Is the statement “**The Sun Orbits the Earth**” false? Not according to professor Richard Muller of the University of California, Berkeley, as I discussed yesterday. Muller argues that Einstein’s theory of general relativity implies that you can view the Sun as orbiting the Earth if you like, or that both the Sun and Earth orbit Venus, or a random point in space, or anything else for that matter. Meanwhile, every science textbook in our kids’ classrooms says that “**The Earth Orbits the Sun**“. But for all of our discussions yesterday on this subject, we did not yet collectively come to any conclusions as to whether Muller is right or wrong. And we can’t hope to find evidence that the Earth orbits the Sun if the reverse is equally true!

When we’re trying to figure out whether a confusing statement is really true or not, we have to speak precisely. Up to this stage, I haven’t been careful enough, and in this post, I’m going to try to improve upon that. There are a few small but significant points of clarification to make first. Then we’ll look in detail at what it means to “change coordinates” in such a way that would put the Sun in orbit around the Earth, instead of the other way round.

### Who Orbits What, Precisely Speaking?

Although this is irrelevant to Muller’s point, I want to be clear about what we mean, in the standard pre-Einstein line of thinking, when we argue that Earth orbits the Sun. We don’t mean it precisely. You can see there’s an issue if you consider two identical objects, pulling on each other via gravity and going round each other. Let’s color them red and blue. Would you expect the red object goes round the blue object? Or vice versa?

Well, if they’re identical (except for their paint), there’s no reason that one of them should be privileged over the other. So what actually happens? The answer is that they go round the point exactly between them.

Suppose now that the red object has 10% less mass than the blue object. Does that mean that the blue object will now be stationary while the red object orbits round it? No; things have only changed a little bit. Instead, as you can see in the second figure, the two objects will now orbit a point that, though between them, is slightly closer to the blue object than the red object. Thus the blue object’s orbit has a slightly smaller radius than that of the red object, and it moves more slowly than the red object.

Finally, suppose that the red object’s mass was a thousand times smaller than that of the blue object. They will still orbit a common point, but that point will be much, much closer to the blue object than the red object. The blue object won’t be stationary, but its orbit will be so tiny that it will be *almost* stationary, as you can see in the third figure.

And so although the red object orbits the common point, the blue object is so close to the common point that to say “the red object orbits the blue object” is very close to true — close enough that we do say it, knowing it’s almost but not exactly right.

It’s in this sense that we say (pre-Einstein) that Earth orbits the Sun. The Sun and all the planets orbit a common point (called the “barycenter”), which tends to lie in the line between the Sun and Jupiter, the planet with the most mass. **But the Sun lies close to that common point, and so the Earth, as it orbits that common point, essentially orbits the Sun.**

This is really what we’re really being taught in school. But everything I’ve said here reflects a Newtonian way of thinking. Einstein’s equations for general relativity are more flexible, and Muller is arguing that they imply that the statements

- “the blue object orbits the red one”
- “the red object orbits the blue one”
- “the two objects orbit their barycenter”

are equivalent, indistinguishable, and thus equally true.

### What We Aren’t Asking

Now, before we return to the issues Muller raises for us, let’s look at a few questions that one might think are on the table, but are not.

First, we’re not asking **whether the Greeks**, in particular Ptolemy and others who thought the Earth lies at the center of the universe, **were correct after all.** They were not, period. Although professor Muller stated in his article that *“the revolution of Copernicus was actually a revolution in finding a simpler way to depict the motion, not a more correct way,“* this is inaccurate. First, the Greek astronomers put the orbits of Mercury and Venus in entirely the wrong place; **the two planets could never pass behind the Sun** from Earth’s perspective. The Copernican revolution included putting these two planets in their correct relative location, and in this sense it * was* indeed a more correct way, not merely a simpler way, of depicting the motion. It was a major step forward in accurate astronomy, not merely a change of perspective.

Moreover, the Greek astronomers (and many post-Copernican astronomers, including Tycho Brahe, mentor to Johannes Kepler) believed the Earth does not spin. Copernicus, by contrast, argued it does spin daily *(and Muller has not gone so far as to claim that even this statement is debatable in general relativity… although he could have.)* So in these two senses, Copernicus was correct and the late Greek astronomers were wrong; it is not merely a matter of the coordinates one chooses.

Second, we are not asking about **rotating coordinate systems**. (This point confused many readers yesterday on the blog and on twitter.) We will indeed compare the usual Sun-centered *(heliocentric)* coordinate system with an Earth-centered *(geocentric)* coordinate system, but the geocentric coordinate system of greatest interest for our initial discussion is one **whose origin moves in a circle but whose axes do not rotate** relative to the heliocentric system. (You’ll see what I mean in a moment.) The stars have nearly-fixed coordinates in this coordinate system, so we need not attribute dramatic coordinate motions to the stars and distant galaxies, as would occur in a rotating coordinate system. This makes the ensuing discussion much simpler and clearer.

