#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #60 on: June 16, 2020, 07:58:31 PM »
Two observers on a rocket accelerating at the same rate
At the same rate relative to what? In what direction? It's impossible to judge the correctness of your statement because it doesn't even come close to being complete. In the specific scenario you proposed (Earth and stars under UA), the Earth and stars are stationary relative to one another as far as UA is concerned. In other words, a=0.

Once again, you talk of speed, distance, and an individual photon (for which the concept of Doppler shifts is meaningless). You need to be talking about waves and wavelengths. Until you do so, you're not even discussing the Doppler effect in any meaningful fashion. There's a reason this failure is significant. As much as your upward acceleration will speed up the rate at which the waves hit you, the upward acceleration of the source will slow that rate down. If the two vectors are identical, the effect will cancel out. This is why the Doppler effect concerns the effect of the relative motion of two bodies on the wavelengths of the wave.

I'm not sure you understand how the Equivalence Principle works where Doppler effects are concerned.

EP says that the gravitational redshifts / blueshifts observed near the surface of the Earth, is the same when observed inside a rocket accelerating with the same magnitude.

For example, if two observers on a rocket that's stood upright on the surface of the Earth were to send light pulses to one another, then the light detected at the top of the rocket will be Redshifted (longer wavelength) and the light received at the bottom will be blueshifted (shorter wavelength). This has been verified by the Pound–Rebka experiment.

The change in wavelength is given by:
Where is the wavelength at emission, and is the height between light emitted and detected.

On the same rocket but now accelerating at 1g in space away from any gravity, light signals sent from the bottom and top of the rocket will be reshifted and blueshifted respectively.

The change in wavelength is given by:

Where is the acceleration, and is the distance between light emitted and received. If the acceleration or distance between observers increases then the Doppler effect will also increase.

Light doesn't travel within reference frames, it propagates through space independently, however all observers in their reference frames will observe the light travelling at c. on the accelerated rocket, when light is emitted from the top of the rocket , then the bottom of the rocket is travelling towards it and thus will see it blueshifted (shorter wavelength).

#### omikun

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##### Re: Doubt in Universal Acceleration
« Reply #61 on: June 18, 2020, 12:51:19 AM »
Two observers on a rocket accelerating at the same rate
At the same rate relative to what? In what direction? It's impossible to judge the correctness of your statement because it doesn't even come close to being complete. In the specific scenario you proposed (Earth and stars under UA), the Earth and stars are stationary relative to one another as far as UA is concerned. In other words, a=0.

I think you have it exactly right. If UA is accelerating everything, the earth and stars are stationary relative to each other, and a=0. Like wise, we should be under the influence of UA, and we should experience a=0.

But when you drop a ball, that ball is not being accelerated by UA. The ground is, and it rushes up to the ball. When you pick a piece of dirt off the ground, that also is not accelerated by UA, so it too will "fall" when let go. In short, nothing that we know of on earth is accelerated by UA, only whatever underneath the known "earth" is accelerated by UA. If that is the case, UA isn't universal.

If UA is universal, we would all be floating about and feel no acceleration, because, as you say, a=0. Just as the stars and earth is stationary relative to each other, you, me, this keyboard, etc, we should all be stationary relative to each other.

#### Pete Svarrior

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##### Re: Doubt in Universal Acceleration
« Reply #62 on: June 18, 2020, 08:55:18 AM »
Freefall at terminal is a very noisy place but there is no sensation of falling at all.  Only pre terminal.
Well, yes, once you stop accelerating, your inner ear stops perceiving the effects of the air's acceleration relative to you. What... what exactly is your objection here?

Like wise, we should be under the influence of UA, and we should experience a=0.
Incorrect. Please, at the very least read the article describing UA before posting here.

On the same rocket but now accelerating at 1g in space away from any gravity
Once again, for those in the back: the stars are not a rocket, and they are not accelerating away from the Earth. They are not moving away from the Earth, or at least are not doing so due to UA. As far as UA is concerned, the height of the stars relative to the Earth's surface remains static.

Your analogy would make some sense if you were to replace your source of gravity with an upward acceleration of a rocket. However, you're not doing that. You're adding more acceleration to the mix and acting surprised that your results have changed.  That's not equivalence, now is it?

