Two-spaceship paradox This solution incorrect

[J.F.]

“Two-spaceship paradox” This solution incorrect

AAPPS Bulletin Vol. 15 No. 5, October 2005
http://www.aapps.org/archive/bulletin/vol15/15-5/15_5_p17p21abs.html

Jong-Ping Hsu
Nobuhiro Suzuki

We demonstrate a resolution to the “two-spaceship paradox” by explicit calculation using coordinate transformations with at least one frame undergoing constant linear acceleration. A metric such as ds2 = (1 + Kx)2dw2—dx2—dy2—dz2 can lead to a coordinate transformation between an inertial frame and a frame moving with a constant linear acceleration. This coordinate transformation reduces to the Lorentz transformation in the limit of zero acceleration.

This solution incorrect!

 

[yuiop]

AAPPS Bulletin Vol. 15 No. 5, October 2005
http://www.aapps.org/archive/bulletin/vol15/15-5/15_5_p17p21abs.html

Jong-Ping Hsu
Nobuhiro Suzuki

We demonstrate a resolution to the “two-spaceship paradox” by explicit calculation using coordinate transformations with at least one frame undergoing constant linear acceleration. A metric such as ds2 = (1 + Kx)2dw2—dx2—dy2—dz2 can lead to a coordinate transformation between an inertial frame and a frame moving with a constant linear acceleration. This coordinate transformation reduces to the Lorentz transformation in the limit of zero acceleration.

This solution incorrect!

It is hard to interpret exactly what they conclude but from this statement

"After acceleration and reaching
a constant velocity a, i.e. setting o = 0 at time wa in (16),
both the distances between two spaceships and the length of a
spaceship have only the usual Lorentz contraction"

it appears they conclude that the distance between the two Bell spaceships (with identical acceleration in the launch frame) length contracts. If that is the case, then their conclusion is wrong.

I think the only justification for their conclusion is that they have altered the original paradox so that both spaceships stop accelerating at the same time as measured in the rest frame of the accelerated rockets. From the launch frame the rear ship stops accelerating after the lead ship and by the time they both stop they are closer together, bringing about the length contraction of the separation distance that they claim. This would require both ships to continuously re-syncronize their onboard clocks, as from their point of view their onboard clocks are continuously going out of sync. If they did not re-sycnronize their onboard clocks after launching and both cut their engines at the same time (t) as measured by their own onboard clocks, they would be the same distance apart (in the launch frame) when they stop accelerating as they were when they launched. In their own rest frame they would appear to be further apart.

The classical Bell two rockets paradox refers to two rockets continuously accelerating with identical constant acceleration as measured by their onboard accelerometers and does not include stopping the engines. Stopping the engines was introduced by this pair of researchers to suit their own purposes and distorts the original paradox.

 

[1effect]

The classical Bell two rockets paradox refers to two rockets continuously accelerating with identical constant acceleration as measured by their onboard accelerometers and does not include stopping the engines. Stopping the engines was introduced by this pair of researchers to suit their own purposes and distorts the original paradox.

If the two rockets do shut off their engines, the correct analysis can be found here

 

[pervect]

In my opinion, the Hsu paper is not well written, but is not actually incorrect.

I mentioned this on the Wiki Talk page for the Bell Spaceship paradox:

ttp://en.wikipedia.org/w/index.php?title=Talk:Bell%27s_spaceship_paradox&oldid=172261153

Let's look a little closer: Hsu & Suzuki (henceforth H&S) say:

Recently, a “two-spaceship paradox” was proposed and discussed in the AAPPs bulletin using special relativity with ds^2 = c^2 dt^2 - dr^2 ,\ for an inertial frame and ds^2 = \left( 1 + \frac {\alpha x}{c^2} \right) c^2 dt^2 - dr^2 [ed: henceforth Rindler metric] for a non-inertial frame with constant linear acceleration, citing Matsuda & Kino****a (henceforth M&K).

However, there isn't any mention of an accelerating frame in M&K, nor do M&K mention the Rindler metric until part 7. There are accelerating spaceships in M&K, but not accelerating frames. Here is what M&K say:

Now, let us imagine two spaceships of the same type, A and B, which stay still at first in the inertial frame S, the distance between the two spaceships being L. At t=0 these spaceships start accelerating in the same direction along the line joining A and B, undergo the same acceleration for the same duration, stop accelerating at the same time and reach a steady speed u, all viewed from S.

...

In order to make the problem simple, let us assume that the time duration of the acceleration is infinitesimally small (but not zero), so that the world lines of the two spaceships are combinations of two straight lines.

So, how are we supposed to handle this editorially? H&S have apparently misunderstood M&K and solved a different problem. One can see this from a close reading of the paper - H&S refer to M&K's article, but their problem statement is not the same. But I have another thought here ...

Perhaps H&S are objecting to M&S's treatment in part 7 of the accelerating observer where they do mention the metric. This section is very short. But part 7 is the only part of the paper where the Rindler metric is even metioned. This would arguably take H&S back out of "erroneous" and back into "they weren't very clear". BTW, I don't think part 7 of M&K is all that clear either, so I can see where maybe H&S felt comments on that part were needed. Here's what I mean by part 7.

7. GENERAL RELATIVISTIC VIEW POINT
The strange experience of the pilot on the spaceship B can be understood easily if a general relativistic viewpoint is introduced. For him gravity emerges suddenly at the time of the acceleration. The line element for an accelerating observer is ...

So I think that Hsu, et al, are objecting to part 7 of the Matsuda et al, paper, and that some people are incorrectly attributing this objection to part 7 as an objection to part 1 of the paper.

 

[yuiop]

From the launch frame the rear ship stops accelerating after the lead ship and by the time they both stop they are closer together, bringing about the length contraction of the separation distance that they claim...