### The Solar System From Two Perspectives

Now, let’s look at the solar system in these two sets of coordinates. First, here is a cartoon of the Solar System, showing the planets from Mercury out to Jupiter in the usual heliocentric coordinate system. It’s a cartoon because I’ve given the planets perfectly circular orbits, made the Sun exactly stationary, and removed the planets’ moons. You can see that each planet orbits at a different rate, but each one’s motion is simple.

Now I want to change coordinates to geocentric ones. The origin of the coordinates is the Earth, but the axes are the same as before, so there is **no rotation of the coordinates**, just a shift that moves in a circle. This is illustrated in the next figure

What does the solar system look like in these coordinates? Here you go: in these coordinates, the Earth is stationary, and the Sun orbits the Earth in a circle.

Meanwhile the other planets orbit in complicated paths that involve loops within loops; these are the famous “epicycles” that are characteristic of geocentric coordinate systems. *[In fact, this viewpoint has some practical value, as it shows us what Earthly astronomers see as they watch the Sun and planets move relative to the fixed stars.] *

It’s clearly true that the planets’ orbits are simpler in the heliocentric coordinate system. But… not so fast. I didn’t put the Moon into my animations. If I had, you’d see the Moon’s orbit is very simple in geocentric coordinates, but quite complicated in heliocentric coordinates. The same would be true for the International Space Station, and for other artificial satellites. So is it really true that heliocentric coordinates make the solar system simpler than geocentric ones? Or does it just look that way because the Earth has only one moon, rather than, say, twelve? Imagine if switching from geocentric to heliocentric coordinates simplified the motion of eight planets and complicated the motion of twelve moons; would you be so sure that heliocentric coordinates were simplest? Let’s not confuse luck with principle.

### Looking Ahead

I hope this post helps provide some intuition and clarification for what we need to understand here. As far as I can see, these are the main questions still to address:

- Is the heliocentric coordinate system actually better than others, and if so, why exactly?
- Is it true (or at least truer) that the Earth orbits the Sun rather than the other way round?
- How are questions (1) and (2) related to each other?
- Given the answer to (3), how much merit is there to Professor Muller’s argument?

Your thoughts? Comments are open.

If you were to define a zero point as the average motion of the nearest hundred stars, then, with respect to that, the earth would show a yearly cyclical motion whereas the sun (or, rather, the barycenter) would not. Thus, in a meaningful sense, the earth orbits the sun, not the other way round.

So far, you’re still the only one to provide a coherent argument. That said, I don’t think your argument is as strong as it can be, because it’s contingent on the existence of the nearby stars and the fact that their motions are sufficiently slow and regular, and tend to average out. Can you make an argument which doesn’t depend on them? Moreover, you still have to explain in what sense Muller is wrong. Anyway, let’s see if everyone agrees with you and whether people can improve your argument.

I agree with this statement, and I think I posted something similar in a previous thread. If we define our system as the earth & sun, then the barycenter is the center of mass of the system. The center of mass can be moving in a reference frame, however it can not accelerate unless it exchange’s momentum with something external to the system.

Using the distant stars as reference frame / coordinate system, if the sun (or, rather, the barycenter) was orbiting the earth, what would it be exchanging momentum with? Now granted, I can not accept the challenge put forth by Dr. Strassler to make an argument that does not depend on the distant stars, and I’m looking forward to the response that someone puts forth not depending on the distant stars.

BTW: one clarification I would like to make, that I believe I may have been vague about in a previous post, is when I refer to Newton’s Third Law, I’m using it as being tantamount to conservation of momentum…not necessarily the requirement for equal and opposite forces.

Michael, the difference between what you posted and what Coel posted is the most crucial part of the whole story. You simply claimed that there is a barycenter, without proof that one can be established. **Coel proposed a mechanism by which one can be determined.** The difference is *everything*, not a minor detail. Your statement “If we define our system as the earth & sun, then the barycenter is the center of mass of the system.” is incomplete and in some ways incorrect in general relativity. See, for example, this long discussion among students of the subject: https://physics.stackexchange.com/questions/92683/is-the-center-of-mass-in-general-relativity-equal-to-the-center-of-mass-in-newto#:~:text=There%20is%20generally%20no%20center,it%20as%20a%20reference%20point.

Moreover, your statement “I’m using [Newton’s third law] as being tantamount to conservation of momentum” — that’s fine, but conservation of momentum does not work the way you think it does in general relativity. For instance, look at this long discussion among students of this question: https://physics.stackexchange.com/questions/62939/general-relativity-and-the-conservation-of-momentum

Furthermore, I don’t think you want to make the claim that the reason professor Muller is wrong is that he doesn’t understand momentum conservation. That doesn’t seem like a very likely explanation as to what is going on.

