Talk:Bell's spaceship paradox

Bell's Spaceship Paradox

Treat the combined system of two spaceships and the tether as a single unit. As seen from an oberver at rest the entire system contracts. As seen on the spaceships, there is no contraction. In either case, the tether cannot break. DrWhoJP 11:24, 18 September 2007 (UTC)

That's not useful. Another way to state that the entire accelerating system contracts is to say that the front accelerates less than the rear in the frame of the launch pad. However, the Bell spaceship example/paradox is about identical rockets that equally accelerate. Harald88 16:27, 30 September 2007 (UTC)

No, it's not about identical rockets that equally accelerate. It's about rockets that accelerate (differently) "such that they remain a fixed distance apart as viewed from the rest frame". That's not identical rockets with identical acceleration. Simanos (talk) 09:11, 5 December 2009 (UTC)

Comments Including a Descriptive Solution

Obviously, to have any meaningful discussion about Bell’s Spaceship Paradox it is important to have that discussion in reference to the correctly stated version of the paradox or, at least, with a common understanding of the paradox. While I didn’t research the wording of the paradox, the precise wording only appears to matter to the extent of stating that the acceleration of each spaceship is “measured by an onboard accelerometer.” According, I did not confirm the wording of the paradox in the Wikipedia article as being historically correct. Appropriately, my comments below are in reference to the paradox as stated and the article as worded on February 10, 2009.

History: The whole point of the problem was to resolve the ambiguity concerning the reality of "Lorentz contraction" & time dilation. Right from the start in 1905, there was constant debate about how "real" the effects were. Although some authors wrongly asserted the effects were always physically real or some that they were merely apparent (a sort of "perspective" effect), the theorists mostly hedged about inbetween.

Dewan & Beran came up with the beautifully simple and elegant "two rockets and string" scenario in order to (as they thought) prove Lorentz contraction to be physically real since apparent effects could obviously not snap strings.

Consequently there are no "onboard accelerometers" in any original statements of the problem (or by Bell 17 years later). They are redundancies wheeled in by some commentators that only serve to confuse the issue. The problem simply requires the two rockets (or spaceships) to be absolutely identical and to have identical, unspecified acceleration programs. Like all seminal relativity problems or paradoxes, the observers should be able to explore the structure of their environment using the classic apparatus of measuring rods clocks and light signals.

For purposes of synchronising clocks and for gauging rocket separation, the travellers must revert to inertial motion. There is no difficulty about this since, in order to preserve the "identicality" of the two rockets, they can simply cut engines either at identical (unadjusted) on-board clock times, or equivalently at identical fuel consumptions. They will then remain symmetrical to stationary ground observers, as required, but of course their own mutual measurements of synchronisation & separation will now differ from ground.

Good to see this initiative as there seems to be a lot of wasted discussion on this page based on inconsistent "definitions" of the problem. From what I read in the article the basic assumption in the definition of the paradox is "that they remain a fixed distance apart as viewed from the rest frame". This is equivalent to the rockets having "identical, unspecified acceleration programs", again as measured from the rest frame of course! The calculation in the analysis section is based on this. It concludes that once the rockets reach terminal velocity the usual Lorentz contraction formula applies as expected. This means that in the frames of reference of the rockets the distance between them must have increased (by a factor gamma). This means that the string must have stretched (or if it is not sufficiently elastic, snapped) along the way regardless of the details of what happened in between. AlexFekken (talk) 14:45, 19 January 2010 (UTC)

My plan is to post the following discussion on the discussion page for feedback. Depending on the feedback I would expect to proceed with one of the following options: (1) revise my discussion, (2) post the solution in a section of the main article under the heading “Solution,” (3) create a new article titled “Solution to Bell’s Spaceship Paradox,” or (4) abandon my discussion because of a lack of any consensus.

The Paradox

Bell’s Spaceship Paradox is considered a paradox because there are seemingly conflicting answers to the question of whether the string will break depending on the approach and the method used to solve the problem. In brief, if the two equally accelerating spaceships and the interconnecting string are considered to constitute three co-moving elements of an accelerated frame of reference, then the expected length compression in the direction of travel would appear to affect all points, elements, and distances within the frame including both the two spaceships and the interconnecting string. Consequently, both the distance separating the two spaceships and the interconnecting string would appear to sustain equal compression, and the string would therefore not break. This view is seemly consistent with the view of an observer on either spaceship, where the observer would seemingly (though incorrectly) not expect to observe any change in the distance between the two spaceships or any change in the length of the interconnecting string.