Where is the acceleration
a is 0. Therefore, . I'll let you crunch the numbers from there.

The Doppler effect occurs when two bodies are in motion relative to one another. Reference frames don't come into this. You can use one of the bodies as a frame of reference to help you simplify the task of drawing a diagram, but you can just as well do it from an external inertial FoR.

I would strongly suggest that you do this, even if just in your imagination. I'll borrow some diagrams from Wikipedia. Consider a source of waves like this:

Now imagine it's moving away from you. Imagine the observer is located in the middle of the left-hand-side y axis. Whether the source accelerates or not is not important, since we're only discussing whether the effect is present at all, not its magnitude. The presence of relative motion will therefore be good enough.

You can, of course, see that a point on the left y axis would now observe a Doppler shift.

Finally, imagine that the point is not stationary on the y axis, but is rather moving in sync with the source. We can draw it from two perspectives. One of them uses the source as a frame of reference. That's just a repeat of the first diagram I've linked. But we can also draw it from an inertial perspective for your benefit.

Even though the source is moving, so is the observer. Since their speeds are matched, the observer picks up the waves' peaks at the exact same rate as if both objects were stationary. Consequently, the wavelength is unchanged compared to the two objects being stationary within the inertial FoR.

The Doppler effect occurs when the source and observer are in motion relative to one another. The Equivalence Principle doesn't come into this. The two bodies are not in motion relative to one another (as far as UA is concerned), so there is no possible Doppler shift to be seen as a result of UA.
« Last Edit: June 18, 2020, 03:59:33 PM by Pete Svarrior »

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#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #63 on: June 18, 2020, 02:25:50 PM »
On the same rocket but now accelerating at 1g in space away from any gravity
Once again, for those in the back: the stars are not a rocket, and they are not accelerating away from the Earth. They are not moving away from the Earth, or at least are not doing so due to UA. As far as UA is concerned, the height of the stars relative to the Earth's surface remains static.

Your analogy would make some sense if you were to replace your source of gravity with an upward acceleration of a rocket. However, you're not doing that. You're adding more acceleration to the mix and acting surprised that your results have changed.  That's not equivalence, now is it?

Where is the acceleration
a is 0. Therefore, . I'll let you crunch the numbers from there.

The Doppler effect occurs when two bodies are in motion relative to one another. Reference frames don't come into this. You can use one of the bodies as a frame of reference to help you simplify the task of drawing a diagram, but you can just as well do it from an external inertial FoR.

I would strongly suggest that you do this, even if just in your imagination. I'll borrow some diagrams from Wikipedia. Consider a source of waves like this:

Now imagine it's moving away from you. Imagine the observer is located in the middle of the left-hand-side y axis. Whether the source accelerates or not is not important, since we're only discussing whether the effect is present at all, not its magnitude. The presence of relative motion will therefore be good enough.

You can, of course, see that a point on the left y axis would now observe a Doppler shift.

Finally, imagine that the point is not stationary on the y axis, but is rather moving in sync with the source. We can draw it from two perspectives. One of them uses the source as a frame of reference. That's just a repeat of the first diagram I've linked. But we can also draw it from an inertial perspective for your benefit.

Even though the source is moving, so is the observer. Since their speeds are matched, the observer picks up the waves' peaks at the exact same rate as if both objects were stationary. Consequently, the wavelength is unchanged compared to the two objects being stationary within the inertial FoR.

The Doppler effect occurs when the source and observer are in motion relative to one another. The Equivalence Principle doesn't come into this. The two bodies are not in motion relative to one another (as far as UA is concerned), so there is no possible Doppler shift to be seen as a result of UA.

Pete, all those diagrams you have used above are Doppler shifts for 'sound waves', so are not relevant for this discussion as we are talking about light waves. I suppose the last diagram is a little bit more on the lines but its still sound waves and the source and observer are travelling at constant speed they are nor accelerating.