I thought I had better correct my error before someone else does. If the two rockets have sycronized their clocks with each other, then the rear ship stops before the front ship as measured in the launch frame. That of course would result in the string stretching and snapping even without length contraction. If that is what they are getting at, then they are missing the point. In the classic Bell paradox the string stretches and snaps even if the two rockets never stop accelerating.

 

[yuiop]

If the two rockets do shut off their engines, the correct analysis can be found here

Quote from the page you linked

"Bell pointed out that length contraction of objects as well as the lack of length contraction between objects in frame S can be explained physically, using Maxwell's laws. The distorted intermolecular fields cause moving objects to contract - or to become stressed if hindered from doing so. In contrast, no such forces act in the space between rockets."

http://en.wikipedia.org/wiki/Bell_spaceship_paradox#Analysis

 

[1effect]

Quote from the page you linked

"Bell pointed out that length contraction of objects as well as the lack of length contraction between objects in frame S can be explained physically, using Maxwell's laws. The distorted intermolecular fields cause moving objects to contract - or to become stressed if hindered from doing so. In contrast, no such forces act in the space between rockets."

http://en.wikipedia.org/wiki/Bell_spaceship_paradox#Analysis

Bell wrote a whole paper on the subject, he was of the opinion that relativity should be taught starting from length contraction (a debatable position). Either way, you jumped over the all important sentence:

"Thus when switching the description to the co-moving frame, the distance between the spaceships appears to increase by the relativistic factor \gamma = 1/\sqrt{1-v^2/c^2}. Consequently, the string is stretched."

Yes, I know. wiki is an aggregation of posts. I already pointed out to another poster to look only at the calculations, they do not use length contraction. Sorry.

 

[J.F.]

The Rockets-and-String and Pole-and-Barn Paradoxes Revisited
J.H.Field
http://arxiv.org/abs/physics/0403094v3
The distinction between the real positions of moving objects in a single reference frame and the apparent positions of objects at rest in one inertial frame and viewed from another, as predicted by the space-time Lorentz Transformations, is discussed. It is found that in the Rockets-and-String paradox the string remains unstressed and does not break and that the pole in the Barn-and-Pole paradox never actually fits into the barn. The close relationship of the Lorentz-Fitzgerald Contraction and the relativity of simultaneity of Special Relativity is pointed out and an associated paradox, in which causality is apparently violated, is noted.
http://arxiv.org/PS_cache/physics/pdf/0403/0403094v3.pdf

 

[1effect]

The Rockets-and-String and Pole-and-Barn Paradoxes Revisited
J.H.Field
http://arxiv.org/abs/physics/0403094v3
The distinction between the real positions of moving objects in a single reference frame and the apparent positions of objects at rest in one inertial frame and viewed from another, as predicted by the space-time Lorentz Transformations, is discussed. It is found that in the Rockets-and-String paradox the string remains unstressed and does not break and that the pole in the Barn-and-Pole paradox never actually fits into the barn. The close relationship of the Lorentz-Fitzgerald Contraction and the relativity of simultaneity of Special Relativity is pointed out and an associated paradox, in which causality is apparently violated, is noted.
http://arxiv.org/PS_cache/physics/pdf/0403/0403094v3.pdf

This is a very interesting paper. I happen to agree 100% with the author's views. Thank you.

 

[Doc Al]

The Rockets-and-String and Pole-and-Barn Paradoxes Revisited
J.H.Field
http://arxiv.org/abs/physics/0403094v3
The distinction between the real positions of moving objects in a single reference frame and the apparent positions of objects at rest in one inertial frame and viewed from another, as predicted by the space-time Lorentz Transformations, is discussed. It is found that in the Rockets-and-String paradox the string remains unstressed and does not break and that the pole in the Barn-and-Pole paradox never actually fits into the barn. The close relationship of the Lorentz-Fitzgerald Contraction and the relativity of simultaneity of Special Relativity is pointed out and an associated paradox, in which causality is apparently violated, is noted.
http://arxiv.org/PS_cache/physics/pdf/0403/0403094v3.pdf

I just skimmed through that paper. (Was it published? Doesn't look like it.) It seems that what they call the "Rockets and String" paradox involves the rockets undergoing constant proper acceleration. This is the opposite of the Bell Spaceship paradox, which involves constant acceleration with respect to the Earth ("stationary") frame. If you keep the acceleration constant with respect to a co-moving frame, then of course the string is unstressed and doesn't break (it maintains its proper length throughout). But the observed length of the string from the Earth frame is L_0/\gamma.

Not impressed with the analysis.

 

[yuiop]

This is the second paragraph of the paper:

"It was pointed out 54 years later by Terrell [3] and Penrose [4] that when other important
physical effects (light propagation time delays and optical aberration) are taken
into account, as well as the LT, the moving sphere considered by Einstein in the 1905 SR
paper would not appear to be flattened, in the direction of motion, into an ellipsoid, as
suggested by Einstein, but rather would appear undistorted, but rotated. Shortly afterwards,
Weinstein [5] pointed out that the LFC of a moving rod is apparent only if it is
viewed in a direction strictly perpendicular to its direction of motion. It appears instead
to be relatively elongated if moving towards the observer, and to be more contracted
than the LFC effect if moving away from him. These effects are a consequence of light
propagation time delays. A review [6] has discussed in some detail the combined effects
of the LT, light propagation delays and optical aberration on the appearence of moving
objects and clocks."

That paragraph shows they are a bit confused or being misleading. The fact that a sphere with relative velocity still appears as a sphere due to light propagation delays does not invalidate Einstein's claim that a moving sphere would be physically flattened into an ellipsoid. Other objects that are not spherically perfect appear distorted and sometimes elongated (optically). It just so happens that the light propagation delays make the physically flattened spheriod look like a sphere. If you take light propagation delays into account you can prove that an object such as a tennis ball is elongated at sub-relativistic speeds by taking a photograph with a slow shutter speed. No one seriously believes objects get physically longer just because they appear longer in a blurred photograph.

