Bell's spaceship paradox

The Bell's spaceship paradox is a paradox to length contraction in the theory of relativity . It was first described and solved by E. Dewan and M. Beran (1959), and gained greater prominence through the description by John Bell (1976). For similar thought experiments see the paradox of contraction of length and Ehrenfest's paradox .

Explanation

Image 1: Above : From the perspective of the observer (S) who is resting in the laboratory, the distance between the spaceships remains constant due to the same acceleration, but the rope contracts and breaks. Below : In the rest system of the spaceships (S ') the distance between the spaceships has increased during the acceleration phase, while the rope length remains unchanged. As a result, there is also a tear here.

The length contraction, also called Lorentz contraction, is a phenomenon of relativistic physics . Every moving scale is shorter in the direction of movement than an equal, stationary scale. This shortening eludes our everyday experience, as it only becomes noticeable at speeds that are significant compared to the speed of light .

Image 2: Variant with vertical acceleration in the sense of Bell. Ship A in S sends signals to B and C, which arrive there at the same time and trigger the identical acceleration of B and C.

Dewan and Beran (1959, Fig. 1) and Bell (1976, Fig. 2) considered the following thought experiment : Two spaceships, as seen by a stationary observer in the inertial system S, begin to accelerate simultaneously from a standing position, in the same direction, in parallel to their connecting line. From the point of view of the stationary observer, the ongoing acceleration is also synchronous until it comes to rest in the new inertial system S '. A rope is stretched between the two and breaks at the slightest stretch.

1. Question : Does the rope tear if its fastening points and every section of the rope are accelerated in exactly the same way, from the point of view of the stationary observer, to the same final speed? Since the attachment points are accelerated equally, their distance remains unchanged for the stationary observer . (This is not in contradiction to the length contraction of the distance between the rockets, because as will become clear in the following question 2, the rest length between the rockets in S 'has increased. This increased rest length is moved from the point of view of S and consequently the Subject to length contraction, whereby it remains the same here.) The rope is also moved, and since its rest length does not change, it has become shorter due to the length contraction . At rest it would therefore have to be longer than to reach from one attachment point to the other. So the rope breaks. ${\ displaystyle L}$ ${\ displaystyle L}$

2. Question : If, from the point of view of the missile crews, the accelerations were the same, then their distance would not change. And since the rope would rest opposite the crews, it would not change its length. Then why does it tear from the missile crew's point of view? The solution to this apparent contradiction is that, from the crew's point of view, both accelerations are not the same due to the relativity of simultaneity . The rear rocket accelerates more slowly for both crews and only reaches its final speed after the front rocket. For example, if each engine is briefly ignited twice and both thrust phases take place simultaneously for the stationary observer, then the second thrust for the crews already moving does not take place at the same time, but earlier for the front rocket than for the rear. From the point of view of the crews, the thrusts occur in the front missile in a shorter time, it is accelerated in a shorter time and therefore in the end further away from the rear missile at the same top speed than before the acceleration. Both crews also see the rope break due to the increased distance between the missiles.

Derivation

Accelerated spaceships

Geometric representation of the Lorentz transformation using a symmetrical Minkowski diagram : The red lines are the world lines of the attachment points A and B, which are accelerated in the same direction in the same way. The blue lines are simultaneous measurements in S, the green lines simultaneous measurements in S '. After the acceleration, the length is significantly greater than the previous length in S ', and also greater than the unchanged length in S. The dashed lines correspond to the broken rope from the perspective of S and S'.

{\ displaystyle L '}

{\ displaystyle L '_ {alt}}

{\ displaystyle L}

Analogous to Ehrenfest's paradox, Bell's spaceship paradox can also demonstrate the increase in the rocket distance in the rest system (where ) or the connection with the Lorentz contraction by applying the Lorentz transformation . ${\ displaystyle L '= \ gamma L}$ ${\ textstyle \ gamma = 1 / {\ sqrt {1-v ^ {2} / c ^ {2}}}}$

Let an inertial system S be given, with the following rest lengths for the rocket-rope ensemble:

L ₁ is the length of both missiles;

L ₂ = B - A , where A and B are the respective attachment points of the rope to the missiles;

L ₃ is the length of the rope.

