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

[Ibix]

So how would you measure that? In the case of the Earth, for example?

With an accelerometer. You weigh the same at sea level everywhere. Precise measurements will tell you that sea level is an oblate spheroid (give or take...). Or you could notice Coriolis forces if you were a meteorologist or artillery gunner.

 

[Nugatory]

So does that mean that there is such a thing as invariant proper rotation?

Yes, and that's one of the ways that general relativity is not Machian (for some definitions of Machian). However, you really want to be thinking in terms of proper acceleration instead; the phenomenon we're dealing with here is centripetal proper acceleration.

[Edit: PeterDonis's #74 below points out some of the difficulties you risk by casually tossing around the word "rotation".]

 

[JohnNemo]

Yes, and that's one of the ways that general relativity is not Machian (for some definitions of Machian). However, you really want to be thinking in terms of proper acceleration instead; the phenomenon we're dealing with here is centripetal proper acceleration.

I'm a bit confused as to how this relates to the equivalence principle - the principle that there are no privileged reference frames. If a rotating object can have invariant centripetal proper acceleration, isn't a reference frame from which it appears to have the same centripetal acceleration sort of privileged?

 

[PeterDonis]

A Foucault pendulum, for example.

With an accelerometer.

Or you could notice Coriolis forces

It's worth noting that these are different notions of "rotation", which do not always correspond. Also, none of them are exactly the same in general as the notion of "rotation relative to the distant stars" (although they do all match up for the "universe containing nothing but a bucket of water" scenario being discussed in this thread), which is the primary notion of "rotation" being discussed in this thread. This is probably opening a can of worms, but I will try to describe briefly the differences.

The first notion is basically measuring the vorticity of a congruence of worldlines (the worldliness of the pieces of the Earth that form the circle around which the endpoints of the pendulum's swing move). This notion differs from "rotation relative to the distant stars" in at least three ways: Thomas precession, de Sitter precession, and Lense-Thirring precession. Depending on exactly where you put the pendulum, you can potentially eliminate one or more of these (for example, a pendulum at the Earth's North or South poles will have zero Thomas and de Sitter precession, but nonzero Lense-Thirring precession). In flat spacetime (for example, the universe containing nothing but a bucket of water), Thomas precession is present, but the other two are not, and in the case of the bucket Thomas precession would be zero, so for that specific case, this notion of rotation matches that of "rotation with respect to the distant stars" (where here "distant stars" means "flat spacetime at infinity").

The second notion is measuring proper acceleration, which might or might not indicate "rotation", depending on the circumstances. If you work through the details to see how you distinguish proper acceleration due to rotation from proper acceleration due to linear acceleration, you will find that proper acceleration due to rotation does not exactly match up with rotation relative to the distant stars in a general curved spacetime. It does, however, in the flat spacetime of the bucket example.

The third notion, as stated, is a coordinate effect, but there is a way of restating it in terms of which trajectories are geodesics and which feel proper acceleration, and how much and in which direction. It then becomes more or less equivalent to the second notion above.

 

[JohnNemo]

With an accelerometer. You weigh the same at sea level everywhere. Precise measurements will tell you that sea level is an oblate spheroid (give or take...). Or you could notice Coriolis forces if you were a meteorologist or artillery gunner.

I am a bit confused about how this relates to the equivalence principle - the principle that there are no privileged reference frames. If an object has invariant proper rotation, isn't a reference frame from which is appears to have the same rotation sort of privileged?

 

[Nugatory]

I'm a bit confused as to how this relates to the equivalence principle - the principle that there are no privileged reference frames. If a rotating object can have invariant centripetal proper acceleration, isn't a reference frame from which it appears to have the same centripetal acceleration sort of privileged?

You have two misunderstandings.
First, the Equivalence Principle (as most people use the term) doesn't say what you're saying. It says that being at rest in a gravitational field is locally (that is, within a region in which tidal effects are negligible) equivalent to uniform proper acceleration.

Second, the centripetal proper acceleration is the same in all frames, so you can't use it to privilege anyone frame. It is the reading on a particular physical device (for example, the accelerometer sitting on the table in front of me) and all observers regardless of their state of motion and the coordinates they choose to label events must agree about the number to which the needle on the dial of that device is pointing.

 

[JohnNemo]

You have two misunderstandings.
First, the Equivalence Principle (as most people use the term) doesn't say what you're saying. It says that being at rest in a gravitational field is locally (that is, within a region in which tidal effects are negligible) equivalent to uniform proper acceleration.

