Of Particular Significance

Earth Around the Sun, or Not? The Earth-Centered Coordinates You Should Worry About

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 05/16/2022

We’re more than a week into a discussion of Professor Richard Muller’s claim that “According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun. And they both orbit together around a place in between. And both the Sun and the Earth are orbiting the Moon.” Though many readers have made interesting and compelling attempts to prove the Earth orbits the Sun, none have yet been able to say why Muller is wrong.

A number of readers suggested, in one way or another, that we go far from the Sun and Earth and use the fact that out there, far from any complications, Newtonian physics should be good. From there, we can look back at the Sun and Earth, and see what’s going on in an unbiased way. Although Muller would say that you could still claim the Sun orbits the Earth by using “geocentric” coordinates centered on the Earth, these readers argued that such coordinates would not make sense in this distant, Newtonian region.

Are they correct about this?

Standard Geocentric Coordinates

Let’s make that last argument more precise. About a week ago, I offered you some geocentric coordinates; see below, and also the last two figures in that previous post. These are non-rotating Cartesian coordinates centered on the Earth. They can be defined in the usual heliocentric (Sun-centered) coordinate system, the one we normally take for granted, by centering a non-rotating grid on the Earth, shown in Figure 1. This figure shows a simplified solar system (the Sun at center, with Mercury, Venus, Earth, Mars and Jupiter in circular orbits), as well as the Earth-centered grid which follows the Earth around in its orbit.

Figure 1: The Sun and the first five planets (Earth is green-blue) in a simplified solar system, with all the planets moving in cricles around the Sun, along with a grid showing my simple Cartesian Earth-centered coordinates used in a previous post.

When we now move to the coordinate system defined by the grid in figure 1, the Earth becomes stationary and the Sun starts moving around it, as shown below. The other planets do some strange loops-within-loops — epicycles, they are called.

Figure 2: In figure 1’s Cartesian Earth-centered coordinates, the Sun orbits the Earth, and the other planets have complex orbits, with Mars (red) and Jupiter (yellow) showing clear epicycles. Distant stars would also show loops in their motions.

The argument against such geocentric coordinates is that it’s not just nearby planets like Jupiter that undergo epicycles. So would all of the distant stars! Each will move in a little loop, once an Earth-year! Now indeed, that sounds bad; why would we accept a coordinate system in which extremely distant stars like Sirius or Vega or Betelgeuse would travel in loops that somehow know how long it takes for the Earth to go around the Sun?

Such complaints seem reasonable. This kind of geocentric coordinate system implicitly stretches the Earth’s influence across the entire cosmos, and that doesn’t seem to make any physical or causal sense.

That said, coordinates are just labels. They don’t have to make physical sense or preserve a notion of causality. Only physical phenomena have to do that. But still, it seems crazy to take coordinates seriously that have this property.

And the claim that readers implicitly made is that if you forbid these coordinates — if you use coordinates in which the distant stars are fixed, or at least traveling not in Earth-year-long loops — then you inevitably will prefer heliocentric coordinates.

General Geocentric Coordinates

But this claim, and any similar one, is wrong. No one said that we have to extend the coordinates out from the Earth in a rigid, Cartesian way. Einstein claimed that physics is unchanged no matter how crazy the coordinate system you might choose to describe it. So let’s take the following coordinate system, which is warped, remains the same as the heliocentric coordinates at very large distances, but is geocentric at and near the Earth.

Figure 3: The Sun and the first five planets (Earth is green-blue) in a simplified solar system, with a grid showing Earth-centered coordinates that are the same as the Sun-centered coordinates far away (in contrast to those of figure 1) but are warped near the Earth so as to put it at the center.

In this system of coordinates, here’s what the motion of the Sun and planets looks like.

Figure 4: In the Earth-centered coordinates of figure 3, the inner planets and Sun behave almost as in figure 2, but Jupiter has mostly lost its epicycles, and distant stars act just the same as in the standard Sun-centered coordinates.

The Sun goes round the Earth. Notice that Mars still moves with a significant epicycle, but the epicycles of Jupiter are almost gone. By the time you get to the distant stars, none of them are doing loops anymore. The stars, in this coordinate system, move completely independently of Earth’s motion. Yet the coordinate system has Earth as its center, with the Sun moving round it.