Well, I was in no way trying to imply that Professor Muller doesn’t understand momentum conservation, I was just explaining my thought process. While General Relativity is the correct explanation of gravity, Newton’s laws have worked well for hundreds of years, and while they may be approximations, they are very good approximations, as such any explanation in general relativity as to what we see, should also be able to be explained in Newtonian mechanics as to what we see. The earth orbiting the sun. Sorry for the confusion, I will not post any more so as to not weigh down the thread.

And, just to be complete, when I speak of Newton’s third law as being tantamount to conservation of momentum, I’m not just considering the momentum of “massive” objects, but also the momentum of photons & fields.

Newton’s laws work fine in many circumstances, but Muller’s point is that they easily mislead you as to what is physically meaningful and what is not. The fact that Newton’s laws view the weightlessness of astronauts in the space station as an accident, while Einstein’s laws view the weightlessness as a principled necessity, is an indication that you can’t merely say that “Since Newton’s laws tell me X is true, and since Newton’s laws are good approximations, X must be true.” Sometimes Newton’s laws lead you badly astray. Newton’s laws are also good approximations to quantum mechanics too, but when they tell you that a particle can only go through one of two slits, or that the outcome of two exactly identical experiments must be exactly identical, or that waves can have any amplitude you like, they are wrong. This is the kind of thing we are dealing with here, and your insistence on going back to Newton’s laws for guidance is keeping you in an conceptual prison. I’m trying, gently at first but now more and more forcefully, to warn you that the harder you try to wall yourself off in that prison, the less likely it is that you will ever escape it.

My dad once (Mel Schwartz) had a long discussion with me about this when I was a kid. He explained that the only reason it makes sense to say that the Earth revolves around the Sun is that the physics is MUCH simpler when you make that assumption. There is no privileged frame of reference, however, so saying the Earth revolves around the Sun is quite arbitrary, if extremely convenient from a math perspective. David Schwartz160 W 71st St Apt 12HNew York, NY 10023

🙂 Well, your dad was a truly great physicist, and it seems he agreed with professor Muller. But I will argue later that he was neither entirely wrong nor entirely correct.

Can we revisit the argument I mentioned before that a very large mass cannot revolve around a small mass. Your answer was that by elementary methods, one does not know that sun is approximately million times as massive as earth. But if you accept the precise measurement of mass of sun by astronomers, will this argument work even accepting GR description of frames of reference and gravity? Also as I asked before has Kepler telescope observed any large mass revolving around a small mass? My guess is that everyone assumes that a small mass revolves around a large mass, so no one has looked into it!

You’re still missing Muller’s point, I think. I can always choose coordinates (as I did above) in which it appears that a very large mass revolves around a very small mass. I have not changed anything in the physical phenomena; I have merely described it differently. I can even choose coordinates in which the Earth revolves around the international space station. ***There is nothing wrong with these coordinates!*** No experimental data is violated if I do that, nor are any conservation laws or other principles. In this, Muller is absolutely correct; this is how general relativity works.

Thanks. May be I was not clear in my question. Do you think Muller or Einstein would disagree with the statement that a smaller body revolving around a bigger one is a simpler picture than the other way.? As we agreed before by introducing fictitious forces you can choose any coordinate system.

Muller says very clearly that the heliocentric system is simpler in the case of the solar system. But he implies that *** a simpler coordinate system, i which a certain fact appears to be true, does not imply that this fact *IS* true, since in other coordinate systems (which are equivalent physically, even if not simpler) it is false.***

No one’s arguing about simplicity. We’re arguing about whether simplicity has any bearing on facts. It’s simpler to say pi = 3, but it’s false. Physicists often use units in which the speed of light is 1, but that doesn’t mean it “is” 1; it just means you an choose it to be 1 if you like. The question here is not what is simple; the question here is what is a *fact*, and what is not.

Actually, what I said is unfair, because I myself raised the question as to whether the heliocentric view is really simpler. But in the case of the two body problem, the right statement is that the barycentric coordinates are simpler in the Newtonian limit of an isolated system. More generally, though, a multibody system in a complex environment may be much more ambiguous, especially when you account for rotation of the objects and gravitational radiation. I’m no expert on the problem of two colliding spinning black holes, but it’s a lot harder than the problem you’re imagining, in *any* coordinate system.

OK! Thanks. So it seems to me that as you suggested previously if someone can develop version of Foucault’s pendulum for orbit, we can give answer to Muller’s point that indeed earth revolves round the sun since one needs fictitious forces!