It seems you are going against the Equivalence principle here Pete. I've just had a look on your wiki and it even states what I'm trying to explain to you, here:

Quote
One can also see the role of the equivalence principle by considering a pulse of light emitted over a distance h along the axis of a spaceship in uniform acceleration g in outer space. The time taken for the light to reach the detector is t = h (we use units G = c = 1). The difference in velocity of the detector acquired during the light travel time is v = gt = gh, the Doppler shift z in the detected light. This experiment, carried out in the gravity-free environment of a spaceship whose rockets produce an acceleration g, must yield the same result for the energy shift of the photon in a uniform gravitational field f according to the equivalence principle. The Pound-Rebka-Snyder experiments can therefore be regarded as an experimental proof of the equivalence principle. ”

In the The Pound-Rebka Experiment light pulses were emitted from source to detector with a height of 22.5 m (note, source and detector are NOT moving relative to one another) and yet they detected a 'Doppler shift' due to gravity. Einstein's EP says that if you carry out the same experiment onboard a rocket accelerating at the same magnitude 1g, then you will see the same Doppler effects. Do you agree?

#### Pete Svarrior

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##### Re: Doubt in Universal Acceleration
« Reply #64 on: June 18, 2020, 03:49:37 PM »
Pete, all those diagrams you have used above are Doppler shifts for 'sound waves', so are not relevant for this discussion as we are talking about light waves.
The Doppler effect applies to all waves. It applies to sound waves, light waves, ripples in a disturbed body of water, vibrations of a piece of string - waves. If you believe that sound waves would behave differently in your scenario, you'll have to state why. If you believe that the diagrams do not apply to the perception of light, please feel free to provide ones of your own which illustrate the same scenario, and highlight any corrections that you believe are necessary.

I suppose the last diagram is a little bit more on the lines but its still sound waves and the source and observer are travelling at constant speed they are nor accelerating.
I already explained why this is irrelevant. One: the bodies are stationary relative to one another - this will not change if you accelerate the entire system. Two: we are discussing the presence of a Doppler shift, not its magnitude over time. As such, we only need to concern ourselves with the relative motion of the two bodies (or lack thereof, as the case may be). The magnitudes of said (non-)motion are irrelevant. If you disagree, you will have to state why. Specifically, you will have to address my position.

It seems you are going against the Equivalence principle here Pete.
Not at all. You made two errors:
• You chose one two scenarios which are not equivalent: gravity, and gravity plus acceleration. This is incorrect. For EP to apply, the gravitational element would have to be absent in the second scenario.
• You assumed that the Doppler effect will occur between two bodies who are stationary to each other, as long as they're in motion relative to some other observer. This is a complete misunderstanding of the Doppler effect.
Your first failure is a problem because of EP. The second one is irrespective of EP. Regardless of which catastophe you choose to fix first, repeatedly crying about EP will not change this.

There is one more failure that I raised early on, which I don't want you to forget just yet:
• You assume that UA is the only source of motion/acceleration of the stars relative to the Earth.
Once you've tidied up your messy claims, resolving this failure will finally eliminate all outstanding contradictions.

The Pound-Rebka Experiment
I already explained this to you several times. You're taking an experiment which concerns a gravity-free environment, thwacking it into an environment that is not gravity-free, and pondering super hard about why your results are not working out for you. You'd have to remove gravity/UA from your scenario to obtain reasonable results.

You cannot discuss RET vs FET with such an abysmal understanding of RET. Pick up a physics book and start learning. In the meantime, I'm not interested in more "nuh-uh!" responses. Either state your logic, or start improving your education.
« Last Edit: June 18, 2020, 04:17:04 PM by Pete Svarrior »

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#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #65 on: June 18, 2020, 10:24:16 PM »
I suppose the last diagram is a little bit more on the lines but its still sound waves and the source and observer are travelling at constant speed they are nor accelerating.
I already explained why this is irrelevant. One: the bodies are stationary relative to one another - this will not change if you accelerate the entire system. Two: we are discussing the presence of a Doppler shift, not its magnitude over time. As such, we only need to concern ourselves with the relative motion of the two bodies (or lack thereof, as the case may be). The magnitudes of said (non-)motion are irrelevant. If you disagree, you will have to state why. Specifically, you will have to address my position.

It seems you are going against the Equivalence principle here Pete.
Not at all. You made two errors:
• You chose one two scenarios which are not equivalent: gravity, and gravity plus acceleration. This is incorrect. For EP to apply, the gravitational element would have to be absent in the second scenario.
• You assumed that the Doppler effect will occur between two bodies who are stationary to each other, as long as they're in motion relative to some other observer. This is a complete misunderstanding of the Doppler effect.