The ensemble is now accelerated and comes to rest in the inertial system S '. Due to the principle of relativity , the chemical and intermolecular binding forces in the material of the rocket and the rope in S 'correspond to those before the start, so their rest lengths have remained unchanged here with L' ₁ = L ₁ and L ' ₃ = L ₃ . However, this does not apply to L ' ₂ , since the launch of the rockets from S' did not take place at the same time and did not end at the same time.

A simple method of determining the length is to use the spatial Lorentz transformation. When the acceleration is over, both rockets rest in S 'and consequently no longer change their coordinates. Therefore the length can be determined by subtracting the transformed x-coordinates. If x _A and x _A + L are the attachment points in S, the corresponding locations in the new rest system S 'result with:

{\ displaystyle {\ begin {aligned} x '_ {A} & = \ gamma \ left (x_ {A} -vt \ right) \\ x' _ {B} & = \ gamma \ left (x_ {A} + L_ {2} -vt \ right) \\ L '_ {2} & = x' _ {B} -x '_ {A} \\ & = \ gamma L_ {2}. \ End {aligned}} }

Another method is to calculate the time difference between the acceleration of A and B (in S simultaneously at ) due to the relativity of simultaneity according to the Lorentz transformation: ${\ displaystyle t_ {A} = t_ {B}}$

{\ displaystyle {\ begin {aligned} \ Delta t '& = t' _ {B} -t '_ {A} = \ gamma \ left (t_ {B} - {\ frac {vx_ {B}} {c ^ {2}}} \ right) - \ gamma \ left (t_ {A} - {\ frac {vx_ {A}} {c ^ {2}}} \ right) \\ & = {\ frac {\ gamma vL_ {2}} {c ^ {2}}}. \ end {aligned}}}

Before the acceleration, the original rest length in S from the point of view of S 'is shortened to according to the contraction formula. If it is assumed that the acceleration from S to S 'that now follows is infinitely fast, B will first stop in S', while A moves away from B with v for the duration of until it also comes to rest. The increase in the distance results as before with: ${\ displaystyle L '_ {alt} = L_ {2} / \ gamma}$ ${\ displaystyle \ Delta t '}$

{\ displaystyle {\ begin {aligned} L '_ {2} & = L' _ {alt} + v \ Delta t '= {\ frac {L_ {2}} {\ gamma}} + {\ frac {\ gamma v ^ {2} L_ {2}} {c ^ {2}}} \\ & = \ gamma L_ {2}. \ end {aligned}}}

This is also in agreement with the observation in the laboratory system S, where all rest lengths of the now moving bodies are shortened according to the contraction formula L = L '/ γ :

{\ displaystyle {\ begin {aligned} L_ {1 \ mathrm {contr.}} & = L_ {1} '/ \ gamma, \\ L_ {2 \ mathrm {contr.}} & = L_ {2}' / \ gamma = \ gamma L_ {2} / \ gamma = L_ {2}, \\ L_ {3 \ mathrm {contr.}} & = L_ {3} '/ \ gamma. \ end {aligned}}}

Since the rope has become smaller while remaining unchanged, the rope breaks in S too. The fact that the distance between the rockets remains unchanged due to the same acceleration ( ) is thus in agreement with the Lorentz contraction of lengths moved, since the length increased by γ in S 'is measured shortened by exactly the same factor in S, which is canceled out. ${\ displaystyle L_ {3 \ mathrm {contr.}}}$ ${\ displaystyle L_ {2 \ mathrm {contr.}}}$ ${\ displaystyle L_ {2 \ mathrm {contr.}} = L_ {2}}$

Importance of length contraction

There are different opinions as to whether this result says something about the "physical reality" of the length contraction or not:

Dewan & Beran and Bell themselves saw the result of the paradox as evidence of the physical reality of length contraction. Bell pointed out, for example, that the length contraction and breakage of the rope is a consequence of Maxwell's electrodynamics , which can be demonstrated using the electromagnetic intermolecular fields and forces that hold the bodies together. From the point of view of S, these fields and forces are deformed due to the movement and lead to the contraction of all bodies moving along in S ', while the distance remains unchanged due to the same acceleration - the rope breaks. In S ', where the bodies rest after the acceleration, these fields and forces are unchanged and ensure a constant rest length of the rope and rockets, and only the distance between the rockets has increased due to uneven acceleration - the rope breaks. The above formulas then result for L _new . (The general connection between electrodynamics and Lorentz transformation was shown, for example, by Richard Feynman , according to which the Lorentz transformation can be derived from the potential of a charge moving at constant speed, the Liénard-Wiechert potential . He also pointed to the historical fact that Hendrik Antoon Lorentz also derived the Lorentz transformation in a similar way, see history of the Lorentz transformation ).

