to describe natural phenomenonae by the method’s simplicity. I think it’s not that “simple”. Consider the epicycle vs Kepler models of the planetary motion. Conceptually the epicycle model can be viewed as simpler and even more intuitive because it uses just one basic geometry figure (a sphere with one parameter, its radius), and improvement of the model’s precision requires just more nested epicycles and not more complexity. On the other hand, each Keplerian orbit requires 6 parameters to fully describe the planet’s motion. Physicists use the Keplerian model due to the fact that it catches physics of the phenomenon, while epicycles don’t. Keplerian orbital elements corresponds to real planetary parameters, while the epicycle radii are pure mathematical, nonphysical quantities. Models that represent physical reality allow making important predictions and generalizations, and can be tested, while the pure mathematical models are often just descriptive and adaptive. The article mentions Newton’s derivation of its law of gravitation. Indeed, it must have been really hard for him to derive the universal gravitation law from the epicycle model.

This is an interesting answer, but I think you’re focusing on the wrong target. In my view, Muller wildly oversteps when he claims there was nothing to the Copernican revolution (and as followed up by Kepler, Galileo, Newton etc.) other than a change in coordinates. This is simply false, historically speaking. Copernicus did in fact make *different* predictions than the Greeks did, and was proven correct in the following century.

But if we forgive Muller for this overstatement, we can see he still raises a serious question. Let’s take the most precise, up-to-date astronomy available today. There is still a question as to whether the statement “the Earth orbits the Sun” has scientific meaning, and whether the statement “the Sun orbits the Earth” is falsifiable, once you account for the extraordinary flexibility of general relativity — the “general covariance” of the mathematics. I’d encourage you to think about how to address this issue.

I wrote a long answer to this, and accidentally deleted it. Ugh…

A shorter version: I don’t entirely disagree with you, but that said, I think you’re focused on the wrong target. Muller is simply wrong about the Copernican revolution; Copernicus actually made predictions that were different from the Greeks. In particular, he changed the orbits of Mercury and Venus in a way which was not merely a change of coordinates, and Galileo demonstrated he was correct. So I think we can disregard this part of Muller’s article.

But looking at the rest of the article, what’s at stake is this: even if we take the most up-to-date astronomical knowledge about the solar system, there is still a question as to whether the right interpretation of general covariance in general relativity is that the statement “the Earth orbits the Sun” is equivalent to the statement “the Sun orbits the Earth”, or whether the two statements are non-equivalent but both true, or whether both statements are meaningless. This really has nothing to do with history (or with Muller’s mistake about the history.) That’s what I think you should probably focus your attention on.

/Models that represent physical reality allow making important predictions and generalizations, and can be tested, while the pure mathematical models are often just descriptive and adaptive./

“Physical reality” is not the phenomena only?

Thank you for a very interesting topic! Is there any way I as a layman could explain the question (and the answer) to my son at his different ages and thus levels of understanding the reality?

1) I could show him the model of two balls, rotate that system and highlight for him the fact that inhabitants of both of the balls would/could claim that it is another one rotating. We could then discuss the idea, that for us it would be impossible to define who’s rotating around who.

2) We could draw the orbits of all celestial bodies in the sun-centered frame as well as in the earth-centered frame and note drastical difference in complexity.

3) We could discuss the inertial and non-inertial nature of sun and earth based frames.

My own understanding fails at this point, but is it all not the matter of defining the word ‘rotates around’?

Just a follow-up. Trying to explain things to my son, I’ve found the following reasoning. We might think of Earth rotating around Sun and Sun rotating around Earth, both true in most senses. The same is true for, say, Pluto. Thus the Sun rotates both around the Earth and around the Pluto, which is kind-a hard to imagine for a child unless Earth and Pluto are close to each other with Sun circling around.

So it all comes the definition of “rotates around” again. The word “around” itself implies a more or less “round” behavior in everyday logic, doesn’t it? When I tried to define “rotates around” to my son as “follows elliptic orbit around”, it all became much clearer. Indeed, all the planets “follow elliptic orbit around” the sun at the same time (and frame), so we have much more ground to treat it that-wise (i’m a layman, remember?).

Yes, one of the key issues here is to be precise about what criterion we’re really trying to impose when we say “the Earth goes round the Sun.” And this is tricky because it requires us to have an anchor point in space relative to which to define the Earth’s path. Lacking that anchor point, how are we even to say what the Earth’s path is? But how are we to find an anchor point when space itself does not provide one, and the Sun is suspect (as we’re trying to check our assumptions about it.)?

So I think you’re asking good questions, though not quite organizing the problem correctly yet. This is not a criticism — the issues are subtle, and even graduate students often don’t quite have them straight. Part of my goal here is to methodically try to tease the issues apart, both for the layperson and the physics student, so I ask you for now to stay tuned as I gradually work my way through it.

The Kolmogorov complexity of a heliocentric model is lower than that of a geocentric model.

In general, it makes sense to use less complex models (all other things being equal) because it makes predicting the system’s behavior easier.