# Free fall acceleration in SR

#### harrylin

This is a spin-off of a parallel discussion, starting from:

The question is what SR predicts that an accelerometer in free-fall will read. This issue may be simply due to different people using a different meaning of "SR", but it could have a deeper cause.

A basic reference for this discussion:
Einstein 1905, http://www.fourmilab.ch/etexts/einstein/specrel/www/
and another one for context:
Langevin 1911, http://en.wikisource.org/wiki/The_Evolution_of_Space_and_Time

As you make your $\delta \tau$ small the SR predicted accelerometer reading becomes large while the actual accelerometer reading remains 0. [..]
[..] SR predicts a very large accelerometer reading during the turnaround, and real free falling accelerometers read 0.
SR uses the inertial frames of classical mechanics; in my opinion it's obvious that SR doesn't predict that an accelerometer in free fall will indicate a large acceleration. That conflicts with the known laws of physics, even of classical mechanics.

Arguments in favor of both opinions may help to clarify this issue.

Related Special and General Relativity News on Phys.org

#### Dale

Mentor
SR predicts that an ideal accelerometer reads:$$a^{\mu}=\frac{d^2 x^{\mu}}{d\tau^2}$$Where x is the worldline of the accelerometer in a SR inertial frame and τ is the proper time along that worldline.

If you claim that a frame can be treated as a SR inertial frame, then the above formula is what SR predicts for the proper acceleration of objects.

The question is what SR predicts that an accelerometer in free-fall will read.
That question cannot even be addressed by SR since SR does not handle gravitation. You cannot generally establish a self-consistent SR inertial frame in the presence of gravity. This question is outside the domain of applicability of SR.

The real question is when is it appropriate to use SR as an approximation in a scenario where there is gravity. The answer to that question is that it is appropriate to do so when the resulting errors for the measured quantities in the scenario are small. That is not the case in a gravitational turn-around twin scenario.

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#### harrylin

SR predicts that an ideal accelerometer reads:$$a^{\mu}=\frac{d^2 x^{\mu}}{d\tau^2}$$Where x is the worldline of the accelerometer in a SR inertial frame and τ is the proper time along that worldline.

If you claim that a frame can be treated as a SR inertial frame, then the above formula is what SR predicts for the proper acceleration of objects. [..]
Your definition does not appear in the provided references which don't even use that term; I can find no reason for expecting such a flagrant error in SR. The first reference defines SR wrt the inertial frames of classical mechanics; wrt such frames the laws of Newton hold in good approximation. An ideal accelerometer that in free fall reads zero would be a colossal erroneous modification of Newton's mechanics. Please provide a reliable reference to back up that claim.
[...] You cannot generally establish a self-consistent SR inertial frame in the presence of gravity. [..]
SR was always assumed to be applicable on Earth (of course, in good approximation!); for example CERN uses it in the presence of gravity.
The real question is when is it appropriate to use SR as an approximation in a scenario where there is gravity. The answer to that question is that it is appropriate to do so when the resulting errors for the measured quantities in the scenario are small. That is not the case in a gravitational turn-around twin scenario.
That's certainly not the topic of this thread, and the preliminary discussion was from about here: https://www.physicsforums.com/showthread.php?p=4281150. I hold that by design the resulting error for that case must be small whatever the details of the turn-around, as it was meant to be of sufficiently short duration to be neglected. It's certainly worthy of its own thread, with a detailed calculation.

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#### stevendaryl

Staff Emeritus
SR predicts that an ideal accelerometer reads:$$a^{\mu}=\frac{d^2 x^{\mu}}{d\tau^2}$$Where x is the worldline of the accelerometer in a SR inertial frame and τ is the proper time along that worldline.
You can say that that's true by definition of "ideal accelerometer", but you can't prove that there is any actual device that is an ideal accelerometer.

The canonical example of an accelerometer that I always use is a cubical box, with a ball suspended in the center by 6 identical springs connected to the center of each wall of the box. You measure acceleration by the deflection of the ball from the center.

But if there were a force that pulled springs, ball and box in proportion to their masses, then acceleration due to this force would not be measurable.

Another alternative for an accelerometer is to use light beams: if the light beam travels straight, then the device producing the beam is not accelerating. If the light beam curves in one direction or another, then that indicates acceleration. Maybe this can be used to measure acceleration? It all depends on whether light itself is affected by forces.

#### Mentz114

Gold Member
harrylin said:
An ideal accelerometer that in free fall reads zero would be a colossal erroneous modification of Newton's mechanics. Please provide a reliable reference to back up that claim.
Did you mean to write this ? You must mean something completely different from what I understand to be 'free fall'.

Mentor

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#### Dale

Mentor
The canonical example of an accelerometer that I always use is a cubical box, with a ball suspended in the center by 6 identical springs connected to the center of each wall of the box. You measure acceleration by the deflection of the ball from the center.

But if there were a force that pulled springs, ball and box in proportion to their masses, then acceleration due to this force would not be measurable.
That is correct. The only such forces within the domain of applicability of SR are fictitious forces, and fictitious forces are not measured by accelerometers. The only other such force in any mainstream theory is Newtonian gravity, which is incompatible with SR and therefore outside its domain of applicability.

