General Relativity & The Sun: Does it Revolve Around Earth?

[JohnNemo]

It's not clear, but probably they are using a #2 definition with coordinates that are convenient for analysing planetary motion in our solar system. That doesn't mean that sun isn't also rotating under the #1 definition, it just means that the author of that wikipedia article (who probably understands less relativity than many of the contributors to this thread) was unaware of or uninterested in the subtleties here.

OK. What I am interested in is whether, under *any* definition of "rotation" there is *anything* which is invariant and, if so, I would like to know more about what is invariant.

 

[PAllen]

OK. What I am interested in is whether, under *any* definition of "rotation" there is *anything* which is invariant and, if so, I would like to know more about what is invariant.

I’ve given you a couple of answers to this. Read the linked material on kinematic decomposition leading to the definition of vorticity tensor.

 

[Nugatory]

OK. What I am interested in is whether, under *any* definition of "rotation" there is *anything* which is invariant and, if so, I would like to know more about what is invariant.

Yes, of course there is. We've mentioned rotation defined in terms of proper acceleration many times already in this thread, most recently in #101 above, (I'm not sure how anything could be clearer than "This property is invariant"); and @PAllen has pointed you at the vorticity tensor.

 

[PeterDonis]

Is there anything invariant about rotation and, if not, how come if there is such a thing as invariant proper acceleration?

The proper accelerations of all the different parts of a rotating object are invariant. So you can look at the pattern of proper accelerations (the variation in direction--all pointing towards the axis instead of all pointing in the same direction) to distinguish rotation from linear acceleration. This is the sort of invariant definition of "rotation" that @Nugatory was getting at in a previous post.

Other invariant effects that are generally said to be due to "rotation" are Thomas precession, de Sitter precession, Lense-Thirring precession, and the Sagnac effect. All of these effects, if properly defined in terms of direct observables, are invariant.

 

[PeterDonis]

I would say the vorticity tensor defines rotation in an invariant sense.

Yes, this is another invariant way of defining "rotation". However, it won't necessarily match up with the others (which won't necessarily all match up with each other either). This is one of the issues with "rotation" in GR: different definitions that, according to our intuitions, ought to all go together, actually don't.

 

[JohnNemo]

The proper accelerations of all the different parts of a rotating object are invariant. So you can look at the pattern of proper accelerations (the variation in direction--all pointing towards the axis instead of all pointing in the same direction) to distinguish rotation from linear acceleration. This is the sort of invariant definition of "rotation" that @Nugatory was getting at in a previous post.

Other invariant effects that are generally said to be due to "rotation" are Thomas precession, de Sitter precession, Lense-Thirring precession, and the Sagnac effect. All of these effects, if properly defined in terms of direct observables, are invariant.

You have identified 5 invariant effects which indicate "rotation". Am I right in thinking that by measuring these effects and determining that they are absent you could determine that a particular object was - in an invariant sense - non-rotating?

 

[pervect]

I believe I made an error in attributing the physical oddness of clocks stopping when $\omega r = c$ in the rotating Born coordinate chart that I mentioned earlier on a coordinate singulalrity. So I withdraw that remark, and struck out the appropriate section on the previous post. The determinant of metric tensor doesn't seem to vanish there.

My current thinking is that we can blame the oddities I noted (such as the behavior of clocks) on the unfamiliarity of interpreting the physical significance of null coordinates. Any coordinate system that makes light appear to stop by assigning a constant coordinate to a light beam will be a null coordinate. There's nothing mathematically wrong with null coordinates, but they cannot be forced into the mold of either a time coordinate or a space coordinate. In tensor language, if x is a coordinate, the sign of the invariant length of the vector $\partial / \partial x$ determines whether or not we call it a time, space, or null coordinate. The case where the length of the vector is zero is the case where x is a null coordinate.

 

[PeterDonis]

Am I right in thinking that by measuring these effects and determining that they are absent you could determine that a particular object was - in an invariant sense - non-rotating?

Not quite, because, as I mentioned in an earlier post, in the general case these effects don't all go together--that is, they aren't all present or absent together. You can have an object in which some effects are present and others are not.

