Bell's spaceship paradox

Bell's spaceship paradox is a thought experiment in special relativity involving accelerated spaceships and strings. The results of this thought experiment are for many people paradoxical. While J. S. Bell's 1976 version cite book | author=Bell, J. S. | title=Speakable and unspeakable in quantum mechanics | location=Cambridge | publisher=Cambridge University Press | year=1987 | id=ISBN 0-521-52338-9 A widely available book which contains a reprint of Bell's 1976 paper ] of the paradox is the most widely known, it was first designed by E. Dewan and M. Beran in 1959 [cite journal| last = Dewan| first = E.| authorlink = | coauthors = Beran, M.| title = Note on stress effects due to relativistic contraction| journal = American Journal of Physics| volume = 27| issue = 7| pages = 517–518| publisher = American Association of Physics Teachers| date = March 20 1959| url = http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=AJPIAS&ONLINE=YES&smode=strresults&sort=chron&maxdisp=25&threshold=0&possible1=Note+on+stress+effects+due+to+relativistic+contraction&possible1zone=article&fromyear=1959&fromvolume=27&OUTLOG=NO&viewabs=AJPIAS&key=DISPLAY&docID=1&page=1&chapter=0| doi = 10.1119/1.1996214| id = | accessdate = 2006-10-06 ] as an argument for the physical reality of length contraction.

Bell's thought experiment

In Bell's version of the thought experiment, two spaceships, which are initially at rest in some common inertial reference frame are connected by a taut string. At time zero in the common inertial frame, both spaceships start to accelerate, with a constant proper acceleration g as measured by an on-board accelerometer. Question: does the string break - i.e. does the distance between the two spaceships increase?

In a minor variant, both spaceships stop accelerating after a certain period of time previously agreed upon. The captain of each ship shuts off his engine after this time period has passed, as measured by an onboard clock. This allows before and after comparisons in suitable inertial reference frames in the sense of elementary special relativity.

According to discussions by Dewan & Beran and also Bell, in the spaceship launcher's reference system the distance between the ships will remain constant while the elastic limit of the string is length contracted, so that at a certain point in time the string should break.

Objections and counter-objections have been published to the above analysis. For example, Paul Nawrocki suggests that the string should not break, [cite journal| last = Nawrocki| first = Paul J.| authorlink = | title = Stress Effects due to Relativistic Contraction| journal = American Journal of Physics| volume = 30| issue = 10| pages = 771–772| publisher = | date = October 1962| url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000030000010000771000001&idtype=cvips&gifs=yes| doi = 10.1119/1.1941785| id = | accessdate = 2006-10-06 ] while Edmond Dewan defends his original analysis from these objections in a reply. [cite journal| last = Dewan| first = Edmond M.| authorlink = | title = Stress Effects due to Lorentz Contraction| journal = American Journal of Physics| volume = 31| issue = 5| pages = 383–386| publisher = | date = May 1963| url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000031000005000383000001&idtype=cvips&gifs=yes| doi = 10.1119/1.1969514 | id = | accessdate = 2006-10-06 (Note that this reference also contains the first presentation of the ladder paradox.)] Bell reported that he encountered much skepticism from "a distinguished experimentalist" when he presented the paradox. To attempt to resolve the dispute, an informal and non-systematic canvas was made of the CERN theory division. According to Bell, a "clear consensus" of the CERN theory division arrived at the answer that the string would not break. Bell goes on to add "Of course, many people who get the wrong answer at first get the right answer on further reflection". Later, Matsuda and Kinoshita [ cite journal | author=Matsuda, Takuya; & Kinoshita, Atsuya | title=A Paradox of Two Space Ships in Special Relativity | journal=AAPPS Bulletin | volume=February | year=2004 | pages=? [http://www.aapps.org/archive/bulletin/vol14/14_1/14_1_p03p07.pdf eprint version] ] reported receiving much criticism after publishing an article on their independently rediscovered version of the paradox in a Japanese journal. Matsuda and Kinshita do not cite specific papers, however, stating only that these objections were written in Japanese.

Analysis

In the following analysis we will treat the spaceships as point masses and only consider the length of the string. We will analyze the variant case previously mentioned, where both spaceships shut off their engines after some time period T.

According to the discussions by Dewan & Beran and also Bell, in the "spaceship-launcher"'s reference system (which we'll call "S" ) the distance "L" between the spaceships ("A" and "B ") must remain constant "by definition".

This may be illustrated as follows. The displacement as function of time along the X-axis of "S" can be written as a function of time f(t), for t > 0. The function f(t) depends on engine thrust over time and is the same for both spaceships. Following this reasoning, the position coordinate of each spaceship as function of time is:

:$x\_A(t)\; =\; a\_0\; +\; f(t)\; qquad\; x\_B(t)\; =\; b\_0\; +\; f(t)$

where :f(0) is assumed to be equal to 0:x_A(t) is the position (x coordinate) of spaceship A:x_B(t) is the position (x coordinate) of spaceship B:a₀ is the position of spaceship A at time 0:b₀ is the position of spaceship B at time 0.

This implies that $x\_A(t)\; -\; x\_B(t)\; =\; a\_0\; -\; b\_0,$ which is a constant, independent of time.

In other words, the distance "L" remains the same. This argument applies to all types of synchronous motion.Thus the details of the form of f(t) are not needed to carry out the analysis. Note that the form of the function f(t) for constant proper acceleration is well known (see the article hyperbolic motion).

