Hafele-Keating Exp: Balancing SR Effects in Plane Ref Frame

[name123]

TL;DR Summary

Question of about how it works with a plane as frame of reference?

I have seen the "Hafele-Keating with the plane as reference frame?" thread (https://www.physicsforums.com/threads/hafele-keating-with-the-plane-as-reference-frame.767913/ ), but the replies do not seem to explain (to me anyway) what when taking a plane as a reference frame, balances the slowing of the clock on the plane due to Special Relativity (SR) as that would no longer be happening (in the plane's frame of reference), and also the slowing of the clock on Earth due to (SR) which would be happening (in the plane's frame of reference), such that the result should be the same. The piece I am looking for is what would balance the result, regardless of whether the experiment was done over 100 hours or 100 years. I have assumed that if all clocks were synched while in the air, the effect of bringing them together would be the same regardless of whether it went for 100 hours or 100 years.

Also does anyone know whether the experiment took the General Relativity time dilation due to the acceleration of the planes flying at constant horizontal speed into account (rather than just the altitude)?

​

 

[russ_watters]

I don't mean to be rude here, but your question is extremely poorly written and referenced. The second part is easily answered from the wikipedia article, which is an extremely low bar for what homework you should have done before posting; yes, the experiment took GR into account:
https://en.wikipedia.org/wiki/Hafele–Keating_experiment

Perhaps there is a language issue here as well, but the first sentence/main part is practically unreadable as written.

 

[Dale]

Summary:: Question of about how it works with a plane as frame of reference?

Also does anyone know whether the experiment took the General Relativity time dilation due to the acceleration of the planes flying at constant horizontal speed into account (rather than just the altitude)?

There is no GR time dilation due to acceleration.

Regarding the remainder of your question: why would you want to use the plane’s frame? The principle of relativity says that you can use any frame and all experimental results are the same. So you use the easiest frame to calculate. Using an inconvenient frame only makes the calculations more difficult but cannot change the outcome.

 

[name123]

I don't mean to be rude here, but your question is extremely poorly written and referenced. The second part is easily answered from the wikipedia article, which is an extremely low bar for what homework you should have done before posting; yes, the experiment took GR into account:
https://en.wikipedia.org/wiki/Hafele–Keating_experiment

Perhaps there is a language issue here as well, but the first sentence/main part is practically unreadable as written.

I do apologise. What I was trying to get at is that as I understood it with the experiment where the frame of reference was the Earth, there would be time dilation on the clock on the plane due to Special Relativity + General Relativity. So let's call say:
time_dilation_plane_clock = SR_Effects_Earth_Ref + GR_Effects_Earth_Ref

I was then assuming that if the plane was the frame of reference that there would instead be dilation on the Earth clock due to Special Relativity, and there would still be General Relativity effects.

So let's say:
time_dilation_plane_clock = SR_Effects_Plane_Ref + GR_Effects_Plane_Ref

I thought the SR_Effects_Earth_Ref were quite different to SR_Effects_Plane_Ref as I thought the former suggest the plane clock is going slower, and the latter the Earth clock. So I was wondering what difference was there between GR_Effects_Earth_Ref and GR_Effects_Plane_Ref that compensated for the difference in the SR time dilations between the two references. Because I had assumed that:
SR_Effects_Earth_Ref + GR_Effects_Earth_Ref = SR_Effects_Plane_Ref + GR_Effects_Plane_Ref

=> SR_Effects_Earth_Ref - SR_Effects_Plane_Ref = GR_Effects_Plane_Ref - GR_Effects_Earth_Ref

Is that any clearer, or do you still not understand what I am asking?

Regarding the wikipedia page: In the overview it splits the effects into Kinematic Effects and Gravitational Effects. Under Kinematic Effects I only noticed it mentioning Special Relativity. I did not notice it mentioning General Relativity. And under Gravitational Effects its states:
"General relativity predicts an additional effect, in which an increase in gravitational potential due to altitude speeds the clocks up. That is, clocks at higher altitude tick faster than clocks on Earth's surface."
Thus I only noticed it mention the altitude effect. Where did you think that it mentions that the General Relativity time dilation due to the acceleration of the planes flying at constant horizontal speed was taken into account (rather than just the altitude)?

