Ehrenfest / rod thought experiment.

[yuiop]

With this artificial speeding up of the clocks nearer the tail, the radar distance measured at the nose would agree with radar distance measured at the tail. Agree?

No. Radar measurements are made using "proper clocks", not "coordinate clocks" and the front-back-front measurement will differ from from the back-front-back measurement. The attached diagram illustrates why. The time taken by the red signal (measured by the red observer) is much longer than the time taken by the blue signal (measured by the blue observer).

Yep, I agreed that radar distance measured at the back would be shorter than the radar distance measured from the front in post #31, where I was conceding the same point made by Passionflower, with the assumption that we are using unsynchronized clocks. However, do you agree that we could arrange a synchronization scheme by suitable fixed multiplication factors of the clock rates, so that all clocks on the rocket appear to be running at the same rate and so that the one way speed of light is the same in both directions, that the front and back radar measurements would then agree with each other?

Do you agree if we only use the natural proper clocks that we use for radar measurements, for all measurements, that the one way speed of light is not isotropic and the proper clocks will be obviously unsynchronised to the observers on the rocket?

P.S. I assume you are saying that "radar distance" unqualified, is a timing of a two way signal made by a "proper clock" by definition. I guess I will have to qualify the radar measurement made by synchronised clocks as "coordinate radar distance" or something. By suitable adjustments this coordinate radar distance can be made to agree with the ruler length in both directions. Whether this coordinated measurement agrees with something that approximates the proper length of the accelerating ship or even the CMIRF measurement of the ship I am not sure about. One problem with the CMIRF measurement is that a CMIRF that is co-moving with the front of the ship is not co-moving with the back of the ship and vice verse so we would have to define a location for the CMIRF somewhere near the middle of the ship.

 

[bcrowell]

This resolution fails, because it introduces the straw man of the "rigid disk". Nowhere in the problem statements does it claim a rigid disk.

The necessity for the disk to be rigid is implicit in the statement of the paradox. For example, the statement of the paradox assumes r=r', but this is obviously false if the disk contorts like a potato chip.

You might want to read the Gron and Dieks papers. All of this has been understood in detail for decades.

 

[yuiop]

The necessity for the disk to be rigid is implicit in the statement of the paradox. For example, the statement of the paradox assumes r=r', but this is obviously false if the disk contorts like a potato chip.

Even r=r' does not have to imply that the disk is made of infinitely rigid material. A competent engineer could design a disk with hydraulic rams that dynamically adjusts its radius to be constant in a non rotating frame. Rephrasing the paradox as "we have an infinitely rigid disk and we apply angular acceleration to it" makes the paradox a non starter because SR is not compatible with infinitely rigid materials.

Einstein's version of the paradox is carefully worded and avoids any implication of a infinitely rigid material:

"In a space which is free of gravitational fields we introduce a Galilean system of reference K (x,y,z,t), and also a system of coordinates K' (x',y',z',t') in uniform rotation relative to K. Let the origins of both systems, as well as their axes of Z, permanently coincide. We shall show that for a space-time measurement in the system K' the above definition of the physical meaning of lengths and times cannot be maintained. For reasons of symmetry it is clear that a circle around the origin in the X, Y plane of K may at the same time be regarded as a circle in the X', Y' plane of K'. We suppose that the circumference and diameter of this circle have been measured with a unit measure infinitely small compared with the radius, and that we have the quotient of the two results. If this experiment were performed with a measuring-rod at rest relative to the Galilean system K, the quotient would be π. With a measuring-rod at rest relative to K', the quotient would be greater than π. This is readily understood if we envisage the whole process of measuring from the stationary'' system K, and take into consideration that the measuring-rod applied to the periphery undergoes a Lorentzian contraction, while the one applied along the radius does not."

He just considers one reference frame with uniform rotation relative to a reference frame that is not rotating. He does not demand that that R =R', but simply compares the ratio of C/R in the non rotating frame with the ratio C'/R' in the rotating frame. The crux of the paradox, if there is one, is why is the circumference to radius ratio in the rotating frame not 2*pi? How the disk got into uniform rotational motion is a red herring and simply stating that it not possible to have a disk in uniform rotation is just bypassing the issue, especially when there is plenty of evidence that it is possible for disks to rotate. We can reproduce the paradox of why is the circumference to radius ratio not equal to 2*pi as measured by rulers in the rotating frame, without requiring an infinitely rigid disk. That is the question of real interest. The fact that the C'/R' ratio is not 2*pi in the rotating frame is a true fact and it is not resolved by stating that that will never be observed, because it require an infinitely rigid disk to get into that situation, because that is simply not true. We can spin up a reasonably (but not infinitely) rigid disk and we do not not need to worry if its radius changes during the spin up. We know that any non rotating disk has a C/R ratio of 2*pi so we do not need a non rotating disk with the same radius as the rotating disk to make comparisons. If it was Ehrenfest's original intention to demonstrate that it is impossible to spin up an infinitely rigid disk, then he did not make that very clear and there are plenty of ways to demonstrate that infinitely rigid anythings are paradoxical in the context of SR. I don't think it Ehrenfest's intention to demonstrate that a disk can not have Born rigid angular acceleration either, because he barely mentions acceleration.