My fault for not saying which EP I was referring to - I meant the EP referenced in #27

Second, the centripetal proper acceleration is the same in all frames, so you can't use it to privilege anyone frame. It is the reading on a particular physical device (for example, the accelerometer sitting on the table in front of me) and all observers regardless of their state of motion and the coordinates they choose to label events must agree about the number to which the needle on the dial of that device is pointing.

I was thinking that if I measure your apparent acceleration from my frame and it agrees with your acceleromater, doesn't that make my frame sort of privileged?

 

[PAllen]

My fault for not saying which EP I was referring to - I meant the EP referenced in #27
I was thinking that if I measure your apparent acceleration from my frame and it agrees with your acceleromater, doesn't that make my frame sort of privileged?

What is described in #27 is not the equivalence principle, but instead what is called general covariance or coordinate invariance.

As to the second, would you really claim that a free fall frame near the Earth surface is privileged compared to frame at rest on the Earth because the former has a body on the Earth having coordinate acceleration matching its accelerometer reading, while the latter does not?

 

[Nugatory]

I was thinking that if I measure your apparent acceleration from my frame and it agrees with your acceleromater, doesn't that make my frame sort of privileged?

The universe is full of other accelerometers that won't agree with your frame-dependent measurement, but will agree with some frame-dependent measurement made using some other frame. So this sort-of-privileged isn't worth much: "Somewhere there might be an accelerometer that happened to read the same as the apparent acceleration I just calculated using this frame" is true for all frames, so all frames have this privilege.

 

[JohnNemo]

The universe is full of other accelerometers that won't agree with your frame-dependent measurement, but will agree with some frame-dependent measurement made using some other frame. So this sort-of-privileged isn't worth much: "Somewhere there might be an accelerometer that happened to read the same as the apparent acceleration I just calculated using this frame" is true for all frames, so all frames have this privilege.

Yes. I can see that when you put it that way.

The idea of invariant proper acceleration is new to me so can I ask a few questions about this.

If you had a laboratory at the North Pole on a rotatable base such that it did not rotate relative to distant stars (even though the Earth does), what would its proper acceleration be?

 

[Nugatory]

If you had a laboratory at the North Pole on a rotatable base such that it did not rotate relative to distant stars (even though the Earth does), what would its proper acceleration be?

One g, pointing straight up (but be aware that that's a somewhat sloppy way of describing it - it would be a good exercise to find a precise and coordinate-independent way of saying "straight up"). In this case, there is no centripetal component to the proper acceleration, as long as we can consider the lab to be arbitrarily small compared with the Earth so tidal effects can be ignored.

Seeing as how the rotating platform is only turning once every 24 hours, even if the lab were fixed to the surface of the rotating Earth the centripetal proper acceleration at the edge of the lab would be very small.

 

[JohnNemo]

One g, pointing straight up (but be aware that that's a somewhat sloppy way of describing it - it would be a good exercise to find a precise and coordinate-independent way of saying "straight up"). In this case, there is no centripetal component to the proper acceleration, as long as we can consider the lab to be arbitrarily small compared with the Earth so tidal effects can be ignored.

Seeing as how the rotating platform is only turning once every 24 hours, even if the lab were fixed to the surface of the rotating Earth the centripetal proper acceleration at the edge of the lab would be very small.

But what about the motion of the Earth round the Sun etc?

 

[PAllen]

Yes. I can see that when you put it that way.

The idea of invariant proper acceleration is new to me so can I ask a few questions about this.

If you had a laboratory at the North Pole on a rotatable base such that it did not rotate relative to distant stars (even though the Earth does), what would its proper acceleration be?

I’m wondering why this concept is new to you. It was part of Newtonian mechanics including Galilean relativity 5 or more centuries ago. Neither SR nor GR changed it. Could it be that what is new to you is that GR did not change it?

Anyway, in our universe, and any cosmology described by homogeneity and isotropy of matter and energy, your proposed polar laboratory would have no proper acceleration. It would be a local inertial frame per GR.

 

[PAllen]

But what about the motion of the Earth round the Sun etc?

The Earth follows an inertial path, and to a very high approximation, the polar frame which would see the distant stars as non rotating would be inertial. However, if you want to speak to arbitrary precision, the non rotating frame would actually be one that sees very slow movement of distant stars due to frame dragging.

