Pound & Rebka Experiment: Earth's Spin Rotation Speed

[BeedS]

Earth has an equatorial circumference of 40030.17Km and diameter of 12742Km, in 24 hours you will travel 40030.17Km or 40030.17Km / 24 = 1667.92Km/h in space from the earths spin rotation.
If you move 1 Km up in a tall building you are now on a circumference of 40036.45Km, diameter of 12744Km and travel 40036.45Km in 24hrs or 40036.45Km / 24 = 1668.18Km/h

I know these measurements are not set to the height differences between the source and receiver in the P&R experiment.

My question, did they account for the speed difference between the source and receiver at different circumferences in the P&R experiment results?

or

Is my logic flawed somewhere?

 

[malawi_glenn]

What is a P and R experiment?

 

[PeterDonis]

did they account for the speed difference between the source and emitter at different circumferences in the P&R experiment results?

I assume you mean the Pound & Rebka experiment?

https://en.wikipedia.org/wiki/Pound–Rebka_experiment

If so, the answer to your question is that there was no "speed difference" that needed to be accounted for: in the rest frame of the experimental apparatus, which is the frame in which the experimenters analyzed the experiment, the source and the receiver (I assume you meant that, not "emitter") were both at rest.

 

[Nugatory]

My question, did they account for the speed difference between the source and emitter at different circumferences in the P&R experiment results?

By "P & R experiment" you mean the Pound-Rebka measurement of gravitational time dilation?

If so, the quick answer to your question is "yes, of course".

The slightly longer answer is an exercise for you. The height difference in this experiment was 22.5 meters. What was the relative transverse velocity between the top and the bottom? What would be the approximate magnitude of the effects of this speed difference (you will get in the right general neighborhood by using the time dilation formula)? How does this effect compare with the other effects that the experimenters had to deal with?

 

[PeterDonis]

the quick answer to your question is "yes, of course".

I'm not sure this is correct, at least not as a description of the analysis that was actually done of the experiment. The original P&R paper is here:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.3.439

I see nothing about relative transverse velocity due to height difference. Indeed, even in more modern textbook treatments I have not seen this come up.

The height difference in this experiment was 22.5 meters. What was the relative transverse velocity between the top and the bottom?

I'm not sure there is an actual nonzero "relative transverse velocity" invariant. In an Earth-centered inertial frame there will be a nonzero difference in transverse velocity between source and receiver, but that is a frame-dependent quantity.

 

[BeedS]

the source and the receiver (I assume you meant that, not "emitter") were both at rest.

Thanks, I edited the post and corrected it.

 

[BeedS]

If so, the answer to your question is that there was no "speed difference" that needed to be accounted for: in the rest frame of the experimental apparatus, which is the frame in which the experimenters analyzed the experiment

They conducted the experiment multiple times at different heights to obtain the differences in frequency measurements to compare? Every time they changed the height the speed changed?

 

[PeterDonis]

They conducted the experiment multiple times at different heights to obtain the differences in frequency measurements to compare? Every time they changed the height the speed changed?

First of all, if you are asking whether they did the experiment at different heights, AFAIK, no, they didn't. They did repeated runs but all of them were in the same place.

Second, your question wasn't about "speed", but about relative speed between the source and the emitter. And my point was that, in the rest frame of the apparatus, which is the frame in which P&R analyzed it, that relative speed is zero. That will be true regardless of the height at which the experiment is performed.

If you object that the statement I just made is frame-dependent, well, so is the "relative speed" in the Earth-centered inertial frame that you described in your OP. I agree that the actual physics involved should be described in terms of invariants, not frame-dependent quantities; but, as I said in response to @Nugatory, I am not aware of any invariant corresponding to a nonzero "relative speed" that could affect the P&R experimental results. Nor did P&R include any such thing in their analysis, as is evident from reading their original paper, which I have linked to.

 

[BeedS]

Basically, I was just wondering if the redshift could have been caused by the speed difference instead of gravitational redshift.

 

[PeterDonis]

I was just wondering if the redshift could have been caused by the speed difference instead of gravitational redshift.

No, it couldn't.

 

[PeterDonis]

I assume you mean the Pound & Rebka experiment?

Moderator's note: Thread title changed to reflect the "yes" answer to this.

 

[Dale]

Basically, I was just wondering if the redshift could have been caused by the speed difference instead of gravitational redshift.

To add a little more detail, the speed difference is the wrong direction. In the Earth centered non-rotating frame the top is moving faster so the top would have slower time.

But the opposite was found. The bottom has slower time.

So the speed difference not only cannot account for the observation, it actually opposes it.

My question, did they account for the speed difference between the source and receiver at different circumferences in the P&R experiment results?

