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

[JohnNemo]

Yes, GR is not machian in this sense.

What does GR predict in this situation?

 

[PAllen]

What does GR predict in this situation?

GR predicts you can have a rotating bucket in an empty universe that experiences centripetal force, contrary to one formulation of Mach's principle.

 

[JohnNemo]

GR predicts you can have a rotating bucket in an empty universe that experiences centripetal force, contrary to one formulation of Mach's principle.

I am interested in this example.

You mention a bucket. Is this a random choice of object?

If it is literally a bucket then it would consist of millions of particles and each particle would be in a state of acceleration relative to particles nearer the axis of rotation.

Does MP really predict no centripetal force in this situation? If so, why?

 

[Nugatory]

You mention a bucket. Is this a random choice of object?

It's the example used most often, and has been ever since Isaac Newton used it in what we would now call a "thought experiment". Googling for "Newton's bucket" will bring up some good references.

If it is literally a bucket then it would consist of millions of particles and each particle would be in a state of acceleration relative to particles nearer the axis of rotation.

Yes, that's the whole point of the exercise.

Does MP really predict no centripetal force in this situation? If so, why?

Mach's principle doesn't predict anything, because it's not a theory that makes predictions. It's more an intuition about how a well-founded theory of gravity "ought to" work. It turns out that the best theory of gravity we have, namely General Relativity, doesn't obviously work that way. However, we don't live in a universe that is completely empty except for a single bucket of water, so there is no way of determining whether GR's prediction for how such a bucket would behave is correct.

 

[JohnNemo]

It's the example used most often, and has been ever since Isaac Newton used it in what we would now call a "thought experiment". .. Mach's principle doesn't predict anything, because it's not a theory that makes predictions. It's more an intuition about how a well-founded theory of gravity "ought to" work. It turns out that the best theory of gravity we have, namely General Relativity, doesn't obviously work that way. However, we don't live in a universe that is completely empty except for a single bucket of water, so there is no way of determining whether GR's prediction for how such a bucket would behave is correct.

Thank you for clarifying what we are referring to. My initial problem was what does rotation mean if you just have a bucket alone in the universe, but if it is a bucket of water then at least we have the bucket and the water which could be rotating relative to each other.

So I understand that this is basically Newton's bucket experiment transplanted into a universe containing nothing else. I also understand that we are treating the bucket and the water as two separate indivisible objects.

In the real universe there are four stages in the experiment

1. Bucket and water are at rest relative to each other and relative to the distant stars - water surface flat

2. Bucket is rotating relative to the water. Water is at rest relative to the distant stars - water surface flat

3. Bucket and water at rest relative to each other and both rotating relative to distant stars - water surface concave

4. Bucket is rotating relative to water (in the opposite direction). Bucket is at rest relative to distant stars - water surface concave


Now if we imagine there are no distant stars we get

1. Bucket and water are at rest relative to each other

2. Bucket is rotating relative to the water.

3. Bucket and water at rest relative to each other.

4. Bucket is rotating relative to water (in the opposite direction)

Now I can understand intuitively that MP would conjecture that the water is flat in all four scenarios - there is no mass in the universe (except for the bucket which is trivial) so why would the water be anything other than flat.

So far I have no difficulty. What I struggle with is why would GR predict anything different? In particular I can't see how GR could predict different results for 1 and 3 as, absent anything else in the universe, 1 and 3 are the same.

 

[Nugatory]

First, a mandatory caution and disclaimer: Mach's Principle cannot be stated precisely, and there's more than way of understanding it. So don't be surprised or confused when you find other formulations...

What I struggle with is why would GR predict anything different? In particular I can't see how GR could predict different results for 1 and 3 as, absent anything else in the universe, 1 and 3 are the same.

We solve the Einstein Field Equations for an empty universe, and then the geodesic equation for the trivial metric that comes out of that solution. #1 and #3 are not the same. In one case the worldline of a droplet of water near the rim of the bucket is a geodesic and in the other it's not, so in one case the droplet experiences proper acceleration and an accelerometer will read non-zero; and in the other there is no proper acceleration.

This is actually just an unusually confusing example of something that we already accept with ordinary garden-variety special relativity: From inside a windowless and sealed room I cannot say anything meaningful about my speed relative to anything else, but an accelerometer will still detect acceleration and rotation.

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.

Machians try to explain the asymmetry by saying that it's the presence of all the other matter in the universe that accounts for the observability of acceleration - informally, "mass out there influences inertia here" or "local physical laws are determined by the large-scale structure of the universe" (both of these quoted via wikipedia). That's a reasonable enough position if you feel that the asymmetry needs more explaining than "that's the way our universe works", but it cannot be tested because we don't have any empty universe in which we can compare cases #1 and #3.

 

[JohnNemo]

This is actually just an unusually confusing example of something that we already accept with ordinary garden-variety special relativity: From inside a windowless and sealed room I cannot say anything meaningful about my speed relative to anything else, but an accelerometer will still detect acceleration and rotation

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

 

[Ibix]

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

Yes.

 

[JohnNemo]

Yes.

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

 

[PAllen]

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

A Foucault pendulum, for example.

 

[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’?