# Two-spaceship paradox This solution incorrect

1. Jan 29, 2008

### J.F.

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

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!

Last edited by a moderator: May 3, 2017
2. Jan 29, 2008

### yuiop

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.

Last edited by a moderator: May 3, 2017
3. Jan 29, 2008

### 1effect

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

4. Jan 29, 2008

### pervect

Staff Emeritus
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:

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.

5. Jan 29, 2008

### yuiop

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.

6. Jan 29, 2008

### yuiop

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."

7. Jan 29, 2008

### 1effect

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.

Last edited: Jan 29, 2008
8. Jan 30, 2008

### 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

9. Jan 30, 2008

### 1effect

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

10. Jan 30, 2008

### Staff: Mentor

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.

11. Jan 30, 2008

### 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.

Last edited: Jan 30, 2008
12. Jan 30, 2008

### 1effect

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.

13. Jan 30, 2008

### 1effect

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.

Last edited: Jan 30, 2008
14. Jan 30, 2008

### ZapperZ

Staff Emeritus
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.

15. Jan 30, 2008

### 1effect

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).

Last edited: Jan 30, 2008
16. Jan 30, 2008

### 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.

Last edited: Jan 30, 2008
17. Jan 30, 2008

### 1effect

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"

18. Jan 30, 2008

### Staff: Mentor

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".

19. Jan 30, 2008

### 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.

20. Feb 1, 2008

### yuiop

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