The problem is — we can’t. Thanks to general relativity. A spinning earth (where the spin is not caused by gravity) is different from an orbiting earth (where the orbit is caused by gravity.) Think about the weightlessness of astronauts in the space-station; is there a Foucault-like pendulum for them, so they can confirm (from inside the space station, without looking outside) that they orbit the Earth?

Another way to say it: the experience you have inside an accelerating rocket ship is that there is a fictitious force that holds you to the back wall of the rocket. This fictitious force feels and acts just like gravity; it makes things fall to the “floor”. Why? Because it *is* gravity.

Thanks! I finally understand the point. Please emphasize this point in your future write ups.

Yes, this is a point that deserves highlighting. The question is how to do it pedagogically to bring along the maximum number of readers… it’s not easy to convey it!

Interesting how the geocentric cartoon clearly shows the retrograde motion of Mars from Earth’s perspective.

Would the Sun be undergoing different acceleration in the geocentric model, and would there be observable consequences of this that we don’t see in reality? And what force would be necessary for the Earth to stay stationary with all that mass revolving around it?

The answer is: no observable consequences, and no force necessary. You are thinking in Newtonian language, and trying to give a Newtonian explanation for what you think you are seeing in my geocentric diagram. But. Einstein says: choose whatever coordinates you like; I will tell you that all motion depends on the shape of space-time, and changing your coordinates doesn’t change that shape at all. And that’s why Muller says — correctly! — that geocentric coordinates are just as correct as heliocentric coordinates. There is absolutely no change in the physics; no measurement will tell the difference between them. ***Nothing physical/measurable/meaningful depends on your choice of coordinates.***

Here I elaborate the previous discussions at the “Earth Goes Around the Sun? What’s Your Best Evidence?” post. Below by heliocentric reference frame I mean coordinates having stationary Solar System barycenter, and by geocentric reference frame – stationary Earth barycenter accordingly. “Stationary” means free fall within the Milky Way gravity field.

Let’s first explore transforming from heliocentric to geocentric reference frame more closely. According to the equivalence principle of general relativity an accelerated frame experiences gravity field which acceleration of free fall is equal in value and opposite in direction to the frame acceleration vector. If the Earth barycenter is postulated stationary, then in the Solar System there exists a pervasive, non-stationary gravity field having such a property that at any moment the acceleration it induces is equal and opposite to the Earth barycenter acceleration in heliocentric frame (i.e. its acceleration is computed assuming the Solar System barycenter is stationary with no additional fields). By taking a closer look at the additional field we note that to satisfy all the observations within the Solar System that field must be uniform. And finally, any non-stationary and uniform field is nonlocal, i.e. implies instant action at a distance, which violates special relativity. Thus we may answer questions 1-3:

1) The heliocentric frame is better than others because it is stationary *and* compatible with special relativity

2) Stationary geocentric frame is not compatible with special relativity, i.e. Earth orbits the Sun assuming special relativity is valid

3) The assumption of special relativity’s validity relates answers 1-2

Answering the question 4) I believe that Professor Muller’s argument tries to illustrate the complexity and counter-intuitivity of the general relativity. There might be other technically true but equally hard to accept statements:

– “All Solar System bodies don’t orbit each other but go straight indefinitely”, referring to their motion along local geodesics

– “Generally, there’s no such thing as global coordinates in general relativity”, citing Einstein who once said about its most famous theory that to express physical laws without coordinates is like “describing thoughts without words”

– And even more extreme: “There’s no motion in general relativity”, because according to the block universe point of view all space-time events are fixed once and forever within a braid-like structure

This is an interesting argument, and I think it’s partially correct. Still, to my mind, there’s something incomplete about it, because I don’t think you’ve quite addressed the issues when it comes to coordinate choices. Muller would probably tell you that requiring stationarity and special relativity for your coordinate choices is arbitrary. Furthermore, I could have chosen a different set of coordinates which aren’t quasi-cartesian, are *asymptotically* stationary, and locally geocentric; how would your argument exclude those? And since your argument for (2) seems to refer to coordinate choices rather than relying on something coordinate-independent, it’s a bit suspect; anything physical ought to be coordinate-independent. But you’re closing in on the issues, I think.

It’s true that anything physical ought to be coordinate-independent, but only locally and in a free-fall. Confined in such a free-falling lab (“Einstein elevator”) any observer can perfectly assume that it’s stationary, whether on the Sun or on the Earth. The GR equations in their general form are also local, and applying them to any physical system requires specifying of the system’s coordinates in form of a metric (excluding probably full-GR numerical simulations). Moreover, GR’s metric is not a mathematical abstraction but rather a real property of space-time, therefore we can’t simply choose metrics arbitrary and expect full equivalence (which may be a counter-argument to Professor Muller’s article). I’m not a specialist to fully understand details of the asymptotically stationary and locally geocentric frame you suggested, but according to astronomical observations our local universe up to 100s My scale is approximately Euclidean with high precision. The Solar System barycenter reference frame is established by observing positions of several dozens quasars (distant extragalactic objects). The Solar System model is fitted to the positions in such a way that excludes any *correlated* positional residuals. Therefore we may also say that the Solar System barycenter’s position near the Sun’s surface and not in the Earth’s vicinity is an experimental fact.