 

[1effect]

I just skimmed through that paper. (Was it published? Doesn't look like it.) It seems that what they call the "Rockets and String" paradox involves the rockets undergoing constant proper acceleration. This is the opposite of the Bell Spaceship paradox, which involves constant acceleration with respect to the Earth ("stationary") frame. If you keep the acceleration constant with respect to a co-moving frame, then of course the string is unstressed and doesn't break (it maintains its proper length throughout). But the observed length of the string from the Earth frame is L_0/\gamma.

Not impressed with the analysis.

I have to disagree.If you get past the not-so-interesting variant of the Bell's paradox, the paper has a very good analysis of length contraction.It gives a very detailed historical view of the treatment of this subject with a lot of surprising details. It also contains a very good criticism of the solution for the "Pole in the barn". The fact that it hasn't been published yet is quite surprising but does not detract from the interesting treatment.

 

[1effect]

This is the second paragraph of the paper:

"It was pointed out 54 years later by Terrell [3] and Penrose [4] that when other important
physical effects (light propagation time delays and optical aberration) are taken
into account, as well as the LT, the moving sphere considered by Einstein in the 1905 SR
paper would not appear to be flattened, in the direction of motion, into an ellipsoid, as
suggested by Einstein, but rather would appear undistorted, but rotated. Shortly afterwards,
Weinstein [5] pointed out that the LFC of a moving rod is apparent only if it is
viewed in a direction strictly perpendicular to its direction of motion. It appears instead
to be relatively elongated if moving towards the observer, and to be more contracted
than the LFC effect if moving away from him. These effects are a consequence of light
propagation time delays. A review [6] has discussed in some detail the combined effects
of the LT, light propagation delays and optical aberration on the appearence of moving
objects and clocks."

That paragraph shows they are a bit confused or being misleading. The fact that a sphere with relative velocity still appears as a sphere due to light propagation delays does not invalidate Einstein's claim that a moving sphere would be physically flattened into an ellipsoid. Other objects that are not spherically perfect are distorted and sometimes elongated. It just so happens that the light propagation delays make the physically flattened spheriod look like a sphere. If you take light propagation delays into account you can prove that an object such as a tennis ball is elongated at sub-relativistic speeds by taking a photograph with a slow shutter speed. No one seriously believes objects get physically longer just because they appear longer in a blurred photograph.

I think that you are completely missing the most interesting part of the paper. The author does not share your views that length contraction is physical, he's quite clear about it. Quite a few others he cites in the beginning of the paper share his view.(Terrell, Weinstein). He makes this clear in the very paragraph you cited.

 

[ZapperZ]

I think that you are completely missing the most interesting part of the paper. The author does not share your views that length contraction is physical, he's quite clear about it. Quite a few others he cites in the beginning of the paper share his view.(Terrel, Weinstein)

But that's like saying the effective mass that we measure for charge particles in matter is "not physical". What does that mean that something is not physical? The fact that we can give "holes" a value for mass and charge? Is that physical? If it isn't, how come it works so well? And when it comes down to it, doesn't the fact that we can say that "it works" should mean something?

"Length contraction" has the same issue as "relativistic mass" and "time dilation". One can discuss this until one is blue to see if these things are "real" or not. However, one cannot deny that they are used! When we have to calculate something and produce something that needs to work, we make use of those concepts. To me, that is always the bottom line. Does it work, or not? That question isn't answerable based simply on a matter of tastes or philosophical inclinations.

I guess that's why I became an experimentalist.

Zz.

 

[1effect]

But that's like saying the effective mass that we measure for charge particles in matter is "not physical". What does that mean that something is not physical? The fact that we can give "holes" a value for mass and charge? Is that physical? If it isn't, how come it works so well? And when it comes down to it, doesn't the fact that we can say that "it works" should mean something?

"Length contraction" has the same issue as "relativistic mass" and "time dilation". One can discuss this until one is blue to see if these things are "real" or not. However, one cannot deny that they are used! When we have to calculate something and produce something that needs to work, we make use of those concepts. To me, that is always the bottom line. Does it work, or not? That question isn't answerable based simply on a matter of tastes or philosophical inclinations.

I guess that's why I became an experimentalist.

Zz.

In the light of the paper that you cited yesterday on the experimental verification of length contraction, I am inclined to agree with you. You are the first to cite such a proof. Not even the FAQ page on experimental verification of Sr contains such papers. Quite the opposite, the authors say that there is no experimental verification of length contraction (of course, there is ample verification of relativistic mass and of time dilation).

 

[yuiop]

In the barn and pole paradox an observer on the ground (A) sees the pole which has a longer rest length than the barn, fit inside the barn when the pole is moving relative to him and the doors are momentarily closed. Observer A says the explanation is that the pole is length contracted. An observer (B) on the pole says that A is confused and that the pole fits in the barn because the doors did not close simultaneously. Observer B demonstrates to A that the pole is not length contracted in B's reference frame and that everything can be explained by clocks. Observer B explains to A that the reference frame of the pole is the true preferred reference frame because length contraction is not required to explain the observations. Observer A then asks B to measure the length of the barn as he passes through it. B does so and discovers that the barn is shorter than the rest length measured by A. B is at a loss to explain the shorter length of the barn by his own measurements, because he has already claimed to be in a preferred reference frame and that everything can be explained by time alone.