In other words, there is no single invariant division between "rotating" and "non-rotating"; these terms do not name natural categories. They're just convenient approximations that work well in many common scenarios, but break down if you try to push them too far.

 

[JohnNemo]

Not quite, because, as I mentioned in an earlier post, in the general case these effects don't all go together--that is, they aren't all present or absent together. You can have an object in which some effects are present and others are not.

In other words, there is no single invariant division between "rotating" and "non-rotating"; these terms do not name natural categories. They're just convenient approximations that work well in many common scenarios, but break down if you try to push them too far.

Are you able to give a feel for how they are related? For example, is it the case that at fast rotational speeds they are all present but at slower speeds you might have some but not others?

 

[PeterDonis]

is it the case that at fast rotational speeds they are all present but at slower speeds you might have some but not others?

No, it's not that simple. It's a matter of spacetime geometry; the relationship between them is different for different spacetime geometries.

 

[PAllen]

I believe I made an error in attributing the physical oddness of clocks stopping when $\omega r = c$ in the rotating Born coordinate chart that I mentioned earlier on a coordinate singulalrity. So I withdraw that remark, and struck out the appropriate section on the previous post. The determinant of metric tensor doesn't seem to vanish there.

My current thinking is that we can blame the oddities I noted (such as the behavior of clocks) on the unfamiliarity of interpreting the physical significance of null coordinates. Any coordinate system that makes light appear to stop by assigning a constant coordinate to a light beam will be a null coordinate. There's nothing mathematically wrong with null coordinates, but they cannot be forced into the mold of either a time coordinate or a space coordinate. In tensor language, if x is a coordinate, the sign of the invariant length of the vector $\partial / \partial x$ determines whether or not we call it a time, space, or null coordinate. The case where the length of the vector is zero is the case where x is a null coordinate.

I’m glad you corrected this, as I was tempted to give one of my favorite examples of a disguised Minkowski spacetime metric:

ds² = da db + da dc + da de + db dc + db de + dc de
This is flat spacetime in all light like coordinates with (+,-,-,-) signature.

 

[pervect]

I’m glad you corrected this, as I was tempted to give one of my favorite examples of a disguised Minkowski spacetime metric:

ds² = da db + da dc + da de + db dc + db de + dc de
This is flat spacetime in all light like coordinates with (+,-,-,-) signature.

Interesting. I'm not familiar with that metric, though I'll think about it some when I get a chance. I was thinking about the one-space one-time case, where we substitute u = x - ct and v=x+ct to turn the Minkowskii metric $-c^2\,dt^2 + dx^2$ into $du\,dv$.

I'm not sure of the best way to put it into words that might be relevant to the thread. . I suppose the short version would be that it's true that as one approaches the speed of light that clocks run slower and slower, and that in the appropriate limit they stop. But all this winds up proving is that we don't necessarily have to represent or describe space-time in ways that involve clocks and rulers.

 

[JohnNemo]

No, it's not that simple. It's a matter of spacetime geometry; the relationship between them is different for different spacetime geometries.

If we take the example from #99 of a spherical region of space where our solar system is but assume there is nothing there except a moderate sized planet which is at rest and non-rotating relative to the distant stars, would you expect the five invariant effects referred to earlier to be minimal or absent?

More generally, is there any kind of correlation between a body being non-rotating relative to the distant stars and the five invariant effects being minimal or absent?

 

[PeterDonis]

If we take the example from #99 of a spherical region of space where our solar system is but assume there is nothing there except a moderate sized planet which is at rest and non-rotating relative to the distant stars, would you expect the five invariant effects referred to earlier to be minimal or absent?

In this particular, highly idealized case, yes, all of them would be absent.

is there any kind of correlation between a body being non-rotating relative to the distant stars and the five invariant effects being minimal or absent?

Not in general, because in general there are other bodies present in the spherical region of space in question, and those other bodies affect the spacetime geometry there.