Authors such as Petkov (2009) and Franklin (2009) interpret this paradox in a slightly different way. They agree with the result that in S 'the rope breaks due to the increase in the distance between the rockets due to uneven acceleration. In contrast, the length contraction in S (and in other reference systems) should not be regarded as the cause of the tension in the rope that also occurs there. Because the length contraction is merely the result of a rotation in a four-dimensional space by means of the Lorentz transformation, and no real physical effects could be generated or made to disappear through mere coordinate transformations. Regardless of the choice of reference system, the reason for the rope tearing is that the course of the world lines of the rockets are changed by the acceleration (see Minkowski diagram above). Because of this, the distance between the rockets is increased in S 'while the rope remains the same length, and in all other inertial systems the measured length of the rope differs from the rocket distance.

History and publications

As early as 1959, E. Dewan and M. Beran correctly described a variant of the underlying problem. The outcome has been questioned in debates that have recurred from time to time. In 1962, PJ Nawrocki published an article that contradicted the analysis of E. Dewan and M. Beran. E. Dewan defended his analysis in 1963. In 1976, John Bell described the problem that has since been called Bell's spaceship paradox. Bell was referring to a discussion of the paradox in the CERN cafe. A well-known experimental physicist contradicted the standard explanation and in the subsequent survey in the theory department of CERN a majority spontaneously believed that the rope would not break. Bell added that the result becomes understandable if one takes into account that the mutual spacing in the rest system of the rockets increases. In 2004 T. Matsuda and A. Kinoshita reported a lively controversy in Japanese physics journals after they had published their standard explanation of the paradox (see above) there. Matsuda and Kinoshita concluded by stating that even after a hundred years of relativity theory, there were still physicists who did not understand the real meaning of length contraction.

Most publications, however, agree that the rope will break, with the paradox represented in various reformulations, modifications and different scenarios. For example by Evett & Wangsness (1960), Dewan (1963), Romain (1963), Evett (1972), Gershtein & Logunov (1998), Tartaglia & Ruggiero (2003), Cornwell (2005), Flores (2005), Semay ( 2006), Styer (2007), Freund (2008), Redzic (2008), Peregoudov (2009), Redžić (2009), Gu (2009), Petkov (2009), Franklin (2009), Miller (2010), Fernflores ( 2011), Kassner (2012). A similar problem has also been discussed with regard to angular accelerations : Grøn (1979), MacGregor (1981), Grøn (1982, 2003).

Individual evidence

^ ^A ^b ^c ^d Edmond M. Dewan, Beran, Michael J .: Note on stress effects due to relativistic contraction . In: American Association of Physics Teachers (Ed.): American Journal of Physics . 27, No. 7, March 20, 1959, pp. 517-518. bibcode : 1959AmJPh..27..517D . doi : 10.1119 / 1.1996214 .

↑ ^a ^b ^c ^d J. S. Bell, How to teach special relativity , Progress in Scientific Culture, 1 (1976)

↑ ^a ^b ^c ^d J. S. Bell, Speakable and unspeakable in quantum mechanics , Cambridge University Press (1987), ISBN 0-521-52338-9 (contains the above article by Bell from 1976)

^ ^A ^b ^c ^d Edmond M. Dewan: Stress Effects due to Lorentz Contraction . In: American Journal of Physics . 31, No. 5, May 1963, pp. 383-386. bibcode : 1963AmJPh..31..383D . doi : 10.1119 / 1.1969514 .

↑ ^a ^b Flores, Francisco J .: Bell's spaceships: a useful relativistic paradox . In: Physics Education . 40, No. 6, 2005, pp. 500-503. bibcode : 2005PhyEd..40..500F . doi : 10.1088 / 0031-9120 / 40/6 / F03 .

