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

[Dale]

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.

OK, so let’s walk though the process of actually doing something like this so that you understand what needs to be done and why nobody wants to do it.

1) you need to start with a mathematical model of your spacetime. The standard model here is the Schwarzschild spacetime.

2) in that spacetime you need to write down an equation representing the clock’s worldline.

3) Here is where you make a choice, either you do things the easy way and simply skip to step 6) or you do things the hard way and continue on to step 4) which is a lot of work and completely unnecessary

4) Then you need to come up with a coordinate transform. This coordinate transform must be smooth and invertible everywhere and the clock’s worldline should have constant spatial coordinates in these new coordinates. It is especially easy to get non-invertible coordinates.

5) take that transformation and apply it to the Schwarzschild metric to get the metric in the new coordinate system

6) calculate the length of the worldline by integrating the metric along the worldline.

The math is set up so that you are guaranteed to get the same result whether you do steps 4) and 5) or not.

 

[name123]

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.

So if the plane flying Eastward was considered at rest, the Earth would be considered to be spinning in the opposite direction and the Sun would be taking whatever path it had to to appear to rise in the East. And that path could be explained as following the laws of physics given the suggested spacetime geometry. Would the stationary plane be in the same rest frame as the centre of the Earth?

 

[PeroK]

So if the plane flying Eastward was considered at rest, the Earth would be considered to be spinning in the opposite direction and the Sun would be taking whatever path it had to to appear to rise in the East. And that path could be explained as following the laws of physics given the suggested spacetime geometry. Would the stationary plane be in the same rest frame as the centre of the Earth?

The first thing to note is that there is nothing physically significant about "considering an object at rest". All we are doing is using a system of coordinates with that object at the spatial origin. There is nothing that can stop you doing this!

For this problem, we know the spacetime geometry near the Earth in Schwarzschild coordinates. The mathematics underpinning GR is complex but very flexible. With a bit of work that spacetime geometry (encapsulated in the metric) could be tranformed from Schwarzschild to your new coordinate system. As could the worldlines of the other two clocks. This process has been outlined by @Dale above.

But, the irony is, that it's much easier to prove the general theorem that the proper time on every clock is an invariant (the same in all coordinates systems) hence the answer must be the same in all coordinate systems. At best, your efforts would confirm that general theorem about coordinate transformations in this particular case!

This is partly why no one would do it: it's easier to prove mathematically that you must get the same answer in all coordinate systems than it is to carry out the actual calculations for a given coordinate transformation.

That's why once you have solved the problem is the simpelst coordinate system the problem is solved! And, in this case, you can compare your calculated theoretical prediction with the results of a real experiment.

Repeating the calculations in several coordinate systems does nothing but test the robust mathematics of coordinate transformations and the invariance of proper time along a timelike path.

 

[name123]

The first thing to note is that there is nothing physically significant about "considering an object at rest". All we are doing is using a system of coordinates with that object at the spatial origin. There is nothing that can stop you doing this!

For this problem, we know the spacetime geometry near the Earth in Schwarzschild coordinates. The mathematics underpinning GR is complex but very flexible. With a bit of work that spacetime geometry (encapsulated in the metric) could be tranformed from Schwarzschild to your new coordinate system. As could the worldlines of the other two clocks. This process has been outlined by @Dale above.

But, the irony is, that it's much easier to prove the general theorem that the proper time on every clock is an invariant (the same in all coordinates systems) hence the answer must be the same in all coordinate systems. At best, your efforts would confirm that general theorem about coordinate transformations in this particular case!

This is partly why no one would do it: it's easier to prove mathematically that you must get the same answer in all coordinate systems than it is to carry out the actual calculations for a given coordinate transformation.

That's why once you have solved the problem is the simpelst coordinate system the problem is solved! And, in this case, you can compare your calculated theoretical prediction with the results of a real experiment.

So if we considered that the plane that was in the experiment considered to be flying eastward, was instead at rest. The Earth would be considered to be spinning in the opposite direction (westward). And common explanations regarding why hurricanes rotated in certain directions would presumably need to be changed, as the spin of the Earth would be in the opposite direction. The Sun would have to take a path to appear to rise in the East (even though the world was spinning westward). And that path would need to be explained as following the laws of physics given the suggested spacetime geometry. And that could all happen.