 

[1effect]

In the barn and pole paradox an observer on the ground (A) sees the pole which has a longer rest length than the barn, fit inside the barn when the pole is moving relative to him and the doors are momentarily closed. Observer A says the explanation is that the pole is length contracted. An observer (B) on the pole says that A is confused and that the pole fits in the barn because the doors did not close simultaneously. Observer B demonstrates to A that the pole is not length contracted in B's reference frame and that everything can be explained by clocks. Observer B explains to A that the reference frame of the pole is the true preferred reference frame because length contraction is not required to explain the observations. Observer A then asks B to measure the length of the barn as he passes through it. B does so and discovers that the barn is shorter than the rest length measured by A. B is at a loss to explain the shorter length of the barn by his own measurements, because he has already claimed to be in a preferred reference frame and that everything can be explained by time alone.

Yes, this is the mess that one gets into when trying to solve problems by the simplistic application of length contraction. I think that the paper that we have just been looking at points this shortcoming in its analysis of "Pole in the barn"

 

[Doc Al]

In the barn and pole paradox an observer on the ground (A) sees the pole which has a longer rest length than the barn, fit inside the barn when the pole is moving relative to him and the doors are momentarily closed. Observer A says the explanation is that the pole is length contracted. An observer (B) on the pole says that A is confused and that the pole fits in the barn because the doors did not close simultaneously. Observer B demonstrates to A that the pole is not length contracted in B's reference frame and that everything can be explained by clocks. Observer B explains to A that the reference frame of the pole is the true preferred reference frame because length contraction is not required to explain the observations. Observer A then asks B to measure the length of the barn as he passes through it. B does so and discovers that the barn is shorter than the rest length measured by A. B is at a loss to explain the shorter length of the barn by his own measurements, because he has already claimed to be in a preferred reference frame and that everything can be explained by time alone.

Yes, this is the mess that one gets into when trying to solve problems by the simplistic application of length contraction. I think that the paper that we have just been looking at points this shortcoming in its analysis of "Pole in the barn"

The only thing "simplistic" is all the talk about "true preferred" reference frames. B is hardly "at a loss" to explain the situation: He disagrees that his pole was ever in the barn with both doors closed. As usual, it's the relativity of simultaneity that resolves this "paradox".

 

[J.F.]

Lorentz contraction and accelerated systems
Angelo Tartaglia,and Matteo Luca Ruggiero.
http://arxiv.org/abs/gr-qc/0301050
http://arxiv.org/PS_cache/gr-qc/pdf/0301/0301050v1.pdf
The paper discusses the problem of the Lorentz contraction in accelerated systems, in the context of the special theory of relativity. Equal proper accelerations along different world lines are considered, showing the differences arising when the world lines correspond to physically connected or disconnected objects. In all cases the special theory of relativity proves to be completely self-consistent


3. Light rays
Let us now consider a situation where two equal rockets are initially placed on the x
axis at a distance l from one another. Every rocket carries on board a scientist to make
measurements, and an engineer to control the thrust of the rocket. The engineers carry
identical (initially) synchronized clocks and have the same instructions for the regime
of the engines. Let us call F the front rocket (and moving observer), and R the rear
rocket, with its observer(see figure 1). F and R are not physically connected, so that
they move exactly with the same proper acceleration at any time. The way used to
monitor the reciprocal positions is the exchange of light rays.

 

[yuiop]

Lorentz contraction and accelerated systems
Angelo Tartaglia,and Matteo Luca Ruggiero.
http://arxiv.org/abs/gr-qc/0301050
http://arxiv.org/PS_cache/gr-qc/pdf/0301/0301050v1.pdf
The paper discusses the problem of the Lorentz contraction in accelerated systems, in the context of the special theory of relativity. Equal proper accelerations along different world lines are considered, showing the differences arising when the world lines correspond to physically connected or disconnected objects. In all cases the special theory of relativity proves to be completely self-consistent
...

In what way do you think the arguments presented are not completely self-consistent?

 

[yuiop]

The only thing "simplistic" is all the talk about "true preferred" reference frames.

I was presenting the viewpoint of a person that refuses to accept the reality of length contraction. Such a person "prefers" to find a reference frame that explains the observations purely in terms of time dilation and the relativity of simultaneity. Observer B is such a person and indeed he finds that pole passes through the barn unhindered, because in his reference frame the doors were not closed simultaneously. B is now happy because he has an explanation that does not require length contraction. B prefers the reference frame that is moving relative to the barn. If B's philosophy is that length contraction is not "real" then he must regard the reason that A requires the false explanation of length contraction is that A is not in a "true" reference frame.

B is hardly "at a loss" to explain the situation: He disagrees that his pole was ever in the barn with both doors closed. As usual, it's the relativity of simultaneity that resolves this "paradox".

The situation that B is at a loss to explain (if he does not accept the reality of length contraction) is not why the pole passes through the barn, but why the length of the moving barn by B's own measurements is less than its rest length.

Say B is initially at rest with barn and records the length of the barn (its proper length). B then accelerates with his pole to some velocity v relative to the barn. Using a clock at the front of his pole B records the time that the front of the barn and the rear of the barn pass him using the same clock. From the time the barn takes to pass him he calculates the length of the barn to be less than the measurement he made when he was at rest with the barn. The challenge for B is explain why the barn appears to be shorter by his own measurements without invoking length contraction as an explanation. B could decide his own clocks are out of sync to explain why the barn appears shorter when moving relative to him. (After all B requires two clocks at different locations on his pole to determine the relative velocity of the barn). However that would contradict B's explanation of why the pole passes through the barn where B stated that it was A's clocks that were out of sync. B can not have it both ways.

 

[Doc Al]

I was presenting the viewpoint of a person that refuses to accept the reality of length contraction. Such a person "prefers" to find a reference frame that explains the observations purely in terms of time dilation and the relativity of simultaneity.

OK. You had me confused as to what you were up to, since it's clear (from other posts) that you understand perfectly well that time dilation, relativity of simultaneity, and length contraction all go together. You can't just pick the ones you like.