 

[name123]

There is no GR time dilation due to acceleration.

I thought there was an equivalence between gravity and acceleration. And that in GR there was time dilation due to gravity/acceleration. If that is correct then I assume you mean there was no acceleration. But in the thread I referenced at the beginning some replies indicated that there was proper acceleration upwards caused by the lift of the wings, even in level flight at constant speed, which couldn't be ignored. Is that incorrect (that there was acceleration that couldn't be ignored)?

 

[PeterDonis]

I thought there was an equivalence between gravity and acceleration. And that in GR there was time dilation due to gravity/acceleration.

No, there isn't time dilation due to "acceleration" in GR. What there is is time dilation due to depth in a potential well. This appears in the analysis of the Hafele-Keating experiment as the "GR time dilation" due to altitude above the Earth--lower altitude, more time dilation.

 

[name123]

No, there isn't time dilation due to "acceleration" in GR. What there is is time dilation due to depth in a potential well. This appears in the analysis of the Hafele-Keating experiment as the "GR time dilation" due to altitude above the Earth--lower altitude, more time dilation.

I assume that you are considering the amount of gravity to be synonymous with "depth in a potential well". If so then are you suggesting that gravity and acceleration are not equivalent, that one leads to time-dilation and not the other?

 

[PeterDonis]

I assume that you are considering the amount of gravity to be synonymous with "depth in a potential well".

No. "Amount of gravity" is not a well-defined term since "gravity" itself is not; it can mean multiple things. You appear to be using it to mean "acceleration due to gravity", which is not the same as "depth in a potential well". That should be obvious since it's true even in Newtonian physics.

are you suggesting that gravity and acceleration are not equivalent, that one leads to time-dilation and not the other?

Not at all. If you are at rest in an accelerating rocket, you can view yourself as being in a potential well; you are deeper in the well at the bottom of the rocket than you are at the top, and there will be a corresponding time dilation. But the time dilation is not due to the acceleration itself; it is due to the difference in depth in the potential well, which in this particular case happens to be "caused" by the rocket's acceleration (whereas in the case of the Earth it is "caused" by the curved spacetime geometry around the Earth).

 

[name123]

If you are at rest in an accelerating rocket, you can view yourself as being in a potential well; you are deeper in the well at the bottom of the rocket than you are at the top, and there will be a corresponding time dilation. But the time dilation is not due to the acceleration itself; it is due to the difference in depth in the potential well, which in this particular case happens to be "caused" by the rocket's acceleration (whereas in the case of the Earth it is "caused" by the curved spacetime geometry around the Earth).

If you are at rest in an accelerating plane, can you view yourself as being in a potential well? If so then let me rephrase the question that was at the end of the initial post. Did the experiment take into account the General Relativity time dilation due to being in a potential well when at rest in an accelerating plane?

 

[PeterDonis]

If you are at rest in an accelerating plane, can you view yourself as being in a potential well?

If you mean a plane accelerating in order to maintain constant altitude above the Earth, yes, you are in the Earth's potential well. Note that there is only one relevant height in the plane itself, so there is no usable potential well inside the plane (like the one inside the accelerating rocket I described); but there is certainly a difference in potential between the plane flying at altitude and the surface of the Earth, and that potential difference gives rise to the "GR time dilation" that appears in the analysis of the Hafele-Keating experiment.

Did the experiment take into account the General Relativity time dilation due to being in a potential well when at rest in an accelerating plane?

Yes. See above (which just restated what I already said in post #6).

 

[name123]

If you mean a plane accelerating in order to maintain constant altitude above the Earth, yes, you are in the Earth's potential well. Note that there is only one relevant height in the plane itself, so there is no usable potential well inside the plane (like the one inside the accelerating rocket I described); but there is certainly a difference in potential between the plane flying at altitude and the surface of the Earth, and that potential difference gives rise to the "GR time dilation" that appears in the analysis of the Hafele-Keating experiment.