Another alternative for an accelerometer is to use light beams: if the light beam travels straight, then the device producing the beam is not accelerating. If the light beam curves in one direction or another, then that indicates acceleration. Maybe this can be used to measure acceleration? It all depends on whether light itself is affected by forces.
I believe that one consequence of the first postulate is that your mechanical and your optical accelerometers will read the same.

Gold Member

#### WannabeNewton

Did you mean to write this ? You must mean something completely different from what I understand to be 'free fall'.
I would like to know what he/she meant by that too because it certainly isn't my understanding of free fall.

#### stevendaryl

Staff Emeritus
Did you mean to write this ? You must mean something completely different from what I understand to be 'free fall'.
I wasn't sure about that, either, but he might mean that according to Newtonian physics, an object in freefall is accelerating under the force of gravity. So an "accelerometer" that measures zero in freefall isn't correctly measuring the Newtonian notion of acceleration. But then, that just means that there is no (localized) device that can measure acceleration in the Newtonian sense.

#### Mentz114

Gold Member
I would like to know what he/she meant by that too because it certainly isn't my understanding of free fall.
I wasn't sure about that, either, but he might mean that according to Newtonian physics, an object in freefall is accelerating under the force of gravity. So an "accelerometer" that measures zero in freefall isn't correctly measuring the Newtonian notion of acceleration. But then, that just means that there is no (localized) device that can measure acceleration in the Newtonian sense.
Yes, I guess that's what he means. In the other thread GregAshmore mentions 'unbalanced forces'. I suppose an accelerometer can be said to measure resistance ( reaction ) to an applied force ( speaking in Newtonian terms ).

It shows how tricky gravity is an why it makes sense to geometrise it.

#### PeterDonis

Mentor
SR doesn't predict that an accelerometer in free fall in flat spacetime will indicate a large acceleration.
You should have included the bolded phrase; as DaleSpam pointed out, it's crucial. SR can't deal with curved spacetime; the spacetime in the "gravitational turnaround" scenario, where the traveling twin is in free fall the whole time, is curved. It has to be, otherwise the traveling twin couldn't be in free fall the whole time.

#### PeterDonis

Mentor
SR was always assumed to be applicable on Earth (of course, in good approximation!); for example CERN uses it in the presence of gravity when the approximation is good enough.
You left out a phrase here too, and again it's crucial. The CERN people don't analyze experiments including the effects of gravity; they analyze experiments in a local inertial frame in which the effects of gravity are negligible. That's why they can use SR for the analysis: because SR is still valid within a local inertial frame.

#### harrylin

Did you mean to write this ? You must mean something completely different from what I understand to be 'free fall'.
Obviously I did not, thanks for pointing that out! Indeed, I meant the inverse, as I stated earlier and even in my first post here. An ideal [mechanical] accelerometer that in free fall doesn't read zero would be a colossal erroneous modification of Newton's mechanics.

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#### harrylin

See these references:
http://physicspages.com/2011/05/25/acceleration-in-special-relativity/
http://www.mth.uct.ac.za/omei/gr/chap2/node2.html [Broken] and http://www.mth.uct.ac.za/omei/gr/chap2/node4.html [Broken]
http://en.wikipedia.org/wiki/Four-velocity#Definition_of_the_four-velocity and http://en.wikipedia.org/wiki/Four-acceleration

The formula is correct. If you disagree, then please provide the formula you believe is correct.
I checked your first two references, which according to my browser do not even contain the word "accelerometer". I didn't look further.

ADDENDUM: I overlooked that you asked me to state the obvious. You suggest(ed?) that SR proposed laws that were known to be erroneous. All parts of an accelerometer fall at the same rate in a homogeneous field; to suggest that according to SR this known fact would not be true not only makes no sense to me, it also doesn't follow from either the postulates or the Lorentz transformations.

BTW, I suddenly notice in post #2 a change of opinion about the SR predicted accelerometer reading; that's good, it means that such discussions have merit. It's very well possible that I'll similarly change my opinion, but I need to see some substantial argument.

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#### harrylin

[..] I believe that one consequence of the first postulate is that your mechanical and your optical accelerometers will read the same.

#### harrylin

You should have included the bolded phrase; as DaleSpam pointed out, it's crucial. SR can't deal with curved spacetime; the spacetime in the "gravitational turnaround" scenario, where the traveling twin is in free fall the whole time, is curved. It has to be, otherwise the traveling twin couldn't be in free fall the whole time.
With "fall" I mean the standard meaning of "falling" in SR and English, of a freely moving object in a gravitational field. That is in contrast to "inertial".

#### harrylin

You left out a phrase here too, and again it's crucial. The CERN people don't analyze experiments including the effects of gravity; they analyze experiments in a local inertial frame in which the effects of gravity are negligible. That's why they can use SR for the analysis: because SR is still valid within a local inertial frame.
The surface of the Earth is not a local inertial frame, but as discussed earlier we fully agree on the fact that we work with approximations. You appear to hold that SR doesn't make any prediction in a gravitational field. However, CERN certainly is in a gravitational field.