 

[Nugatory]

But what about the motion of the Earth round the Sun etc?

I was ignoring them because they are so small - there's more understanding to be gained by idealizing the situation to a rotating Earth surrounded by distant fixed stars than by including all the tugs and pulls from the rest of the solar system - these just obscure the fundamentally simple physics with unnecessary complications.

 

[JohnNemo]

As far as GR is concerned, just considering it as a physical theory, the equivalence has always been there and has never been seriously doubted.

However, it's worth noting that there is a long-standing debate over how "Machian" GR is, which often involves examples like the one we are discussing. Some people might misinterpret this as a debate about whether the equivalence really is generally accepted. It's not a debate about that. It's more of a philosophical debate about what different people think a theory "should" look like, and whether GR looks like that, and if not, what a more comprehensive theory that includes GR as a special case within its domain of applicability might look like.

I have been reading a paper here http://www.pitt.edu/~jdnorton/papers/decades.pdf which examines Einstein's development of GR historically, how he originally hoped it would be Machian and how over time he seemed to accept that it was not, and how others viewed his theory at various stages of development.

I am particularly interested in section 5 entitled "Is general covariance physically vacuous?" (page 817) in which the author describes objections from Kretschmann with which the author seems to agree (In the title "is general covariance physically vacuous?" the word "physically" is used literally and the word "vacuous" is used metaphorically).

The argument (as I understand it) is that the fact that you can choose any frame, including an accelerating rotation frame, as your reference frame and all the laws of physics still work, means no more than that you have some good mathematical tools - it does not make any useful explanatory statement about physical reality.

This would be contrasted (in my understanding - my example) with, say, the Lorentz transformation, which is a mathematical transformation but goes hand in hand with certain assertions about physical reality - that there is no such thing as absolute velocity, that the speed of light is invariant, etc.

To what extent would you agree with this analysis?

 

[PeterDonis]

The argument (as I understand it) is that the fact that you can choose any frame, including an accelerating rotation frame, as your reference frame and all the laws of physics still work, means no more than that you have some good mathematical tools - it does not make any useful explanatory statement about physical reality.

This is how I understand Kretschmann's argument, yes. Basically it says that you can give any theory a tensor formulation, so saying "a valid theory must have a tensor formulation" doesn't place any restrictions on theories, and so doesn't tell you anything useful about the reality that theories are supposed to represent.

This would be contrasted (in my understanding - my example) with, say, the Lorentz transformation, which is a mathematical transformation but goes hand in hand with certain assertions about physical reality - that there is no such thing as absolute velocity, that the speed of light is invariant, etc.

Sort of. The problem with this as you state it is that the Lorentz transformation only works for a certain restricted class of physical situations and coordinate choices. It doesn't work if gravity is present, and it doesn't work if you choose non-inertial coordinates.

Lorentz invariance, however, is indeed an assertion about physical reality. (More precisely, local Lorentz invariance, since this remains valid in the presence of gravity.) But Lorentz invariance doesn't depend on how you formulate your theory or how you express your laws of physics or what coordinates you choose. It can be tested for directly in experiments. For a review of these sorts of tests, see this paper:

https://arxiv.org/abs/gr-qc/0502097

 

[JohnNemo]

This is how I understand Kretschmann's argument, yes. Basically it says that you can give any theory a tensor formulation, so saying "a valid theory must have a tensor formulation" doesn't place any restrictions on theories, and so doesn't tell you anything useful about the reality that theories are supposed to represent.

I am coming to the conclusion that I have rather misunderstood the scope of GR. I had thought that under GR all acceleration was purely relative but, as I understand it, although it was Einstein's initial hope that he could develop such a theory - and that is why it is named General Relativity - the theory he was able to develop was more limited and, as I understand it, is purely about how spacetime is curved by the presence of (and to a degree by the movement of) objects with mass (and that this accounts for what we think of as gravity which, in GR, is not actually a force).

Have I got this basically right now? I'm not seeking to oversimplify the finer details of GR but just trying to mentally see where it fits.

 

[PeterDonis]

Have I got this basically right now?

Not really, no. You are missing a key distinction between two types of acceleration: coordinate acceleration and proper acceleration.