I did not understand why there is only one relevant height in a plane. There is near the ceiling, on the floor etc.

In post #8 you talked about a potential well in a rocket being "caused" by acceleration. But in space the rocket would to a certain extent be in potential wells of planets. Why could such a rocket have a potential well "caused" by acceleration, but not a rocket deeper in the well of a certain planet?

 

[PeterDonis]

In post #8 you talked about a potential well in a rocket being "caused" by acceleration.

Yes; notice that I put "caused" in scare quotes. That's because the potential well is not really a separate thing; it's just a different way of describing the motion of the rocket or the spacetime geometry around a gravitating mass like the Earth.

in space the rocket would to a certain extent be in potential wells of planets.

But it would generally be in a free-fall orbit, which means it would have zero proper acceleration. so it would not be "causing" any potential well of its own. Also, since the rocket would be moving in the potential well of a planet, you would have to also include time dilation due to its motion (just as the analysis of the Hafele-Keating experiment did for the airplanes, since they were moving relative to the center of the Earth--this is the "SR" time dilation that appears in the analysis).

Why could such a rocket have a potential well "caused" by acceleration

It most likely wouldn't since it would be in a free-fall orbit. See above.

but not a rocket deeper in the well of a certain planet?

What is this referring to?

 

[name123]

Yes; notice that I put "caused" in scare quotes. That's because the potential well is not really a separate thing; it's just a different way of describing the motion of the rocket or the spacetime geometry around a gravitating mass like the Earth.

That's fine.

But it would generally be in a free-fall orbit, which means it would have zero proper acceleration. so it would not be "causing" any potential well of its own.

We were discussing accelerating rockets not ones in free fall. You seemed to state that an accelerating rocket would "cause" a potential well of its own, when it would to a certain extent be in potential wells of planets. And that there would be time dilation (post #8). And yet you also seemed to state that an accelerating plane deeper in the potential well of a certain planet (say Earth) would not "cause" a potential well of its own. I just wondered why the difference.

I can understand why it wouldn't make a difference in the experiment though, as the clock on the Earth would be undergoing the same acceleration, and thus the same time dilation.

 

[name123]

Thanks for your patience, and I was wondering if you could look at post #4 and explain why
SR_Effects_Earth_Ref - SR_Effects_Plane_Ref = GR_Effects_Plane_Ref - GR_Effects_Earth_Ref
or where I had gone wrong in my reasoning?

 

[PeterDonis]

You seemed to state that an accelerating rocket would "cause" a potential well of its own, when it would to a certain extent be in potential wells of planets.

In cases where the potential is weak enough (i.e., when both spacetime curvature and accelerations are small--note that a 1 g acceleration is "small" by this criterion), and where the potentials "line up" (basically this means the direction of acceleration of the rocket would have to be the same as the local direction of "acceleration due to gravity" of the planet), you can just add them, as you would in Newtonian physics. So in this approximation, yes, you could view the accelerating rocket as having a potential well, and the planet as having a potential well, and the total potential at a given point would be the sum of the two.

However, as should be evident from my description above, the cases where this works are very limited. If the conditions above are not met (the main one, of course, being that the rocket's acceleration has to line up just the right way), the "potential" concept breaks down and cannot be used.

you also seemed to state that an accelerating plane deeper in the potential well of a certain planet (say Earth) would not "cause" a potential well of its own

I said no such thing. I said there was only one relevant height in a plane.

I did not understand why there is only one relevant height in a plane.

Two reasons: first, everyone in a plane (unless it's a Boeing 747 or an Airbus A380, which have two levels) is on the same level, so there are no height differences.

Second, the plane's acceleration is so small that, even for the planes I mentioned above that have two levels instead of one, the time dilation due to the potential difference from its top to its bottom is too small to be measurable with our current technology. So as far as we can currently measure, everyone in the plane can be treated as being at the same height.