What I still don't understand is whether the stationary plane would be in the same rest frame as the centre of the Earth? Perhaps hovering over the equator. The reason I ask, is because if it was, then wouldn't its clock dilation only be expected to show the GR effects of altitude compared with one in the centre of the Earth (or perhaps on the North Pole)?

 

[Dale]

So if we considered that the plane that was in the experiment considered to be flying eastward, was instead at rest. The Earth would be considered to be spinning in the opposite direction (westward). And common explanations regarding why hurricanes rotated in certain directions would presumably need to be changed, as the spin of the Earth would be in the opposite direction. The Sun would have to take a path to appear to rise in the East (even though the world was spinning westward). And that path would need to be explained as following the laws of physics given the suggested spacetime geometry. And that could all happen.

Yes. The math would be a needlessly complicated mess, but you would have fictitious forces and gravity causing everything that needs to happen to explain all observables.

What I still don't understand is whether the stationary plane would be in the same rest frame as the centre of the Earth? Perhaps hovering over the equator. The reason I ask, is because if it was, then wouldn't its clock dilation only be expected to show the GR effects of altitude compared with one in the centre of the Earth (or perhaps on the North Pole)?

That is completely arbitrary. You could make the coordinate system such that the center of the Earth is at rest or not, whatever you choose. You can do that even if the physical distance between the center of the Earth and the airplane changes, as long as they don't intersect. It just makes an already complicated task even more needlessly complicated.

Regardless of how complicated you make your coordinates, the result will always be the same. That is guaranteed by the mathematical tools we use.

 

[jbriggs444]

What I still don't understand is whether the stationary plane would be in the same rest frame as the centre of the Earth?

At the levels of experimental accuracy achieved for Hafele-Keating, there is no inertial frame in which the center of the Earth is at rest which extends far enough to accurately cover a plane that is "stationary" at the airport. Such a plane has a one gee upward proper acceleration and cannot be at rest in any inertial frame for more than a moment.

As has been noted, Earth-centered Schwarzschild coordinates do cover the plane nicely.

 

[PeroK]

And common explanations regarding why hurricanes rotated in certain directions would presumably need to be changed,

There is no need to change any explanations for anything. Do you have to change your explanations for everything when you sit on a plane and consider yourself at rest? If you toss a coin, you don't take the Sun, Moon, spinning Earth and speed of the plane into account. You think about what's happening in the rest frame of the plane and that is right for that experiment. But, if you are asked about hurraicanes, you choose a frame where the Earth is spinning.

You are massively over-thinking the significance of choosing the simplest frame of reference to study a problem. There is no physical significance in a choice of reference frame.

What I still don't understand is whether the stationary plane would be in the same rest frame as the centre of the Earth?

This makes no sense. Reference frames in GR are local. I've said that several times now in this thread.

So, we really should be talking about coordinate systems. Given that the distance between the plane and the centre of the Earth is fixed, you could choose a coordinate system where both are at fixed. points. But, again, that signifies nothing beyond the fact that they remain a fixed distance apart.

There's nothing strange or dodgy about that. We do that all the time in problems where all that matters is the altitude of the plane. If you are sitting on a plane you can easily imagine you and the centre of the Earth being fixed.

 

[name123]

So if we considered that the plane that was in the experiment considered to be flying eastward, was instead at rest. The Earth would be considered to be spinning in the opposite direction (westward). And common explanations regarding why hurricanes rotated in certain directions would presumably need to be changed, as the spin of the Earth would be in the opposite direction.

There is no need to change any explanations for anything. Do you have to change your explanations for everything when you sit on a plane and consider yourself at rest? If you toss a coin, you don't take the Sun, Moon, spinning Earth and speed of the plane into account. You think about what's happening in the rest frame of the plane and that is right for that experiment. But, if you are asked about hurraicanes, you choose a frame where the Earth is spinning.

But the explanation for the hurricanes requires the Earth to be spinning in a certain direction. If it wasn't, the explanation would not work. It is one thing to use whatever frame of reference you like, but it seems to me to be another to suggest that actually it is just as true that the world is spinning westwards as it is the world is spinning eastwards. Because spinning eastwards for example explains why hurricanes spin as they do in the Northern and Southern hemispheres. Spinning westwards would it seems to me require the explanation to change. But I accept it is usually me not understanding the issue properly.