I was mainly responding to 1effect's response to your post, but I probably misread what he meant as well.

My bad.

 

[J.F.]

In what way do you think the arguments presented are not completely self-consistent?

I think that for a “Two- spaceship paradox” Logunov metric (12.12)
[see p.167]
http://arxiv.org/PS_cache/physics/pdf/0408/0408077v4.pdf
[see Eq.11, p.146]
http://dbserv.ihep.su/~pubs/tconf99/ps/genk.pdf
more related than Moller metric. But from Logunov metric proper distace between to spaceship does not changes.

 

[DrGreg]

The Rockets-and-String and Pole-and-Barn Paradoxes Revisited
J.H.Field
http://arxiv.org/abs/physics/0403094v3

This is a self-published paper and I'm confident it will never be published in a peer-reviewed journal because it is wrong.

Field calculates a length by two different methods and gets two different answers; he calls one the "real" length and the other the other the "apparent" length. In fact, both answers are correct answers to two different problems. To compound the confusion, Field then incorrectly claims that the "apparent" length is due to aberration i.e. it is the length you would see with your eyes, distorted due to light-propagation delays.

The error occurs in the largest paragraph on page 5. He considers two separated objects, O₁ and O₂, both initially at rest on the x-axis of inertial frame S. The objects are both accelerated along the x-axis until they are both rest in another inertial frame S'.

Both objects accelerate "by applying identical acceleration programs". This is where the confusion begins.

On the one hand, he considers that both objects begin with synced clocks (relative to S) and subsequently, whenever both clocks show the same elapsed proper time they undergo the same proper acceleration. He correctly deduces that the objects remain a fixed distance apart relative to S, and gives a correct mathematical illustration of this in the special case of constant and equal proper accelerations, on page 6. He calls this fixed distance their "real" separation measured in S.

On the other hand, he states that both objects have the same acceleration simultaneously "in their common rest frame" (i.e. in each co-moving inertial frame, he claims), and so they are both at rest in this frame, and so their separation in this frame is constant. In particular, he says, their separation in their final rest frame S', which he calls the "real" separation in S', is identical to their separation in the initial rest frame S. So, he claims, the "real" separations in both S and S' are identical. The above argument is flawed because it confuses "at the same proper time" (which depends on past motion) with "simultaneously" (which depends only on current motion). In fact, relative to S', the two objects stop accelerating at different times, so their separation in S' cannot be constant.

Having claimed the two "real" separations in the two frames are the same, Field then uses the Lorentz transform to obtain a contracted length. He calls this the "apparent" separation and incorrectly asserts this is an optical (aberration) effect. In fact, for the scenario as originally described, Field's "apparent separation" in S' is the true separation in S', and his "real separation" in S' is fiction.

The rest of the paper is based on this false premise so I haven't bothered to read it.

 

[Doc Al]

This is a self-published paper and I'm confident it will never be published in a peer-reviewed journal because it is wrong.

Agreed. (Notice how long it's been "under revision".) Lots of self-"published" crap on arXiv, so caveat lector.

 

[yuiop]

I think that for a “Two- spaceship paradox” Logunov metric (12.12)
[see p.167]
http://arxiv.org/PS_cache/physics/pdf/0408/0408077v4.pdf
[see Eq.11, p.146]
http://dbserv.ihep.su/~pubs/tconf99/ps/genk.pdf
more related than Moller metric. But from Logunov metric proper distace between to spaceship does not changes.

In the second paper Eq(11) "gives the expression of the metrics of a uniformly
accelerated FR. This is a bit vague as there are many different ways to accelerate something. Do they mean all points accelerated with constant and equal proper acceleration or do they mean born-rigid acceleration? Born-rigid acceleration is the acceleration of a system in such a way that the components of the system maintain the same proper separation. This is a complicated form of acceleration as different parts of the system have to be accelerated at different rates to maintain the required constant proper separation. This is not the acceleration used in the Bell's two spaceship thought experiment. In the Bell experiment the rockets maintain the same proper acceleration.

Say we have an inertial reference frame S with two rockets separated by some distance. The front pilot is instructed to accelerate with constant proper acceleration. The rear pilot is instructed to maintain the same proper separation from the the front rocket at all times. This would be a form of Born-rigid transportation. In ref frame S the rockets would appear to get closer together while in S' they maintain constant proper separation (as instructed). The rear pilot has to accelerate harder than the front pilot to maintain the constant spatial relationship so the two pilots do not experience the same proper acceleration. In the Bell's experiment the two rockets are instructed to maintain the same proper acceleration and so the proper separation does not remain constant and in fact increases. Any theory, paper or metric that claims that proper separation remains constant between two objects undergoing constant and equal proper acceleration is not consistent with Special Relativity, but is describing something else and is probably incorrect.

The second paper http://dbserv.ihep.su/~pubs/tconf99/ps/genk.pdf appears to be referring to an absolute or preferred reference frame they call the "Galilee Frame of Reference RF-1" and they appear to be trying to prove that an absolute reference frame is detectable. I do not think their theory is entirely consistent with Special Relativity although I can not be absolutely certain as their arguments are a little difficult to follow ;)

http://dbserv.ihep.su/~pubs/tconf99/ps/genk.pdf

 

[J.F.]

In the second paper Eq(11) "gives the expression of the metrics of a uniformly
accelerated FR. This is a bit vague as there are many different ways to accelerate something. Do they mean all points accelerated with constant and equal proper acceleration

Yes, they mean the all points accelerated with constant and equal proper acceleration by law

x(t,s)=s+\frac{c^{2}}{w}\sqrt{1+(wt/c)^{2}}-
-\frac{c^{2}}{w}

0{\leq}s{\leq} L

 

[yuiop]

Yes, they mean the all points accelerated with constant and equal proper acceleration by law

x(t,s)=s+\frac{c^{2}}{w}\sqrt{1+(wt/c)^{2}}-
-\frac{c^{2}}{w}

0{\leq}s{\leq} L

What is the w parameter? I can not find any desciption of w anywhere in that paper.

