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

[Orodruin]

From the Earth's reference frame, what accounts for the stars orbiting the Earth? I'm assuming that the Earth's gravity has a negligible effect on distant stars. Is it some kind of frame dragging effect which makes the stars circle the Earth?

The Coriolis force. Note that the Coriolis force actually gives you twice the force that would be required, but half of that is countered by the centrifugal force.

 

[JohnNemo]

The Coriolis force. Note that the Coriolis force actually gives you twice the force that would be required, but half of that is countered by the centrifugal force.

Sorry, my question was badly worded. By using the word "accounts" it sounded like I was asking a question about how Newtonian mechanics accounts for it by using fictitious forces.

What I meant to ask was: "In GR what is the cause (explanation) of the circular rotation of the stars"

 

[PeroK]

From the Earth's reference frame, what accounts for the stars orbiting the Earth? I'm assuming that the Earth's gravity has a negligible effect on distant stars. Is it some kind of frame dragging effect which makes the stars circle the Earth?

In one sense nothing needs to account for it. If you bounce up and down on a trampoline you could ask what accounts for everyone else bouncing down and up in your frame?

Whatever it is it's not the trampolinist's gravity!

Note that there are other rotating planets, with different angular velocity from the Earth. So, what forces are the distant stars supposed to feel?

And someone living on a distant planet will have no way to measure the effect of the Earth's rotation on them. Because there is none.

This is where is trying to grasp the I or A level answer you lose touch with the B level answer.

 

[PeroK]

Sorry, my question was badly worded. By using the word "accounts" it sounded like I was asking a question about how Newtonian mechanics accounts for it by using fictitious forces.

What I meant to ask was: "In GR what is the cause (explanation) of the circular rotation of the stars"

In GR the equations of motion come from the differential description of spacetime. In one reference frame the line element has a certain form, which leads to the geodesic equations, which leads to a solution for a particle or light ray in those coordinates.

If we change coordinates, we change the form of the line element, hence the geodesic equations hence get a different solution in those coordinates.

But, unless the maths has gone wrong, the two coordinate solutions must describe the same physical solution.

There is no cause in one frame or the other except that particles are following the natural paths through spacetime. Only the coordinate description of that spacetime and the associated geodesics changes.

 

[PeterDonis]

What I meant to ask was: "In GR what is the cause (explanation) of the circular rotation of the stars"

There is none. Causes don't change when you change frames, and something that doesn't even exist in one frame can't have a cause in another frame.

What you should be asking is, what is the cause of the relative rotation of the Earth and the distant stars. And the answer to that is the simple mundane answer: the geometry of spacetime and the worldlines of the matter inside the Earth and of the distant stars within that geometry. A further answer would go into the history of how those various pieces of matter came to have those worldlines, and how the spacetime of the universe and in the particular regions in question came to have its geometry.

 

[Vanadium 50]

This seems to have taken a meandering trip to somewhere other than enlightenment.

Consider a coordinate system centered on New York. In this coordinate system, an object has a certain set of coordinates describing its position. Now consider a coordinate system centered on Los Angeles. In this coordinate system, the same object has a different set of coordinates describing its position. When switching coordinates, did the object move?

That's the same thing happening here.

 

[JohnNemo]

There is none. Causes don't change when you change frames, and something that doesn't even exist in one frame can't have a cause in another frame.

I think I was wrong to use the word "cause" in my question. I should have stuck to the word "explanation".

Explanations comes in all shapes and sizes and some are more helpful in some situations than others. And it does seem to me that some explanations can be more or less reference-frame specific in that their explanatory power may be greater in one frame and less (or nil) in another.

For example if I leave my arms loose and spin round very fast, my arms will stretch out horrizontally. A typical explanation of why this is would be to say that because I am rotating (relative to the Earth's reference frame) my arms are accelerating towards my body and that is why they stretch out.