What I still don't understand is whether the stationary plane would be in the same rest frame as the centre of the Earth?

This makes no sense. Reference frames in GR are local. I've said that several times now in this thread.

So, we really should be talking about coordinate systems. Given that the distance between the plane and the centre of the Earth is fixed, you could choose a coordinate system where both are at fixed. points. But, again, that signifies nothing beyond the fact that they remain a fixed distance apart.

There's nothing strange or dodgy about that. We do that all the time in problems where all that matters is the altitude of the plane. If you are sitting on a plane you can easily imagine you and the centre of the Earth being fixed.

I'm sorry, I just meant to ask whether it would appear to not be moving relative to the centre of the Earth. If so then presumably there would be no "SR" element involved in calculating its time dilation compared with a clock in the centre of the Earth or a clock on the North Pole.

The issue was that the result in the experiment wasn't just the altitude, there was an "SR" element too. Also the Earth clock, and the other plane's clock would seem to have the same GR altitude dilation, but their velocities relative to the centre of the Earth would also be different, so I wasn't clear on why the results would still match the experiment result?

 

[PeroK]

But the explanation for the hurricanes requires the Earth to be spinning in a certain direction.

Which it is, relative to an inertial reference frame - or, as near as we can get to one. This is important if you study air currents in the Earth's atmosphere.

But, if you are studying a problem where the air currents are irrelevant, then you can choose the surface of the Earth as your rest frame. There is no obligation when studying one problem to re-assess and redo the rest of physics in your chosen reference frame. A reference frame is chosen to suit the problem under consideration.

You are putting far too much physical significance on the simple idea of choosing a suitable reference frame for a given problem.

 

[Dale]

There is no physical significance in a choice of reference frame.

@name123 I would like to highlight this comment by @PeroK. This is the meaning of the general principle of relativity. Reference frames have no physical significance, they are just conveniences and so we strive to use the most convenient ones.

 

[name123]

Which it is, relative to an inertial reference frame - or, as near as we can get to one. This is important if you study air currents in the Earth's atmosphere.

But, if you are studying a problem where the air currents are irrelevant, then you can choose the surface of the Earth as your rest frame. There is no obligation when studying one problem to re-assess and redo the rest of physics in your chosen reference frame. A reference frame is chosen to suit the problem under consideration.

You are putting far too much physical significance on the simple idea of choosing a suitable reference frame for a given problem.

OK, thank you. I hadn't taken on the point that "There is no physical significance in a choice of reference frame." And I have noticed Dale point it out too.

But in a coordinate system where the plane remains at the centre of the coordinate system wouldn't the plane have 0 velocity with respect to the centre of the Earth in that coordinate system? If it did then I am not sure why the time dilation would appear as it did in the experiment, as while the GR altitude element would seem the same, the SR element based on relative velocity would seem different in the different coordinate system.

Or is it simply that an intertial reference frame (or close to one) must be chosen?

 

[Dale]

But in a coordinate system where the plane remains at the centre of the coordinate system wouldn't the plane have 0 velocity with respect to the centre of the Earth in that coordinate system?

As I already said in post 40 that is completely arbitrary. You could have the velocity wrt the center of the Earth be 0 or not, whatever you choose.

If it did then I am not sure why the time dilation would appear as it did in the experiment, as while the GR altitude element would seem the same, the SR element based on relative velocity would seem different in the different coordinate system.

What makes you think that the GR element would seem the same?

You would have to work out the messy math to find that out. You are guaranteed that it would work, but the details would be messy. There really is no easy way to answer that question. I have outlined the hard way to answer it, but I am not willing to do it for you, so if you really want to know you will have to work through it on your own.

Else, you can just use the principle of relativity to ensure that the result must be the same regardless of how complicated and artificial you make the explanation. That is my preference

 

[PeroK]

But in a coordinate system where the plane remains at the centre of the coordinate system wouldn't the plane have 0 velocity with respect to the centre of the Earth in that coordinate system? If it did then I am not sure why the time dilation would appear as it did in the experiment, as while the GR altitude element would seem the same, the SR element based on relative velocity would seem different in the different coordinate system.

The idea of splitting differential ageing into "gravity" and "velocity" components is purely coordinate dependent. That's the way it splits in Schwarzschild coordinates.