Whether or not they accounted for the effect is a slightly different question from whether or not the effect can explain it. Their analysis of the experiment was performed in the frame in which both are at rest. In this frame the effect you mention is a small centrifugal force which slightly alters the local gravitational acceleration, $g$. To account for the effect then would only require using a locally measured value for $g$ instead of a standard value. I do not remember if they did so.

 

[vanhees71]

I'm not sure this is correct, at least not as a description of the analysis that was actually done of the experiment. The original P&R paper is here:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.3.439

I see nothing about relative transverse velocity due to height difference. Indeed, even in more modern textbook treatments I have not seen this come up.I'm not sure there is an actual nonzero "relative transverse velocity" invariant. In an Earth-centered inertial frame there will be a nonzero difference in transverse velocity between source and receiver, but that is a frame-dependent quantity.

I think this is simply too small an effect to be of relevance here. In principle of course you can analyze the entire experiment in curved spacetime. As a first better approximation you can try the Schwarzschild space time describing approximately the gravitational field of the Earth.

 

[Vanadium 50]

This is just too painful to watch. My eyes are melting!

This question could be answered immediately by plugging in some numbers. I will leave this as an exercise to the OP. But I will share the answer.

Given a tower of height h, and the Earth's radius R and mass M, the gravitational time dilation will be of order (GMh/R²) which is measured in parts per triillion. Of course there is a velocity difference between the bottom and top of the tower -the top covers more distance per day, by of order (h/R)(v/c), where v is the Earth;s rotational velocity. This is a part per trillion too, but its squared, so the effect is a trillion times smaller.

The OP is free to work out all the factors of 2 and π, but of course it won't change the conclusions.

 

[PeterDonis]

I think this is simply too small an effect to be of relevance here

My question is, what effect? Show me an invariant that governs this effect. The "relative velocity" between top and bottom is not an invariant; it's frame dependent. In the rest frame of the apparatus it is zero.

 

[PeterDonis]

Of course there is a velocity difference between the bottom and top of the tower -the top covers more distance per day

As I have already pointed out, this "velocity difference" is frame-dependent; in the rest frame of the apparatus, which is the frame in which P&R did their analysis, there is zero "velocity difference"; the entire apparatus is at rest in this frame.

So if you, or anyone, wants to claim that there is some "velocity" effect going on here, you need to show me what invariant governs this effect. So far nobody has done so.

 

[Dale]

if you, or anyone, wants to claim that there is some "velocity" effect going on here, you need to show me what invariant governs this effect. So far nobody has done so

The invariant is the difference in proper acceleration between a hovering worldline and a rotating worldline.

 

[Sagittarius A-Star]

My question, did they account for the speed difference between the source and receiver at different circumferences in the P&R experiment results?

No, that was not needed. The speed difference between the source and receiver with reference to the ECI frame is neglectable for $h<<R$ and therefore also it's influence on their difference in gravity-potential in the ECEF frame.

The gravity of Earth, denoted by g, is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).

Source:
https://en.wikipedia.org/wiki/Gravity_of_Earth

 

[PeterDonis]

The invariant is the difference in proper acceleration between a hovering worldline and a rotating worldline.

Hm. That makes sense since the proper acceleration appears directly in the analysis, in the formula $g h / c^2$. But the correction here from the non-rotating to the rotating case is not due to the relative velocity between the source and the emitter; it's due to the difference in $g$ between two stationary congruences, the non-rotating one and the rotating one. "Stationary" means that the source and emitter are at rest relative to each other in both cases (and here "at rest relative to each other" has an invariant meaning, namely, that both worldlines are integral curves of the same timelike Killing vector field).

If this viewpoint is correct, then what we need to check is what value for $g$ Pound & Rebka used in their analysis. Unfortunately I don't think any of the numbers in their paper, or in other discussions I have seen, are given to a high enough accuracy to tell; the difference is only about 0.3 percent (and that's at the Earth's equator--at the latitude of Harvard, where P&R did their experiment, it would be even smaller), and I only see numbers given to two significant figures (I just checked MTW chapter 38 and it only gives numbers to that accuracy as well).

 

[PeterDonis]

The speed difference between the source and receiver with reference to the ECI frame is

...a frame-dependent quantity, and therefore not something you can base a physical argument on.

 

[Sagittarius A-Star]

...a frame-dependent quantity, and therefore not something you can base a physical argument on.

I think in this case the Newtonian approximation can be used to calculate the centrifugal acceleration.

 

[Dale]

If this viewpoint is correct, then what we need to check is what value for Pound & Rebka used in their analysis. Unfortunately I don't think any of the numbers in their paper, or in other discussions I have seen, are given to a high enough accuracy to tell

Yes, I agree.

 

[Nugatory]

...a frame-dependent quantity, and therefore not something you can base a physical argument on.