For example, a satellite orbiting the Earth exhibits all three of the precessions I referred to--which are actually best referred to simply as "rotational precession" or something like that, since in the general case there is no invariant way to separate them out. The overall effect is that "non-rotating" relative to the local spacetime geometry--i.e., the absence of the pattern of proper accelerations referred to earlier, and the absence of the Sagnac effect--is not the same as "non-rotating" relative to the distant stars (which would be the absence of the precession).

(Btw, I'm not sure whether the proper acceleration pattern and the Sagnac effect always go together; I don't think they do, but I can't come up with a counterexample at the moment.)

 

[JohnNemo]

@PeterDonis Thank you for all your explanations thus far. They are very useful and must have taken some considerable time to write in total – you are by far the most prolific writer on this thread.

I am aware that I have not responded to many posts by others on this thread. This is partly the result of the way discussion threads go – you get into a sort of dialogue with some people and not others – but I think it is also caused by the fact that I am trying to grapple with what are for me difficult concepts, and the idiosyncrasies of the individual learner influence which answers they find easiest to follow and follow up on. This is a very individual thing - an answer which objectively is both accurate and pertinent may leave one learner cold while being very illuminating to a different learner. So I would like to thank everyone who has taken the time to write on-topic posts with the intention of trying to answer my questions.

 

[JohnNemo]

Thinking about what I have learned about GR I think I find it mentally useful to think of spacetime as a kind of aether. Thinking about it this way helps to emphasise that spacetime is a ‘real thing’ and not just ‘space’ (in the non-technical meaning of the word ‘space’).

The word ‘aether’ is not in vogue, I suppose because everyone knows that Einstein showed that Lorentz’s aether was an unnecessary postulate about 100 years ago. I have also noticed that Lorentz Ether Theory is the bette noire of the forums, no doubt with good reason. But if we can get beyond the allergy to this word it seems to me that the general idea of an aether gives the right general mental picture of something real which things can be measured against to determine invariant proper acceleration and the various invariant quantities we associate with rotation. Of course this GR aether is not the same as that envisaged by Lorentz – it is four dimensional and has a geometry determined by the distribution of mass and energy.

Do you think this is a reasonable way of looking at it?The idea of thinking about spacetime as an aether was suggested to me by Einstein’s writings. He gave a speech in 1920 entitled Ether and the Theory of Relativity which is here https://en.wikisource.org/wiki/Ether_and_the_Theory_of_Relativity an extract of which follows:

“Certainly, from the standpoint of the special theory of relativity, the ether hypothesis appears at first to be an empty hypothesis. In the equations of the electromagnetic field there occur, in addition to the densities of the electric charge, only the intensities of the field. The career of electromagnetic processes in vacua appears to be completely determined by these equations, uninfluenced by other physical quantities. The electromagnetic fields appear as ultimate, irreducible realities, and at first it seems superfluous to postulate a homogeneous, isotropic ether-medium, and to envisage electromagnetic fields as states of this medium.

But on the other hand there is a weighty argument to be adduced in favour of the ether hypothesis. To deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view. For the mechanical behaviour of a corporeal system hovering freely in empty space depends not only on relative positions (distances) and relative velocities, but also on its state of rotation, which physically may be taken as a characteristic not appertaining to the system in itself. In order to be able to look upon the rotation of the system, at least formally, as something real, Newton objectivises space. Since he classes his absolute space together with real things, for him rotation relative to an absolute space is also something real. Newton might no less well have called his absolute space "Ether"; what is essential is merely that besides observable objects, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.

The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities, but helps to determine mechanical (and electromagnetic) events...

What is fundamentally new in the ether of the general theory of relativity as opposed to the ether of Lorentz consists in this, that the state of the former is at every place determined by connections with the matter and the state of the ether in neighbouring places, which are amenable to law in the form of differential equations; whereas the state of the Lorentzian ether in the absence of electromagnetic fields is conditioned by nothing outside itself, and is everywhere the same...

Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”

 

[PeterDonis]

The viewpoint described in the Einstein lecture you refer to seems reasonable to me. The common allergic reaction to the word "ether" is to the word, not to the underlying idea that Einstein describes. I personally would just use the word "spacetime", and express the idea in simple form as "spacetime is a physical thing". (Note that Einstein uses the word "space", not "spacetime", but he really means the latter.)