Referring to the space-time diagram (above right), we can see that both spaceships will stop accelerating at events A` and B`, which are simultaneous in the launching frame S.

We can also see from this space-time diagram that events A` and B` are not simultaneous in a frame comoving with the spaceships. This is an example of the relativity of simultaneity.

From our previous argument, we can say that the length of the line segment A'B' equals the length of the line segment AB, which is equal to the initial distance L between spaceships before they started accelerating. We can also say that the velocities of A and B in frame S, after the end of the acceleration phase, are equal to v. Finally, we can say that the proper distance between spaceships A and B after the end of the acceleration phase in a comoving frame is equal to the Lorentz length of the line segment A`B``. The line A`B`` is defined to be a line of constant t', where t' is the time coordinate in the comoving frame, a time coordinate which can be computed from the coordinates in frame S via the Lorentz transform:

:$t\text{'}\; =\; left(\; t\; -\; v\; x\; /\; c^2\; ight)\; /\; sqrt\{1-v^2/c^2\}$

Transformed into a frame comoving with the spaceships, the line A`B`` is a line of constant t` by definition, and represents a line between the two ships "at the same time" as simultaneity is defined in the comoving frame.

Mathematically, in terms of the coordinates in frames S and S', we can represent the above statements by the following equations::$t\_\{B\text{'}\}\; =\; t\_\{A\text{'}\},$

results into:

:$x\_B\; -\; x\_A\; =\; x\_\{B\text{'}\}-x\_\{A\text{'}\}\; =\; L,$ :$x\_\{B"\}\; -\; x\_\{B\text{'}\}\; =\; v\; left(\; t\_\{B"\}\; -\; t\_\{B\text{'}\}\; ight)$ :$t\text{'}\_\{B"\}=\; t\text{'}\_\{A\text{'}\},$ implies::$t\_\{B"\}\; -\; frac\{v\}\{c^2\}\; x\_\{B"\}\; =\; t\_\{A\text{'}\}\; -\; frac\{v\}\{c^2\}\; x\_\{A\text{'}\}$ In frame S', since both ends of the rope are marked simultaneously :

:$overline\{A\text{'}B"\}\; =\; x\text{'}\_\{B"\}-x\text{'}\_\{A\text{'}\},$

where:

:$x\text{'}\_\{B"\}\; =\; gamma*(x\_\{B"\}\; -\; v*t\_\{B"\}),$ :$x\text{'}\_\{A\text{'}\}\; =\; gamma*(x\_\{A\text{'}\}\; -\; v*t\_\{A\text{'}\}),$

so:

:$x\text{'}\_\{B"\}-x\text{'}\_\{A\text{'}\}\; =\; (x\_\{B"\}\; -\; x\_\{A\text{'}\})*\; frac\{1\}\{gamma\}$

Calculate:

:$x\_\{B"\}-x\_\{A\text{'}\}\; =(x\_\{B"\}\; -\; x\_\{B\text{'}\})+(x\_\{B\text{'}\}-x\_\{A\text{'}\})=(x\_\{B"\}-x\_\{A\text{'}\})*(frac\{v\}\{c\})^2+L\; ,$

so:

:$x\_\{B"\}-x\_\{A\text{'}\}\; =L*\{gamma\}^2\; ,$

therefore:

:$overline\{A\text{'}B"\}\; =\; L*\{gamma\},$

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.

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.

The Bell spaceship paradox is very rarely mentioned in textbooks, but appears occasionally in special relativity notes on the internet.

An equivalent problem is more commonly mentioned in textbooks. This is the problem of Born rigid motion. Rather than ask about the separation of spaceships with the same acceleration, the problem of Born rigid motion asks "what acceleration profile is required by the second spaceship so that the distance between the spaceships remains constant in their proper frame". The accelerations of the two spaceships must in general be differentcite book | author=Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | id=ISBN 0-7167-0344-0 | pages=165] cite journal | last = Nikolić |first = Hrvoje | title = Relativistic contraction of an accelerated rod | journal=American Journal of Physics | volume=67 | issue = 11| year=1999 | pages = 1007–1012|publisher = American Association of Physics Teachers| date = 6 April 1999 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000011001007000001&idtype=cvips&gifs=yes|doi = 10.1119/1.19161 |accessdate = 2006-10-07 [http://arxiv.org/abs/physics/9810017 eprint version] ] In order for the two spaceships, initially at rest in an inertial frame, to maintain a constant proper distance, the lead spaceship must have a lower proper acceleration.

See also

* Ehrenfest paradox
* Physical paradox
* Supplee's paradox
* Rindler coordinates
* Twin paradox
* Born rigidity
* Hyperbolic motion (relativity)

External links

* Michael Weiss, [http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html Bell's Spaceship Paradox] (1995), USENET Relativity FAQ
* Austin Gleeson, [http://www.ph.utexas.edu/~gleeson/NotesChapter13.pdf Course Notes Chapter 13 ] See "Section 4.3"

References

Further reading

*cite journal | author=Romain, J. E. | title=A Geometric approach to Relativistic paradoxes | journal=Am. J. Phys. | volume=31 | year=1963 | pages=576–579 | doi=10.1119/1.1969686

*cite journal | author=Hsu, Jong-Ping; & Suzuki | title = Extended Lorentz Transformations for Accelerated Frames and the Solution of the "Two-Spaceship Paradox" | journal = AAPPS Bulletin | volume=October | year=2005 | pages=? [http://www.aapps.org/archive/bulletin/vol15/15-5/15_5_p17p21%7F.pdf eprint version]