 

[yuiop]

Length X remains constant by definition but what does that actually mean?

It means the length of the accelerating rocket as measured by observers onboard the rocket using rulers or radar measurements, does not vary over time for as long as the rocket maintains constant acceleration.

It still is contracted relative to other frames correct?

Yep it is measured to be length contracted in non accelerating RF's (except maybe the instantaneous CMIRF) and getting smaller all the time.

In fact it appears that to some extent rulers also have a different length at the front and the back even if the difference is negligable so proper distance seems to be a problematic concept in all ways or am I missing something here, Again?

Dalespam mentions that proper distance is only well defined locally in an accelerating RF. Over extended distances. It can be problematic defining proper distance.

 

[bcrowell]

Einstein's version of the paradox is carefully worded and avoids any implication of a infinitely rigid material:

Einstein wasn't stating a paradox. Ehrenfest was.

 

[Passionflower]

However, do you agree that we could arrange a synchronization scheme by suitable fixed multiplication factors of the clock rates, so that all clocks on the rocket appear to be running at the same rate and so that the one way speed of light is the same in both directions, that the front and back radar measurements would then agree with each other?

As soon as someone is going to say yes I will jump on it and ask for the formula as this would be highly interesting.

However, I am just shooting from the hip here, I do not readily see how such a scheme could be possible.

Take a rocket with three clocks, one stationed at the ceiling, one in the middle and one on the floor of the rocket. The radar distances will be obviously be different for all paths between the clocks. So how would one adjustment per clock come out with the results you are looking for?

Do you agree if we only use the natural proper clocks that we use for radar measurements, for all measurements, that the one way speed of light is not isotropic

I would certainly agree with that. And to anticipate the responses from the 'measured locally police' here is an argument they would perhaps love: Because the speed of light is c at the point of measurement and the radar distance is not equal to the ruler distance the speed of light away from that point must be different!

I assume you are saying that "radar distance" unqualified, is a timing of a two way signal made by a "proper clock" by definition. I guess I will have to qualify the radar measurement made by synchronised clocks as "coordinate radar distance" or something. By suitable adjustments this coordinate radar distance can be made to agree with the ruler length in both directions. Whether this coordinated measurement agrees with something that approximates the proper length of the accelerating ship or even the CMIRF measurement of the ship I am not sure about.

Also in this case I would respond with "show me the formula", as that is obviously highly interesting.

 

[Austin0]

Take a particle's worldline, pick some event on that worldline, and find the tangent vector at that event. Then form the hyperplane which is orthogonal to the tangent vector. Find the event that forms the intersection of a neighboring particle's worldline with that hyperplane. Then calculate the spacetime interval between the two events. That is the proper distance between two neighboring particles. I should mention that it needs to be a differential distance, not a large distance.

A point of confusion here. The hyperplane being orthogonal to the tangent vector.
Wouldn't the hyperplane be at an angle similar to the tangent vector ? A mirror image relative to a light null geodesic drawn through the same point?
Or maybe I am just not understanding what you mean by orthogonal in this case.

In any case this seems to be a refinement of the original definition. But if there is a difference in periodicity between neighboring particles then isn't the restriction to infinitesimal distance just kind of a way to make the spacetime innaccuracy seem to go away? There would still be an infinitesimal difference between neighboring particles at the front and the back, wouldn't there?
And any intermediate measurement would be adding up these small differences??

Born rigidity is a kinematic condition, not a dynamic condition. I.e. it is a statement about the motion of each particle, where it is at any given instant in time relative to its neighbors. It is not a statement in any way about forces acting on the particles. I have hopefully consistently used the word "strain" and avoided the words "stress" or "force".

I understood that the basis for the whole concept was founded in forces.
That without a particular controlled acceleration throughout the system that the forces of acceleration and Lorentz contraction would either create an internal contraction of proper length or invoke a disruptive expansion??
T hat without greater acceleration at the back there would be contraction of proper length and with equal proper acceleration through the system there would be a resulting stretching and disruption through the opposition of this and the Lorentz contraction.
Is this not the case?

 

[Passionflower]

I understood that the basis for the whole concept was founded in forces.
That without a particular controlled acceleration throughout the system that the forces of acceleration and Lorentz contraction would either create an internal contraction of proper length or invoke a disruptive expansion??

No that is completely wrong.

 

[Austin0]

Originally Posted by Austin0
Length X remains constant by definition but what does that actually mean?

It means the length of the accelerating rocket as measured by observers onboard the rocket using rulers or radar measurements, does not vary over time for as long as the rocket maintains constant acceleration.

Well if the system, relative to an inertial frame is contracting and the clocks are slowing down over time
then wouldn't this seem to mean that the radar distance would also decrease over time and if it didn't how would this work out for local agreement between accelerating observers and inertial observers at colocated points?


Originally Posted by Austin0
It still is contracted relative to other frames correct?

Yep it is measured to be length contracted in non accelerating RF's (except maybe the instantaneous CMIRF) and getting smaller all the time.

Originally Posted by Austin0
In fact it appears that to some extent rulers also have a different length at the front and the back even if the difference is negligable so proper distance seems to be a problematic concept in all ways or am I missing something here, Again?