I mostly agree. I don’t agree with the first sentence, because anything physical is coordinate-independent, period. But beyond that, I’ll be making analogous points in the posts that are to come next week and the following. The key point is that there are quantities that don’t depend on the choice of coordinates. In particular, the result of a measurement, expressed in terms of other measurements, can’t depend on the coordinates. In the last part of your answer you are extremely close to saying this, and your conclusion is either precisely correct or approximately so; I still have to do a little more calculating to see which is true. One can certainly use properties of the spacetime at large distances from the solar system to nail down something that one can call its “center”, and the Sun is closer to that center than the Earth. In this sense I disagree with Muller.

Probably my wording isn’t precise. By coordinate in the first sentence I mean “reference frame”, and I suppose that by anything physical is strictly coordinate-independent you mean space homogeneity, correct?

I’m not sure I understand what you’re asking. By analogy, any properties of objects on the Earth’s surface are necessarily independent of what map projection we choose to use; that’s what I mean here.

Thanks, that’s clear now. Using the map analogy, I understood the discussed coordinate independence as such that any physical experiment sufficiently isolated from its environment gives exact the same result anywhere on the map.

Ok, we know that all inertial frames are equivalent, but from observations we concluded that Sun and Earth locked in a mutual circular motion. My point is that since the heliocentric frame and geocentric frame can’t therefore be *both* inertial, we’re forced to decide which frame really is (or more so).

Yeah, that’s not what “coordinate independence” means. Getting the same result anywhere on the map is “translation invariance” — that the laws of nature are the same in all locations. That’s not the same as saying that the laws of nature are the same in all coordinate systems, including warped ones. The point is that the ratio of Greenland’s area to Africa’s area is independent of the map you use, even if the map makes Greenland *look* larger than Africa. The length of all great circles is the same, even if on the map they look very different from one another. Etc.

You say that the heliocentric frame and geocentric frame can’t be *both* inertial. Let’s consider the approximation where we treat the Earth on short time scales as the base point for a Cartesian inertial frame. You’d then say that the international space station’s interior can’t possible be an inertial frame. Well, it’s true that in Newton’s theory of gravity, the weightlessness of the astronauts in the station is due to an accident, in which Earth’s gravity is perfectly cancelled by a fictitious force that would be absent in an inertial frame. But in Einstein’s theory, the weightlessness of the astronauts in the station results from a *principle*. There is no gravity there. So Einstein doesn’t agree with your assumptions.

>> …So Einstein doesn’t agree with your assumptions.

Wordpress doesn’t allow more nested replies apparently, therefore I reply here. I completely agree with your ISS example and understand that Earth as a massive planet follows an inertial trajectory with high precision. Let me elaborate on what I mean by “heliocentric vs geocentric inertial frame”. We need to decide which reference frame follows inertial trajectory in the Milky Way’s gravity field. By observing distant objects we basically test this trajectories, and find that Earth follows a wavy trajectory (i.e. non-inertial), while Sun goes straight (i.e. inertial).

I agree something like this is the best available argument. The question is whether this is unambiguous. I think Muller will argue it is not. You are assuming (as I did, for simplicity, in my discussion of geocentric coordinates) that the we choose a Cartesian-like grid across the solar system based on the flat space far away. This need not be the case (I’ll address that in some future post). Once we start considering time-varying curved coordinate systems within the solar system, ones that gradually become ordinary Cartesian coordinate systems as we go to greater and greater distance, we will have to confirm that there is no ambiguity in the interpretation of observations. In other words, if we consider all possible *asymptotically-inertial* coordinate systems (inertial far from the solar system, but not necessarily near the Earth) we will have to check if the statement “Earth moves in a wavy line, Sun’s motion is straight” is an unambiguous interpretation of the observations.

Again here’s the counter-argument from GR. Suppose we found such an coordinate system (metric). But 1) I think that there’s not nearly enough matter-energy content in the Solar System to generate the required curvature. 2) Even if we are able to construct a self-gravitating (vacuum) solution, curvature itself has matter-energy content, and anyway will wreak complete havoc there.

We are in absolute agreement it has nothing to do with curvature. Curvature (in particular something like the Kretschmann invariant) is coordinate-independent. But I am not talking about that. Remember that in GR, when we change coordinates, we change gravity too. Gravitational effects will appear and disappear along with coordinate transformations. This, despite there being no change in any curvature invariant.

And gravitational effects can bend light and change apparent positions, as well as create redshifts and time delays.

