Bell's spaceship paradox

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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 Bell's 1976 version [1] of the paradox is the most widely known, it was first designed by E. Dewan and M. Beran in 1959 [2] as an argument for the physical reality of length contraction.

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[edit] 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 ideal clock (that is to say, a clock designed to a hypothetical standard of perfection) carried on board his ship. 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, [3] while Edmond Dewan defends his original analysis from these objections in a reply. [4] 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". [1]. Later, Matsuda and Kinoshita [5] 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.

[edit] 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 of their engines after some time period T.

The world lines (navy blue curves) of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dotted lines are "lines of simultaneity" for observer A. Is the spacelike line segment A′B″ longer than the spacelike line segment AB?
The world lines (navy blue curves) of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dotted lines are "lines of simultaneity" for observer A. Is the spacelike line segment A′B″ longer than the spacelike line segment AB?

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 = a_0 + f(t) \qquad x_B = b_0 + f(t)

where

f(0) is assumed to be equal to 0
xA is the position (x coordinate) of spaceship A
xB 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 - x_B = a_0 - b_0\, which is a constant, independent of time. 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 wikipedia 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' = \left( t - v x / c^2 \right) / \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. Because the Lorentz interval is a geometric quantity which is independent of the choice of frame, we can compute its value in any frame which is computationally convenient, in this case frame S.

Mathematically, in terms of the coordinates in frame S, we can represent the above statements by the following equations:

t_{B'} = t_{A'}\,
x_B - x_A = x_{B'}-x_{A'} = L\,
x_{B''} - x_{B'} = v \left( t_{B''} - t_{B'} \right)
t_{B''} - \frac{v}{c^2} x_{B''} = t_{A'} - \frac{v}{c^2} x_{A'}
\overline{A'B''} = \sqrt{ \left( x_{B''}-x_{A'} \right)^2 - c^2 \left( t_{B''} - t_{A'} \right)^2 }

By introducing the auxiliary variables

H = t_{B''} - t_{B'} = t_{B''} - t_{A'}\,
W = x_{B''} - x_{B'}\,

and noting that

W+L = x_{B''} - x_{B'} + x_{B'} - x_{A'} = x_{B''} - x_{A'}\,

the above equations can be re-written as

W = v H \qquad H = \frac{v}{c^2} \left(W + L \right) \qquad \overline{A'B''} = \sqrt{\left(W+L\right)^2 - c^2 H^2}

and solved to find

\overline{A'B''} = \frac{L}{\sqrt{1-\frac{v^2}{c^2}}}

Thus, the distance between the spaceships has increased by the relativistic factor \gamma = 1/\sqrt{1-v^2/c^2}.

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 different [6] [7] 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.[7]

[edit] See also

[edit] External links

[edit] References

  1. ^ a b Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press. ISBN 0-521-52338-9.  A widely available book which contains a reprint of Bell's 1976 paper
  2. ^ Dewan, E.; Beran, M. (March 20 1959). "Note on stress effects due to relativistic contraction". American Journal of Physics 27 (7): 517-518. DOI:10.1119/1.1996214. Retrieved on 2006-10-06. 
  3. ^ Nawrocki, Paul J. (October 1962). "Stress Effects due to Relativistic Contraction". American Journal of Physics 30 (10): 771-772. DOI:10.1119/1.1941785. Retrieved on 2006-10-06. 
  4. ^ Dewan, Edmond M. (May 1963). "Stress Effects due to Lorentz Contraction". American Journal of Physics 31 (5): 383-386. DOI:10.1119/1.1969514. Retrieved on 2006-10-06.  (Note that this reference also contains the first presentation of the ladder paradox.)
  5. ^ Matsuda, Takuya; & Kinoshita, Atsuya (2004). "A Paradox of Two Space Ships in Special Relativity". AAPPS Bulletin February: ?.  eprint version
  6. ^ Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman, 165. ISBN 0-7167-0344-0. 
  7. ^ a b Nikolić, Hrvoje (6 April 1999). "Relativistic contraction of an accelerated rod". American Journal of Physics 67 (11): 1007-1012. DOI:10.1119/1.19161. Retrieved on 2006-10-07. eprint version

[edit] Further reading

  • Romain, J. E. (1963). "A Geometric approach to Relativistic paradoxes". Am. J. Phys. 31: 576-579. 
  • Hsu, Jong-Ping; & Suzuki (2005). "Extended Lorentz Transformations for Accelerated Frames and the Solution of the "Two-Spaceship Paradox"". AAPPS Bulletin October: ?.  eprint version
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