For those of you who suggested that it’s obvious (or near-obvious) that Earth orbits the Sun, these are the coordinates that Muller can ask you about. The only effect of these geocentric coordinates is near the Earth and Sun. No hint remains, by the time you get to the distant stars, that anything is different from heliocentric coordinates. And so, if you assumed implicitly or explicitly that because the distant stars are in nearly flat space, you could extend good heliocentric coordinates all the way down to the Sun and apply quasi-Newtonian reasoning, these curved geocentric coordinates raise challenging questions that you need to answer. Does your argument, whatever it was, truly survive the use of a coordinate systems like this one? And why can’t Muller use them to show the Sun orbits the Earth?

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36 Responses

  1. My argument remains the same: the Kolmogorov complexity of the equations describing a geocentric coordinate system is higher than the Kolmogorov complexity of the equations describing a heliocentric coordinate system. There’s thus a (theoretical, since Kolmogorov complexity is not a Computable Function) choice function for which description to use. There’s still an arbitrary choice (prefer simpler or prefer more complex), but simpler descriptions are easier to work with.

    1. But the argument isn’t about what coordinate system to use, is it? Muller’s claim is that “The Earth Orbits the Sun” isn’t a contentful, meaningful, coordinate-system-independent statement, and that “The Sun Orbits the Earth” is equally true. Even Muller agrees that heliocentric coordinates gives simpler equations and that one would generally want to use them in describing the solar system. But he’s making an additional statement that you’re not addressing here.

  2. I don’t see this post affecting my previous arguments. It applies if you’re going to reduce an orbit merely to coordinates certainly, but I’ve offered other definitions than that. The professor’s argument boils down to there being a symmetry between earth-moving-around-sun and sun-moving-around-earth. Likewise when I do a handstand I hoist the entire world above me. Which is a cute bit of technical jargon that works fine as pen and ink.

    1. This issue really affects Newtonian thinkers; as I recall, it doesn’t affect your arguments. But one step at a time; I don’t think you’ve yet appreciated all the issues either.

  3. I am not sure to understand why the coordinate definition would have an impact of the definition of who orbit who (I will restrict to the case of 2 bodies for simplicity).

    I think that the definition of who orbit who should be defined by the asymmetry of motion between the two bodies (to match the classical sense we have using newtonian physics).

    I would argue that to be meaningful, this asymmetry should be defined in a way that must be independent of the coordinate system chosen. For example, the coordinate system chosen have no impact of the observation I can do (for example looking at the electromagnetic spectrum), and so have no physical meaning. I only see them as a way to compute things by assigning numerical values.
    To give meaning to what I want to express: I can take a picture of the world, and then rearrange the pixels with any chosen bijective transformation and make the picture look more or less anything: it has no bearing on the interactions of the objects in the world.

    In the newtonian settings, I would say the asymmetry is the asymmetry of velocity of the two body against a perfectly flat background with a orthonormal coordinate system (the “go far and look” proposal).
    I do not have enough understanding of GR to be able to translate this from the newtonian settings.

    But I think I can propose something that should be coordinate independent in GR and that at least correlate with it (i.e. give the same answer in the newtonian limit of low gravitational field):
    – with two bodies, there exists a lagrange point L1 (the one between the two bodies).
    – I think that this L1 point can be defined only using the fact that the curvature is flat at this point (and not around in the direction of the two bodies, a local minimal/maximum I think). I think this is coordinate independent.
    – I would then use the asymmetry of distance with the two bodies and the L1 point to defined who orbit who.
    – Now here I am unsure, but I think in GR you can define a proper distance that is independent of the coordinate system that could be used for this comparison (I am unsure if this will work with very strong gravitational field).

    Unrelated: I wanted to thank a lot you for all your work on this web site. I was very very glad when I stumble upon it a few years back :). I learned a lot from your articles. There are technical but well explained, and I appreciate a lot the scientific honesty (like what is ‘mainstream/well accepted’, what is ‘exploratory/conjecture’, when there are simplifications, etc).

