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

[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.

 

[JohnNemo]

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. 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.

Thus far we have been talking about rotation with inverted commas and it has been said that it is an imprecise term. I am thinking that the difficulty consists in the fact that rotation involves acceleration (which is invariant) and velocity (which is not) and that although the five invariant indicators suggest rotation, the finer details of the rotation - how many revolutions per unit time, orbital point - are not invariant. Am I thinking along the right lines?

I have read somewhere that the common centre of mass about which the orbits we are currently considering occur, is inside the Sun:

1. Is that right?

2. Is that always right, irrespective of frame of reference, or might the centre about which the orbit takes place be outside the Sun in some frames?

 

[Dale]

I am thinking that the difficulty consists in the fact that rotation involves acceleration

The difficulty consists in that the English language is vague and the word “rotation” can refer to several different physical states.

 

[PeterDonis]

the fact that rotation involves acceleration

If you define "rotation" as "indicator 1 is present", yes. But the whole point is that there are multiple indicators of rotation and they don't always go together. So, as @Dale said, the ordinary language word "rotation" does not refer to one unique physical thing. That's why we don't do physics using ordinary language; we do it using math, where we can precisely specify what we are talking about.

although the five invariant indicators suggest rotation, the finer details of the rotation - how many revolutions per unit time, orbital point - are not invariant

It's generally correct that many commonly used parameters of rotation are not invariant, yes. AFAIK the barycenter of the system--the "center" about which all the objects are revolving--is invariant, though; it's marked out by a particular worldline in spacetime.

I have read somewhere that the common centre of mass about which the orbits we are currently considering occur, is inside the Sun

That's correct if we are just considering the Sun and the Earth in isolation. If we are considering the entire solar system, the barycenter is sometimes inside the Sun and sometimes outside, depending on how the planets are aligned.

Is that always right, irrespective of frame of reference

It depends on what you mean. The worldline that describes the barycenter of the solar system is invariant. However, its "spatial location" at a given "time" depends on your choice of coordinates (this should be obvious from the words I put in quotes).

 

[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.)

What would be the case where

(D) The same as C but the rotation of the Sun about its own axis is what it actually is?

 

[PeterDonis]

(D) The same as C but the rotation of the Sun about its own axis is what it actually is?

Then indicators (4) and (5) would be Yes for the Sun.

 

[Nugatory]

I have read somewhere that the common centre of mass about which the orbits we are currently considering occur, is inside the Sun:

Careful... The "common center of mass" and the point "about which the orbits we are currently considering occur" are not necessarily the same thing, and both are frame-dependent. But with that said, if we simplify the problem down to just the Earth and the sun, no perturbations from the other planets...

1. Is that right?

Yes, if we choose to use a frame in which the fixed stars are at rest. This result comes from ordinary plain-vanilla Newtonian physics, no relativistic thinking needed - google for "gravity two-body problem". The center of the orbit is also inside the sun (although it's a different point) if we choose to use a frame in which the sun is at rest. Both of these frames are unusually convenient for analyzing the motion of objects within the solar system, so are often used - but it's still an arbitrary choice of frame.

2. Is that always right, irrespective of frame of reference, or might the centre about which the orbit takes place be outside the Sun in some frames?

An obvious counterexample is the frame in which the Earth is at rest; the center of mass is inside the sun but the center of the sun's orbit is inside the earth.

For some examples of the complexity of defining the center of mass in an invariant way, you might try https://en.wikipedia.org/wiki/Center_of_mass_(relativistic)

 

[JohnNemo]

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

I have a query about (5). In an earlier post you said that there was nothing special about a frame of reference which was non-rotating relative to the distant stars.

I can see that in practice the large scale distribution of mass and energy in the universe is not about to suddenly change but that is not exactly the same as invariant, is it? So (5) seems to be in a different category from (1) to (4).

Did you include (5) simply because, rotation not being a precisely defined term, it is a useful thing to compare with when trying to visualise the results of the different examples?

 

[PeterDonis]

I can see that in practice the large scale distribution of mass and energy in the universe is not about to suddenly change but that is not exactly the same as invariant, is it?

"Invariant" means "independent of your choice of coordinates". It doesn't mean "never changing". Spacetime includes time, so "changes" in invariant quantities are perfectly possible; it just means those invariant quantities have different values at different points of spacetime. But those values won't depend on your choice of coordinates.

 

[JohnNemo]

Careful... The "common center of mass" and the point "about which the orbits we are currently considering occur" are not necessarily the same thing, and both are frame-dependent. But with that said, if we simplify the problem down to just the Earth and the sun, no perturbations from the other planets...
Yes, if we choose to use a frame in which the fixed stars are at rest. This result comes from ordinary plain-vanilla Newtonian physics, no relativistic thinking needed - google for "gravity two-body problem". The center of the orbit is also inside the sun (although it's a different point) if we choose to use a frame in which the sun is at rest. Both of these frames are unusually convenient for analyzing the motion of objects within the solar system, so are often used - but it's still an arbitrary choice of frame.
An obvious counterexample is the frame in which the Earth is at rest; the center of mass is inside the sun but the center of the sun's orbit is inside the earth.

For some examples of the complexity of defining the center of mass in an invariant way, you might try https://en.wikipedia.org/wiki/Center_of_mass_(relativistic)

So both ‘common centre of mass’ and the ‘point about which the orbits occur’ are frame dependant and not invariant.

But they co-incide if the frame is at rest and non-rotating relative to the distant stars. Can you help me to understand what it is about this particular frame which causes this result?

 

[Dale]

Can you help me to understand what it is about this particular frame which causes this result?

That particular frame is inertial