In your new coordinates, you would have removed the relative velocity between the centre of the Earth and the plane, but the transformed spacetime metric would be such that the same proper times emerge for both clocks for the expeiment. The mathematics of coordinate transformations ensures this.

The explanation might look different, but that's what happens when you change coordinates. The important thing is that the result is the same.

 

[name123]

So the transformation would make the "gravity" component higher?

 

[jbriggs444]

It is one thing to use whatever frame of reference you like, but it seems to me to be another to suggest that actually it is just as true that the world is spinning westwards as it is the world is spinning eastwards.

Indeed.

If you write down the laws of physics based on a coordinate system in which the rocky crust of the Earth is stationary, you get some wonky terms. For instance, a term for "Coriolis force". You use this frame, look up in the sky and see that the stars are rotating westward at about one rotation per day.

If you write down the laws of physics based on a coordinate system in which the rocky crust of the Earth is rotating but in which the stars and the sky are [mostly] fixed you will find that the "Coriolis force" goes away. The laws of physics are simpler when stated in this frame.

If you write down the laws of physics based on a coordinate system in which the rocky crust of the Earth is rotating westward at about one rotation every 24 hours you can still write down the laws of physics. In this frame, the stars in the sky are rotating at about two rotations every 24 hours and the "Coriolis force" is twice as strong as when the rocky crust was at rest.

All three frames work. We choose the one that makes our life easiest. Since we like to work in a frame where Chicago is at rest and winds average near zero, we usually accept the Coriolis force and predict weather accordingly. We could use a frame in which both Chicago and the wind are moving eastward at some 700 miles per hour, but that's too much work.

But none of that is what you are talking about. You want to make a point about what is "really" happening.

With General Relativity, there is no such thing as what is "really" happening. All three descriptions are on equal footing. We "just" (as if this is an easy thing) have to express the "metric" so that the laws of physics work correctly in the chosen coordinate system.

If the laws of physics work, the frame is as real as it needs to be.

 

[PeroK]

So the transformation would make the "gravity" component higher?

The transformation would transform the spacetime metric AND the worldlines of all the clocks. I don't think it's helpful to think of the gravity component when looking at general coordinate transformations. I said early that the mathematics of GR was complex and flexible. Complexity is the price you pay for flexibility. You can choose any coordinates you like (flexible), but you have a compelxity in terms of how to explain what's happening.

In addition, you are potentially going down the wrong road altogether here by insisting on a physical understanding of the changes, when in fact nothing physical is changing. It's the same experiment with the same result whatever frame of reference you use to analyse it.

Especially when it comes to GR this is the wrong way to think about things.

 

[name123]

Thanks to everyone that answered, I think I roughly get it now, even though I'm ignorant of the complexities. Thanks all for your patience.

 

[vanhees71]

With General Relativity, there is no such thing as what is "really" happening. All three descriptions are on equal footing. We "just" (as if this is an easy thing) have to express the "metric" so that the laws of physics work correctly in the chosen coordinate system.

If the laws of physics work, the frame is as real as it needs to be.

I'd rather say, in GR what's "really happening" is described by coordinate independent, i.e., generally invariant, quantities.

 

[PeterDonis]

I was thinking of time dilation "caused" by something accelerating vs something not undergoing proper acceleration.

You are misleading yourself by focusing on acceleration. See below.

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.

Yes, there would, because the plane is at a higher altitude than the clock on Earth, and the different in altitudes means a difference in potential, which does cause time dilation. As I posted multiple times earlier in the thread, that time dilation is the "GR time dilation" that appears in the analysis of the Hafele-Keating experiment.

 

[PeterDonis]

the explanation for the hurricanes requires the Earth to be spinning in a certain direction.

Not in any absolute sense. The "spinning in a certain direction" is relative to a local "non-spinning" frame in which the center of the Earth is at rest, where "non-spinning" is defined by Fermi-Walker transport of the spatial basis vectors along the worldline of the center of the Earth (think of a set of imaginary gyroscopes located at the center of the Earth and defining three mutually perpendicular directions in space). If you choose a different frame, the "spinning" of the Earth will be different, but so will a lot of other coordinate-dependent things that all change in concert to keep all of the physical invariants, like the presence of a hurricane, the same.