Dalespam mentions that proper distance is only well defined locally in an accelerating RF. Over extended distances. It can be problematic defining proper distance.

So in effect as long as the distances are ,in practice, too small to measure,, proper distance is constant
, yes? sorry. couldn't resist

 

[Austin0]

No that is completely wrong.

Well OK which part. Being called wrong is neither helpful or informative.

 

[Austin0]

As far as I can tell, the line of simultaneity pivoting about the origin is only true, if the clocks of the accelerating rocket nearer the origin are artificially sped up relative to the clocks nearer the nose.

True.

A) I thought the difference in periodicity between the clocks at the front and the back was a result of the difference in proper acceleration and not an artificial contrivance ??
B) That the convergence of L's of S at the pivot event was due to the clocks at the back being naturally dilated relative to the front , not sped up either natrually or artificially . Is this incorrect?

If the clocks are left to do their own thing so that they agree with local natural processes, the natural line of simultaneity would be tilted in the opposite direction and would not pass through the origin, no?

Yes, although now "simultaneity" would bear no relationship to Einstein synchronisation.

With this artificial speeding up of the clocks nearer the tail, the radar distance measured at the nose would agree with radar distance measured at the tail. Agree?

Here you seem to be talking about the natural clocks being retarded in the rear , the opposite of the first part above . SO I am confused.
But it does seem that if the clocks were artificially synchronized as to rate and proper time reading, the radar measurements would then agree.

No. Radar measurements are made using "proper clocks", not "coordinate clocks" and the front-back-front measurement will differ from from the back-front-back measurement. The attached diagram illustrates why. The time taken by the red signal (measured by the red observer) is much longer than the time taken by the blue signal (measured by the blue observer).

SO here you seem to mean natural clocks that are not artificially calibrated when you say proper clocks ,,Is this correct?

 

[Passionflower]

Well OK which part. Being called wrong is neither helpful or informative.

If you want help then take notice what people say, instead of continuously questioning everything.

Dalespam is completely right:

Born rigidity is a kinematic condition, not a dynamic condition. I.e. it is a statement about the motion of each particle, where it is at any given instant in time relative to its neighbors. It is not a statement in any way about forces acting on the particles. I have hopefully consistently used the word "strain" and avoided the words "stress" or "force".

Then you write pretty much the opposite of what Dalespam writes.

So what is your attitude, do you actually want to learn something from others in this forum?

 

[yuiop]

As soon as someone is going to say yes I will jump on it and ask for the formula as this would be highly interesting.

However, I am just shooting from the hip here, I do not readily see how such a scheme could be possible.

Take a rocket with three clocks, one stationed at the ceiling, one in the middle and one on the floor of the rocket. The radar distances will be obviously be different for all paths between the clocks. So how would one adjustment per clock come out with the results you are looking for?
...
Also in this case I would respond with "show me the formula", as that is obviously highly interesting.

I am a bit short of time at the moment, so I will have to come back to this later and hopefully come up with a formula. Off hand, without checking, I think you will end up with something like Rindler coordinates. That there is a single multiplication factor for each clock that should do the trick, comes from the near certain assumption that onboard the rocket with constant proper acceleration, the clocks appear to be running at different frequencies, but the relative ratio of those frequencies is not changing over time.

 

[Passionflower]

I am a bit short of time at the moment, so I will have to come back to this later and hopefully come up with a formula. Off hand, without checking, I think you will end up with something like Rindler coordinates. That there is a single multiplication factor for each clock that should do the trick, comes from the near certain assumption that onboard the rocket with constant proper acceleration, the clocks appear to be running at different frequencies, but the relative ratio of those frequencies is not changing over time.

Here is a simple case:

The rocket has three clocks A, B and C.

$$\Delta t_A \, AB$$

$$\Delta t_B \, AB$$

$$\Delta t_B \, BC$$

$$\Delta t_C \, BC$$

The problem seems to be how to adjust clock B. If we adjust B so that A-B works out we seem to have a problem with B-C and vice versa. No?

 

[Austin0]

If you want help then take notice what people say, instead of continuously questioning everything.

I take notice of everything people say. I read and reflect on many posts and threads and neither question or comment but simply learn.
But if things peple say are either unclear or raise questions what is the point of the forum if those questions are repressed?

Originally Posted by Dalespam
Born rigidity is a kinematic condition, not a dynamic condition. I.e. it is a statement about the motion of each particle, where it is at any given instant in time relative to its neighbors. It is not a statement in any way about forces acting on the particles. I have hopefully consistently used the word "strain" and avoided the words "stress" or "force".

Dalespam is completely right: Then you write pretty much the opposite of what Dalespam writes.
So what is your attitude, do you actually want to learn something from others in this forum?

I have tremendous respect for DaleSpam , not only for his depth and range of knowledge but also because he consistently responds to actual points and questions, not with unspecific insinuations of my incompetence, but with information. Consequently I have learned much from him and am very appreciative.
Here again I did not say I thought he was wrong but simply stated my understanfing of Born rigidity that I have gained from reading and "learned" from others in this forum. Seeking clarification.
But you did not address either of the points in any sort of productive way , only with unspecific negative assessments and questioning my attitude.

