Bell's spaceship paradox
The Bell's spaceship paradox is a paradox to the length contraction in relativity theory. It was created by E. Dewan and M. Beran (1959 ) described and solved for the first time, and gained a greater awareness by the description of John Bell ( 1976). For similar thought experiments see paradox of length contraction and Ehrenfest paradox cal.
Explanation
The length contraction, also called Lorentz contraction is a phenomenon of relativistic physics. Moving each scale is less than one equivalent, static scale in the direction of movement. This shortening beyond our everyday experience, as it is noticeable only at speeds that fall compared to the speed of light in weight.
Dewan and Beran (1959 ), and later John Bell ( 1976) considered to the following thought experiment: Two spaceships begin, seen by a stationary observer in the inertial frame S, while accelerating from a standstill, and indeed in the same direction, parallel to its connection line. Between the two a rope is stretched, the tears at the slightest strain.
Question 1: Tear down the rope when his attachment points and each part of the rope in exactly the same way, are accelerated from the point of view of the observer at rest, up to the same terminal velocity? Since the mounting points are the same speeds, their distance to the observer at rest remains unchanged. The cable is moved and because of the length contraction is shorter than at rest. At rest, it must therefore be longer than to the attachment point to range from one to the other. The rope breaks.
Question 2: If the accelerations are the same from the perspective of the missile crews, then their distance does not change. And since the rope against the crews rested, it does not change its length. Why is it still pulls from the perspective of the missile crew? The resolution of this apparent contradiction is that from the perspective of both crews accelerations due to the relativity of simultaneity are not the same flat. For both crews the rear rocket accelerates slower and reached only after the front rocket their terminal velocity. For example, each engine ignited briefly twice, and find both thrust phases for the stationary observer rather than simultaneously, then place the second boost to the then already moving crews do not take place simultaneously, but in front of the rocket earlier than in the rear. Relapses occur from the perspective of the crews at the front rocket in less time, it is accelerated in a shorter time and therefore end up farther with the same final velocity of the rocket rear than before the acceleration. Also, both crews see tear the rope.
Derivation
Rotating disk
Since the ultimate purpose of this paradox to speed in any inertial frame, the end points of a body in the same way, it is similar to the Ehrenfest 's paradox. ( The reverse case, the Born rigidity condition is, according to which a body is to be accelerated so that its resting length remains unchanged in a mitbeschleunigten reference system. ) Even Albert Einstein (1916, 1919) described as a consequence of Ehrenfest 's thought experiment that standards on a uniform rotating disk are contracted from the perspective of an inertial frame. It follows that to measure the circumference U disk, the standards need to be applied more often. This is interpreted as a consequence of an increase of the contrast in the periphery of disc co-rotating reference system, according to the relationship:
Therefore, it is impossible to put a disk in rotation with relativistic high angular velocity without would occur ( in addition to the usual material burdens ) additional stresses or deformations.
Accelerating spaceships
Analog to Ehrenfest 's paradox the magnification of the missile gap in the rest frame can ( where ) and the connection with the Lorentz contraction are demonstrated by an application of the Lorentz transformation also in Bell's spaceship paradox.
Let there be given an inertial frame S, with the following rest lengths for the rocket rope ensemble:
The ensemble is now accelerating and comes in the inertial frame S ' to rest. Due to the principle of relativity correspond to the chemical and intermolecular bonding forces in the material of rockets and the rope in S ' which. Before the start, so their rest lengths are here remained unchanged with L' 1 = L1 and L 3 = L3 But not this applies to L' 2, since the launch of the missile from the perspective of S ' was not the same and ended up not at the same time.
A simple method for determining the length is now applying the spatial Lorentz transformation. When the acceleration is completed, both rocket rest in S 'and therefore no longer change their coordinates. Therefore, the length can be determined by subtracting the transformed x - coordinates. If xA and xA L are the attachment points in S, the corresponding locations result in new rest frame S ' with:
Another method is to calculate the time difference between the acceleration A and B ( in S at the same time ) as a result of the relativity of coincidence according to the Lorentz transformation:
Before acceleration, the original resting length in S is from the perspective of S ' according to the contraction formula shortened. If we assume that the ensuing acceleration from S to S 'infinite runs fast, is first B in S' stop while A, so that it also comes with v for a period of from B to rest. The increase in the distance that is obtained with as before:
This is also in agreement with the observation in the laboratory system S, where all the rest lengths of the now moving body according to the contraction formula L = L '/ γ are shortened:
As the rope and thus become smaller, while remained unchanged in S breaks the rope. The fact that the distance between the vehicles due to the same acceleration remains unchanged ( ), is thus in accordance with the Lorentz contraction of moving lengths, as measured shortened in S ' to γ increased length by exactly the same factor in S, which cancels itself.
