Ehrenfest / rod thought experiment.

[yuiop]

WHy would the radar length with rotation be longer than the ruler length?
WHy would the radar length agree with rulers with linear acceleration?
Aren't the clock rates different at the front and the back in a Born rigid rocket?
Radar distance longer from the Front to Back to Front , than from B->F->B ??

Sorry yuiop, it has been a long time since I have seen such a demonstration of trying to maintain an untenable position by obfuscation and redefining.

Facts:

• A Born rigid rocket with length X,remains X during constant proper acceleration by definition!
• The radar distance measured by an observer at the front and at the back is different, the observer at the front would measure the distance longer and the observer at the back would measure a distance shorter compared to the rocket distance in an inertial frame.

Actually Austin;s probing questions made me realize I have made a complete hash with claims about radar length versus ruler length in posts #7 and #11. (I still stand by my claims in the OP, but made a mess of the red herrings that have thrown in since). So yes my claims about radar length versus ruler length are untenable and I realized that while I was away from my PC Basically I had been shooting from the hip and missed my target completely. My foot is in bandages right now. I should have used a trick I have learned on this forum that when someone who usually knows what they are talking about asks for your exact position on a subject, the best defense is not to state it clearly!

So basically ignore my comments in #7 and #11 that are way off the mark, except for one (I think valid observation) that by definition Born rigid acceleration can not take any object from of not being accelerated to a state of being accelerated. I made that observation when reflecting on Jesse's observation that an linearly accelerating is not necessarily stress free even with Born rigid acceleration and is in a state of equilibrium stress. However, the motion can be described as "strain free" in the sense that its length is not changing over time as long as the rocket always has constant acceleration. This is clearly not true for the the disc with angular acceleration where the radar length of a short segment of the perimeter is always increasing according to an observer on the disc at one end of the segment.

To Passionflower, I agree with your statement of *facts* in the post quoted above. Sorry for the confusion I caused in posts #7 and #11.

 

[yuiop]

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.

Dalespam, you are probably just the person to ask (although anyone should feel free to answer) about this statement in mathpages that I think might be incorrect:

This is shown clearly in the figure above. The line of simultaneity for the accelerating particles simply rotates about the pivot event

from http://www.mathpages.com/home/kmath422/kmath422.htm

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. 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? 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?

This is similar to applying a correction factor to the frequency of clocks on GPS satellites so that they agree with clocks in land based receiving equipment.

 

[DrGreg]

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.

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?

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).

 

[bcrowell]

That you didn't calculate the proper distances between neighboring particles before and after.

I see. It's an instantaneous impulse, so the positions of the particles are the same before and after, in the lab frame. Therefore the lab-frame distances between them are also the same before and after. What does change discontinuously is the frames of the comoving observers, so it is necessary to check that the comoving-frame distances also stay the same after the Lorentz boost -- but that's what I did explicitly.

What I probably didn't make very clear was that the impulses I was describing were only the impulses that would provide the angular acceleration. There is a separate set of steady radial forces required as well, even when the ruler doesn't have any angular acceleration. I wasn't describing those.

If you read the Gron paper, it should be pretty clear why the issues he describes go away completely when you have a one-dimensional object.

 

[yuiop]

FAQ: How is Ehrenfest's paradox resolved?

As described in [Einstein 1916], the relativistic rotating disk was an example that was influential in leading Einstein to describe gravity in terms of curved spacetime. Einstein writes:

"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."

Einstein's friend Paul Ehrenfest posed the following paradox [Ehrenfest 1909]. Suppose that observer L, in the lab frame, measures the radius of the disk to be r when the disk is at rest, and r' when the disk is spinning. L can also measure the corresponding circumferences C and C'. Because L is in an inertial frame, the spatial geometry does not appear non-Euclidean according to measurements carried out with his meter sticks, and therefore the Euclidean relations C=2πr and C'=2πr' both hold. The radial lines are perpendicular to their own motion, and they therefore have no length contraction, r=r', implying C=C'. The outer edge of the disk, however, is everywhere tangent to its own direction of motion, so it is Lorentz contracted, and therefore C' is less than C.

Call the above the problem statement of the Ehrenfest paradox. The proposed resolution is:

The resolution of the paradox is that it rests on the incorrect assumption that a rigid disk can be made to rotate. If a perfectly rigid disk was initially not rotating, one would have to distort it in order to set it into rotation, because once it was rotating its outer edge would no longer have a length equal to 2π times its radius. Therefore if the disk is perfectly rigid, it can never be set into rotation.

This resolution fails, because it introduces the straw man of the "rigid disk". Nowhere in the problem statements does it claim a rigid disk. We know that in practice in everyday life, non rigid disks such as the wheels of a car can set into rotation.

Thorough modern analyses are available,[Grøn 1975,Dieks 2009] and in particular it is not controversial that, as claimed in [Einstein 1916], C/r is measured to be *greater* than 2π by an observer in the rotating frame.

If this is non controversial then it is clear that Einstein did not intend the spinning disk to be perfectly rigid and the "accepted resolution" is not valid.

We *can* get a non rigid disk to rotate. If observers on the spinning disk measure the circumference they will find it to be greater than 2*pi*R, where R is the radius of the spinning disk as measured by observers in a non rotating frame at rest with the rotation axis of the disk.

A maybe interesting question to ask, is what the observers on a disk with constant angular velocity, measure the radius R' of the spinning disk to be. It is often glibly stated that the radial measurement is not parallel to the motion and therefore not subject to length contraction and so R=R'.

This assumes a ruler measurement of the radius by the observers on the disk, but a ruler with non zero mass will itself be subjected to stresses and strains by the angular motion and is not a reliable measurement. If observers on the perimeter of the disk send a signal to the center which is reflected back, i.e a radar measurement of R', then they find that due to the time dilation of their own clocks, that R' = \gammaR. Therefore when they compare their radar measurements of the circumference C' of the spinning disk to the their radar measurement's of the spinning disk's radius R', they conclude that C' = 2*pi*R' and so by radar measurements alone, the circumference/radius relationship agrees with the Euclidean equation, (but not by ruler measurements). However, the geometry of the disk as measured by the observer's on the disk is not Euclidean in general, because light paths that are not parallel to the radius will follow highly curved paths in some cases when compared to a ruler map of the disk.

 

[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