I’m not skilled enough to discuss further, but sound interesting, like shadow gravity. Thanks and looking forward to the post on that.

I don’t know if it will be one post… I haven’t yet completely solved the technical problems here, much less the pedagogical ones. This may take some time.

Consider two free-falling, non-rotating observers positioned some distance apart, say a light-year or so in their own frame of reference, and with the solar system between them.

One observer points a telescope at the solar system, keeping the other observer in the center of the telescope’s field of view, and observes the solar system over an extended period, say a hundred years or so. We will assume that the velocity of both observers relative to the solar system is low enough that the solar system remains in the telescope’s field of view over this period.

Typically, what that observer will see is that the Sun moves slowly across the field of view, with the Earth orbiting much more rapidly around it.

I think the key point here is that out of all the possible pairs of observers that could hypothetically view the solar system in this way, there are many such that would see the Earth orbiting the Sun, but only in an extremely contrived scenario would such an observer see the Sun orbiting the Earth.

You’re definitely getting closer here. You’ve moved us away from requiring the background stars, which is very good. You’re making some hidden (though reasonable) assumptions here and I think it will be important to tease them out. But in the end my own suggestions about how to approach this will not be very different from yours and yours provides an interesting alternative. Your use of two observers on opposite sides of the Sun is a nice touch that I hadn’t thought about.

It would also still be good to clarify why you and Muller would disagree.

All the Informations are available at the surface area ie.. circles of earth and Sun, (holographic) where the light travel at constant speed in space, the subtle difference in spacetime curvature which contains the information about, first the Energy and thus the Mass of earth and Sun (length contraction), so that which orbit which is less pronounced?

Thus gravity emerging making the Universal acceleration constant (p = 0), but not at the large scale?

Suppose I choose to say that everything revolves around the earth. Next, I go to the Moon or to Mars (or wherever). After that, what reason do I have not to change viewpoint, and say that everything now revolves around the Moon, or Mars. So choosing earth, the Moon, or Mars as the center, seems fairly arbitrary. – You know what, the truth is actually that everything revolves around ME, wherever I am. And the rest of you don’t matter.

🙂 Yes, there is of course solipsism in thinking the Earth’s the center of the universe, or even in thinking the Sun’s the center, or that the galaxy’s core is the center. The universe does not revolve around us in any sense. But this is a red herring…the question at hand has nothing to do with where *we* are, but rather, with the actual dynamics of the Sun, planets and moons. We could ask exactly the same question about another, distant system of star, planets and moons; if we take, for instance, TRAPPIST-1, https://en.wikipedia.org/wiki/TRAPPIST-1, can we say that its planets orbit its star? Or is that statement not meaningful, or at least, no more true than the statement that the star orbits one of the planets (as Muller would argue).

This is totally out there, but thinking about this from an analytical point of view, one invariant that we can construct is the action.

Ignoring the back-reaction on the geometry, perhaps we can treat the system like a pair of point particles:

S[x_{sun}, x_{earth}] = S_free[x_{sun}] + S_free[x_{earth}] + S_int[x_{sun}, x_{earth}]

S_free[x(\tau)] = m \int d\tau \sqrt{ g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} }

S_int[x(\tau), y(\tau)] = – m_x m_y \int d\tau (\sqrt{ g_{\mu \nu} (x – y)^{\mu} (x – y)^{\nu} })^{-1}

where \tau is the proper time. I’m totally not certain about the interaction term, just spitballing here. Also the metric would be Schwarzschild, but centered around the sun, which is also a little tricky to deal with.

We vary to get the EL eoms and then solve them. We can then evaluate the action on the solutions, i.e. get the on-shell value of the action.

Since each term in the action is invariant by itself, we could take the ratios of the on-shell actions as follows:

S_free[x_{sun, eom}] / S[x_{sun, eom}, x_{earth, eom}]

and

S_free[x_{earth, eom}] / S[x_{sun, eom}, x_{earth, eom}]

Perhaps we then claim that whichever ratio is greater, corresponds to the particle being orbited — or at least, there is an asymmetry. This might end up being identical to your other commenter’s suggestion about heavy particles and light particles.

N.b. The reason I’m trying this approach is to relate this to the Hydrogen atom system in QM. Of course, after a certain level it makes no sense to ask whether a proton or electron orbit one another, we are describing wave functions [incidentally, we usually do this by using the standard 2-body formalism with the “COM particle” (m_p + m_e) and the “reduced particle” (m_e m_p/(m_p +m_e)]. But I *think* one can ask if a particle is behaving more “quantum mechanically” than another, in the sense that we can compare their actions to the scale of h-bar.

I believe a particle-worldline approach (as you’re suggesting here) isn’t sufficiently invariant to match the invariance of general relativity. To do it properly, you need to introduce the interactions between the particles not with the interaction you’ve suggested but with the full Einstein equation, where the Sun and Earth are included in that equation through the energy-momentum tensor.