    1. Thanks for your kind words about the website.

      Yes, the Lagrange points’ existence and location — their distance from the Sun and Earth can be measured in a coordinate-independent way, using light round-trip travel time — between a satellite at L1 and the two bodies — do indeed serve as coordinate-independent evidence that there is some meaning to who-orbits-who. More evidence against Muller, I agree… but it needs a GR expert to give it a technical formulation.

  4. I’ve tried commenting this a few times, but it doesn’t seem to be showing up on my computer—not sure why. I figured I’d give it one final shot as a stand-alone comment instead of a reply, but sorry if this is actually a triple-post!
    I’m replying to the following comment from Dr. Strassler:
    Ok, now you are indeed moving the discussion in the right direction, I think. I’m sympathetic with this approach, though I’ve been trying to understand how to make it precise. Yes, the space-time for a two-body system is quasi-periodic, and we need to use that. But you have to adjust your language so that you put two clocks on two different trajectories that start and end at the same point; otherwise you cannot compare them; I don’t think that’s a problem, just a necessary refinement. Second, let me pose a more subtle issue. Suppose we replace the Sun with a black hole of the Sun’s mass, and replace the Earth with a black hole of the Earth’s mass. The orbits are essentially the same. But where exactly would we put the clocks? The issue is to have a formulation of the proposal that doesn’t depend on the Sun and Earth having surfaces (or interiors), or in other words, to deal with your notion of “comoving” in a more precise way.
    My reponse:
    My first thought on your point about replacing the Earth and Sun with surfaceless black holes would be to have the clocks orbit them. If you wanted to choose a nice intrinsic-feeling orbit you could take the innermost stable circular orbit at r = 6M, at least in the Schwarzschild geometry. Then there are the added benefits of the orbit being closed and symmetric from point to point, so the time dilation effect would be constant for each clock—at least in the absence of any Earth-Sun orbital motion. I’m pretty confident it would be relatively easy to calculate what this “no-orbit” time dilation effect would be and (in theory) compare it to the “experimentally measured” effect, at which point my above argument would apply. What do you think?

    As regards the specifics of making sure the clocks begin and end in the same place, I’ve honestly never really understood how this works in Relativity. Could the answer be as simple as synchronizing the clocks “at infinity”, moving them in to the Earth-Sun system, placing each clock in orbit about its respective body, waiting for one full period, and, once that period is complete, drawing the clocks back out “to infinity” to be compared? This strikes me as too basic to be the answer.

    1. Got it this way, at least. If it shows up elsewhere I may delete the copies.

      I’m not sure I like the “clocks in orbit at a fixed multiple of the Schw radius”, but I agree it’s an option. There might be a better one. I’ll be thinking about this too because it’s a point I want to understand better anyway, maybe we can improve it.

      Regarding synchronizing clocks — it can be almost as simple as you state, but the problem is that the result is path-dependent. It matters how you move the clocks apart, and how you draw them back together — and, as we see in the black hole replacement approach, where you put them while you are waiting for them to count off time. And “waiting for one full period” could be tricky to define too. So you have to make sure that these ambiguities which will arise from the path-dependence and definitions of “period” and so forth, are subleading compared to what you want to measure. I think you can prove that, but it’s technical.

      1. Not to rehash this, but I’ve been Mulling things over and would like to update my response. What if, instead of fixing the clocks in circular orbits whose radii Re and Rs are a constant multiple of the Schwarzschild radii, we fixed them in “complimentary orbits” whose radii add to precisely the Earth-Sun distance and therefore form two “kissing circles”—one centered on the Earth and one on the Sun? We could synchronize the clocks at the “kissing point” and then send them around their respective orbits. In general, they would have different orbital periods, but (ignoring the probably strongly non-negligible three-body nature of this problem) we could use Kepler’s Third Law for orbits about the Earth and Sun to find the specific ratio Re / Rs which equalizes these periods. I’m pretty confident an analogous calculation can be done in GR, where we can make the proper-time orbital periods equal.