 

[name123]

I said no such thing. I said there was only one relevant height in a plane.

I think my misunderstandings took us a bit off track. I was thinking of time dilation "caused" by something accelerating vs something not undergoing proper acceleration. But now realize that the clock on Earth would be accelerating at the same speed as a plane at constant altitude in the experiment, and so there would be no relative time dilation.

Thanks for your patience. If you had the time I was wondering if you could look at post #4 and explain why
SR_Effects_Earth_Ref - SR_Effects_Plane_Ref = GR_Effects_Plane_Ref - GR_Effects_Earth_Ref
or where I had gone wrong in my reasoning?

 

[PeroK]

I can understand why it wouldn't make a difference in the experiment though, as the clock on the Earth would be undergoing the same acceleration, and thus the same time dilation.

Acceleration does not cause time dilation!

The equivalence principle is saying something else entirely and the soundbite "acceleration is equivalent to gravity" is highly misleading.

This is a common misconception. Take two clocks moving past you. The first is moving at constant speed $v$, and the second is moving instantaneously at speed $v$ as it passes you but is accelerating at some signficant acceleration. The time dilation that you measure for both clocks the instant they pass you is the same, with a gamma factor of $\frac 1 {\sqrt{1 - v^2/c^2}}$.

The gamma factor for the clock moving at constant speed will be constant. The gamma factor for the accelerating clock will, of course, change over time, but it always depends only on the instantaneous speed and never on the magnitude of the instantaneous acceleration.

In the Hafele-Keating experiment, the simplest calculation is to use a hypothetical clock a long way from Earth and compare the three clocks in the experiment with that:

1) The clock on Earth, at a certain gravitational potential, and with a certain rotational speed.

2) The eastwards traveling clock, at a slightly higher gravitational potential and greater rotational speed.

3) The westwards traveling clock, at the higher gravitational potential and lower rotational speed.

Altitude and speed are the only factors. There is no additional factor for centripetal or other acceleration, because acceleration does not cause time dilation.

Once all these clocks are co-located again at the end of the experiment, you can compare their readings against the reading on the hypothetical clock and hence compare their readings against each other.

Using one of the experimental clocks as the main reference clock is possible, of course, but it just makes the calculations more difficult.

 

[name123]

Acceleration does not cause time dilation!

The equivalence principle is saying something else entirely and the soundbite "acceleration is equivalent to gravity" is highly misleading.

This is a common misconception. Take two clocks moving past you. The first is moving at constant speed $v$, and the second is moving instantaneously at speed $v$ as it passes you but is accelerating at some signficant acceleration. The time dilation that you measure for both clocks the instant they pass you is the same, with a gamma factor of $\frac 1 {\sqrt{1 - v^2/c^2}}$.

The gamma factor for the clock moving at constant speed will be constant. The gamma factor for the accelerating clock will, of course, change over time, but it always depends only on the instantaneous speed and never on the magnitude of the instantaneous acceleration.

Thank you for making it clear.

In the Hafele-Keating experiment, the simplest calculation is to use a hypothetical clock a long way from Earth and compare the three clocks in the experiment with that:

1) The clock on Earth, at a certain gravitational potential, and with a certain rotational speed.

2) The eastwards traveling clock, at a slightly higher gravitational potential and greater rotational speed.

3) The westwards traveling clock, at the higher gravitational potential and lower rotational speed.

Altitude and speed are the only factors. There is no additional factor for centripetal or other acceleration, because acceleration does not cause time dilation.

Once all these clocks are co-located again at the end of the experiment, you can compare their readings against the reading on the hypothetical clock and hence compare their readings against each other.

Using one of the experimental clocks as the main reference clock is possible, of course, but it just makes the calculations more difficult.

Thank you for your response. I was wondering if you had the time whether you could look at post #4 and explain why
SR_Effects_Earth_Ref - SR_Effects_Plane_Ref = GR_Effects_Plane_Ref - GR_Effects_Earth_Ref
or where I had gone wrong in my reasoning?