Proper acceleration is what you feel as weight or measure with an accelerometer. It is not relative, and nobody, including Einstein, ever thought it was. How much weight a given observer feels, or the reading on a particular accelerometer, is invariant--all observers will agree that a particular observer feels a particular weight or a given accelerometer reads a particular value. This has to be true because these things are direct observables.

Coordinate acceleration is the second derivative of your spatial coordinates with respect to coordinate time. This description makes it obvious that it depends on your choice of coordinates. When Einstein talked about making a theory in which acceleration would be relative, he was talking about coordinate acceleration; and he succeeded, because in GR, coordinate acceleration is indeed relative--you can always find coordinates in which it vanishes for a given object.

What Einstein might have missed, at least in his early attempts to formulate GR, is that coordinate acceleration is also relative in SR. But to see this, you have to realize that you can use non-inertial coordinates in SR. Early treatments of SR did not recognize this, or at least did not explicitly acknowledge it, and formulated SR in terms of inertial frames, giving them a privileged status. But more modern treatments of SR recognize that the key property that distinguishes SR from GR is not inertial frames but spacetime being flat instead of curved. You can use non-inertial coordinates in flat spacetime, and by doing so, you can always find coordinates in which the coordinate acceleration of a particular object vanishes, showing that coordinate acceleration is indeed relative in SR as well as GR.

 

[JohnNemo]

Not really, no. You are missing a key distinction between two types of acceleration: coordinate acceleration and proper acceleration.

Proper acceleration is what you feel as weight or measure with an accelerometer. It is not relative, and nobody, including Einstein, ever thought it was. How much weight a given observer feels, or the reading on a particular accelerometer, is invariant--all observers will agree that a particular observer feels a particular weight or a given accelerometer reads a particular value. This has to be true because these things are direct observables.

I see that but I suppose I’m not really counting that as a ‘big’ part of GR because that is what everyone thought before Einstein - so no change.

Coordinate acceleration is the second derivative of your spatial coordinates with respect to coordinate time. This description makes it obvious that it depends on your choice of coordinates. When Einstein talked about making a theory in which acceleration would be relative, he was talking about coordinate acceleration; and he succeeded, because in GR, coordinate acceleration is indeed relative--you can always find coordinates in which it vanishes for a given object.

OK but isn’t Kretschmann right here that it is very useful mathematics but ‘physically vacuous’?

 

[PeterDonis]

I’m not really counting that as a ‘big’ part of GR because that is what everyone thought before Einstein

Not quite. I think it's true that it was generally understood that proper acceleration was not relative. However, I don't think it was generally understood that proper acceleration and coordinate acceleration are distinct concepts.

isn’t Kretschmann right here that it is very useful mathematics but ‘physically vacuous’?

It's true that saying you can always choose coordinates in which the coordinate acceleration of a given object vanishes does not distinguish different physical theories. You can do it with Newtonian physics as well as relativity. But that also was not fully understood at the time Einstein discovered GR. For example, the general understanding of Newtonian physics was that in a non-inertial frame, the laws of physics do not look the same: you have extra things like "centrifugal force" and "Coriolis force" that don't appear in inertial frames. It wasn't until Cartan developed his tensor-based formulation of Newtonian physics, which was, IIRC, in the 1920s, that it was fully realized that you could write the Newtonian laws so that they would look the same in any coordinates, the way Einstein showed you could do with relativity.

The way I would describe the modern understanding is similar to Einstein's response to Kretschmann's objection: while it's true that you can write any theory's laws in tensor form, so they look the same in any coordinates, some sets of laws look simpler in this form than others. For example, GR looks simpler than Newtonian gravity in this form. So there is still heuristic value in writing laws in this form; even if, strictly speaking, you can't rule out any theories this way, you can still compare different theories and see which ones look simpler.

 

[JohnNemo]

It's worth noting that these are different notions of "rotation", which do not always correspond. Also, none of them are exactly the same in general as the notion of "rotation relative to the distant stars" (although they do all match up for the "universe containing nothing but a bucket of water" scenario being discussed in this thread), which is the primary notion of "rotation" being discussed in this thread. This is probably opening a can of worms, but I will try to describe briefly the differences.

Is it the case that the Earth rotates once approximately every 24 hours (the different notions agreeing to a close approximation) and that this is an objective absolute property and not just a rotation relative to the distant stars?