I will try to explain (again) how we can see Doppler shifts when two observers are stationary relative to one another in a uniform accelerating frame.
Einstein's elevator, have a look at this:

If the Pound-Rebka experiment was carried out on the accelerating elevator away from any gravity then according to Einstein they would've had the same results with a Doppler shift z of

Using the Doppler shift formula in my earlier post you can see that the expected value for a Doppler shift in an accelerating frame where both observer are 22.6 m apart and do not move relative to one another is:

plug in the values to give

Not only would both observers see Doppler shifts, they would also see the effects of time dilation, the clock at the top of the elevator will tick faster that the one at the bottom one, which is same effects in gravity close to the surface of the Earth.

Are you winding me up Pete or what?

« Last Edit: June 20, 2020, 06:56:18 PM by Groit »

#### Pete Svarrior

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##### Re: Doubt in Universal Acceleration
« Reply #66 on: June 20, 2020, 11:47:08 AM »
I will try to explain (again) how we can see Doppler shifts when two observers are stationary relative to one another in a uniform accelerating frame.
You don't need to keep re-explaining it. You need to fix the errors in your claims. I even provided you with a handy list. Are you going to get started, or are we done here?

Once again, in case you forgot: Your failure is not in thinking that UA should produce an identical shift to RET's gravity. Your failure is in thinking that it doesn't, or, to be more precise, that it's not being outweighed by factors external to UA. Your abysmal misuse of formulae doesn't help, but it's your lack of basic understanding that's leading you there in the first place. Start unraveling the failures I've listed and it'll click. If you don't want to, well, I can't force you.
« Last Edit: June 20, 2020, 12:01:25 PM by Pete Svarrior »

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#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #67 on: June 20, 2020, 07:41:17 PM »
Pete, it looks like I've had a problem with embedding images on the forum. The images were showing for me and not everyone
I have just edited 4 of my posts in this thread that should have had diagrams. Here's my last post with the diagram that i thought was included before, and if you would like to go back to my previous posts (and any others interested) then it should become much clearer. Sorry about this.

I suppose the last diagram is a little bit more on the lines but its still sound waves and the source and observer are travelling at constant speed they are nor accelerating.
I already explained why this is irrelevant. One: the bodies are stationary relative to one another - this will not change if you accelerate the entire system. Two: we are discussing the presence of a Doppler shift, not its magnitude over time. As such, we only need to concern ourselves with the relative motion of the two bodies (or lack thereof, as the case may be). The magnitudes of said (non-)motion are irrelevant. If you disagree, you will have to state why. Specifically, you will have to address my position.

It seems you are going against the Equivalence principle here Pete.
Not at all. You made two errors:
• You chose one two scenarios which are not equivalent: gravity, and gravity plus acceleration. This is incorrect. For EP to apply, the gravitational element would have to be absent in the second scenario.
• You assumed that the Doppler effect will occur between two bodies who are stationary to each other, as long as they're in motion relative to some other observer. This is a complete misunderstanding of the Doppler effect.

I will try to explain (again) how we can see Doppler shifts when two observers are stationary relative to one another in a uniform accelerating frame.
Einstein's elevator, have a look at this:

If the Pound-Rebka experiment was carried out on the accelerating elevator away from any gravity then according to Einstein they would've had the same results with a Doppler shift z of

Using the Doppler shift formula in my earlier post you can see that the expected value for a Doppler shift in an accelerating frame where both observer are 22.6 m apart and do not move relative to one another is:

plug in the values to give

Not only would both observers see Doppler shifts, they would also see the effects of time dilation, the clock at the top of the elevator will tick faster that the one at the bottom one, which is same effects in gravity close to the surface of the Earth.

Are you winding me up Pete or what?

« Last Edit: June 20, 2020, 07:43:21 PM by Groit »

#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #68 on: June 20, 2020, 10:37:19 PM »
Not at all. You made two errors:
• You chose one two scenarios which are not equivalent: gravity, and gravity plus acceleration. This is incorrect. For EP to apply, the gravitational element would have to be absent in the second scenario.

Where did i say gravity plus acceleration?

• You assumed that the Doppler effect will occur between two bodies who are stationary to each other, as long as they're in motion relative to some other observer. This is a complete misunderstanding of the Doppler effect.