 

[PeroK]

Thank you for making it clear.
Thank you for your response. I was wondering if you had the time whether you could look at post #4 and explain why
SR_Effects_Earth_Ref - SR_Effects_Plane_Ref = GR_Effects_Plane_Ref - GR_Effects_Earth_Ref
or where I had gone wrong in my reasoning?

I don't understand where that equation comes from. If we consider the clock on Earth, the first point is that it not moving inertially. Leaving gravity out of things for a moment, that clock is moving in a circular path relative to an inertial clock at the north pole, centre of the Earth etc.

Imagine a clock hovering at the equator and not rotating with the Earth. Each time the Earth clock comes round, it will be the Earth clock that has lost time compared to the hovering clock. There is no symmetric time dilation in this case. What you have, in fact, is differential ageing.

You cannot treat the Earth clock as an inertial clock and simply apply time dilation to any clock that is moving relative to it. The same applies to the traveling clocks.

What you could do is compare all three clocks with the hovering clock - which would be inertial (ignoring gravity). That's how I would tackle that problem using SR.

Once we add gravity, it's simplest conceptually to take our hovering clock far away from the Earth so that it becomes a reference for gravitational time dilation as well.

 

[name123]

I don't understand where that equation comes from. If we consider the clock on Earth, the first point is that it not moving inertially. Leaving gravity out of things for a moment, that clock is moving in a circular path relative to an inertial clock at the north pole, centre of the Earth etc.

Thank you again for clearing things up for me. It had been a while since I had read it, and returning to it I had just glanced over and thought they had used the clock on the Earth as a frame of reference, and they hadn't it was the centre of the Earth.

Imagine a clock hovering at the equator and not rotating with the Earth. Each time the Earth clock comes round, it will be the Earth clock that has lost time compared to the hovering clock. There is no symmetric time dilation in this case. What you have, in fact, is differential ageing.

You cannot treat the Earth clock as an inertial clock and simply apply time dilation to any clock that is moving relative to it. The same applies to the traveling clocks.

Thank you I did not realize that.

When you earlier wrote: "Using one of the experimental clocks as the main reference clock is possible, of course, but it just makes the calculations more difficult." Is Special Relativity no longer used because the clock is not inertial, or is it still used, but some adjustment made?

 

[PeroK]

When you earlier wrote: "Using one of the experimental clocks as the main reference clock is possible, of course, but it just makes the calculations more difficult." Is Special Relativity no longer used because the clock is not inertial, or is it still used, but some adjustment made?

The solution essentially must use GR to take account of gravity. The difference in the clock readings can be decomposed into a gravitational factor and a factor due to motion. The second factor can be calculated using SR as I mentioned above - and by far the simplest way is to imagine an additional inertial reference clock.

Using one of the experimental clocks gets into the compexities of accelerating reference frames in SR.

 

[name123]

The solution essentially must use GR to take account of gravity. The difference in the clock readings can be decomposed into a gravitational factor and a factor due to motion. The second factor can be calculated using SR as I mentioned above - and by far the simplest way is to imagine an additional inertial reference clock.

Using one of the experimental clocks gets into the compexities of accelerating reference frames in SR.

Earlier you mentioned that "acceleration does not cause time dilation" so I assumed the acceleration due to lift can be ignored if the planes are flying at a constant height, as it won't affect the time dilation. And the experiment could presumably be done where the planes and the clock on the Earth are all synched while the planes are in orbit, and all checked at some later point when they are in orbit (to cut out the accelerations involved in taking off and landing). Could you perhaps explain some of the complexities in such a situation?

 

[PeroK]

Earlier you mentioned that "acceleration does not cause time dilation" so I assumed the acceleration due to lift can be ignored if the planes are flying at a constant height, as it won't affect the time dilation. And the experiment could presumably be done where the planes and the clock on the Earth are all synched while the planes are in orbit, and all checked at some later point when they are in orbit (to cut out the accelerations involved in taking off and landing). Could you perhaps explain some of the complexities in such a situation?