That might be an overstatement? We routinely use frame-dependent quantities to calculate the value of physical invariants. For example, in the twin paradox I can calculate the proper time for both twins using a frame in which the earth is not at rest - the elapsed times and distances going into the calculation are of course frame-dependent, but the invariant emerges as expected.

Here the value of the frame-dependent relative velocity is that is an easily calculated proxy for how much the non-inertial rotating rest frame in which the experiment is conducted differs from an idealized Einstein's elevator frame. That is sufficient to determine whether the difference will be small enough to ignore.

 

[PeterDonis]

I think in this case the Newtonian approximation can be used to calculate the centrifugal acceleration.

The issue is not what approximation is valid for calculating a numerical answer. The issue is what physical quantities are involved in determining the redshift, and whether any of them can be described as "the relative velocity between the source and the emitter". For reasons which should be evident from my posts in this thread, I think the answer to the latter question is "no".

 

[PeterDonis]

We routinely use frame-dependent quantities to calculate the value of physical invariants.

That's true as a matter of calculation, but that's simply because you can calculate an invariant in any frame you wish, using any quantities you wish, by hook or by crook, as long as the final result is an invariant.

But that doesn't change the fact that anything you're going to give as a physical cause of something, such as the observed P&R redshift, must be an invariant. "The relative velocity between source and emitter in an Earth-centered inertial frame" is not an invariant, so it can't be a valid contributing cause to the observed redshift. Whether it can figure in a calculation of a valid invariant is a separate question (but as should be evident from my post in response to @Dale's suggestion, I think the answer to that is still "no").

 

[PeterDonis]

Here the value of the frame-dependent relative velocity is that is an easily calculated proxy for how much the non-inertial rotating rest frame in which the experiment is conducted differs from an idealized Einstein's elevator frame.

I disagree. I think @Dale's answer here is correct: the relevant invariant is the proper acceleration $g$, and the "effect of rotation of the Earth" is the difference between the $g$ of the actual experimental apparatus, which is rotating with the Earth, and what $g$ would be in for a "hovering" apparatus at the same altitude (and latitude and longitude), but not rotating with the Earth. There is nothing like a "relative speed between source and emitter" appearing anywhere in such a calculation.

 

[Vanadium 50]

As I have already pointed out, this "velocity difference"

...and as I have pointed out makes a change in the twelfth decimal place. (I think it's actually the eleventh), Is that really the hill you want to charge up?

 

[PeterDonis]

as I have pointed out makes a change in the twelfth decimal place.

Sure, this is a good argument for this physical effect being way too small to matter in this experiment, if you have already established that it's a valid physical effect.

But if it's not a valid physical effect to begin with (and I have given reasons why it isn't, which nobody has given a valid argument against), then the number you are quoting is simply a meaningless number, no matter what formula you plugged numbers into to get it.

If @Dale is correct (and as I've said, I think he is) that the proper way to capture the effect of the Earth's rotation is in the difference in $g$ between the rotating and "hovering" cases, then that difference is much, much larger than your number in the quote above: it's roughly a tenth of a percent, i.e., the third decimal place, not the twelfth. So your number quoted above is not only physically meaningless, but actively misleading, since it has led you to overlook a valid physical effect that is eight or nine orders of magnitude larger.

 

[Sagittarius A-Star]

The issue is what physical quantities are involved in determining the redshift, and whether any of them can be described as "the relative velocity between the source and the emitter". For reasons which should be evident from my posts in this thread, I think the answer to the latter question is "no".

In the OP it does not say "the relative velocity between the source and the emitter", but ...

the speed difference between the source and receiver at different circumferences in the P&R experiment

... and the calculation in the OP makes clear, that the reference frame for this statement is the ECI.

In the ECI, you can calculate invariant quantities, if you use the Newtonian approximation, then argue, that the centrifugal acceleration of sender and receiver is approximately the same $R\omega^2$ (with $h/R = 22.5 m / 6371 km$), show, that the centrifugal contribution is significantly smaller than the gravitational contribution, and then continue locally in the restframe of the floor of the building with SR.

 

[Vanadium 50]

which nobody has given a valid argument against

Except Dale.

But I am bailing out of this thread. I've been here long enough to see the outcome of people who argue with you.

 

[PeterDonis]

In the OP it does not say "the relative velocity between the source and the emitter"

It says "speed difference", yes. That's just a quibble over terminology. You can go back and make that substitution in all of my posts; it won't change anything of substance.

In the ECI, you can calculate invariant quantities

Sure, but the "speed difference" is not one of them.

 

[PeterDonis]

Except Dale.

As far as I can tell, @Dale agrees with me. If I'm mistaken about that, he can correct me.

 

[PeterDonis]

Thread closed for moderation.