Pete, two observers that are stationary to each other in an accelerating frame, away from any gravity, really do see the Doppler effect. Its part of the equivalence principle. Have a look at the diagrams i just recently added. If you are saying this is wrong, then you are saying Einstein is wrong also.

There is one more failure that I raised early on, which I don't want you to forget just yet:
• You assume that UA is the only source of motion/acceleration of the stars relative to the Earth.

Once you've tidied up your messy claims, resolving this failure will finally eliminate all outstanding contradictions.

Lets forget about the stars for now and just concentrate on the Earth. In FET the Earth is accelerating upwards at 1g correct? If this was true then not only would we observe Doppler shifts equivalent to a gravitational redshift which has been verified in many experiments such as the Pound-Rebka experiment, but we would also observe a redshift / blueshift drift that will be time dependent.

This is one difference between gravity and acceleration and the 'equivalence principle' only holds from the first moments of acceleration.

Here's a diagram showing the redshift drift for uniform accelerating frames, and also a link to the paper which is from a good source. The math gets pretty intense for further reading, because it talks about Rindler coordinates etc and i don't think any of us on here are experts in GR are we? anyway the first section should be enough to get the drift.

https://arxiv.org/pdf/1907.06332.pdf

[/list][/list]
« Last Edit: June 20, 2020, 10:38:53 PM by Groit »

#### Pete Svarrior

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##### Re: Doubt in Universal Acceleration
« Reply #69 on: June 21, 2020, 12:03:40 AM »
Where did i say gravity plus acceleration?
Every time you describe "a rocket accelerating away from any gravity". In other words: all the time, everywhere, without respite.

Pete, two observers that are stationary to each other in an accelerating frame, away from any gravity, really do see the Doppler effect.
Right, but that's not the signifciant factor in the shift you're observing from the stars. I pointed this out very early on, so there's no way you could possibly still be rambling about that. You're talking about the shift that supposedly contradicts observation. Hint: it doesn't, and as soon as you've tidied up your mess, that much will be obvious.

Lets forget about the stars for now and just concentrate on the Earth.
No, let's not. I'm not interested in your excuses and diversions. Fix your messy claims. If you do not want to take responsibility for your failures, then stop wasting our time.
« Last Edit: June 21, 2020, 12:22:30 AM by Pete Svarrior »

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#### Groit

##### Re: Doubt in Universal Acceleration
« Reply #70 on: July 04, 2020, 10:44:40 PM »
Lets forget about the stars for now and just concentrate on the Earth.
No, let's not. I'm not interested in your excuses and diversions. Fix your messy claims. If you do not want to take responsibility for your failures, then stop wasting our time.

We don't need to use the stars, as we could perform an experiment at the Earth's surface to determine whether we are accelerating due to 'gravity' or 'UA'.
In fact the Pound-Rebka (and more recent) experiments have done just that.

These experiments were carried out to test GR and show that light will lose or gain energy in a gravitational field depending which way the photons are emitted and received. For the Pound-Rebka experiment, photons were dropped from a height of 22.6 m towards the surface of the Earth and they successfully detected a blueshift close to the predicted value.

This experiment was repeated many times over long periods and always gave the same results with a blueshift (z) of:

If the Pound-Rebka experiment was carried out in an accelerating frame (without gravity) such as a rocket or in this case the Earth accelerating upwards (UA), then the results would be somewhat different. This is due to the 'blueshift drift' effect for accelerating frames. The formula for the expected z values is given by:    which is time dependent.

The plot below shows blueshift (z) against time, if the same experiment was carried out in gravity (red line) and an accelerating frame such as UA (blue line). For the accelerating frame, we have blueshift drift, and for gravity no blueshift drift would be detected. The time axis is in seconds and goes to six months.

The Pound-Rebka (and other) experiments never detected any blueshift drift over time, therefore we must be in a gravitational field.

Using the Equivalence Principle as evidence to support UA is fine for things like how objects fall, projectile motion etc... but when it comes to the nature of light and how we observe its Doppler shifts over periods of time then it no longer holds. This flaw within the Equivalence Principle is a way to distinguish between an accelerating frame and gravity.