The initial and final acceleration or deceleration of each plane is over short time. Rather than having a constant speed of $1000km/h$, you have a variable speed (and variable altitude) for relatively short intervals . But, these complexities are negligible compared to the total differential ageing.

A quick Internet search reveals that a plane reaches its cruising altitude after about 10 minutes and takes about 30 minutes to land. If you wanted extreme precision in the experiment, you would have to factor this in, but it's generally not significant enough.

 

[name123]

The initial and final acceleration or deceleration of each plane is over short time. Rather than having a constant speed of $1000km/h$, you have a variable speed (and variable altitude) for relatively short intervals . But, these compexities are negligible compared to the total differential ageing.

A quick Internet search reveals that a plane reaches its cruising altitude after about 10 minutes and takes about 30 minutes to land. If you wanted extreme precision in the experiment, you would have to factor this in, but it's generally not significant enough.

But as I said the initial synching of the clocks, and the clock comparison could all be done after the planes have taken off and before the planes landed. So there would be no variable speed or altitude. Were they the complexities you were talking about?

 

[PeroK]

But as I said the initial synching of the clocks, and the clock comparison could all be done after the planes have taken off and before the planes landed. So there would be no variable speed or altitude. Were they the complexities you were talking about?

These days, the GPS satellite clock synchronisation is like a continuous Hafele-Keating experiment. The compexities in that case become the calculations for light signal transmission during the synchronisation process.

Hafele-Keating was done in 1971 with the technology available at that time. Synchronising two atomic clocks once they were on planes at cruising altitude was probably not an option.

 

[name123]

These days, the GPS satellite clock synchronisation is like a continuous Hafele-Keating experiment. The compexities in that case become the calculations for light signal transmission during the synchronisation process.

Hafele-Keating was done in 1971 with the technology available at that time. Synchroning two atomic clocks once they were on planes at cruising altitude was probably not an option.

Fair enough, I am just trying to establish what added complexity there is using a clock on the Earth as a frame of reference compared to using the centre of the Earth as a frame of reference.

 

[vanhees71]

I think the entire confusion of the OP is due to this unfortunate split of kinematical effects in "special relativistic" and "general relativistic" ones. Special relativity applies when the influence of any gravitational effect is negligible and then you have kinematic effects of special relativity in the description (relativity of simultaneity, time dilation, length contraction, and Wigner rotations). If gravity plays a significant role, then you have to use general relativity and this includes all the kinematical effects including gravitational ones. In the Hafele-Keating experiment you have to take into account gravitational effects, which were significant to explain the outcome of the experiment, i.e., you need GR to describe this outcome.

 

[PeroK]

Fair enough, I am just trying to establish what added complexity there is using a clock on the Earth as a frame of reference compared to using the centre of the Earth as a frame of reference.

In GR, reference frames tend to be local constructs. There are, for example, no global inertial reference frames in curved spacetime. The main technique for solving problems in GR generally involves finding the best coordinate system in which to study the problem and then using this to understand measurements, which are generally local.

In this case, I have been implicitly using Schwarzschild coordinates to model the entire problem. In these coordinates we can model the proper time of all three clocks and compare their readings whenever they are colocated.

There is no need to emphasise the reference frame of each individual observer/clock. In fact, it's not a good idea to try to do GR problems that way.

If you really want to study the problem from the Earth clock's perspective, then you need to transform Schwarzschild coordinates to a coordinate system where the Earth clock is at rest. And, transform the worldlines of the other two clocks into this coordinate system. As mentioned before, it's much simpler to stick with Schwarzschild in this case.

 

[name123]

Thank you. But even if I knew how (and I don't), there would presumably still be an SR element would there not, or would it all be GR?

 

[vanhees71]

GR includes all effects, including the purely kinematical SR effects and the effects of the gravitational interaction, if you insist on splitting the effects in SR and GR effects at all.