 

[Nugatory]

Is it the case that the Earth rotates once approximately every 24 hours (the different notions agreeing to a close approximation) and that this is an objective absolute property

It sounds as if you are trying to use "objective" as a replacement for "frame-independent". Don't do that - one of those terms has a precise meaning that can be used to clearly describe the physics, and the other does not.

But if I'm understanding your question properly, then for a suitable definition of "rotates" the answer is yes. "It is really rotating" are the sloppy natural-language words that we use to describe something when the proper accelerations of its various parts are related in a particular way; because these proper accelerations are frame-invariant that sloppy description will work in all frames.

...and not just a rotation relative to the distant stars?

That question is well and thoroughly meaningless, because the distant stars are present in our universe. We could try rewording it some:

If we had a ball of rock floating in any otherwise empty universe with two jet engines on opposite sides and pointing in opposite directions, if we fire the engines for a while and then turn them off... Will points on the surface experience a different centripetal acceleration after the engines are turned off than before they were turned on?​

General relativity predicts that they will, and based on these proper accelerations we would say "it is rotating". You are free to leverage that into an assertion that the rotation of the Earth in our universe is "not just" a rotation relative to the distant stars... but if it's "not just" that in our universe, then what is it?

 

[JohnNemo]

It sounds as if you are trying to use "objective" as a replacement for "frame-independent". Don't do that - one of those terms has a precise meaning that can be used to clearly describe the physics, and the other does not.

But if I'm understanding your question properly, then for a suitable definition of "rotates" the answer is yes. "It is really rotating" are the sloppy natural-language words that we use to describe something when the proper accelerations of its various parts are related in a particular way; because these proper accelerations are frame-invariant that sloppy description will work in all frames.

Sorry for the sloppy language.

I understand that proper acceleration is frame independent.

Since rotation involves centripetal acceleration, I'm assuming that rotation is frame independent also, but you seem to be saying that it may not be that simple.

I can see that if there is nothing else in the universe then talking about rotation might be questionable, but if there is at least something else present in the universe then is rotation frame independent or is it still not that simple?

 

[PeterDonis]

Is it the case that the Earth rotates once approximately every 24 hours (the different notions agreeing to a close approximation) and that this is an objective absolute property and not just a rotation relative to the distant stars?

"Objective absolute property" is vague ordinary language, not physics.

As @Nugatory said, when we say that the Earth is "rotating", if we want to translate that into actual physics, we have to talk about invariants--things like the proper accelerations of all the different parts of the Earth and how they are related. The question then becomes, what determines the values of those invariants? And the answer GR gives is, the geometry of spacetime. The geometry of spacetime tells you what the proper acceleration will be for any worldline you like. And what determines the geometry of spacetime? The answer GR gives is, the distribution of stress-energy in the universe.

So we basically have two cases to compare: (1) the distribution of stress-energy in our actual universe; (2) the distribution of stress-energy in a hypothetical universe that just contains the Earth and nothing else. GR can give us reasonably well-defined spacetime geometries for both of these distributions of stress-energy. And it just so happens that, as far as the vicinity of the Earth is concerned, the two geometries are basically the same. That is because, in case #1, the stress-energy in the rest of the universe is spherically symmetric about the Earth, outside a certain distance from the Earth, to a good approximation, and there is a theorem that says that a spherically symmetric distribution of stress-energy outside a certain distance has no effect on the spacetime geometry within that distance--more precisely, that it makes the spacetime geometry within that distance flat, just as if there were no stress-energy at all in the rest of the universe.

So from the standpoint of the spacetime geometry near the Earth, the effect of the rest of the universe is basically the same as if there were nothing else in the universe. This is why it's easy to mistakenly think that the rest of the universe is irrelevant to the Earth's rotation. It isn't, but the real issue isn't that the rest of the universe makes the spacetime geometry near the Earth be non-flat; it's that if there were no stress-energy anywhere else in the universe, the flat geometry would ultimately have to come from a boundary condition at infinity, which would have to be put into the model "by hand" instead of coming from some law of physics. Whereas in our actual universe, there is no "infinity"--no boundary condition is required; everything comes from the actual distribution of stress-energy and the laws of GR, nothing extra has to be put in "by hand".

 

[JohnNemo]

OK, returning to the question of does the Sun go round the Earth or does the Earth go round the Sun, obviously if we take the Sun as our reference frame, the Earth is in orbit, whereas if we take the Earth as our reference frame the Sun is in orbit. However if we take a reference frame which is non-rotating relative to the distant stars, it looks very much like the Earth is orbiting the Sun and not the other way around. But is there something special about a reference frame which is non-rotating relative to the distant stars?