Importance of the length contraction
There are different views about whether this result says something about the " physical reality" of length contraction or not:
Dewan & Beran and Bell himself saw the result of the paradox as evidence for the physical reality of length contraction. Bell for example, pointed out that the length contraction and the ripping of the rope is a consequence of Maxwell's electrodynamics, which can be the basis of the electromagnetic fields and intermolecular forces that hold together the body demonstrated. From the perspective of S these fields and forces are deformed due to the movement and lead to the contraction of all in S ' co-moving body, while the distance due to the same acceleration remains unchanged - the rope breaks. In S ' where the body rest after the acceleration, these fields and forces are unchanged and ensure a constant resting length of rope and missiles, and only the distance between the missile has increased due to non-simultaneous acceleration - the rope breaks. This then results in the above formulas for Lnew. ( The general relationship between electrodynamics and Lorentz transformation was, for example, pointed out by Richard Feynman, after which the Lorentz transformation from the potential of moving with constant velocity charge, the Liénard -Wiechert potential, can be derived. He also pointed to the historical fact out that Hendrik Antoon Lorentz also herleitete the Lorentz transformation in a similar way, see History of the Lorentz transformation).
Authors such as Petkov (2009) and Franklin ( 2009) interpret this paradox, however, in a slightly different way. They agree with the conclusion that in S ' rips the rope due to the increase of the distance between the rockets because of unequal acceleration. In contrast, the length contraction in S ( and in other reference systems ) should not be considered as a cause of the tensions there also occur in the rope. Because the length contraction is merely the result of a rotation in a four-dimensional space by means of the Lorentz transformation, and by mere coordinate transformations no real physical effects could be produced or made to disappear. Regardless of the choice of the reference system is the reason for the breaking of the rope that the courses of the world lines of the rockets are changed by the acceleration ( see above Minkowski diagram ). Therefore, in S 'is the distance between the missile increases while the rope remains the same length, and in all other inertial frames, the measured length of the rope from the missile distance differs.
History and Publications
Back in 1959 described E. Dewan and M. Beran, a variant of the underlying problem correctly. The result was put into emerging again from time to time debates in question. 1962 PJ Nawrocki published an essay which disagreed with the analysis of E. Dewan and M. Beran. E. Dewan defended his analysis 1963. 1976 John Bell described the problem that has since been called the Bell's spaceship paradox. Bell was referring to a discussion of the paradox at the CERN Cafe. The standard explanation contradicted a renowned experimental physicist and also in the subsequent survey in the Theory Division of CERN, a majority was spontaneously think the rope will not break. Bell added that the result will be understood when it is considered that the mutual distance in the rest frame of the rocket is greater. T. Matsuda and A. Kinoshita reported in 2004 by a Japanese rain controversy in physics journals, after there by them also represented standard explanation of the paradox (see above) were published. Matsuda and Kinoshita concluded by noting that there is still a physicist even after a hundred years of relativity theory, which would not have understood the real meaning of length contraction.
However, most publications agree that the rope will break, the paradox is represented in various reformulations, modifications and different scenarios. For example, by Evett & Wang Ness (1960), Dewan (1963 ), Romain (1963 ), Evett (1972 ), Gershtein & Logunov (1998), Tartaglia & Ruggiero (2003), Cornwell (2005), Flores ( 2005), Semay ( 2006), Styer (2007), friend (2008), Redzic (2008), Peregoudov (2009), Redžić (2009), Gu (2009), Petkov (2009), Franklin (2009), Miller (2010 ), remote Flores ( 2011), Kassner (2012 ). A similar problem was also discussed in terms of angular accelerations: Green (1979 ), MacGregor (1981 ), Grøn (1982, 2003).