        If you’ll temporarily grant me the somewhat heinous simplifications I made above, then this setup seems to solve a lot of issues. For one, in the classical calculation the orbital period of the clocks is nearly identical to that of the Earth and the clocks naturally return to the same kissing point after a year (at least to within a few minutes). After synchronizing them, then, we could send them on their way and simply wait for them both to return. This also seems to suggest a solution to the problem of “waiting one full period”, as the definition of a period could be built around “both clock orbits intersecting again”. Finally, I expect that the Earth-orbiting clock will run slightly slower, but even if the Sun’s clock is the one running slower we can be sure that it has nothing to do with gravity and everything to do with orbital motion since the clocks were rigged up to have the same proper-time orbital periods.

        The biggest problem I can see with this approach is that I don’t think the orbits I’m describing actually exist—but maybe there is a more complicated pair of orbits with analogous properties which do? I’m also not sure how to deal with the clock orbits failing to intersect due to time dilation and the slight difference between their periods and the Earth’s. Even still, it seemed like a significant improvement over my previous response, so I wanted to hear people’s thoughts!

        PS: Thank you for your comments on synchronizing clocks—they have helped clear up a lot of my confusions!

    1. 😀 Joke’s on you, bro. If you can’t tell the difference between flat-earther balderdash and a serious discussion about what is and isn’t meaningful in general relativity, you’ve got a lot to learn.

  5. Richard Mueller is making a straw man argument by reducing the meaning of “orbiting” to its coordinate description and then arguing that coordinates have no objective reality.

    We generally take the meaning of orbiting as the geodesic path constrained inside a gravitational well. To be constrained inside taken to mean there exists a minimum value of the curvature invariants for all points of the object world-line (e.g. R^{abcd}R_{abcd} has a minimum value). In Newtonian terms, orbiting objects have a maximum distance away from the center of mass.

    The curvature invariants constructed from the Weyl curvature tensor give a coordinate independent description of the gravitational field, i.e. there is a physical center of mass with an observer independent 4-momentum. The world-line (tube) defined by system 4-momentum itself defines a preferred foliation on which for any spatial hypersurface the orbiting objects have an average spatial separation. For all observers the solar system center of mass is inside the Sun and the Earth orbits well outside of it.

    It is true that if we had an astronaut tumbling in space, say in-between the Earth-Mars orbit that they don’t have the wherewithal to determine if they’re orbiting the Sun or vice verse, and quite unlikely that their first thoughts would be “well, suppose I set up a system of clocks along a radial line conjoining the Sun, me, and asymptotic infinity…” but this ignorance should be what determines if there’s any meaningful sense of having one body orbit another.

    1. I completely agree that Muller is full of crap, for the reason you state. The entire discussion of coordinates is irrelevant to the question at hand (but notice how very few readers have yet raised this point…)

      I’m not convinced your technical argument is entirely correct or sufficiently clear yet. You can probably improve it with some more thought. But there’s more: Muller has the advantage that he sounds simple. How simple an argument can we construct that shows that the statement “the earth goes round the sun” is, in fact, meaningful and coordinate independent? It would be good to know a clear, complete technical argument *and* a convincing non-technical one.

      1. In that context we need concede the argument to Muller as there’s another issue to contend with, namely, there is no agreed upon convention for the meaning of “to orbit.”

        If “to orbit” is taken to be synonymous with “to go around” then it is trivially true that the Sun goes around the Earth. We can in principle draw up some metric tensor with the Earth at the center that is a solution to Einstein’s field equations.

        What is unsatisfactory about such a description is that it is not what is usually meant by “orbiting.” For example Sgr A* resides at (near) the center of our galaxy but we don’t say the galaxy or our solar system is in orbit of Sgr A*, rather the content of the galaxy is orbit about the galactic center of mass.

        In this context we would say the Earth and Sun orbit the center of mass of our solar system, that is, the word “orbit” invokes a preferred frame in which this is unambiguous. In all reference frames for any spatial slice of the spacetime, the center of mass of the solar system resides on the interior of the world-volume of the Sun.

        Even in the context of an air-tight definition of “orbiting” where the Earth unambiguously orbits the Sun, no one is under the obligation to accept this definition, Mr. Muller in particular, irrespective of how well it accords with our everyday use of the word.

        1. Good, this is another point which has not been given enough attention. Yes, “to go around X” is not the same as “to orbit X due to the influence of X”, and until someone defines the meaning of ” to orbit” clearly, our discussion is flying through clouds. I think you’re the first commenter to address the issue.