In The Evolution of Physics (1938) - available at https://archive.org/details/evolutionofphysi033254mbp - Einstein wrote

"Can we formulate physical laws so that they are valid for all CS, not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!" (page 224)

which seems to indicate that he thought that there was nothing special at all about a reference frame which was non-rotating relative to the distant stars ("The struggle... between the views of Ptolemy and Copernicus would then be quite meaningless") but, when he wrote this, was he hoping that GR would be more Machian than it turned out to be?

 

[Nugatory]

But is there something special about a reference frame which is non-rotating relative to the distant stars?
...
which seems to indicate that he thought that there was nothing special at all about a reference frame which was non-rotating relative to the distant stars ("The struggle... between the views of Ptolemy and Copernicus would then be quite meaningless") but, when he wrote this, was he hoping that GR would be more Machian than it turned out to be?

You do realize that you are using the word "rotation" in this post with a completely different meaning than the proper-acceleration-based invariant meaning used in some of the previous posts? But because you've switched back to coordinate rotation the answer to the first question is "no" and the answer to the second question is "Probably not, because Machian principles aren't involved in assigning coordinates to events"

 

[pervect]

OK, returning to the question of does the Sun go round the Earth or does the Earth go round the Sun, obviously if we take the Sun as our reference frame, the Earth is in orbit, whereas if we take the Earth as our reference frame the Sun is in orbit. However if we take a reference frame which is non-rotating relative to the distant stars, it looks very much like the Earth is orbiting the Sun and not the other way around. But is there something special about a reference frame which is non-rotating relative to the distant stars?

In The Evolution of Physics (1938) - available at https://archive.org/details/evolutionofphysi033254mbp - Einstein wrote

"Can we formulate physical laws so that they are valid for all CS, not only those moving uniformly, but also those moving quite arbitrarily, relative to each other? If this can be done, our difficulties will be over. We shall then be able to apply the laws of nature to any CS. The struggle, so violent in the early days of science, between the views of Ptolemy and Copernicus would then be quite meaningless. Either CS could be used with equal justification. The two sentences, 'the sun is at rest and the Earth moves', or 'the sun moves and the Earth is at rest', would simply mean two different conventions concerning two different CS. Could we build a real relativistic physics valid in all CS; a physics in which there would be no place for absolute, but only for relative, motion? This is indeed possible!" (page 224)

which seems to indicate that he thought that there was nothing special at all about a reference frame which was non-rotating relative to the distant stars ("The struggle... between the views of Ptolemy and Copernicus would then be quite meaningless") but, when he wrote this, was he hoping that GR would be more Machian than it turned out to be?

I'll give the super short answer that I would up writing at the conclusion to this rather long post first, in the hopes it will avoid the too long, didn't read issue. Then comes the bulk of the post, which seems to have grown quite a bit over my original intent.

The short version: When we talk about objects never moving faster than "c", we are not using tensor language. When we talk about objects having time-like worldlines, we are using tensor language. The intent is basically the same, only the semantics are different. However, the tensor language statements won't necessarily be recongizable to people who are not familiar with tensors.

The longer disucssion, in a good-news, bad-news format.

Good news: Using tensor methods, we can indeed express the laws of physics in a coordinate system (CS) where the sun orbits the Earth - but there are some potential misunderstandings and limitatoins here, see below.

Bad news #1. One doesn't generally learn tensor methods until one is in graduate school. The methods one learns in high school physics will NOT allow one to think of the Sun as orbiting the Earth. Assuming that the tensor methods needed work in the ways that one is (presumably) familiar with from high school leads to misunderstandings.

Bad news #2. The coordinate systems in which the Sun orbit the Earth do not necessarily cover all of space-time. There are limits on the size of accelerating frames, for instance. There is a bit more below.

Bad news #3. The relationship between the coordinates and physically measurable quantities becomes considerably less straightforwards in general coordinates.

It's helpful to consider a specific example, which we will take to be a rotating frame of reference (such as the rotating Earth) using tensor methods. For convenience we will omit gravity, and just talk about a rotating frame of reference in a space-time without gravity. This relates to the title question of this thread as well, though the omission of gravity makes it not quite the same.