Now if we consider my own reference frame, clearly this explanation has no explanatory power since in my own reference frame I am not rotating. So we might look for another explanation for what is happening in this reference frame. One explanation I have heard is that the distant stars spinning round me create a gravitational field which pulls my arms out. I would be interested to know what you think of this explanation but the main point for present purposes is that some explanations work better than others in particular reference frames. This does not mean that every single reference frame has its own unique explanation, and there may be some explanations which are useful over many, perhaps all, reference frames, but at least some types of explanation are more useful in some reference frames than others. So it seems to me that it is not wrong to ask if there is an explanation in the context of a particular reference frame. Of course the answer to any specific question of this nature may be "sorry- I can't think of one (other than a very general mathematical explanation applicable to all reference frames)". But it seems to me that it is not wrong to ask.

To develop this point a little more: it seems to me that the very idea of equivalence necessarily assumes that some explanations are more natural and useful in some reference frames rather than others. 100 years ago we discovered that gravitation and acceleration were equivalent, but we didn't ditch the word "acceleration" and just use the word "gravity" from then on. We continued to use the word "gravity" for those situations and reference frames for which it appeared to be the most useful word to use, and continued to use "acceleration" where that appeared more natural, always remembering, that they were equivalent in the sense of ultimately referring to the same underlying reality.

So my intention when asking the question was not to suggest that reality could be different for different frames of reference, but simply to ask if there was some useful explanation (beyond the very general) which could be given for the phenomenon of stars orbiting the Earth (in the Earth's frame of reference). If there isn't, fair enough, but I wanted to make clear what I was intending to ask.

 

[PeroK]

@JohnNemo post #34 didn't help?

 

[JohnNemo]

@JohnNemo post #34 didn't help?

Nothing wrong with what you said but a bit general. My fault for using the word "cause" which was ambiguous.

 

[PeroK]

Nothing wrong with what you said but a bit general. My fault for using the word "cause" which was ambiguous.

What level of physics do you understand? If I said light follows null geodesics, where $ds^2 = 0$, do you understand what that means?

 

[PeterDonis]

I think I was wrong to use the word "cause" in my question. I should have stuck to the word "explanation".

IMO this doesn't make a difference, because a valid explanation has to be made in terms of valid causal factors. More precisely, a valid explanation of an actual physical phenomenon, as opposed to an artifact of your choice of coordinates, needs to be made in terms of valid causal factors.

Consider @Vanadium 50 's example of an object whose coordinates change when you change from a New York-centered coordinate chart to a Los Angeles-centered coordinate chart. As far as physics is concerned, there is nothing to "explain" because nothing happened. If you ask why the object's coordinates changed, the answer won't involve any kind of physical cause; the answer will be "because you changed your coordinates". Whereas, if the object collides with another object and gets damaged, any explanation of why the object got damaged will have to be in terms of the collision and its parameters, which don't change when you change coordinates.

it does seem to me that some explanations can be more or less reference-frame specific in that their explanatory power may be greater in one frame and less (or nil) in another.

I don't agree with this either, for the reason just given: valid causal factors don't change when you change frames.

For example if I leave my arms loose and spin round very fast, my arms will stretch out horrizontally. A typical explanation of why this is would be to say that because I am rotating (relative to the Earth's reference frame) my arms are accelerating towards my body and that is why they stretch out.

I don't know if this kind of explanation is "typical", since you haven't given any references, but a correct explanation of why this is would be to say that there are two causal factors acting on your arms: their inertia, and the forces exerted on them by the rest of your body. What happens to your arms is determined by the combined effect of these two causal factors. These causal factors are the same no matter which frame you choose.

One explanation I have heard is that the distant stars spinning round me create a gravitational field which pulls my arms out.

And this explanation can work, if you realize that the term "gravitational field" (like the term "Coriolis force", which came up in an earlier post in this thread) is really a way of referring to the "inertia" causal factor--or, if you want to put it in more GR-like terminology, the "geometry of spacetime" causal factor. But notice that this causal factor alone does not explain what actually happens to your arms: if this causal factor were the only one operating, your arms would fly away and not stay attached to your body. So to explain what actually happens to your arms, you need both causal factors I mentioned: inertia/spacetime geometry, plus the internal forces between your arms and the rest of your body. Only the combination of the two explains what actually happens to your arms.