          I would argue, though, that rather than this being a reason to concede something to Muller, it is a reason to be somewhat critical of him. After all, he did not define his terms… and if a writer is going to make a strong claim, it’s the writer’s responsibility to state the context in which the claim applies. To be fair, he wrote this little diatribe first in an answer to a Quora question, and it was then reprinted in Forbes… it wasn’t exactly published in a philosophy journal with extremely high definitional standards.

          But still, he makes it clear: he is making a claim that something about the Earth-Moon-Sun system is true in general relativity that is not true in Newtonian gravity. It’s not just about coordinates, because I can change coordinates in Newtonian contexts too, at the price of “fictitious” forces… is that such a high price, if I know what I’m doing? Muller’s claim is much stronger than that. He is claiming that “there can be no coordinate-independent definition of `X orbits Y’ in general relativity.” I think that is correct in a general metric, but not in the metrics characteristic of systems with long-lived orbits. It’s a little bit like saying “a special notion of time cannot be defined in general relativity”; true in general, but in some metrics you can define a physical quantity which acts like time (e.g. the CMB temperature in an FRW universe.)

          Still thinking out loud about this though, since I don’t know my GR well enough to state this precisely yet.

          1. If we take the usual meaning of orbit then Muller is wrong and your animations are somewhat misleading.

            It might be worth clarifying the “usual meaning.” It is standard whether in we’re in chapter 3 of Goldstein’s mechanics or determining photon orbits in the Kerr metric that orbiting is taken to mean the bound trajectories inside a gravitational well, so technically objects orbit a center of mass. It is standard usage that if the center of mass resides within one the objects we say that the other objects orbit the central object. Since the center of mass of the solar system resides inside the solar radius we say the Earth orbits the Sun, even if defining the center of mass in GR is no easy task.

            In your top animation the Sun is at the center of the coordinates and in the lower animation the Earth is at the center of the coordinates, but in both animations the center of mass is inside the large orange dot. To say in the lower animation that the center of mass of the solar system is orbiting the Earth is to use the word “orbit” in a way that is not intended.

            I would be cautious about taking cosmic time, which is any arbitrary global time coordinate or world-time like any other, to hold some special significance above being a convenient choice. However, it might be significant if our universe is an evolving block universe, but that is a discussion for another time.

            1. The solar system barycenter is always close to the Sun but often outside what we see as the surface of the Sun. For example, it’s outside the Sun since 2017 and won’t reenter it until 2027.

            2. I am not sure you have the best definition of “orbit” yet. I don’t think that “X orbits Y” has meaning on a general spacetime, and the “usual meaning” doesn’t address this.

              As for cosmic time — my point is this: in some metrics, there is something physical that acts like a clock, in that it can be used as a reference point across space. That doesn’t mean I am using it as a time-coordinate; instead it is a physical reference point *which acts like* a time-coordinate. We can choose our time coordinate to match that physical reference point, or not; that part’s arbitrary. But the existence of the physical reference point is not arbitrary. The inflaton sometimes plays this role in inflation.

              In the same sense: the rate at which a Foucault pendulum rotates serves as a latitude-like physical reference, independent of what coordinates I choose on the Earth. Were the Earth non-rotating, or rotating irregularly due to it having a complex shape, that physical reference point would be absent. So “X is more equatorial than Y” has meaning on Earth, not because “more equatorial than” can *always* be defined, but because it can be defined on spaces that have a latitude-like physical reference.

              So the possibility of defining “X orbits Y” in a physical, coordinate-independent way could certainly be dependent on the class of space-time geometries in question. What Muller is missing, I think, is that all planetary systems have such geometries — or they’d be unstable.

  6. Possible solution:

    GR, and Newton, both have translational and rotational symmetry so the laws of physics are covariant under any translations or rotations of the coordinate system. But, in classical physics, you prefer the coordinate system that, through global translation and rotation, eliminates as many forces as possible. The heliocentric system is preferred to geocentric because it eliminates the “fictional” centrifugal and epicyclic forces leaving only gravity.

