**Bell's spaceship paradox** is a

thought experimentA thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...

in

special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

involving accelerated spaceships and strings. The results of this thought experiment are for many people paradoxical. While

J. S. BellJohn Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...

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

length contractionIn physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...

.

## 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, in such a way that they remain a fixed distance apart as viewed from the original rest frame. Question: Does the string break (i.e. does the distance between the two spaceships increase in the reference frame of either spaceship)?

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 relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

. Note that one ship will observe the other accelerating differently from itself if the two ships are to accelerate identically in the original rest frame, because of the relativity of simultaneity.

According to discussions by Dewan & Beran and also Bell, in the spaceship launcher's reference frame 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, while Edmond Dewan defends his original analysis from these objections in a reply. 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

CERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...

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 reported receiving much criticism after publishing an article on their independently rediscovered version of the paradox in a Japanese journal. Matsuda and Kinoshita 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 a 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 a function of time is

where

- f(0) is assumed to be equal to 0
- x
_{A}(t) is the position (x coordinate) of spaceship A at time t
- x
_{B}(t) is the position (x coordinate) of spaceship B at time t
- a
_{0} is the position of spaceship A at time 0
- b
_{0} 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 motionHyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.The proper acceleration...

).

Referring to the space-time diagram on the 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 simultaneityIn physics, the relativity of simultaneity is the concept that simultaneity–whether two events occur at the same time–is not absolute, but depends on the observer's reference frame. According to the special theory of relativity, it is impossible to say in an absolute sense whether two events occur...

.

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

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 equation

which results in

In frame S′, since both ends of the rope are marked simultaneously,

where

so

Calculate

so

Therefore

Thus, when switching the description to the comoving frame, the distance between the spaceships appears to increase by the relativistic factor

Consequently, the string is stretched.

Alternatively, one can arrive at the same last equation with a bit less effort starting from the equation

Remembering that

(the two ships stop accelerating simultaneously in the "launching" frame S), and using the above formula

and the definition of L, one easily gets

Bell pointed out that the 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 on the space between objects.

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 rigidBorn rigidity, proposed by and later named after Max Born, is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics....

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
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry...

- Physical paradox
A physical paradox is an apparent contradiction in physical descriptions of the universe. While many physical paradoxes have accepted resolutions, others defy resolution and may indicate flaws in theory...

- Supplee's paradox
In relativistic physics, Supplee's paradox arises when considering the buoyant force exerted on a relativistic bullet immersed in a fluid subject to an ambient gravitational field...

- Rindler coordinates
In relativistic physics, the Rindler coordinate chart is an important and useful coordinate chart representing part of flat spacetime, also called the Minkowski vacuum. The Rindler chart was introduced by Wolfgang Rindler. The Rindler coordinate system or frame describes a uniformly accelerating...

- Twin paradox
In physics, the twin paradox is a thought experiment in special relativity, in which a twin makes a journey into space in a high-speed rocket and returns home to find he has aged less than his identical twin who stayed on Earth...

- Born rigidity
Born rigidity, proposed by and later named after Max Born, is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics....

- Hyperbolic motion (relativity)
Hyperbolic motion is the motion of an object with constant proper acceleration in special relativity. It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram.The proper acceleration...

## External links

## Further reading

eprint version
- Redžić D.V.(2010)"Relativistic length agony continued" }}
- Foukzon J., Podosyonov S.A., Potapov A.A.,(2009),"Relativistic length expansion in general accelerated system revisited".
- Podosyonov S.A., Foukzon J. and Potapov A.A.,(2010) "A Study of the Motion of a Relativistic Continuous Medium", Gravitation and Cosmology, 2010,Vol.16,No.4,pp. 307–312.ISSN 0202-2893,