 

[PeroK]

Thank you. But even if I knew how (and I don't), there would presumably still be an SR element would there not, or would it all be GR?

It's all GR, because it's curved spacetime. SR, strictly speaking, is flat spacetime or the local limit in curved spacetime. The differential ageing between the clocks has an element due to their relative motion and that is similar to an SR problem if you ignore gravity.

It's really gravity + motion; not GR + SR.

 

[name123]

It's all GR, because it's curved spacetime. SR, strictly speaking, is flat spacetime or the local limit in curved spacetime. The differential ageing between the clocks has an element due to their relative motion and that is similar to an SR problem if you ignore gravity.

It's really gravity + motion; not GR + SR.

So it could be said that the experiment just proved GR?

Does GR allow for the conception that the Sun rotates the Earth or does the spacetime geometry determine which way it will be?

 

[PeroK]

So it could be said that the experiment just proved GR?

Does GR allow for the conception that the Sun rotates the Earth or does the spacetime geometry determine which way it will be?

SR is a special case of GR. E.g. in a particle accelerator gravity is negligible and may be ignored. The model of particle scattering uses SR.

Hafele-Keating tests GR in the particular case of clocks moving relative to each other at different altitudes in the Earth's gravitational field.

Note that although the Sun's gravity affects the Earth, relative to the Sun the entire experiment takes place at approximately the same distance from the Sun and with the centre of the Earth providing a common solar orbital speed. That's why we can approximate things by ignoring the Sun and use the simple model of Schwarzschild coordinates centred on the Earth.

 

[name123]

SR is a special case of GR. E.g. in a particle accelerator gravity is negligible and may be ignored. The model of particle scattering uses SR.

Hafele-Keating tests GR in the particular case of clocks moving relative to each other at different altitudes in the Earth's gravitational field.

Note that although the Sun's gravity affects the Earth, relative to the Sun the entire experiment takes place at approximately the same distance from the Sun and with the centre of the Earth providing a common solar orbital speed. That's why we can approximate things by ignoring the Sun and use the simple model of Schwarzschild coordinates centred on the Earth.

As I understand it with SR an inertial object can be considered to be at rest. But is it just as arbitrary with GR when there is spacetime geometry? Thus could the Earth be considered to spinning, but not orbiting the Sun. That instead the Sun orbited the Earth. Or does the spacetime geometry determine which way the relative motion is?

With the Hafele-Keating experiment the centre of the Earth is considered to be at rest even though the Earth is rotating. But could one of the planes also be considered to be at rest, with the direction the planet is spinning in depending on which plane was considered to be at rest? If it could, then would it be in the same rest frame as the centre of the Earth?

My query isn't about how certain parts of the equations can be usefully applied, it is to do with the arbitrariness of which objects were at rest.

Thanks for your patience here btw, hoping that other people on the more basic levels might also not be clear.

 

[PeroK]

As I understand it with SR an inertial object can be considered to be at rest. But is it just as arbitrary with GR when there is spacetime geometry? Thus could the Earth be considered to spinning, but not orbiting the Sun. That instead the Sun orbited the Earth. Or does the spacetime geometry determine which way the relative motion is?

In SR you have special frames of reference called inertial reference frames where "the laws of Newtonian mechanics apply". These are global, in that they cover all spacetime. Any object that is moving inertially has an inertial rest frame.

Any object may be considered at rest: whether its rest frame is inertial or not is the key thing.

In GR there are no global inertial reference frames. If an object is inertial, then it moves along a geodescic of the spacetime, but its inertial rest frame is only a local concept.

All we have done in this problem is ignore all influences that do not sufficiently affect the difference between the times on the three clocks. To a good enough approximation the only things influencing the clocks are the Earth's gravity and the motion of the clocks relative to the Earth. We can ignore both the Sun and Moon's gravity unless we wanted an answer to extreme precision.

There is no big conceptual idea here about what's orbitting what. We are, like we always have done in physics, simplifying the problem by removing factors that have negligible affect on the answer.