One way to understand this is to go back to the hydrogen atom (or positronium) but do the full relativistic field theory, where you treat the particles using worldlines with the potential A_mu (instead of g_mu,nu) and you also use the Maxwell equations d_mu F^mu,nu = J^nu , where J includes the particles. Now try to solve for F (and thus, in some gauge, for A) and plug that back into the worldline particle actions to see what interaction term you get. But now think about how gauge invariance works… you had to fix a gauge to get an answer, so how do you show the result is gauge invariant?

If you can understand all that, you may be able to copy it into the gravitational case. But you’ll see that maintaining or confirming the diffeomorphism invariance is challenging — and that’s the invariance that whose implications we’re trying to understand here.

Meanwhile, the notion that you could use quantum physics to say who is lighter is clever and would work in principle, though not in practice. But it still may not answer the question, because GR says (according to Muller) that you cannot say if A orbits B or if B orbits A, no matter which of A and B has smaller mass. So we really have to probe this claim, and your method won’t do that, I think.

Hmm… I’ll have to think a bit in the morning about the gauge invariance of the worldline description, as in the EM case I would have in the action the term:

\int d\tau A_{\mu} \dot{x}^{\mu} = \int d\tau A_{\mu} dx^{\mu}

which I believe is gauge invariant, as A -> A + d\lamba would give you a total derivative. But perhaps what you are saying is along the lines of having to introduce Fadeev-Popov ghosts to maintain gauge invariance, or a R_{\xi} like term. I was ignoring the backreaction anyway, and including a Ricci scalar term would be complicated. This review https://arxiv.org/abs/1102.0529 describes what I had in mind but for the case of a single particle – though of course my concept is still half-baked.

I did have another thought if you’d indulge me – wouldn’t the gravitational redshifts as viewed by observers on the Earth and on/near the Sun be opposite in sign? I.e. if both observers accepted a standard reference of some spectral line, then the observer on Earth would see that line in the sun’s spectrum redshifted (on average, not accounting for the sun’s rotation about its own axis) and the observer on/near the Sun would see that line blue-shifted, if we were able to remove the transverse doppler effect? (Actually, looking it up now, the transverse Doppler effect for objects in circular motion looks to act in the opposite direction to the gravitational effect I’m talking about here: https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#One_object_in_circular_motion_around_the_other , so hopefully it is smaller in magnitude…)

Yes, the problem is that what you really need to do is “integrate out” the gauge field or the metric in order to write a two-particle non-local interaction term. And you can’t do that without inverting the Maxwell equation or the Einstein equation, which can’t be done without gauge fixing. I’m not sure where your proposal for the two-particle interaction came from; it’s not the full gravitational one.

Gravitational redshifts may help you map out the curvature, but I don’t think you need them if that’s your goal; you can simply take a tour around the system and measure tidal forces, I believe. You have to be very careful with kinematic and gravitational redshifts; they aren’t separately invariants. You might want to consider how things work in flat space if you use Rindler coordinates and examine how accelerated observers see the world; it’s an exercise I haven’t completed but I think it’s important to do it.

I had two proposals.

The first one has already been proposed above: going very far out so that the residual gravity is low enough to be ignored (for some times) and observe the motion of the sun and the earth. The earth will have a higher velocity and that could be used to say that earth orbit the sun.

The second one is more a thought experiment. For the whole solar system, if you were able to remove one body, it would not affects the motions of the other bodies with the same intensity (on the medium/long term).

For example, if you remove an earth satellite, the motion of the earth is nearly unchanged (even on long time scale). But on the other hand if you remove the earth, the effect on the motion of the satellite is huge.

Maybe that difference of effect could be used to tell (or define) who orbit who ?

One question I was asking myself: what do we mean by ‘who orbit who’ ? I mean: is it simply a matter of definition ? practicality ? (if so, for what ?) or are we looking at a real physical difference ? (if so, which one ?)

The first proposal is about as good as you can do, but still very subtle. We’ll have to examine it more closely soon. The second proposal, the thought experiment, one could actually perform in principle by causing one of the bodies to explode. [You can’t remove an object by hand, because that would cause energy-momentum non-conservation and would result in inconsistencies in Einstein’s equations.] But this might not be necessary; we should be able to tell which object is dominant in the solar system by mapping out the curvature, which is coordinate-independent. As for your final questions, yes, these are subtle, and real care is required.

When we study the signaling between objects we can form an average central coordinate system which is the most isotropic in relation to red- and blueshifts of the signals.

In the ‘optical view’ we then see how the Earth mostly orbits the Sun, not vice versa.