[EDIT] I'm guessing that w is the proper acceleration?

 

[J.F.]

What is the w parameter? I can not find any desciption of w anywhere in that paper.

Henri Poincare and Relativity Theory
Authors: A. A. Logunov

http://arxiv.org/abs/physics/0408077
desciption of w(=a) started from subsection:
12. Relativistic motion with constant acceleration. p.165
http://arxiv.org/PS_cache/physics/pdf/0408/0408077v4.pdf

 

[J.F.]

What is the w parameter? I can not find any desciption of w anywhere in that paper.

[EDIT] I'm guessing that w is the proper acceleration?

Yes of course, w is the proper acceleration.

 

[yuiop]

Yes, they mean the all points accelerated with constant and equal proper acceleration by law

x(t,s)=s+\frac{c^{2}}{w}\sqrt{1+(wt/c)^{2}}-
-\frac{c^{2}}{w}

0{\leq}s{\leq} L

Ok, they are essentially using the same formula used by Baez in "The Relativistic Rocket Equations" See http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

I quote from Baez:

"First of all we need to be clear what we mean by continuous acceleration at 1g. The acceleration of the rocket must be measured at any given instant in a non-accelerating frame of reference traveling at the same instantaneous speed as the rocket (see relativity FAQ on accelerating clocks). This acceleration will be denoted by a. The proper time as measured by the crew of the rocket (i.e. how much they age) will be denoted by T, and the time as measured in the non-accelerating frame of reference in which they started (e.g. Earth) will be denoted by t. We assume that the stars are essentially at rest in this frame. The distance covered as measured in this frame of reference will be denoted by d and the final speed v. The time dilation or length contraction factor at any instant is the gamma factor γ."

His formula is d = (c2/a) (sqrt[1 + (at/c)2] - 1).

Say we have two rockets initially at rest in frame S. We take one rocket r1 to be at the origin of frame S (x=0) and the other rocket r2 is at x = L. Both accelerate with constant and equal proper acceleration (a) in the x direction. At time t as measured in the frame S they will be a distance d2-d1 apart

= [L + (c2/a) (sqrt[1 + (at/c)2] - 1)] - [((c2/a) (sqrt[1 + (at/c)2] - 1)]

= L.

An observer in an inertial frame that is momentarily comoving with accelerating rockets, measures them to be a distance L/sqrt(1-(v/c)^2), which is of course greater than their initial separation.

 

[J.F.]

An observer in an inertial frame that is momentarily comoving with accelerating rockets, measures them to be a distance L/sqrt(1-(v/c)^2), which is of course greater than their initial separation.

But inertial frame S' that is momentarily comoving with both accelerating rockets is impossible. Inertial frame S' that is momentarily comoving only with one from two rockets is possible.
An observer in an inertial frame S' that is momentarily comoving only with one from two rockets measures them to be a distance L(t') by law:
L(t^{'}) = (2/\sqrt{1-t^{'}^{2}})-1
w=c=1

 

[yuiop]

But inertial frame S' that is momentarily comoving with both accelerating rockets is impossible. Inertial frame S' that is momentarily comoving only with one from two rockets is possible.
An observer in an inertial frame S' that is momentarily comoving only with one from two rockets measures them to be a distance L(t') by law:
L(t^{'}) = (2/\sqrt{1-t^{'}^{2}})-1
w=c=1

I disagree. Two rockets accelerating with constant equal proper acceleration starting simultaneously from rest in Frame S will be accelerating with equal acceleration from the point of view of someone in frame S and they will maintain constant separation and equal velocity at all times according to the observer in frame S. The separation distance measured by either rocket pilot will appear to be increasing if they make the measurement by radar signals because of light travel times and time dilation. If they have equal velocity at any infinitesimal instant in frame S then it is easy to show that there is an inertial reference frame that is instantaneously co-moving with both rockets simultaneously.

Taking the view that the rockets do not maintain the same constant separation in frame S requires assuming the motion of the front rocket somehow affects the motion of the rear rocket. How do you propose that happens if the two rockets are not physically connected?

 

[J.F.]

I disagree. Two rockets accelerating with constant equal proper acceleration starting simultaneously from rest in Frame S will be accelerating with equal acceleration from the point of view of someone in frame S and they will maintain constant separation and equal velocity at all times according to the observer in frame S. The separation distance measured by either rocket pilot will appear to be increasing if they make the measurement by radar signals because of light travel times and time dilation. If they have equal velocity at any infinitesimal instant in frame S then it is easy to show that there is an inertial reference frame that is instantaneously co-moving with both rockets simultaneously.

Taking the view that the rockets do not maintain the same constant separation in frame S requires assuming the motion of the front rocket somehow affects the motion of the rear rocket. How do you propose that happens if the two rockets are not physically connected?

I disagree. You claims in contradiction with Lorentz trasforms.

 

[Doc Al]

I disagree. You claims in contradiction with Lorentz trasforms.

Where's the contradiction? Are you saying that the Lorentz transforms prevent two rockets (moving at the same speed and thus in the same frame) from accelerating equally (constant proper acceleration) and thus maintaining equal separation as measured in their own frame? I don't think so!

FYI: There's no problem whatsoever in defining a co-moving inertial frame at any moment.

 

[J.F.]

Where's the contradiction? Are you saying that the Lorentz transforms prevent two rockets (moving at the same speed and thus in the same frame) from accelerating equally (constant proper acceleration) and thus maintaining equal separation as measured in their own frame? I don't think so!

FYI: There's no problem whatsoever in defining a co-moving inertial frame at any moment.