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#### PeterDonis

Mentor
You appear to hold that SR doesn't make any prediction in a gravitational field.
No, I said that SR can't explain how the traveling twin in the Langevin version of the twin paradox doesn't feel any force during the turnaround. You can still derive the difference in proper time between the stay-at-home twin and the traveling twin using SR, by making the inertial legs very long compared to the turnaround; but that doesn't explain how the turnaround can be made in free fall. And you can't finesse this by saying "well, it's all in a local inertial frame", because the free-fall turnaround cannot be made within a single local inertial frame. This is a crucial difference between that case and the CERN case; CERN experiments can be analyzed within a single local inertial frame because they happen so fast and over such a short span of distance compared to the size of the Earth.

#### Dale

Mentor
I checked your first two references, which according to my browser do not even contain the word "accelerometer". I didn't look further.
The definition I gave and the pages I linked to give the formula for the four velocity and explain it's relationship to proper acceleration and the coordinate acceleration in the momentarily comoving inertial frame. I am sure that you are aware that proper acceleration is the acceleration measured by an accelerometer.

What I asked you to state is the formula that you believe SR uses to predict the reading on an accelerometer, which, with characteristic evasiveness, you have failed to do.

This has become a pattern with you. I refer to some unambiguous mathematical expression, you claim it is wrong, I ask you to provide what you believe to be the correct mathematical expression, and you fail to do so, over, and over, and over, ...

#### harrylin

No, I said that SR can't explain how the traveling twin in the Langevin version of the twin paradox doesn't feel any force during the turnaround.
SR merely makes predictions. This thread examines the repeated claim elsewhere that SR predicts a large accelerometer reading in free fall, which is contrary to the known facts at the time of SR's inception.
You can still derive the difference in proper time between the stay-at-home twin and the traveling twin using SR, by making the inertial legs very long compared to the turnaround; but that doesn't explain how the turnaround can be made in free fall. And you can't finesse this by saying "well, it's all in a local inertial frame", because the free-fall turnaround cannot be made within a single local inertial frame. This is a crucial difference between that case and the CERN case; CERN experiments can be analyzed within a single local inertial frame because they happen so fast and over such a short span of distance compared to the size of the Earth.
Similarly, Langevin's example could be analyzed within one, resp. two universal inertial frames (the ones of SR) because the turn-around was supposed to happen so fast and over such a short time span compared to the duration of the trip that this should be irrelevant for the calculation; however both for CERN as well as Langevin we can only be certain if we do a GR estimation, and it is off-topic here. If you would like to do so, please start it as a new topic; that will be interesting indeed!

#### harrylin

The definition I gave and the pages I linked to give the formula for the four velocity and explain it's relationship to proper acceleration and the coordinate acceleration in the momentarily comoving inertial frame. I am sure that you are aware that proper acceleration is the acceleration measured by an accelerometer.
In the other thread we discussed different definitions of "proper acceleration", and which are irrelevant for a prediction about an accelerometer reading in free fall. Here in the lab we have set up several student experiments with accelerometers, which is why I'm particularly attentive to such things.
What I asked you to state is the formula that you believe SR uses to predict the reading on an accelerometer, which, with characteristic evasiveness, you have failed to do. [..]
The reading of a standard and good accelerometer (mechanical) in free fall can only be zero; I know no postulate of SR that changes that fact. So if you really want this stated as an equation: the following should be approximately true in general and exactly true in free fall for a mechanical accelerometer according SR:
Fbending = m(g-a) = 0 -> adisplayed = "0".

And once more: I did not consider what prediction SR would make for an optical accelerometer; that's an interesting variant.
Addendum: After short consideration I expect SR to predict that an optical accelerometer will indicate acceleration in a gravitational field.

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#### PAllen

An interesting historical note to this thread, which I hope does not distract the discussion, is that in 1911, the same year as Langevin's paper, Einstein had already predicted that any theory of gravity consistent with relativity must include gravitational time dilation. If Langevin had any inkling of this, he certainly made no mention of it in his 1911 paper.

#### harrylin

An interesting historical note to this thread, which I hope does not distract the discussion, is that in 1911, the same year as Langevin's paper, Einstein had already predicted that any theory of gravity consistent with relativity must include gravitational time dilation. If Langevin had any inkling of this, he certainly made no mention of it in his 1911 paper.
I also thought about that fact; however in my thinking (post #21) he had sufficient reason not to mention that detail. More about this would indeed distract from the discussion here.

#### Dale

Mentor
the following should be approximately true in general and exactly true in free fall for a mechanical accelerometer according SR:
Fbending = m(g-a) = 0 -> adisplayed = "0".
Please define your terms. Assuming that g is the acceleration due to gravity from some gravitating masses, then to me this equation seems obviously incompatible with SR. Specifically, there is no way to calculate g in SR.

Please provide a reference for this formula. Specifically, a reference or combination of references which shows that this formula is the formula for SR. Or, at a minimum, a reference for g in SR.

Addendum: After short consideration I expect SR to predict that an optical accelerometer will indicate acceleration in a gravitational field.
If this were true, then mechanical systems and optical systems would have different sets of inertial frames. Which would violate the first postulate.

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