However, if the two equally accelerating spaceships are considered to constitute two individual and separate accelerated frames of reference, then the length compression in the direction of travel would appear to affect each spaceship as a separate entity, but not the distance between them. Because the interconnecting string is initially being accelerated by the forward-positioned spaceship by virtue of being pulled (as opposed to being pushed by the rear-positioned spaceship), it will also exhibit length compression and therefore stretch and eventually break. This view is seemingly consistent with the view of a stationary observer where the observer views parallel trajectories for each spaceship maintaining a constant distance between them, but views compression in the length of the string. Accordingly, the relevant question in the paradox is which of these two opposing views, if either, is correct.

The Solution

Because the wording of the problem states that the acceleration of each spaceship is measured by an onboard accelerometer, the correct solution is that the string will stretch and eventually break. However, had the wording been different regarding what constituted equal acceleration, the answer could have been different. In the case as stated, because each spaceship is accelerated independently, the distance between them remains fixed as observed by a stationary observer, who also views the compression in the length of string and thereby logically concludes that the string will break. In contrast, observers on the two spaceships observe the distance between the two spaceships to increase thereby also providing a logical (but different) explanation for why they also view the breakage of the string.

Regarding onboard clocks, to a stationary observer the onboard clocks on both spaceships would remain in synchronization to each other. Furthermore, if the two spaceships are in radio communication, both the stationary observer and onboard observers would observe each spaceship to have a corresponding red shift or blue shift during their corresponding periods of acceleration. However, in contrast to the stationary observers, onboard travelers would observe their radio communication to become skewed and out of synchronization. The periods of observed red shift and blue shift would not be equal and signals would indicate that the clock on the other spaceship had shifted in time during the period of mutual spaceship acceleration. Stationary observers would also view this as an observation of the travelers thereby observing the travelers to view a loss of synchronization while the stationary observers would view synchronization of time being preserved.

An example is as follows: A traveler in the forward spaceship would observe radio or light transponder signals sent and received from the forward spaceship to indicate that the onboard clock on the rear spaceship was running slower (during the period of acceleration) thereby lagging or reading behind in time. In contrast, a traveler in the rear spaceship would observe radio or light transponder signals sent and received from the rear spaceship to indicate that the onboard clock on the forward spaceship was running faster (during the period of acceleration) thereby leading or reading ahead in time. Furthermore, all of this would appear logical to the stationary observer, but could appear nonsensical to traveling observers unless they rationalized the event from the perspective of the stationary observer and then transferred the event to their own frame of reference. This aforementioned skew is a consequence of the travelers experiencing acceleration.

Thus far the explanation is broadly correct. The subsequent consideration of how to accelerate to preserve the string or clock synchronisation is largely irrelevant. What ought to be considered is how string breaking could occur if we repeat the scenario from a new inertial launch station that was co-moving with the rockets at some point in their travel. If two new identical rockets are simultaneously launched in the opposite direction to before from this moving inertial platform, the string must now get longer as the rockets accelerate and their "absolute" velocity decreases.

As Dewan & Beran pointed out in their original 1959 paper, the reason the string must also now break is not due to physical contraction, but because the re-synchronised clocks used to launch "simultaneously" will actually launch the rocket pulling the string, slightly before the tail end rocket.

This shows that although the "absolute" frame of reference cannot be identified in SR, it must exist and all reciprocal "contraction" and "time dilation" between relatively moving observers are an unquantifiable mixture of physically real, and merely apparent effects.

Ultimately, it is obvious this must be so, as it is clearly mathematically impossible (and therefore also physically impossible) for both A>B and A<B to hold. Put simply, impossible for two rods to be physically shorter than each other, or two clocks to run slower than each other.

Furthermore, if the pilot of the forward spaceship adjusts the acceleration of the forward spaceship to compensate for the compression in length to thereby not break the string, he would have to accelerate at a lower rate and for a longer period of time (to achieve the same final velocity). To travelers onboard they would observe the onboard accelerometer displaying a lower rate of acceleration and the onboard clock displaying a longer period of acceleration. However, these observations would be completely consistent with observers who view the two spaceships as stationary in a uniform gravitational field. And, if educated in the general theory of relativity, the travelers would be aware that in the up-field spaceship the onboard accelerometer would be expected to read lower and the onboard clock would be expected to operate faster and thereby read ahead.

Accordingly, to preserve synchronization between the onboard clocks for both spaceships, the event must include two provisions not stipulated in the paradox. First, the forward spaceship must accelerate at a lesser rate than the rearward spaceship by an amount corresponding to that required to account for the length compression in the direction of travel. Second, the onboard clock on the forward spaceship must be slowed to a lower rate than the onboard clock on the rearward spaceship by an amount corresponding to that required for the different clock rates that occur in an equivalent gravitational field corresponding to the spaceship accelerations. If the two above provisions are accomplished, then the event as observed by the travelers would appear as if both spaceships were in a common gravitation field and the upper-positioned spaceship was simply adjusting to the known and understood gravitational effects of general relativity.