Which frame? Inertial frame a co-moving only with one roket, or with both rockets simultaneously?

 

[yuiop]

I disagree. You claims in contradiction with Lorentz trasforms.

Hi J.F.

First, an apology. Upon further reflection you appear to be right to claim that two rockets undergoing constant and equal proper acceleration would not have a common instantaneous co-moving inertial frame at any time. At any given instant in the momentary reference frame of one rocket, the other rocket is moving away from it and so they do not consider themselves to be stationary with respect to each other so there is no common instantaneous inertial reference frame for both rockets.

To avoid talking at cross purposes I think we should try and set some clear definitions for the types of accelerations methods applied.

Type 1 acceleration:

The rockets are not connected and are instructed to launch simultaneously in the same direction and maintain constant proper acceleration (as measured by accelerometers onboard each rocket. This is the acceleration method used in Bell’s original two rocket paradox except that a flimsy string (that snaps) connects the two rockets. It is better to replace the string with a tape measure that has drum that can feed out extra tape to maintain constant tension in the tape.

As the rockets accelerate in this scheme an observer on either rocket notices that the rockets appear to be getting further apart as measured by bouncing light signals off each other and as confirmed by the tape measure bailing out extra tape to span the gap while maintaining constant tension on the tape.

The velocity and distance of each rocket in the launch frame is given the relative rocket equations (See Baez http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html ) where a is the same for each rocket. At all times the rockets maintain constant separation as measured by an observer in the launch frame.

On a Minkowski diagram the paths of the rockets are parallel As seen from the launch frame and a line of simultaneity drawn from the origin to one rocket does not pass through the other rocket.


Type 2 acceleration:

The rockets are connected by a strong (but not perfectly rigid) rod. The two rockets are instructed to maintain constant proper separation at all times. To a certain extent this will happen naturally under gentle acceleration with only one of the rockets providing thrust and the rod undergoing natural length contraction. This acceleration scheme is sometimes know as Born-rigid transportation.

As the rockets accelerate according to this scheme the separation distance remains constant (by definition) so they are at rest with respect to each other. However the two rockets do not experience the same proper acceleration. This is the equivalence principle. To first order approximation they appear to be stationary in a gravitational field, the rear rocket “feeling” a greater gravitational force than the front rocket.

On a Minkowski diagram the paths converge and a line of simultaneity drawn from the origin to one rocket passes through the other rocket.

-------------------------------------------------------------------------------------

In summary, if both rockets experience constant and equal proper acceleration then they do not maintain constant and equal proper separation and if they measure constant and equal separation they do not measure equal proper acceleration.
In summary, if both rockets experience constant and equal proper acceleration then they do not maintain constant and equal proper separation and if they measure constant and equal separation they do not measure equal proper acceleration.

 

[Doc Al]

Oops... I forgot to correct my error here (which kev has already explained) and add my own apology.

Where's the contradiction? Are you saying that the Lorentz transforms prevent two rockets (moving at the same speed and thus in the same frame) from accelerating equally (constant proper acceleration) and thus maintaining equal separation as measured in their own frame?

The rockets can certainly arrange to maintain constant proper separation (the situation I had in mind), but as kev points out that does not mean that the rockets have equal proper accelerations. (My bad.)

FYI: There's no problem whatsoever in defining a co-moving inertial frame at any moment.

For the case I had in mind--constant proper separation--this is true.

Which frame? Inertial frame a co-moving only with one roket, or with both rockets simultaneously?

You are correct that if the two rockets have the same proper acceleration, then there is no single inertial frame that they share. My apologies!

 

[jcsd]

Two rockets starting with the same velocity with the same proper accelartion do share the same mometraily comoving inertial frame.

At any given instant they have the same velocity in the original inertial frame S. (Ignoring translation) which comoving inertial frame they are in is purely a function of their velocity in a given inertial frame and as their velocity is equal in an inertial frame they are in the same momentrailry comoving inertial frame at all times.

Lorentz contraction is the difference between the 'same' lengths measured in different inertial frames. The proper length between them changes because they are changing inertial frames.

 

[Doc Al]

At any given instant they have the same velocity in the original inertial frame S.

At any given instant according to frame S they have the same speed, but not according to each other. From the rear rocket's viewpoint, the front rocket is getting further away. (The definition of "same instant" is different for the moving rockets than for frame S.)

Lorentz contraction is the difference between the 'same' lengths measured in different inertial frames. The proper length between them changes because they are changing inertial frames.

Not sure I understand this statement. You agree that the proper separation really does change, right?

 

[jcsd]

At any given instant according to frame S they have the same speed, but not according to each other. From the rear rocket's viewpoint, the front rocket is getting further away. (The definition of "same instant" is different for the moving rockets than for frame S.)

Yes, but they are not in an inertial frame.

Take an arbitary inertial frame S. Any frame traveling at constant velocity in S is also an inertial frame. Each velocity in S defines a unique inertial frame. This is true in Newtonian physics (it's a postulate/defintion) and it's true in special relativity.

Our two spaceships are at any given instant moving with the same velcoity in an inertial frame (we cna see this just by considering the orignal frame), so at any given instant they have the same comoving inertial frame.

Not sure I understand this statement. You agree that the proper separation really does change, right?

What I'm trying to illustarte is that length contraction is something that happens when you compare lengths in different inertial frames. The proper length at any given instant is the length between the ships in the comoving inertial frame, as they change comoving inertial frame, the proper length changes.

 

[Doc Al]

Yes, but they are not in an inertial frame.

Exactly my point.

Take an arbitary inertial frame S. Any frame traveling at constant velocity in S is also an inertial frame. Each velocity in S defines a unique inertial frame. This is true in Newtonian physics (it's a postulate/defintion) and it's true in special relativity.

This is true. But the rockets are not moving with constant velocity, they are accelerating.

