page_title: Spacetime
section_title: Dewan–Beran–Bell spaceship paradox
hierarchical_section_title: Spacetime / Beyond the basics / Acceleration / Dewan–Beran–Bell spaceship paradox
caption_attribution_description: English: A thought experiment related to one discussed by J. S. Bell (see Bell's spaceship paradox). Two observers in Minkowski spacetime accelerate with constant magnitude k {\displaystyle k} acceleration for proper time &#x03C3; {\displaystyle \sigma } (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. As Bell pointed out, in Minkowski geometry, the length of the spacelike line segment A&#8242;B&#8243; turns out to be greater than the length of the spacelike line segment AB. Русский: Мысленный эксперимент, связанный с рассмотрением Дж. С. Беллом (см. Парадокс Белла). Два наблюдателя в пространстве-времени Минковского ускоряются с постоянной величиной ускорения k {\displaystyle k} за собственное время &#x03C3; {\displaystyle \sigma } (ускорение и прошедшее время измеряются самими наблюдателями, а не сторонним инерциальным наблюдателем). Они являются сопутствующими и инерциальными до и после фазы ускорения. Как отметил Белл, в геометрии Минковского длина пространственноподобного отрезка A&#8242;B&#8243; "оказывается больше длины пространственноподобного отрезка AB.
context_page_description: In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive where and when events occur differently.
  Until the 20th century, it was assumed that the 3-dimensional geometry of the universe was independent of one-dimensional time. However, in 1905, Albert Einstein based a work on special relativity on two postulates:
  The laws of physics are invariant in all inertial systems
  The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
  The logical consequence of taking these postulates together is the inseparable joining together of the four dimensions—hitherto assumed as independent—of space and time.
context_section_description: The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.
  In Fig. 4‑4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration. Will the string break?
  When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.
  To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance L' = γL in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.
  Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.
  The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.
  A spacetime diagram (Fig. 4‑5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude  acceleration for proper time   (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length of the spacelike line segment  turns out to be greater than the length of the spacelike line segment .
  The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 4‑5, the acceleration is finished, the ships will remain at a constant offset in some frame  If  and  are the ships' positions in  the positions in frame  are:
  The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame . As shown in Fig. 4‑5, Bell's example asserts the moving lengths  and  measured in frame  to be fixed, thereby forcing the rest frame length  in frame  to increase.
caption: Figure 4-5. The blue lines represent the world lines 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 line is a line of simultaneity for either observer after acceleration stops.
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