 

[Austin0]

Just some thoughts on the question of calibrated clock rates in a Born rigid system.

Using the clock at the midpoint of the system it might be productive to think about calibrating the other clocks from this point, on the basis of the Rindler gamma factor for the relative location of the clocks from there to the front and to the back. Incrementally increasing the rate of the clocks approaching the rear and reciprocally slowing the rates approaching the front.
On the basis of the midpoint CMIRF, it would seem that a continuous synch signal, [proper time of this clock at midpoint, broadcast through the system] , would result in radar agreement long range through the system as well an isotropic c. [one way synchroniuzation as usual, on the assumption of isotropicly equal and constant proper distance from the middle to other points in both directions.]
ALthough this adjustment might still leave local radar distances off wrt local rulers
Perhaps a combination of the two techniques?

 

[yuiop]

As soon as someone is going to say yes I will jump on it and ask for the formula as this would be highly interesting.

However, I am just shooting from the hip here, I do not readily see how such a scheme could be possible.

Take a rocket with three clocks, one stationed at the ceiling, one in the middle and one on the floor of the rocket. The radar distances will be obviously be different for all paths between the clocks. So how would one adjustment per clock come out with the results you are looking for?
...

My initial look at the problem suggests the equation v=atanh(aT) where a is the proper acceleration at a given point on the rocket and T is the elapsed proper time of the clock at that point. The proper time and proper acceleration can both be measured locally on the rocket, so the each section of the rocket can calculate its velocity relative to the initial inertial RF at any instant. The instantaneous velocity of all the points on the "simultaneity line" pivoting around the origin is equal, so setting the coordinate time to v should give a coordinate elapsed time that is synchronised. I still have to resolve the objections you have raised and at this point I am not 100% sure they can be resolved. More work to do!

Just some thoughts on the question of calibrated clock rates in a Born rigid system.

Using the clock at the midpoint of the system it might be productive to think about calibrating the other clocks from this point, on the basis of the Rindler gamma factor for the relative location of the clocks from there to the front and to the back. Incrementally increasing the rate of the clocks approaching the rear and reciprocally slowing the rates approaching the front.
On the basis of the midpoint CMIRF, it would seem that a continuous synch signal, [proper time of this clock at midpoint, broadcast through the system] , would result in radar agreement long range through the system as well an isotropic c. [one way synchroniuzation as usual, on the assumption of isotropicly equal and constant proper distance from the middle to other points in both directions.]
ALthough this adjustment might still leave local radar distances off wrt local rulers
Perhaps a combination of the two techniques?

This is basically along the lines I am thinking of investigating.

 

[yuiop]

Here is a simple case:

The rocket has three clocks A, B and C.

$$\Delta t_A \, AB$$

$$\Delta t_B \, AB$$

$$\Delta t_B \, BC$$

$$\Delta t_C \, BC$$

The problem seems to be how to adjust clock B. If we adjust B so that A-B works out we seem to have a problem with B-C and vice versa. No?

Yes, I think you are right. There appears to be no single adjustment that can be made to clock B that can get the radar distance BAB to agree with BCB if the ruler lengths AB and BC are equal. So it seems my suggested scheme is doomed and there is no way to come up with a synchronisation scheme that can make the speed of light isotropic over extended distances in a reference frame undergoing Born rigid acceleration. Basically radar signals sent out simultaneously from B in both directions to A and C, do not return to B simultaneously and there is no calibration procedure that can make those signals from different directions arrive simultaneously. Looks like your shooting from the hip hit the target

 

[Dale]

Sorry, this thread kind of ran away from me yesterday.

A point of confusion here. The hyperplane being orthogonal to the tangent vector.
Wouldn't the hyperplane be at an angle similar to the tangent vector ? A mirror image relative to a light null geodesic drawn through the same point?
Or maybe I am just not understanding what you mean by orthogonal in this case.

Orthogonal means that the Minkowski dot product is 0. So if you transform to the inertial frame where the tangent vector is at rest (space components all = 0) then it is the plane of simultaneity, i.e. all points with the same time coordinate. In a coordinate system where the tangent vector is not at rest then, by the relativity of simultaneity, the orthogonal plane will be "tilted" as you describe ("angle similar to the tangent vector" "A mirror image"). The reason for expressing it as an orthogonal plane is simply to give a coordinate-independent geometric definition.

I understood that the basis for the whole concept was founded in forces.
That without a particular controlled acceleration throughout the system that the forces of acceleration and Lorentz contraction would either create an internal contraction of proper length or invoke a disruptive expansion??
T hat without greater acceleration at the back there would be contraction of proper length and with equal proper acceleration through the system there would be a resulting stretching and disruption through the opposition of this and the Lorentz contraction.
Is this not the case?

No, it is the other way around. Born rigidity defines the geometry, the strains throughout the system. That in turn implies some specific motion for each particle. Then, from the motions you can solve for the forces if desired.

 

[Austin0]

Originally Posted by yuiop
As far as I can tell, the line of simultaneity pivoting about the origin is only true, if the clocks of the accelerating rocket nearer the origin are artificially sped up relative to the clocks nearer the nose.