 

[JohnNemo]

Some people find this asymmetry between speed (always relative, meaningless for an isolated body) and changes in speed (meaningful even for an isolated body) to be ugly and disturbing, but it is an experimental fact that that's how the universe we live in behaves - and that universe really doesn't care much whether we like it.

I understand (because it has been stated on this thread) that proper acceleration can be measured against the geometry of the local spacetime whereas velocity cannot. But can you give me an easy way of visualising why this is the case - i.e. what is it about the geometry of spacetime which means that velocity cannot be measured against it?

 

[Nugatory]

I understand (because it has been stated on this thread) that proper acceleration can be measured against the geometry of the local spacetime whereas velocity cannot. But can you give me an easy way of visualising why this is the case - i.e. what is it about the geometry of spacetime which means that velocity cannot be measured against it?

The universe doesn't have to behave in ways that you find easy to visualize, so there may not be any satisfactory answer. But here's one that you can try on for size...

Something must be subject to a non-zero force if it is to have non-zero proper acceleration. By Newton's third law, if there is a force on something, then there must be an equal and opposite force on something else, so there is always something to measure against. Even in the extreme situation (object spinning about its own axis in an otherwise completely empty universe) that inspired this thread, strain gauges embedded in the object will detect the internal forces between outer and inner layers that hold the object together and keep all parts of it rotating at the same rate. And if we can always detect the force, then we can also always detect the proper acceleration.

 

[JohnNemo]

Not in general, because in general there are other bodies present in the spherical region of space in question, and those other bodies affect the spacetime geometry there.

If we take our own solar system and, for simplicity, imagine that the Sun is not rotating about its own axis, and imagine that the Earth is not rotating about its own axis, and imagine that there is nothing else, apart from the Sun and Earth, in the solar system:

1. Would the five invariant indicators of 'rotation' be present for the Earth?

2. Would the five invariant indicators of 'rotation' be present for the Sun?

 

[PeterDonis]

If we take our own solar system and, for simplicity, imagine that the Sun is not rotating about its own axis, and imagine that the Earth is not rotating about its own axis, and imagine that there is nothing else, apart from the Sun and Earth, in the solar system:

1. Would the five invariant indicators of 'rotation' be present for the Earth?

2. Would the five invariant indicators of 'rotation' be present for the Sun?

First, per a recent post of mine, all three of the "precession" indicators should really be lumped together, since there is no invariant way of separating them. Also, per some other earlier posts, vorticity is another separate indicator. So we really have four indicators: (1) pattern of proper acceleration; (2) precession; (3) Sagnac effect; (4) vorticity.

Second, since all of these indicators do not necessarily correlate, when you set up a scenario, it's not sufficient to say whether an object is "rotating" or "not rotating", since we don't know which, if any, of the indicators you are referring to! So really you have things backwards: you don't say an object is "rotating" or "not rotating" (or "not rotating about its axis, but revolving about something else"), and then ask which indicators are present. You first have to specify which indicators are present, and then determine from those whether you want to describe the object as "rotating" or "not rotating". Or you can specify "rotating" by some other criterion, such as "not rotating relative to the distant stars"; but you have to be explicit about that. (In fact, "rotating relative to the distant stars" can be treated as a fifth indicator.)

So I can't answer your question as you ask it, because there is more than one way to rephrase your question in terms of what "rotation" means. Here are a couple of examples:

(A) Imagine that the Sun is not rotating with respect to the distant stars. Imagine that the Earth is orbiting the Sun, but is also not rotating with respect to the distant stars. Then the indicators will be as follows (at least, these are my best quick intuitive guesses--I have not done the detailed math):

For the Sun: (1) No (2) No (3) No (4) No (5) No

For the Earth: (1) No (2) Yes (3) Yes (4) No (5) No

(B) (Since all of the indicators correlate for the Sun, we'll keep its specification the same for all of the examples.) Imagine that the Earth is orbiting the Sun, and is also tidally locked to the Sun--i.e., it always keeps the same face turned towards the Sun, so it is "not rotating" with respect to the Sun. Then the indicators will be as follows:

For the Earth: (1) Yes (2) Yes (3) Yes (4) Yes (5) Yes

 

[JohnNemo]

So I can't answer your question as you ask it, because there is more than one way to rephrase your question in terms of what "rotation" means.

