Bell's spaceship paradox

In relativistic physics, Bell's spaceship paradox is a problem that was first proposed and discussed by E. Dewan and M. Beran in 1959. but was brought to wider attention by J. S. Bell in 1976. It is also known as the "Two Space-Ship paradox" or the "Rockets and String Paradox".

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 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 their rebuttals have been published to the above analysis. For example, Paul Nawrocki suggests that the string should not break, while Edmond Dewan directly rebuts this analysis.

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?

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

x_A is the position (x coordinate) of spaceship A

x_B 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:

$t'=\left(t-vx/c^{2}\right)/{\sqrt {1-v^{2}/c^{2}}}$

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 auxillary 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

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

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 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, Bell's Spaceship Paradox (1995), USENET Relativity FAQ
• Austin Gleeson, Course Notes Chapter 13 See Section 4.3

References

The list of papers which discuss spaceship and string paradoxes is too large to enumerate here, but we cite some relevant papers.

• Romain, J. E. (1963). "A Geometric approach to Relativistic paradoxes". Am. J. Phys. 31: 576–579.

• Matsuda, Takuya; & Kinoshita, Atsuya (2004). "A Paradox of Two Space Ships in Special Relativity". AAPPS Bulletin. February: ?.{{cite journal}}: CS1 maint: multiple names: authors list (link) eprint version

• Hsu, Jong-Ping; & Suzuki (2005). "Extended Lorentz Transformations for Accelerated Frames and the Solution of the "Two-Spaceship Paradox"". AAPPS Bulletin. October: ?.{{cite journal}}: CS1 maint: multiple names: authors list (link) eprint version

Bibliography

The following book contains a reprint of Bell's 1976 paper discussing his version of the "paradox":

• Bell, J. S. (1987). Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press. ISBN 0-521-52338-9.