I think those "provisions" need not be included or stipulated as they are a consequence of the problem statement that the distance as measured in the rest frame must remain the same. They merely spell out in more detail what the calculation of the Lorentz contraction at terminal velocity already shows us: that there must be relative motion between the rockets in their own reference frames and as a result that they cannot always be in the same frame of reference. The relative motion causes the string to snap. Incorrectly assuming that the rockets are always in the same frame of reference causes the paradox. AlexFekken (talk) 15:27, 19 January 2010 (UTC)

Lorentz Transformations

Regarding problems in using the Lorentz Transformations in solving the paradox, the Lorentz Transformations are simply inadequate to provide a solution to the paradox because they are inadequate to represent the transitions during periods of acceleration. Fundamentally, the transformations map distance and time for only two frames with one common point in time and one common point of position between them (normally origin). The subject paradox is represented by three frames having a coincident point in time, but two different points of position. Although the two spaceships have the same velocity (by one definition), they do not share the same moving frame in terms of t and x. Accordingly, although you can plug numbers into the Lorentz Transformations, discontinuities result. In fact, instantaneous changes in velocity cause corresponding instantaneous changes in positions and clock time in the transformed frame. Even large accelerations over small distances or small accelerations over large distances can cause huge discontinuities.

The Article

I found the Wikipedia article to be confusing. From my perspective the article appears to have a greater interest and focus in reporting the opinions of Dewan, Beran, Nawrocki, Matsuda, and Kinoshita than in clarifying the paradox directly. The article appears to provide the opinion that the solution to the paradox is unknown and that prominent physicists do not agree, hence the references to Dewan, Beran, Nawrocki, Matsuda, and Kinoshita.

Regarding my understanding of the paradox, although I certainly do not have any prominence or notoriety in physics, I didn’t find the paradox overly difficult to resolve (assuming that my explanation is correct). I also found numerous articles, tutorials, q and a’s, and class notes written by physics professors and physicists including Ph.D.’s that conclude what I explained above to be correct. Regarding the opinions of Dewan, Beran, Nawrocki, Matsuda, and Kinoshita, they may or may not have significance. I found that the references to them were typically journal articles to which I didn’t have cost free access, so I was unable to make any further assessment.

The Discussion

I am also confused by the long, detailed, and repeated discussions in this discussion section, especially between parties identified as Rod Bell and EMS, but also by Pjacobi, Harold88, and Pervect. I attempted to read and follow some of these discussions, but I simply did not have the time required. In brief, what I viewed were discussions based in some confusion over frames of reference, the consequences of acceleration, and an inability to apply the principles of the general theory of relativity. I simply do not have time to add clarification to each of these discussions. I did respond to a comment by Rick Crawford on 24 May 2008 in the section headed “Constant distance argument faulty!?”

Bill Wolf (talk) 12:55, 17 February 2009 (UTC)

Hi Bill I'm just "passing by" (I rarely look at Wikipedia discussions anymore), but I can comment on an essential point. I see that you wrote "From my perspective the article appears to have a greater interest and focus in reporting the opinions of Dewan, Beran, Nawrocki, Matsuda, and Kinoshita than in clarifying the paradox directly. " Well, that is the essence of what wikipedia is meant to do - only provide facts of reputed opinions, and not do any WP:OR. If you want to "clarify the paradox directly" in Wikipedia, then you are required to first get a paper published in a reputed journal, and preferably wait until someone cites it (so that its notability is evident). Then your opinion is good enough to be included in Wikipedia. If this is not clear, please inform yourself with the info on the left of this page. Harald88 (talk) 09:51, 1 July 2009 (UTC)

Analysis

This is a funny analysis.

1. You prove that that as observed from both spaceships, their distance L is always the same. This is right.

No this is totally wrong. Perhaps you misunderstood the problem: the distance in the rest frame is constant, what happens in the moving frames is the question. AlexFekken (talk) 15:41, 19 January 2010 (UTC)

1. You draw a spacetime diagram which illustrates this and includes an additional event B'' which obviously shows where the second spaceship will be some time after B' has occured.
2. You draw a line connecting the events A' and B'' and assert that the earth is flat - oops, sorry - that this line shows the spaceships' distance as observed from a comoving frame, i.e. that in the comoving observer's spacetime the string is stretched.