Can I just check that I haven't misunderstood something even more fundamental?

If A is orbiting B, I am thinking of that as in itself rotation (irrespective of any other rotation there may or may not be of any object around its own internal axis). Have I got this wrong?

 

[Nugatory]

If A is orbiting B, I am thinking of that as in itself rotation (irrespective of any other rotation there may or may not be of any object around its own internal axis). Have I got this wrong?

It depends on what you mean by "rotating". One object in orbit around another because of gravitational forces (earth orbiting the sun for example) is a different situation than a rock whirling around on the end of a string.

 

[JohnNemo]

It depends. One object in orbit around another because of gravitational forces (earth orbiting the sun for example) is a different situation than a rock whirling around on the end of a string.

I'm thinking of an object in orbit due to gravity.

 

[Nugatory]

I'm thinking of an object in orbit due to gravity.

In that case, none of the indicators of rotation mentioned in this thread will be present. Note especially that it will not be rotating relative to the distant stars.

 

[PeterDonis]

If A is orbiting B, I am thinking of that as in itself rotation

As has already been pointed out, that depends on how you define "rotation". In general, if A is orbiting B, the precession indicator, at the very least, will be there. I think the Sagnac effect indicator will be there as well. That was the basis for my response to example (A) in post #126.

In that case, none of the indicators of rotation mentioned in this thread will be present.

I don't think that's quite correct. See above.

 

[JohnNemo]

In that case, none of the indicators of rotation mentioned in this thread will be present. Note especially that it will not be rotating relative to the distant stars.

I thought that the centre of the orbiting object would be on a geodesic but that the outer and inner parts would not be on a geodesic and that this would produce stress in the object just as there is stress in an object rotating about its own axis. Have I got this wrong?

 

[JohnNemo]

As has already been pointed out, that depends on how you define "rotation". In general, if A is orbiting B, the precession indicator, at the very least, will be there. I think the Sagnac effect indicator will be there as well. That was the basis for my response to example (A) in post #126.

In this example, why is the precession indicator not present for the Sun? I know the Sun has vastly greater mass, but will there not be at least a small precession effect?

 

[PeterDonis]

I thought that the centre of the orbiting object would be on a geodesic but that the outer and inner parts would not be on a geodesic and that this would produce stress in the object just as there is stress in an object rotating about its own axis. Have I got this wrong?

You're correct that even if the center of an object like the Earth is moving on a geodesic, other parts of it will not be. (This is true whether the object is orbiting another one or not.) However, the proper acceleration indicator of rotation we have been talking about here is not just "proper acceleration is present"; it's "proper acceleration is present in a particular pattern that indicates rotation". That pattern is not the same as the pattern of proper acceleration due to the object being held together by hydrostatic equilibrium between its self-gravity and pressure (like the Earth is).

For example, consider the Earth itself. The "acceleration due to gravity" on the Earth's surface--which means the proper acceleration required to be at rest on the surface--is not the same everywhere on the Earth. Of course the Earth is not spherical (though this itself is largely due to the Earth's rotation on its axis), but even if we take this into account, the proper acceleration at the surface is not exactly equal to what you would predict just based on the Earth's radius and its mass. There is an extra component due to "centrifugal force" (or whatever you want to call it, depending on which frame of reference you want to adopt). That extra component is the proper acceleration indicator of rotation.

 

[PeterDonis]

In this example, why is the precession indicator not present for the Sun?

Because in that example, the Sun is not rotating relative to the distant stars, and is not orbiting any other body.

 

[JohnNemo]

Because in that example, the Sun is not rotating relative to the distant stars, and is not orbiting any other body.