Part of the confusion here is that our ordinary language is not relativistic, in the sense that it gives different names to the same causal factor when viewed from different frames. In the frame in which the distant stars are at rest and the person is rotating, the "spacetime geometry" causal factor is usually called "inertia", whereas in the frame in which the person is at rest and the distant stars are rotating, the same causal factor is called "gravitational field" or "Coriolis force" (combined with "centrifugal force"). The different language makes it appear that it's a different causal factor, when in fact it's the same one.

it seems to me that the very idea of equivalence necessarily assumes that some explanations are more natural and useful in some reference frames rather than others

No, it's the opposite: it's the idea that those different words used in different frames are just different names for the same explanation--the same causal factor--not different explanations.

 

[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.

Can you elaborate on the "longstanding debate over how 'Machian' GR is"?

I thought that Mach's Principle was the very idea of equivalence which you have said is an uncontroversial part of GR, but I must be missing something and it can't be that simple if there is such a longstanding debate.

 

[PeterDonis]

Can you elaborate on the "longstanding debate over how 'Machian' GR is"?

The short answer is that it's a disagreement over whether and to what extent your very next sentence...

I thought that Mach's Principle was the very idea of equivalence which you have said is an uncontroversial part of GR

...is true.

 

[JohnNemo]

The short answer is that it's a disagreement over whether and to what extent your very next sentence...
...is true.

Is the area of debate concentrated upon rotating frames?

 

[PeterDonis]

Is the area of debate concentrated upon rotating frames?

That seems to be the scenario that is most frequently used as an example in the debate, probably because ordinary experience shows us that rotation works differently from linear motion. It's fairly easy to grasp that linear motion is relative because we don't feel any force due to linear motion. It's much harder to grasp that rotation is relative because we feel a force due to rotation. The obvious intuitive explanation of that force is that rotation is "absolute" in a way that linear motion is not.

 

[JohnNemo]

That seems to be the scenario that is most frequently used as an example in the debate, probably because ordinary experience shows us that rotation works differently from linear motion. It's fairly easy to grasp that linear motion is relative because we don't feel any force due to linear motion. It's much harder to grasp that rotation is relative because we feel a force due to rotation. The obvious intuitive explanation of that force is that rotation is "absolute" in a way that linear motion is not.

Can I ask a question which I thought I had asked before but which I now see I had asked before in more restrictive terms than I had intended:

Has the truth of the equivalence principle in GR been seriously doubted in, say, the last 50 years? By 'truth' I mean correspondence between what GR predicts and reality (so far as we are able to measure it).

 

[PeterDonis]

Has the truth of the equivalence principle in GR been seriously doubted in, say, the last 50 years?

No. The debate I referred to about how "Machian" GR is is not a debate about whether the EP is true. And btw, it's not just "true in GR", i.e., it's not just theoretical; it's been tested to very high accuracy. The Wikipedia page gives a decent overview of the experiments that have been done:

https://en.wikipedia.org/wiki/Equivalence_principle

 

[JohnNemo]

No. The debate I referred to about how "Machian" GR is is not a debate about whether the EP is true. And btw, it's not just "true in GR", i.e., it's not just theoretical; it's been tested to very high accuracy. The Wikipedia page gives a decent overview of the experiments that have been done:

https://en.wikipedia.org/wiki/Equivalence_principle

In the Middle Ages before Copernicus it was generally believed that the Earth was fixed immovable at the centre of the universe, and it was everything other than the Earth which was moving. Copernicus proposed that the Sun was fixed immovable at the centre of the universe, and it was everything other than the Sun which was moving. And Copernicus' view gained wide acceptance for the next couple of centuries until people worked out that if there was a centre of the universe it definitely was not either the Sun or the Moon.