DrGreg...True.

A) I thought the difference in periodicity between the clocks at the front and the back was a result of the difference in proper acceleration and not an artificial contrivance ??
B) That the convergence of L's of S at the pivot event was due to the clocks at the back being naturally dilated relative to the front , not sped up either natrually or artificially . Is this incorrect?.

Could you explain a little more about what you and DrGreg are talking about in the above?

Originally Posted by yuiop
If the clocks are left to do their own thing so that they agree with local natural processes, the natural line of simultaneity would be tilted in the opposite direction and would not pass through the origin, no?

DrGreg,,,,Yes, although now "simultaneity" would bear no relationship to Einstein synchronisation.

Ditto

Yes, I think you are right. There appears to be no single adjustment that can be made to clock B that can get the radar distance BAB to agree with BCB if the ruler lengths AB and BC are equal. So it seems my suggested scheme is doomed and there is no way to come up with a synchronisation scheme that can make the speed of light isotropic over extended distances in a reference frame undergoing Born rigid acceleration. Basically radar signals sent out simultaneously from B in both directions to A and C, do not return to B simultaneously and there is no calibration procedure that can make those signals from different directions arrive simultaneously. Looks like your shooting from the hip hit the target

If B=back M=middle and F= front with D being proper distance wouldn't:

Radar time M-->F-->M be dt=(dx_MF+dx_FM/c)
M-->F=
dx_MF= D+dt_MF*v+0.5 a_F dt_MF²
F-->M=
dx_FM=D-(dt_FM*v +0.5 a_M dt_FM ²)

Radar time F-->M-->F then being
dt=(dx_MF+dx_FM/c)

F-->M dx_FM=
D-(dt_FM*v +0.5 a_M dt_FM ²)

M-->F=
dx_MF=
D+dt_MF*v+0.5 a_F dt_MF²

It looks like the round trip, distance and time for F-M-F and M-F-M would be equivalent as far as the acceleration is concerned doesn't it??
This assuming the time dilation differential is compensated for.
It would seem to follow that M-F-M would be equivalent to M-B-M also as far as differential acceleration is concerned so the only difference would seem to be differential contraction between the front and the back which should be very negligable, Am I far astray here?
It does seem possible to make it possible for one way isotropic measurements of c
Just some thoughts anyway

 

[Passionflower]

so the only difference would seem to be differential contraction between the front and the back which should be very negligable, Am I far astray here?

Differential contraction?
Didn't several people try to explain to you that the lengths do not change for Born rigid objects?

Am I far astray here?

It certainly looks so.

 

[Passionflower]

I think it actually is possible to impart an angular acceleration to a Born-rigid one-dimensional object, just not to a Born-rigid two-dimensional object. The argument that it's impossible to have an angular acceleration is given in Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975), and I think it depends on the assumption that the object encloses some area.

Perhaps that is true, I am not sure about it. The topic of rotation in SR is rather shady (to me at least).

 

[Austin0]

Originally Posted by Austin0
so the only difference would seem to be differential contraction between the front and the back which should be very negligable, Am I far astray here?

Differential contraction?
Didn't several people try to explain to you that the lengths do not change for Born rigid objects?
It certainly looks so.

Once again here, I posted several questions and ideas, which you ignored with no helpful input, only to home in on the one point you could take out of context to demonstrate how I am wrong and can't understand.
I started out from the beginning with a complete awareness of the definition of Born rigidity and the constant proper length in that definition and context. Nothing I have said at any point has been an argument against this definition. It has only been a question of how this could be practically determined within such a system given the unequal dilation.
The last couple of posts with yuiop have not been addressed to the Born system as such, but rather, as far as I understand , to the possible implementation within such a system of measurement consistent with a singular comoving inertial frame. Equal isotropic radar measurements,synchronization etc.
This would seem to mean duplicating a system that would appear to other frames to have the same degree of clock desynchronization , periodicity etc. as a single inertial frame.
IN this context i.e. as observed from outside , a Born system would seem to appear to have a differential contraction. DIfferent at the front and the back, or do you disagree with this??
The difference in clock dilation (rate) that would be observed from other frames ,between front and back , could be compensated for by controlled calibration ,as has been discussed.But it seems that the difference in ruler length could not.
This is all I was referring to.

 

[Passionflower]

It has only been a question of how this could be practically determined within such a system given the unequal dilation.
The last couple of posts with yuiop have not been addressed to the Born system as such, but rather, as far as I understand , to the possible implementation within such a system of measurement consistent with a singular comoving inertial frame. Equal isotropic radar measurements,synchronization etc.
This would seem to mean duplicating a system that would appear to other frames to have the same degree of clock desynchronization , periodicity etc. as a single inertial frame.

Let me ask you this: what problem are you trying to solve? As far as I am concerned Born rigid motion is solved and uncontroversial.

IN this context i.e. as observed from outside , a Born system would seem to appear to have a differential contraction. DIfferent at the front and the back, or do you disagree with this??

A contraction between which locations?

The difference in clock dilation (rate) that would be observed from other frames ,between front and back , could be compensated for by controlled calibration ,as has been discussed.But it seems that the difference in ruler length could not.

Again what do you think the problem is? Why would you need to compensate?