    GR goes a step further by pointing out that you can eliminate gravity as well, by applying a nonlinear coordinate transform to spacetime, leaving every body in the solar system in uniform motion in a straight line through spacetime. As a bonus, this transform is uniquely specified by the particular spacetime distribution of energy. The geocentric coordinate systems are valid choices in both classical mechanics and GR, but they are not preferred because of all the unecessary unexplained forces they require.

    The classical case should be fairly easy to demonstrate using observations of the sun and at least one planet. You could (try and) set up the equations of motion, or just draw some force diagrams, then choose different coordinate systems until a minimum number of forces were required. Drawing force diagrams in four dimensions would be a bit tricky for the case of GR, but you could do it using only two spatial dimensions and time.

  7. Matt, have you read: “General covariance and the foundations of general relativity:eight decades of dispute” by Dr John D Norton, Department of History and Philosophy of Science, University of Pittsburgh?

    It’s an interesting historical account of the debate on whether general covariance should be uniquely associated with GR, involving Einstein himself. The physical content, as you well know, comes from the equivalence of inertial and gravitational mass, but this doesn’t alter your excellent, eye-opening article above.

    1. No, I have not. At some point I should take a look. What do you mean by “uniquely associated”? GR is certainly not the only generally covariant theory of gravity.

  8. “Einstein claimed that physics is unchanged no matter how crazy the coordinate system you might choose to describe it.”

    This is not correct, at least not in the way you imply. See, e.g., centrifugal force in a rotating coordinate system. Your time-varying coordinate system would result in horrendous governing equations for force, gravity, acceleration, etc., in the vicinity of the Earth and Sun. So, it remains silly to choose these coordinates.

    1. No, it is correct, and I believe it is you who are mistaken.

      I can equally well ask the question in electromagnetism, where I am free to choose whatever gauge I like, no matter how crazy. The fact that a crazy gauge leads to nasty equations is not the point. (1) The basic equations are gauge invariant, so they simply don’t care; and (2) the fact that there are only two polarization states of a photon, rather than three, and the whole consistency of the theory, *demands* complete gauge invariance, and not restriction to gauges which make the equations easy. All physics must be independent of the choice of gauge.

      The same statements are true of general relativity. All coordinate choices *must* be allowed, and any physics *must* be independent of the choice of coordinates.

      Note added: by the way, regarding rotating coordinates — again, in gauge theory there are some issues with whether one should impose certain boundary conditions on gauges, and there are similar subtleties with certain coordinate choices in general relativity. However, you will note that my coordinate choice for figures 3 and 4 has no such issues — which is why I chose it.

  9. Isn’t the message of GR that there are no privileged frames of reference? That any coordinate system you choose is completely arbitrary? Saying that the Earth revolves around the Sun is useful, for all sorts of reasons, but not “true” in any absolute sense. Or am I misunderstanding?

    1. This is precisely the issue under discussion. Yes, GR says there are no privileged coordinate systems and that what you choose for coordinates is arbitrary. [By the way, is a “coordinate system” the same as a “frame of reference” in GR?] But is “earth revolves around sun” a meaningless statement? a useful one? a true one? or something else? That is exactly what we’re trying to determine here.

      Most readers of this blog would say it is “true” or something close to that, and quite a few gave arguments in favor of this statement. You, and professor Muller, are taking the opposite view. The underlying purpose of this series of posts is to tease apart the issues that matter from those that don’t — and to connect it with even more important issues that trouble theoretical physics today.

      So along those lines, let me ask you a question: there are no privileged coordinate systems on this planet, either. I can use any map I want (which is a coordinate system that can be projected onto a plane) or I can put cucumber coordinates or peanut coordinates or pear coordinates, and put their origins at any point on the planet — and I can even make them time-dependent. So: is the statement “The earth is approximately a sphere” a meaningless statement? a useful one? a true one? or something else?

      1. I think the problem is not that it’s not meaningful/useful/true, but that it’s framed as an invariant statement. It’s not: geodesics depend on the coordinate system and metric. Like saying that an object has 10 J of kinetic energy: true in some frames, not in others.

        As for the planet, with whatever coordinates you choose, you can calculate the intrinsic curvature of the surface and see how well it compares to 1/R^2 to decide if the statement is useful/true.