In popular histories of science it is often said that Copernicus was 'right' and you sometimes get the impression that even some scientists today feel that Copernicus was 'more right' than his predecessors, but from what you say about equivalence it seems that there is no basis for thinking that Copernicus was 'more right' that his predecessors.

Formally both Copernicus and his predecessors were wrong because nothing is 'absolutely' at rest - the concept having to meaning. And even if we are kind to them and overlook that and take them to be referring to relative motion, then they would both the equally right. So neither 'more right' than the other.

 

[PeroK]

In the Middle Ages before Copernicus it was generally believed that the Earth was fixed immovable at the centre of the universe, and it was everything other than the Earth which was moving. Copernicus proposed that the Sun was fixed immovable at the centre of the universe, and it was everything other than the Sun which was moving. And Copernicus' view gained wide acceptance for the next couple of centuries until people worked out that if there was a centre of the universe it definitely was not either the Sun or the Moon.

In popular histories of science it is often said that Copernicus was 'right' and you sometimes get the impression that even some scientists today feel that Copernicus was 'more right' than his predecessors, but from what you say about equivalence it seems that there is no basis for thinking that Copernicus was 'more right' that his predecessors.

Formally both Copernicus and his predecessors were wrong because nothing is 'absolutely' at rest - the concept having to meaning. And even if we are kind to them and overlook that and take them to be referring to relative motion, then they would both the equally right. So neither 'more right' than the other.

Copernicus's ideas were a major advancement in scientific thinking. To conceive that the Earth might be moving round the Sun was a significant step.

Moreover, in finding a model for the relative motion within the solar system it was a critical breakthrough. The elliptic orbits could only be found by looking at the solar system as heliocentric.

Copernicus took the right step, as did Galileo and Newton. You can't judge the progress of science by a simple test or whether someone was completely right or not.

 

[Nugatory]

...So neither 'more right' than the other.

I should get a share of the revenues, I've posted this link so often... http://chem.tufts.edu/AnswersInScience/RelativityofWrong.htm

I mentioned in your other thread that the popular conception that modern physics has somehow proved classical mechanics wrong is itself wrong. That's true, and more generally most laypeople do not understand the extent to which science advances by extension and improvement rather than wholesale replacement.

 

[phinds]

I should get a share of the revenues, I've posted this link so often... http://chem.tufts.edu/AnswersInScience/RelativityofWrong.htm

You beat me to it (not that I've posted it as often as you, but it occurred to me as a response to this thread's post #48)

 

[PeterDonis]

In popular histories of science

We're not discussing popular histories of science, we're discussing science.

 

[JohnNemo]

Copernicus's ideas were a major advancement in scientific thinking. To conceive that the Earth might be moving round the Sun was a significant step.

Moreover, in finding a model for the relative motion within the solar system it was a critical breakthrough. The elliptic orbits could only be found by looking at the solar system as heliocentric.

Copernicus took the right step, as did Galileo and Newton. You can't judge the progress of science by a simple test or whether someone was completely right or not.

If we are to judge not by an objective standard but by what contribution a development (whether a correct development or incorrect development or a relatively correct develioment) made to the overall progress of scientific understanding, don't we need a counterfactual in order to make that evaluation?

If, purely by way of example, the status of fixed unmovable centre of the universe had not been transferred to the Sun, maybe the concept would have been dropped earlier rather than maintain an etherial existence until the early 20th Century slowing down the development of relativity.

Or maybe not... but my point is that we can't properly evaluate without first identifying a counterfactual.

 

[PeroK]

If we are to judge not by an objective standard but by what contribution a development (whether a correct development or incorrect development or a relatively correct develioment) made to the overall progress of scientific understanding, don't we need a counterfactual in order to make that evaluation?

If, purely by way of example, the status of fixed unmovable centre of the universe had not been transferred to the Sun, maybe the concept would have been dropped earlier rather than maintain an etherial existence until the early 20th Century slowing down the development of relativity.