The difficulty I have in framing the question is that I want to ask about the Sun and Earth as they actually are but excluding any 'rotation about own axis' effects (because otherwise it is just too complicated). I don't want to postulate the Earth orbiting, and the Sun not orbiting, if that is not an accurate description of what is actually happening. I'm assuming that they are both orbiting as both masses affect the spacetime geometry.

 

[PeterDonis]

I don't want to postulate the Earth orbiting, and the Sun not orbiting, if that is not an accurate description of what is actually happening. I'm assuming that they are both orbiting as both masses affect the spacetime geometry.

Ah, I see. Yes, that does change the indicators; so let me try another rephrasing of your example:

(C) Relative to the distant stars, the Sun and Earth are each orbiting their common center of mass on geodesics. They are not undergoing any other motion, relative to the distant stars, than that implied by orbiting their common center of mass.

Then the indicators will be as follows (again, this is my best quick answer, I have not done the detailed math):

For the Sun: (1) Yes (2) Yes (3) Yes (4) No (5) No

For the Earth: (1) Yes (2) Yes (3) Yes (4) No (5) No

For (1) (note that I think I should have said Yes to this one for the Earth in the previous versions as well), the magnitudes will be (I think) very small for both the Sun and the Earth (since the period of rotation is one Earth year). For (2), the relative magnitudes will (I think) be much larger for the Earth than for the Sun (because the semi-major axis of the orbit is much larger for the Earth). For (3), I'm not sure about the relative magnitudes for the Sun and Earth.

(For reference, the indicators are: (1) pattern of proper acceleration; (2) precession; (3) Sagnac effect; (4) vorticity; (5) rotating relative to the distant stars.)

 

[Nugatory]

I thought that the centre of the orbiting object would be on a geodesic but that the outer and inner parts would not be on a geodesic and that this would produce stress in the object just as there is stress in an object rotating about its own axis.

You are right about that. My post should have carried the additional qualifier "As long as the size of the object is sufficiently small compared with the diameter of the orbit" so that these tidal stresses are negligible.

But do note the same tidal stresses would appear if the Earth were at rest and not rotating relative to the distant stars while the sun was circling the earth; so their existence is a rather unsatisfactory way of demonstrating that the Earth is in orbit around the sun.

 

[JohnNemo]

Ah, I see. Yes, that does change the indicators; so let me try another rephrasing of your example:

(C) Relative to the distant stars, the Sun and Earth are each orbiting their common center of mass on geodesics. They are not undergoing any other motion, relative to the distant stars, than that implied by orbiting their common center of mass.

Then the indicators will be as follows (again, this is my best quick answer, I have not done the detailed math):

For the Sun: (1) Yes (2) Yes (3) Yes (4) No (5) No

For the Earth: (1) Yes (2) Yes (3) Yes (4) No (5) No

For (1) (note that I think I should have said Yes to this one for the Earth in the previous versions as well), the magnitudes will be (I think) very small for both the Sun and the Earth (since the period of rotation is one Earth year). For (2), the relative magnitudes will (I think) be much larger for the Earth than for the Sun (because the semi-major axis of the orbit is much larger for the Earth). For (3), I'm not sure about the relative magnitudes for the Sun and Earth.

(For reference, the indicators are: (1) pattern of proper acceleration; (2) precession; (3) Sagnac effect; (4) vorticity; (5) rotating relative to the distant stars.)

In this example, which of the three “Yes” indicators enable us to identify the point they are orbiting? I am assuming at least (1) because the lines of the pattern of proper acceleration should, if extended, meet at the point of the orbit.

 

[PeterDonis]

In this example, which of the three “Yes” indicators enable us to identify the point they are orbiting?

None of them. To identify the point they are orbiting, you have to look at the actual worldlines; just looking at rotation indicators is not enough. In fact, the "point" itself is not a point in spacetime, it's a worldline.

I am assuming at least (1) because the lines of the pattern of proper acceleration should, if extended, meet at the point of the orbit.

You are assuming there is an absolute way of dividing up spacetime into space and time. There isn't. The "lines of the pattern of proper acceleration" you are talking about would be lines in space, and space is not an invariant.