#### Pete Svarrior

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##### Re: Doubt in Universal Acceleration
« Reply #71 on: July 04, 2020, 10:46:35 PM »
We don't need to use the stars
No, no, let's not change your claims. Let's simply correct the errors within. You were provided with a list. Get on with it.

And no, your core failure continues to become more pronounced every time you post. You assume that the only source of Doppler shitfts in light is UA/gravity, or that you can somehow isolate it within an Earth-bound observation. It isn't, and you can't.
« Last Edit: July 04, 2020, 10:48:20 PM by Pete Svarrior »

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#### Jeb Kermin

• 10
##### Re: Doubt in Universal Acceleration
« Reply #72 on: July 13, 2020, 11:14:07 PM »
If the Pound-Rebka experiment was carried out in an accelerating frame (without gravity) such as a rocket or in this case the Earth accelerating upwards (UA), then the results would be somewhat different. This is due to the 'blueshift drift' effect for accelerating frames. The formula for the expected z values is given by:    which is time dependent.

The plot below shows blueshift (z) against time, if the same experiment was carried out in gravity (red line) and an accelerating frame such as UA (blue line). For the accelerating frame, we have blueshift drift, and for gravity no blueshift drift would be detected. The time axis is in seconds and goes to six months.

Using the Equivalence Principle as evidence to support UA is fine for things like how objects fall, projectile motion etc... but when it comes to the nature of light and how we observe its Doppler shifts over periods of time then it no longer holds. This flaw within the Equivalence Principle is a way to distinguish between an accelerating frame and gravity.

Hmm, but I thought the idea of the EP, was that there was no local experiment you can do which differentiated between free fall and an accelerating frame relative to some observer who is not accelerating or experiencing free fall.

Now the catch here is "local", because as gravity falls off with the square of the distance, the acceleration is not constant, so your experiment/measurement has to take this into account.  Was that the point of the experiment ?

This leads to another hypothetical experiment, which might be a bit easier to understand.  I was reading the Wiki on UA.  Apparently the "gravitational anomalies" as determined by things like scale measurements are dismissed out of hand because the scales aren't re calibrated (which is kind of the point, but lets ignore that for now)

I don't see a claim whehter UA is predicting an acceleration which is invariant with altitude, or makes some attempt to account for it, if it can be observed.

Edit:  Sorry, there does appear to be a reference to "Celestial gravitation" which causes the acceleration to fall off with altitude, so apparently some celestial bodies exhibit gravitational pull but apparently it does not apply to Earth, at least not the same way).   However, I can't find how this would be quantified.   Gravitational theory predicts that g should fall off with altitude, with the square of the distance between the two bodies.

I'm wondering if an experiment whereby free fall was measured, with say with a ball bearing in a vacuum tube, perhaps suspended by an electromagnet, released with electronics which was tied to a precise timer which is demagnetizes and stops when a sensor at the bottom is hit.

In constant acceleration, the velocity v =  a * t, assuming initial velocity 0.  Which means the position, we'll call height h, is the antideriviative, so is h =  0.5 * a * t^2, again assuming initial height of 0.  Solving for t, we get t = sqrt(h / (0.5 * a))

If the tube were 1 m, then at the surface of the Earth (g = 9.806 m/s^2), the free fall time should be the sqrt (1 m /(0.5 * 9.806 m/s^2)) = 0.4516 s, or 451.6 ms.   At an altitude of 10 km, which is typical of airlines flight, g is predicted to be closer to 9.776 m/s^2, which yields a drop time of 1/(0.5 * 9.776 m/s^2) =   452.3 ms, so a difference of 0.7 ms.  Not exactly easy to measure, but with decent electronics, shouldn't be hard at all, as its only a frequency of ~1500 Hz.

Of course we would have to be sure the  plane is not accelerating, and that the tube was perfectly level.
« Last Edit: July 13, 2020, 11:40:31 PM by Jeb Kermin »

#### ljman

• 1
##### Re: Doubt in Universal Acceleration
« Reply #73 on: July 14, 2020, 09:17:15 PM »
Quote
Hmm, but I thought the idea of the EP, was that there was no local experiment you can do which differentiated between free fall and an accelerating frame relative to some observer who is not accelerating or experiencing free fall

The local aspect of the equivalence principle is mentioned, but is pretty glossed over.  I'd be interested in how flat earthers define "locally"