Or maybe not... but my point is that we can't properly evaluate without first identifying a counterfactual.

There is plenty of historical evidence that the Roman Inquisition's suppression of Galileo's ideas led to the shift in scientific, industrial and economic power to Northern Europe.

That evidence is clear. Moreover, if Galileo had been presented with a copy of a 20th century physics text, I assume he would have been amazed and inspired. The Roman inquisition would have burned it.

Science progresses by trial and error, not by absolute dogma.

 

[PAllen]

Also, while GR allows a consistent description of the solar system in coordinates where the Earth is at rest at the origin, the computations in such coordinates are horrendously more complex than one that takes the sun as the origin. This is one objective basis for saying we prefer to consider the Earth orbiting the sun. Even more, even in Earth centered coordinates, we can say the sun follows a near geodesic centered in the approximate symmetry of the spacetime (which is coordinate independent), while the Earth follows a geodesic of higher potential in the stationary killing vector field. This can be translated to 'earth orbits sun' as a coordinate independent statement.

 

[JohnNemo]

... the extent to which science advances by extension and improvement rather than wholesale replacement.

I agree that (looked at, say, over the last 1000 years in the West) science has advanced and has done so incrementally, but I would add two caveats:

1. Not all increments are of the same size

2. Whilst looked at overall the progression is in the direction of less wrong (as Asimov attractively argues in the link you gave) within that overall picture there are many individual developments which may be more wrong or just as wrong, so the fact that some particular aspect of a theory is chronologically later does not automatically mean that it is more right.

@PeroK comments made me realize that my using Copernicus as a shorthand for the belief that the Sun was fixed immovable at the centre of the universe, can be confusing. In my defence I can point to the fact that no lesser man than Einstein used it in this way. For example in The Evolution of Physics (1938) he wrote "Can we formulate physical laws so that they are valid for all CS (=coordinate systems), 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!"

But if we are looking at the historical Nicolai Copernicus we have to be careful IMO not to elide:

1. The usefulness of calculating the motions of the planets relative to the Sun - i.e. its simplicity and explanatory power leading to the spotting of ellipses etc.

2. The belief that the Sun is fixed immovable at the centre of the universe.Copernicus' major work, De revolutionibus orbium coelestium - On the Revolutions of the Heavenly Spheres (first edition 1543) sets out 1. and specifically disassociates Copernicus from any suggestion of 2.

So IMHO our admiration for Copernicus in relation to 1 is not really relevant to whether the theory which (wrongly) bears his name - i.e. that the Sun is fixed immovable at the centre of the universe - is more right that the alternative theory that the Earth is fixed immovable at the centre of the universe.

 

[JohnNemo]

No. The debate I referred to about how "Machian" GR is is not a debate about whether the EP is true.

I understand that Mach's Principle is to some extent ill-defined but can you give an example of where the area of dispute is?

I suppose I am thinking that if the EP in GR covers accelerating frames including rotating frames, what more could Mach's Principle require?

 

[PAllen]

I understand that Mach's Principle is to some extent ill-defined but can you give an example of where the area of dispute is?

I suppose I am thinking that if the EP in GR covers accelerating frames including rotating frames, what more could Mach's Principle require?

One variant of Mach’s principle, for example, says that it is impossible have a rotating object in a completely empty universe experience centripetal acceleration because there is nothing to rotate relative to. That, instead, distant stars are necessary for rotation to generate inertial forces. There are other formulations as well. None are the same as the principle of equivalence.

 

[JohnNemo]

One variant of Mach’s principle, for example, says that it is impossible have a rotating object in a completely empty universe experience centripetal acceleration because there is nothing to rotate relative to. That, instead, distant stars are necessary for rotation to generate inertial forces. There are other formulations as well. None are the same as the principle of equivalence.

Does GR predict something different for the example you give?

 

[PAllen]

Does GR predict something different for the example you give?

Yes, GR is not machian in this sense.