Never seen a space-time diagram with different frames of reference? Please check the textbooks. AlexFekken (talk) 15:46, 19 January 2010 (UTC)

1. Then by some proper mathematics and presuming the assertion that the earth is flat - oops, sorry - that A'B'' is the spaceships' distance as observed from a comoving frame, you prove that this means the string is streteched in the comoving observer's spacetime.
2. From this you conclude that the string will break.

Hey, are you kidding?

You assert something which is equivalent with "the string will break" (3), you prove that the assertion is equivalent to "the string will break" (4), and from this you conlude that your assertion has been right (5)?

THINK:

An "observer in a comoving frame" is the same like an astronaut going off-board and hovering near his ship with no relative movement to the ship. Now you say: As long as he is on-board, he sees the other ship at distance L, and as soon as he goes off-board, he suddenly sees it at Lɣ ? Oh dear, this is not Wikipedia, this is a voodoo show.

After acceleration, a second-ship-comoving observer has exactly the same future world line as the second spaceship! He is only displaced in y and/or z dimensions, but neither in x or in t! He will observe the first ship in synchronous time and at the same distance like someone on-board of the second ship, e.g. at an event A'' which is in the diagram at the same time and left hand of B''.

This whole analysis is nothing more but circular conclusions based on a wrong (or misunderstood) diagram. What is the source of this analysis? A textbook, or just proprietary WikiPhilosophy? Wikipedia should only include common knowledge, not private speculations. --217.87.150.69 (talk) 11:29, 14 November 2009 (UTC)

I've always found these graphs to be a wonderful way to completely duck things that can withstand a proof. That doesn't make this wrong, but throwing in the Lorentz Gamma for a length doesn't make overly much sense. If you have two objects, attached by a third object, that accelerate at a uniform rate, it is effectively a single body with a single acceleration rate. So long as the velocities used are reasonable (micrometeorite shields, for example, work on the principle that when moving faster than the speed of sound in its component materials an object tends to spend its energy in shattering rather than in elastic or even plastic deformation).

The fact that the same acceleration provides the same velocity means both are in the same reference plane, and as such no relativistic effect is seen and no relativistic factors are required.

Even were the observations of the position of the other craft incorrect, an observation of position is not the position. Because an ambulance sounds faster coming than going doesn't mean it actually is going faster.

If you took the theory that the bodies could be made into disjoint elements, then the strain of the change in length (where only a few microinches per inch will cause failure) would destroy everything at speed.

Given the hand-waving of Cerenkov Radiation and the need to drop to graphs to make any sort of statement about these systems, it seems our understanding is in its infancy and as such blanket statements of correctness are simply conjecture and theory until a suitable proof (mathematical or experimental) can be formed, rigorously tested, and defended against all better options.

Now, GPS satellites are already based on relativistic theory, if the production companies could be convinced to have their systems talk to other in-space systems in different orbits and have them share the data, then there would be empirical data to test against these theories, although the low velocity probably wouldn't give us enough resolution. Michael F (talk) 19:29, 22 April 2011 (UTC)

Improvements to make?

There are some discussions here above that suggest that the article lacks clarity. Perhaps with minor editing the text can be improved.

For example the first, rather technical picture in the "Analysis" section has nothing to do with the basic and easy to understand elaboration of Dewan and Beran's analysis. That's confusing to say the least, so I'll move that picture lower to near to the corresponding text.

What other improvements should be made?

Also, someone (Evanh2008?) put an "Original Reasearch" tag above the article with reference to this Talk page, but as far as I can tell, there's no motivation for it given here. In particular which parts of the text appear to lack references? Harald88 (talk) 11:00, 18 October 2011 (UTC)

As nobody backup up the motivation behind that tag (to the contrary, there was criticism about citing reliable sources instead of doing original research), I will now remove it. Harald88 (talk) 12:44, 19 January 2012 (UTC)

So do the ships accelerate equally or not? And is the observer moving with the ships?

I think those two things should be made more clear, the way it is now it is too easy to overlook details like this and get conflicting answers. --TiagoTiago (talk) 07:17, 5 November 2011 (UTC)

An observer moving or not moving with the ships can't affect if the string breaks! Perhaps you mean that the different analyses should more clearly state from what perspective they are made? Apart of that, the article starts with:

"both spaceships start to accelerate, in such a way that they remain a fixed distance apart as viewed from the original rest frame."

Is that not clear enough? Note that "equal acceleration" is an observer-dependent statement. Harald88 (talk) 12:53, 19 January 2012 (UTC)

Too many other things to read?

There us currently a complaint against the last section of this article, advising: "removing excessive, less relevant or many publications with the same point of view". Is this request reasonable in the case of a paradox that seems not to be resolved to general satisfaction? Or will readers need to see, in more detail, how people with expertise have tried to deal with the problem?P0M (talk) 16:15, 20 January 2012 (UTC)