Our two spaceships are at any given instant moving with the same velcoity in an inertial frame (we cna see this just by considering the orignal frame), so at any given instant they have the same comoving inertial frame.

That would be true if they were moving at constant velocity, but they are not. To see this, imagine that the acceleration happens in bursts. Initially, have the rockets moving at some speed v. At the same moment in the original inertial frame S, we give each rocket a small burst of speed \Delta v. But according to the frame of the moving rockets, the rocket in front received its burst of speed before the rocket in back. As long as the rockets continue with the same acceleration, they will continue to separate. In order to maintain constant proper separation, the rear rocket must have greater acceleration than the lead rocket.

What I'm trying to illustarte is that length contraction is something that happens when you compare lengths in different inertial frames. The proper length at any given instant is the length between the ships in the comoving inertial frame, as they change comoving inertial frame, the proper length changes.

Again, I'm not sure I understand what you are saying here. I agree that length contraction is a change in perspective due to switching frames, but there's more going on here that mere length contraction. Since this is the setup for the Bell spaceship "paradox" (with the flimsy rope connecting the rockets), would you say that the rope breaks or not? (If it breaks, that's not just a disagreement between reference frames.)

 

[J.F.]

AAPPS Bulletin Vol. 15 No. 5, October 2005
http://www.aapps.org/archive/bulletin/vol15/15-5/15_5_p17p21abs.html

Jong-Ping Hsu
Nobuhiro Suzuki

We demonstrate a resolution to the “two-spaceship paradox” by explicit calculation using coordinate transformations with at least one frame undergoing constant linear acceleration. A metric such as ds2 = (1 + Kx)2dw2—dx2—dy2—dz2 can lead to a coordinate transformation between an inertial frame and a frame moving with a constant linear acceleration. This coordinate transformation reduces to the Lorentz transformation in the limit of zero acceleration.

This solution incorrect!

I think, that a Møller metric such as
ds^{2}=(1+kx)^{2}dw^{2}-dx^{2}-dy^{2}-dz^{2}
is not completely adequate for the “two-spaceship paradox”.
To the given problem more corresponds uniformly accelerated frame:

ds^{2}=c^{2}dt^{2}-\left{[dx-\frac {wtdt} {\sqrt{1+(wt/c)^{2}}} \right]^{2}-
dy^{2}-dz^{2}

This metrics turns out as result of a shift
x\to x-(c^2/w){\sqrt{1+(wt/c)^{2}
from the Minkowski frame:
ds^2=c^{2}dt^{2}- dx^{2}-dy^2-dz^2

 

[yuiop]

I think, that a metric such as
ds^{2}=(1+kx)^{2}dw^{2}-dx^{2}-dy^{2}-dz^{2}
is not completely adequate for the “two-spaceship paradox”.
To the given problem more corresponds uniformly accelerated frame:

ds^2=c^{2}dt^{2}- \left[dx-\frac {wtdt} {\sqrt{1+(wt/c)^{2}} \right]^2-dy^2-dz^2

This metrics turns out as result of a shift
x\to x-(c^2/w){\sqrt{1+(wt/c)^{2}
from the Minkowski frame:
ds^2=c^{2}dt^{2}- dx^{2}-dy^2-dz^2

Your point is ...?

Do you agree that a string connecting two rockets undergoing the Type 1 acceleration method described in post #37 will snap due to length contraction?

 

[J.F.]

Your point is ...?

Do you agree that a string connecting two rockets undergoing the Type 1 acceleration method described in post #37 will snap due to length contraction?

Hi kev.
Let's consider a case when the ships are accelerated according to the laws:

x_{1}(t)= \int_{0}^{t} v( \tau )d\tau

x_{2}(t)= L +\int_{0}^{t} v( \tau )d\tau

v(t)\to v=const

Finally, both of the ship, does not get on the Minkowski frame:

ds^2=c^{2}dt^{2}- dx^{2}-dy^2-dz^2

As consequence Lorentz length contraction is broken

 

[yuiop]

Hi kev.
Let's consider a case when the ships are accelerated according to the laws:

x_{1}(t)= \int_{0}^{t} v( \tau )d\tau

x_{2}(t)= L +\int_{0}^{t} v( \tau )d\tau

v(t)\to v=const

Finally, both of the ship, does not get on the Minkowski frame:

ds^2=c^{2}dt^{2}- dx^{2}-dy^2-dz^2

As consequence Lorentz length contraction is broken

Hi J.F.

Sorry I did not see your post earlier. I take it that you have concluded that if two rockets with initial separation L in the launch frame, take off simultaneously with equal proper acceleration, then their mutual proper separation will be increasing while their separation measured by an observer that remains in the launch frame remains constant. That, I think is the whole point of the "two spaceship paradox" and is the reason why Bell concluded (correctly in my opinion) that a string connecting the two rockets will snap.

 

[J.F.]

Hi J.F.

Sorry I did not see your post earlier. I take it that you have concluded that if two rockets with initial separation L in the launch frame, take off simultaneously with equal proper acceleration, then their mutual proper separation will be increasing while their separation measured by an observer that remains in the launch frame remains constant. That, I think is the whole point of the "two spaceship paradox" and is the reason why Bell concluded (correctly in my opinion) that a string connecting the two rockets will snap.

Agreed. But I wish to discuss only a method offered Jong-Ping Hsu.
Hsu has started own method from the transformations:(1)-(3) see URL:
http://www.geocities.com/jaykovf/a127.JPG

The transformations:(1)-(3) preserve the metric (4) see URL:

http://www.geocities.com/jaykovf/a124.JPG

But the metric (4) does not concern to a case when a two rockets with initial separation L in the launch frame, take off simultaneously with equal proper acceleration w.From what it suddenly Hsu postulates, what rockets after the beginning of movement have dropped in on FR (4)?