        1. There are some correct threads in this answer, and I am pretty confident I know what you were trying to say; but your wording has some ambiguities and some statements that, as phrased, aren’t really correct. Do you want to try again, and maybe do so in a way that more readers could follow your logic? I haven’t used the words “geodesic” here, or “intrinisic curvature”, so a lot of readers may not understand your thinking.

          And if you are claiming that “Earth goes round the Sun” cannot be stated in a coordinate-invariant way, are you sure about that?

          1. I’m not Joey Nielsen, but I’d like to take a stab at answering “are you sure ‘Earth goes around the sun’ cannot be coordinate-invariant?” (I paraphrase). If I measure the lapse of (proper) time during, say, one complete orbit according to (A) a non-rotating clock co-moving with the Earth and compare that to (B) the same measurement taken with a non-rotating clock co-moving with the Sun, I believe I could use those measurements to determine which is orbiting the other. Both clocks will run slow in a perfectly predictable way due to the gravitational time dilation near the Earth and Sun, respectively. If I correct for these factors—which anyway should be small—then (to my knowledge) the only significant remaining time dilation effect is due to physical motion. So (after correction) whichever clock runs slower is moving faster—i.e. covering more (orbital?) distance in the same time. I suggest that the slower-running clock is on the “orbiting body”, while the faster-running clock is on the “stationary body”.

            I’m sure I’m missing some complexity here, but I’d love to hear your thoughts. Thank you for giving me something extremely interesting to think about!

            1. Ok, now you are indeed moving the discussion in the right direction, I think. I’m sympathetic with this approach, though I’ve been trying to understand how to make it precise. Yes, the space-time for a two-body system is quasi-periodic, and we need to use that. But you have to adjust your language so that you put two clocks on two different trajectories that start and end at the same point; otherwise you cannot compare them; I don’t think that’s a problem, just a necessary refinement. Second, let me pose a more subtle issue. Suppose we replace the Sun with a black hole of the Sun’s mass, and replace the Earth with a black hole of the Earth’s mass. The orbits are essentially the same. But where exactly would we put the clocks? The issue is to have a formulation of the proposal that doesn’t depend on the Sun and Earth having surfaces (or interiors), or in other words, to deal with your notion of “comoving” in a more precise way.

          2. My 1st point is that it’s not controversial that the mathematical description of an object’s motion in GR is coordinate dependent. There is no absolute standard of rest, but there is also no absolute standard of motion. If we can agree on this, then I think it’s very difficult to reject the idea that “Earth orbits the Sun” is a coordinate-dependent statement. Then the question is what *can* we say? They’re certainly bound in all frames… maybe this is about as far as we can go in GR?

            Regarding the sphere, my point was that the question of whether or not a surface is spherical doesn’t depend on the coordinate system. It’s a meaningful and well-defined question whose invariant answer you can calculate in all frames of reference (in contrast to the orbit question).

            1. Agree with the second point (and in fact I plan to cover it soon) but the first point — I agree with the premise but question the conclusion, as does Jacob Kuntzleman just above your response. So I think there’s still more to discuss.

              1. You say you question the conclusion, but on what basis? What would it mean to say the Earth orbits the Sun in the rest frame of the Earth?

                In the frame of reference where Earth is stationary, the Sun’s clock will tick slowly relative to Earth’s due to the transverse Doppler effect, and Jacob’s plan will reveal the Sun to be the orbiting body, no? If you ask each clock about the elapsed proper time over one orbit, the answer will be one year.

                1. I am suggesting that the statement “the Earth orbits the Sun” can be stated in a way that should be frame-independent. It’s not a matter of how things look (the way Greenland looks bigger than Africa on some maps) but a matter of how things are (the fact that Greenland is smaller than Africa independent of what map you choose to displace them.) I have not found a proof of this statement, which reflects my limitations as a general relativist. But if I am correct, then the proof of the statement will answer your first question.

                  As for the clocks, it’s something to check. We need not consider Earth and Sun. Is it true that if you took two clocks and brought one up to the space station, took it around on N orbits, and then brought it back down (or better, brought the second clock up, for symmetry) and compared them, that they would read the same?

      2. “to connect it with even more important issues that trouble theoretical physics today” I don’t mean to spoil things, but are you aiming for a popularized version of the “no local observables in quantum gravity” argument? If so, looking forward to it!

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