At some distance, the object at rest in these coordinates has what we call a null worldline. Having a null-worldline is a coordinate independent tensor-language statement that's roughly equivalent to the coordinate dependent statement "moving at the speed of light".

In tensor language, we would talk about the Born coordinate chart, <<link>>, and we'd concisely specify the coordinates by giving the metric tensor in the form of its line element:

$$ds^2 = -\left( 1- \frac{\omega^2 r^2}{c^2} \right) \mathrm dt^2 + 2 \omega r^2 \mathrm dt \, \mathrm d\phi + \mathrm dz^2 + \mathrm dr^2 + r^2 \mathrm d\phi^2$$

Physicisits familar with the methods generally regard the specification of such a metric as a complete description of a coordinate system, because they know how to calculate anything they need to calculate about the physics just from being given this mathematical expression. The issues with the size of the coordinate system, by the way, shows up in the above line element because $\left( 1- \omega^2 r^2 / c^2 \right)$ vanishes when $\omega r$ = c, making the tensor singular at this point. This is called a coordinate singularity. So we can see some differences between the rotating coordinates and the non-rotating coordinates, the former has a coordinate singularity, and the later doesn't.

Basically, the physics doesn't change, just the language changes, and people who haven't learned the tensor methods generally don't understand the tensor language. So we use language that is hopefully familiar to them instead.

If we take a physical experiment like that documented in "The Ultimate Speed", <<link>> we don't get any different results. Electrons (in this particular experiment) still have a limiting speed slower than c no matter how much energy we give them. We just use slightly different language to describe the results.

Rather than talking about the steps needed to make velocities (not tensors) into four-velocities (true tensors), I'll take a different approach to coordinate independence for this experiment. Regardless of coordinates, if we compare a light pulse and a pulse of relativistic electrons, the light pulse will move faster, than the electrons. For instance, if we send both pulses out at the same time (and make sure the electron beam is not deflected by any stray fields), the light pulse will arrive at the agreed-on destination first, the electrons will arrive later. This is true regardless of how much energy we give the electrons. We could perform a similar experiment on a rotating platform if we really wanted to. We might notice the electron beam taking a different path than the light beam in this case unless we could raise the energy of the electron beam high enough to make the differences experiemntally unmeasurable. But we'd never see the electron beam beating the light beam to the destination. The best we could do is make the difference experimentally so tiny that we can't measure it reliably.

I haven't really covered the issue of the physical significance of the coordinates, but it's somewhat important, so I'll try to give it a brief exposition. Basically, the "t" coordinate in the above rotating coordinate system doesn't have any direct relationship to what clocks read. In particular, as clocks approach the critical radius $\omega r -> c$, the clocks slow down more and more in terms of the time coordinate t. In the limit, the clocks stop. This isn't the fault of the clocks actually stopping, it's just due to our choice of coordinates. We can figure this out by noting that the clocks don't stop in the inertial coordinates, while they do stop in the rotating coordinates. . Basically, the rotating coordinates are poorly behaved, they have coordinate singularities. The mathematical issue of $g_{00}$ disappearing is the same issue as the clocks stopping in my less formal exposition.

Using tensors, there isn't any problem with using generalized coordinates as long as they are well behaved. Guaranteeing that coordinates are well behaved and interpreting the physical significance of the coordinates is not as trivial as one might assume without some experience and practice actually using generalized coordinates.

 

[PeterDonis]

is there something special about a reference frame which is non-rotating relative to the distant stars?

No. But the distribution of matter and energy can make things look simpler in a particular reference frame. You are confusing yourself by focusing on frames instead of on physics.

Consider a spherical region of space centered on the solar system, with a radius large enough to contain all the planets. As I said in a previous post, the average distribution of matter and energy in the universe is, to a good approximation, spherically symmetric about this region; that means that, if there were no matter and energy at all inside the region, spacetime in the region would be flat.

But there is matter and energy inside the region: the Sun and the planets. (There is other stuff too--satellites, asteroids, comets, etc.--but we can ignore it here.) So spacetime in the region is not actually flat; but because of the theorem I mentioned in a previous post, when figuring out the spacetime geometry within the region, we only need to consider the matter and energy inside the region.

And more than 99 percent of that matter and energy is contained in the Sun; that means that the spacetime geometry within the region of the solar system is, to a good approximation, the geometry of a single source of gravity, the Sun, in which all the other objects move on geodesics. That being the case, the simplest frame in which to describe motion in the region of the solar system is a frame in which the Sun is at rest.