 

[Austin0]

. However, do you agree that we could arrange a synchronization scheme by suitable fixed multiplication factors of the clock rates, so that all clocks on the rocket appear to be running at the same rate and so that the one way speed of light is the same in both directions, that the front and back radar measurements would then agree with each other?

Do you agree if we only use the natural proper clocks that we use for radar measurements, for all measurements, that the one way speed of light is not isotropic and the proper clocks will be obviously unsynchronised to the observers on the rocket?

P.S. I assume you are saying that "radar distance" unqualified, is a timing of a two way signal made by a "proper clock" by definition. I guess I will have to qualify the radar measurement made by synchronised clocks as "coordinate radar distance" or something. By suitable adjustments this coordinate radar distance can be made to agree with the ruler length in both directions. Whether this coordinated measurement agrees with something that approximates the proper length of the accelerating ship or even the CMIRF measurement of the ship I am not sure about. One problem with the CMIRF measurement is that a CMIRF that is co-moving with the front of the ship is not co-moving with the back of the ship and vice verse so we would have to define a location for the CMIRF somewhere near the middle of the ship.

My initial look at the problem suggests the equation v=atanh(aT) where a is the proper acceleration at a given point on the rocket and T is the elapsed proper time of the clock at that point. The proper time and proper acceleration can both be measured locally on the rocket, so the each section of the rocket can calculate its velocity relative to the initial inertial RF at any instant. The instantaneous velocity of all the points on the "simultaneity line" pivoting around the origin is equal, so setting the coordinate time to v should give a coordinate elapsed time that is synchronised. I still have to resolve the objections you have raised and at this point I am not 100% sure they can be resolved. !.

Originally Posted by Austin0
Just some thoughts on the question of calibrated clock rates in a Born rigid system.

Using the clock at the midpoint of the system it might be productive to think about calibrating the other clocks from this point, on the basis of the Rindler gamma factor for the relative location of the clocks from there to the front and to the back. Incrementally increasing the rate of the clocks approaching the rear and reciprocally slowing the rates approaching the front.
On the basis of the midpoint CMIRF, it would seem that a continuous synch signal, [proper time of this clock at midpoint, broadcast through the system] , would result in radar agreement long range through the system as well an isotropic c. [one way synchroniuzation as usual, on the assumption of isotropicly equal and constant proper distance from the middle to other points in both directions.]
ALthough this adjustment might still leave local radar distances off wrt local rulers
Perhaps a combination of the two techniques?

This is basically along the lines I am thinking of investigating.

Let me ask you this: what problem are you trying to solve? As far as I am concerned Born rigid motion is solved and uncontroversial.


A contraction between which locations?


Again what do you think the problem is? Why would you need to compensate?

See above

 

[Passionflower]

Perhaps you should start with a definition of such a reference frame (CMIRF). In an accelerating system we can have an instantaneously co-moving observer. Or is it specifically a frame what you mean? Such a frame obviously applies to one instant of the acceleration, the next instant there will be a completely different frame. Or are you thinking about Fermi-Walker transporting such a frame?

 

[Austin0]

Perhaps you should start with a definition of such a reference frame (CMIRF). In an accelerating system we can have an instantaneously co-moving observer. Or is it specifically a frame what you mean? Such a frame obviously applies to one instant of the acceleration, the next instant there will be a completely different frame. Or are you thinking about Fermi-Walker transporting such a frame?

As I understand the problem, it is in effect an attempt to duplicate a dynamic series of CMIRFs using the clocks and rulers of the accelerating frame.
Using accelerometers, calculators. synch signals or whatever to artificially calibrate the system to simulate and update a quasi -inertial frame [dynamic]
A variation on your idea of using such means to artificially keep a home clock going and updated based on computations derived from acceleration/time calculations.
I will have to look up Fermi-Walker transport, I seem to remember their names from my seach of the Ehrenfest question but never found the papers mentioned.

 

[Passionflower]

As I understand the problem, it is in effect an attempt to duplicate a dynamic series of CMIRFs using the clocks and rulers of the accelerating frame.
Using accelerometers, calculators. synch signals or whatever to artificially calibrate the system to simulate and update a quasi -inertial frame [dynamic]
A variation on your idea of using such means to artificially keep a home clock going and updated based on computations derived from acceleration/time calculations.

Well but in which way, what is the artificial clock supposed to measure?

I already mentioned it is possible to measure coordinate time, or the longest possible time between the start and clock measuring event. So what third option are you looking for?

Personally I would be very interested in comparing a "straight line" clock on a spatial path from A to B with a helical path but this seem very complicated to calculate. In case we have someone here make the claim it is easy then please provide the formula.

 

[yuiop]

The problem seems to be how to adjust clock B. If we adjust B so that A-B works out we seem to have a problem with B-C and vice versa. No?