And when the signaling study was done with all the elementary particles in the system, we have the time curvature field that tells the same story: all local flat coordinate frames are falling to the central sink.

Your procedure is fine, but the question is whether Muller is wrong. He would say your procedure is an arbitrary choice, and no more physical than other choices which would give us a different answer.

How interaction signaling could be an arbitary choice rather than one a unique frame separated physically to parts of curved time and flat asymptotic space – spatial isotropia having only one center of rotations as the common mass center? I cannot imagine any other as physical as interactions.

Einstein was pretty imaginative.

Muller seems imaginative with no arguments for physical continuum – which Einstein strictly adhered to.

I offer three arguments depending on how we’re defining an orbit here, from the specific to the general.

First, ‘An orbit is a straight line through curved space’ or somesuch. In that case, what is the spacial curvature of the system? If I shine beams of light about, what are their paths? If an orbit is defined as due to curvature that should give us our answer right there.

But maybe that’s too specific or esoteric. Other forces besides gravity can make paths bend. So perhaps we define an orbit as something like ‘A path one object influences another to take’. In that case we must determine which body is vital for the orbit (if any). To claim one body orbits another is to claim one body’s influence is dominant. Otherwise it’s just THERE while the second body moves however it likes.

Right now a fly is buzzing around my head. I might say it orbits me, it might say I orbit it. You take out your proton decay ray and evaporate me into gamma rays. The fly continues its lazy buzzing, unaffected. I was not vital to its motion, I had no effect on it. Likewise if you blast the fly. In its (now empty) reference frame my motion continues unabated, it just happened to be sitting there while I looped about.

If we look at the Earth and sun, if the sun goes then the Earth departs from its former frame, it zooms away at speed and its old orbit remains empty. If we take the sun’s position and evaporate the Earth, there IS an effect but it’s pretty minuscule. It’s evident which body has the greater effect.

We can even use this to determine WHEN the geocentric frame is valid. If we put the moon into your little cartoon its motion is greatly affected by Earth’s presence, while the planets aren’t. This puts it in a different class of objects ones that orbit the Earth (And not coincidentally whose motions are simplest in the geocentric model.) We can toss out arguing ‘Which is simplest, all things orbiting the sun or (and ONLY or) all things orbiting the Earth?’ And instead ask ‘WHICH objects orbit the Earth and sun (if any).’ BOTH systems have value and physical meaning and being forced to choose between them is a second place option.

But maybe even THAT is too specific for you. Perhaps you want to define an orbit as any motion whatsoever. The fly DOES orbit me (or vice versa?) We only care about the motion that’s happening, we need no forces or influence. Wave your hand, that’s an orbit, swat a fly, same thing.

In this most broad of cases both me and the fly have two conflicting opinions. How do we resolve this? By consensus. We poll ALL possible reference frames, ALL possible observers and ask them who is moving fastest, on average, relative to them. We care only about movement after all, and so he that moves most must be the orbiter.

Now, for each point in space we can share the same reference frame as me or the fly, a 50-50 split. BUT there are also infinite (Or at least very many) other frames that exist. And in most of those my motion will be the lesser.

More simply it’s easy for us ON Earth to think that our viewpoint is general, that all that matters is US vs SUN, an even match. But there exist viewpoints on Mars, Saturn, all the asteroids, Alpha Centauri… Aliens light years away would not likely be sitting there going ‘Yes, that planet is perfectly still.’ Having the sun’s motion be the greater requires a contrived, rare reference frame. For all Copernicus made us unexceptional there is one remaining thing, our viewpoint is rare, the rest of the universe does not share it.

Not sure if this has even a shred of relevance to the present discussion, but wouldn’t the sun wobble substantially in its orbit about the Earth? (based on the fact that the center of mass of the solar system does not coincide with the center of mass of the sun). While planets would appear to travel in epicycles, I don’t believe they would wobble while revolving about the Earth. You might say that the sun has a “self-contained” epicycle.

Aren’t we slicing or threading into a 1+3 resp. 3+1 decomposition the moment we talk about “goes around”, and without saying so outright, relying on one or two (or a small handful) of a family of time axes? That is, we’re ignoring ultraboosted or otherwise *strongly* non-Eulerian observers and thinking about which of two families of (quasi-)Lagrangian observers (“fixed” on Earth or on the Sun) is “right” about the solar system flow.

aka touches on these decompositions and is at least a little relevant if you want to be firmly post-Newtonian.

I realize it’s more than a little strident to argue that the answer in general relativity is just blockworld, but if we start there we can think about various formalisms and whether they might produce useful insights as they’re applied to the cartoon solar system.

Well, that seems to have eaten the URLs, so Int. J. Geom. Meth. Mod. Phys. Vol. 15 (2018) 1850047 gr-qc 1510.02359 “A tale of two velocities: Threading vs Slicing”.