But there are multiple possible frames in which the Sun is at rest--frames with different rates of rotation relative to the distant stars. Which one makes motion in the region of the solar system look simplest? You can probably guess the answer: the frame that is not rotating relative to the distant stars. (One way to see why this is the case is to imagine that the solar system was not there and the spherical region we have been talking about was empty; then spacetime in that region would be flat, and a frame not rotating relative to the distant stars would correspond to a standard inertial frame in Minkowski spacetime, which is the simplest frame in which to describe geodesic motion in flat spacetime.)

None of this means that a frame in which the Sun is at rest, and which is not rotating relative to the distant stars, is "special" in the sense of being picked out by the laws of physics. The laws of physics look the same in any frame. But the practical description of motion in the region of the solar system--what you get when you work out the particular solution of the laws of physics that describes the matter and energy and spacetime geometry in this region--looks simplest in the frame in which the Sun is at rest and which is not rotating relative to the distant stars, because of the particular distribution of matter and energy and the particular spacetime geometry it gives rise to.

 

[JohnNemo]

None of this means that a frame in which the Sun is at rest, and which is not rotating relative to the distant stars, is "special" in the sense of being picked out by the laws of physics. The laws of physics look the same in any frame. But the practical description of motion in the region of the solar system--what you get when you work out the particular solution of the laws of physics that describes the matter and energy and spacetime geometry in this region--looks simplest in the frame in which the Sun is at rest and which is not rotating relative to the distant stars, because of the particular distribution of matter and energy and the particular spacetime geometry it gives rise to.

I started off thinking that, in GR, acceleration was all relative but I now understand that there is such a thing as invariant proper acceleration, so, having realized that I had started under a misapprehension, I am trying to get clear in my mind what else might be invariant in GR, and I am now concentrating on rotation but struggling a bit...

I know that where you have a rotating object, every part of it is accelerating in the direction of the axis of rotation so, since there is such a thing as invariant proper acceleration I am thinking that there might be something invariant about rotation. OTOH I know that rotation is a bit special because, unlike linear acceleration, the direction of motion is perpendicular to the direction of acceleration.

So... struggling to formulate a question which is not too woolly... I suppose my question is...

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

 

[Nugatory]

Is there anything invariant about rotation?

It depends on what you mean when you say "rotation". Throughout this thread, you have used that one word with two different meanings. Sometimes we've been able to work out which one you're intending at the moment from the context, but other times it is quite ambiguous.

This is one of those times when it is quite ambiguous, so I'll try an answer for both meanings.
1) By "rotation" you might mean that the proper accelerations of the various parts of an object bear a particular relationship to one another. This property is invariant, because the proper accelerations are invariant (although there are some subtleties here that we don't need to go into now).
2) By "rotation" you might mean that in some coordinate system the spatial coordinates of some objects are constant while the spatial coordinates of other objects are changing in a particular way. This property is not invariant, as is to be expected of anything that depends on the choice of coordinate system.

 

[JohnNemo]

It depends on what you mean when you say "rotation". Throughout this thread, you have used that one word with two different meanings. Sometimes we've been able to work out which one you're intending at the moment from the context, but other times it is quite ambiguous.

This is one of those times when it is quite ambiguous, so I'll try an answer for both meanings.
1) By "rotation" you might mean that the proper accelerations of the various parts of an object bear a particular relationship to one another. This property is invariant, because the proper accelerations are invariant (although there are some subtleties here that we don't need to go into now).
2) By "rotation" you might mean that in some coordinate system the spatial coordinates of some objects are constant while the spatial coordinates of other objects are changing in a particular way. This property is not invariant, as is to be expected of anything that depends on the choice of coordinate system.

https://en.wikipedia.org/wiki/Solar_rotation tells me that the Sun rotates about once a month. Is Wikipedia talking about 1 or 2?

 

[PAllen]

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

https://en.m.wikipedia.org/wiki/Con...atical_decomposition_of_a_timelike_congruence

 

[Nugatory]

https://en.wikipedia.org/wiki/Solar_rotation tells me that the Sun rotates about once a month. Is Wikipedia talking about 1 or 2?

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.

 

[PAllen]

I would say the discussion (in the referenced Wikipedia article on solar rotation) on use and inferences from helioseismology would translate readily to a vorticity tensor model, and are thus invariant descriptions of rotation.