Yes, I think you are right. There appears to be no single adjustment that can be made to clock B that can get the radar distance BAB to agree with BCB if the ruler lengths AB and BC are equal. So it seems my suggested scheme is doomed and there is no way to come up with a synchronisation scheme that can make the speed of light isotropic over extended distances in a reference frame undergoing Born rigid acceleration. Basically radar signals sent out simultaneously from B in both directions to A and C, do not return to B simultaneously and there is no calibration procedure that can make those signals from different directions arrive simultaneously. Looks like your shooting from the hip hit the target

I may have to rescind my possibly premature capitulation here, as I have thought of way around the objection you have raised. I have up to now been assuming that that the distances between clocks have to be regularly spaced in the instantaneously co-moving frame. Here is a method htat might work. We have a multitude of clocks, A,B,C etc with A at the nose. We place B at location further back in the rocket so that radar distance A,B,A as measured from A is say 1 light second. We adjust the rate of clock B so that B agrees that the radar distance B,A,B is also one light second. Now we do the same with clock C. Clock C is placed at a distance further back where the radar distance of C measured by B is also 1 light second. Clock C is adjusted so that observer C also agrees that the radar distance is also 1 light second using the adjusted clock. We can continue adding clocks like this so that they all the agree that the coordinate radar distance between any two adjacent clocks is one light second and that the the coordinate distance between any N number of clocks is N-1 light seconds, measured in any direction. Now, while the observers on the accelerating rocket consider the clocks to placed at uniform distance intervals as measured by coordinate radar distances, any instantaneous co-moving frame will see the clocks towards the back as getting more bunched together. Obviously I will need to check these conclusions and provide some equations, but what do you think of the outline of the method so far. Any immediate objections?

 

[Passionflower]

I may have to rescind my possibly premature capitulation here, as I have thought of way around the objection you have raised. I have up to now been assuming that that the distances between clocks have to be regularly spaced in the instantaneously co-moving frame. Here is a method htat might work. We have a multitude of clocks, A,B,C etc with A at the nose. We place B at location further back in the rocket so that radar distance A,B,A as measured from A is say 1 light second. We adjust the rate of clock B so that B agrees that the radar distance B,A,B is also one light second. Now we do the same with clock C. Clock C is placed at a distance further back where the radar distance of C measured by B is also 1 light second. Clock C is adjusted so that observer C also agrees that the radar distance is also 1 light second using the adjusted clock. We can continue adding clocks like this so that they all the agree that the coordinate radar distance between any two adjacent clocks is one light second and that the the coordinate distance between any N number of clocks is N-1 light seconds, measured in any direction. Now, while the observers on the accelerating rocket consider the clocks to placed at uniform distance intervals as measured by coordinate radar distances, any instantaneous co-moving frame will see the clocks towards the back as getting more bunched together. Obviously I will need to check these conclusions and provide some equations, but what do you think of the outline of the method so far. Any immediate objections?

Hmmm.

The relationship between the clocks at different locations A, B, C etc is:

$$\frac{\Delta \tau_A}{\alpha_A} = \frac{\Delta \tau_B}{\alpha_B} = \frac{\Delta \tau_C}{\alpha_C}$$

While the accelerations are:

$$\alpha = \frac{c^2}{x_A + h}$$

The sum of the roundtrip time AB (e.g A's clock + B's clock) is larger than BC, which is larger than CD etc. So how are you going to make that fit?

Note that this is not the case in a homogenic gravitational field where the sum is always twice the inertial radar distance. So there it seems easier

 

[bcrowell]

As far as I can tell, the last ~30 posts have been about whether Einstein synchronization is possible aboard an accelerating rocket. There is a good discussion of Einstein synchronization here: http://en.wikipedia.org/wiki/Einstein_synchronization

Or is the question really whether or not the observers in the rocket are static observers in a static spacetime?

Einstein synchronization is not possible because the no-redshift criterion given in the WP article is not satisfied.

The spacetime is static because staticity is coordinate-independent, and the observers aboard the rocket are just seeing flat spacetime in a different set of coordinates. I believe they are also static observers, because they don't observe any time-variation of the geometry (the geometry is always flat), and they aren't rotating (they see a vanishing Sagnac effect).

Actually, is there a standard definition of a static observer? The books I have seem to define a clear notion of a static spacetime and of static coordinates used to describe a static spacetime (i.e., coordinates in which the metric is diagonal and time-independent). But defining a property of coordinates isn't quite the same as defining a property of an observer, since observers are local and coordinates are global.

 

[yuiop]

Hmmm.

The relationship between the clocks at different locations A, B, C etc is:

$$\frac{\Delta \tau_A}{\alpha_A} = \frac{\Delta \tau_B}{\alpha_B} = \frac{\Delta \tau_C}{\alpha_C}$$

While the accelerations are:

$$\alpha = \frac{c^2}{x_A + h}$$

The sum of the roundtrip time AB (e.g A's clock + B's clock) is larger than BC, which is larger than CD etc. So how are you going to make that fit?

Note that this is not the case in a homogenic gravitational field where the sum is always twice the inertial radar distance. So there it seems easier

Not quite sure what you are doing here. The roundtrip time (eg A to B and back to A) is measured by a single clock at A.

I have uploaded some diagrams of the method of synchronising accelerating clocks that I described earlier. https://www.physicsforums.com/showpost.php?p=2882964&postcount=64 The first diagram is the radar distances as measured by unsynchronised clocks moving with Born rigid motion and the second diagram is with clocks synchronised so that the radar distance between any 2 clocks is 1 light second and the distance between any N accelerating clocks is N-1 light seconds. This method of synchronisation has most of the properties required of it that bcrowell mentioned:

The problem is whether this synchronisation does really succeed in assigning a time label to any event in a consistent way. To that end one should find conditions under which
(a) clocks once synchronized remain synchronized,
(b1) the synchronisation is reflexive, that is any clock is synchronized with itself (automatically satisfied),
(b2) the synchronisation is symmetric, that is if clock A is synchronized with clock B then clock B is synchronized with clock A,
(b3) the synchronisation is transitive, that is if clock A is synchronized with clock B and clock B is synchronized with clock C then clock A is synchronized with clock C.

Condition (a) is met, but it requires that the raw output of the proper clocks has a constant correction factor of $$a_i/a_m$$ where $$a_i$$ is the proper acceleration of the i'th clock under consideration and $$a_m$$ is the proper acceleration of the "master clock". For constant proper acceleration and Born rigid motion, this correction factor does not change over time.

Conditions (b1), (b2) and (b3) are also met, without any difficulty. Radar distances are the same measured in either direction and over long distances on the accelerating rocket with this method. Constant distant intervals in the accelerating frame appear progressively more length contracted towards the rear of the rocket in the inertial momentarily co-moving frame and this is perhaps the "differential length contraction" that austin0 was intuitively referring to.

 

[Passionflower]

Not quite sure what you are doing here. The roundtrip time (eg A to B and back to A) is measured by a single clock at A.

Of course it is.

I was merely giving additional information that if we add the roundtrip times measured by both clock A (ABA) and clock B (BAB) then also this is not constant, unlike in a homogenic gravitational field where the total time remains constant (e.g. double the inertial rountrip time).

I have uploaded some diagrams of the method of synchronising accelerating clocks that I described earlier.

Nice, but I prefer to see the formulas or the raw data. I like to verify it for myself.

 

[yuiop]

Hmmm.

The relationship between the clocks at different locations A, B, C etc is:

$$\frac{\Delta \tau_A}{\alpha_A} = \frac{\Delta \tau_B}{\alpha_B} = \frac{\Delta \tau_C}{\alpha_C}$$

While the accelerations are:

$$\alpha = \frac{c^2}{x_A + h}$$

The sum of the roundtrip time AB (e.g A's clock + B's clock) is larger than BC, which is larger than CD etc. So how are you going to make that fit?

Note that this is not the case in a homogenic gravitational field where the sum is always twice the inertial radar distance. So there it seems easier

We have the equation for the instantaneous velocity v of an accelerating clock with constant proper acceleration $$\alpha$$ given as:

$$v = tanh(\alpha \Delta T)$$

Since v is the same for all clocks on the "line of simultaneity" described in mathpages that pivots about the origin, we can say:

$$tanh(\alpha_A \Delta T_A) = tanh(\alpha_B \Delta T_B) = tanh(\alpha_C \Delta T_C)$$

$$\alpha_A \Delta T_A = \alpha_B \Delta T_B = \alpha_C \Delta T_C$$

$$\Delta T_A = \frac{\alpha_B}{\alpha_A} \Delta T_B = \frac{\alpha_C}{\alpha_A} \Delta T_C$$

So as long as we apply a correction factor of $${\alpha_B}/{\alpha_A}$$ to the raw output of clock B and $${\alpha_C}/{\alpha_A}$$ to the raw output of clock C and so on, the processed outputs of the clocks are guaranteed to remain in sync with master clock A that is arbitrarily chosen. By suitable positioning of the clocks, the upstream radar signal returns simultaneously with downstream radar signal for any given clock. This coordinate system/scheme gives a speed of light and radar distances that are constant/isotropic/symmetric/transitive over long distances (and consistent with ruler distances) in an accelerating reference frame with Born rigid acceleration.

 

[Passionflower]

So as long as we apply a correction factor of $${\alpha_B}/{\alpha_A}$$ to the raw output of clock B and $${\alpha_C}/{\alpha_A}$$ to the raw output of clock C and so on, the processed outputs of the clocks are guaranteed to remain in sync.

I just tested this and while the numbers are very close they are not exactly right!

For instance for:

$$\alpha = 1$$

$$\tau_{h_0} = 1$$

$$h_0= 1/0.1$$

$$h_A= 1/0.3$$

$$h_B= 1/0.2$$

We get:

$$\alpha_A = 0.83333$$

$$\tau_A = 0.83333$$

$$\alpha_B = 0.90909$$

$$\tau_B = 0.90909$$

$$d_{ABA} = 0.20833$$

$$d_{BAB} = 0.19091$$

Then if we take:

$$d_{BAB} * \frac{\alpha_B}{\alpha_A}$$

We get: 0.20826 which is close but not equal to 0.20833 (and it is certainly not due to rounding errors as my precision is much higher).

However, as I suspected a little bit, if we do the same calculation in a homogeneous gravitational field your adjustment seems to work. I think this is due to the fact the the roundtrip time between A and B from A plus the roundtrip time between A and B from B is always double the inertial roundtrip time in a homogeneous gravitational field but not in the case of a Born rigid system having a constant proper acceleration.