↑ ^a ^b ^c ^d Franklin, Jerrold: Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity . In: European Journal of Physics . 31, No. 2, 2010, pp. 291-298. arxiv : 0906.1919 . bibcode : 2010EJPh ... 31..291F . doi : 10.1088 / 0143-0807 / 31/2/006 .

↑ ^a ^b ^c Vesselin Petkov (2009): Accelerating spaceships paradox and physical meaning of length contraction, arxiv : 0903.5128 , published in: Veselin Petkov: Relativity and the Nature of Spacetime . Springer, 2009, ISBN 3-642-01962-5 .

^ Feynman, RP: The Feynman Lectures on Physics . tape 2 . Addison-Wesley Longman, 1970, ISBN 0-201-02115-3 , 21-6: The potentials for a charge moving with constant velocity; the Lorentz formula .

^ Paul J. Nawrocki: Stress Effects due to Relativistic Contraction . In: American Journal of Physics . 30, No. 10, October 1962, pp. 771-772. bibcode : 1962AmJPh..30..771N . doi : 10.1119 / 1.1941785 .

↑ Bell Speakable and unspeakable in quantum mechanics , p. 68. Bell added: Of course many people who gave this wrong answer at first get the right answer on further reflection.

↑ Matsuda, Takuya; and Kinoshita, Atsuya: A Paradox of Two Space Ships in Special Relativity . In: AAPPS Bulletin . 14, No. 1, 2004, pp. 3-7.

↑ Evett, Arthur A .; Wangsness, Roald K .: Note on the Separation of Relativistically Moving Rockets . In: American Journal of Physics . 28, No. 6, 1960, pp. 566-566. bibcode : 1960AmJPh..28..566E . doi : 10.1119 / 1.1935893 .

↑ Romain, Jacques E .: A Geometrical Approach to Relativistic Paradoxes . In: American Journal of Physics . 31, No. 8, 1963, pp. 576-585. bibcode : 1963AmJPh..31..576R . doi : 10.1119 / 1.1969686 .

↑ Evett, Arthur A .: A Relativistic Rocket Discussion problem . In: American Journal of Physics . 40, No. 8, 1972, pp. 1170-1171. bibcode : 1972AmJPh..40.1170E . doi : 10.1119 / 1.1986781 .

↑ Gershtein, SS; Logunov, AA: JS Bell's problem . In: Physics of Particles and Nuclei . 29, No. 5, 1998, pp. 463-468. doi : 10.1134 / 1.953086 .

↑ Tartaglia, A .; Ruggiero, ML: Lorentz contraction and accelerated systems . In: European Journal of Physics . 24, No. 2, 2003, pp. 215-220. arxiv : gr-qc / 0301050 . doi : 10.1088 / 0143-0807 / 24/2/361 .

↑ Cornwell, DT: Forces due to contraction on a cord spanning between two spaceships . In: EPL (Europhysics Letters) . 71, No. 5, 2005, pp. 699-704. bibcode : 2005EL ..... 71..699C . doi : 10.1209 / epl / i2005-10143-x .

↑ Semay, Claude: Observer with a constant proper acceleration . In: European Journal of Physics . 27, No. 5, 2006, pp. 1157-1167. arxiv : physics / 0601179 . doi : 10.1088 / 0143-0807 / 27/5/015 .

↑ Styer, Daniel F .: How do two moving clocks fall out of sync? A tale of trucks, threads, and twins . In: American Journal of Physics . 75, No. 9, 2007, pp. 805-814. bibcode : 2007AmJPh..75..805S . doi : 10.1119 / 1.2733691 .

↑ Jürgen Freund: The Rocket-Rope Paradox (Bell's Paradox) . In: Special Relativity for Beginners: A Textbook for Undergraduates . World Scientific, 2008, ISBN 981-277-159-X , pp. 109-116.

↑ Redžić, Dragan V .: Note on Dewan Beran Bell's spaceship problem . In: European Journal of Physics . 29, No. 3, 2008, pp. N11-N19. bibcode : 2008EJPh ... 29 ... 11R . doi : 10.1088 / 0143-0807 / 29/3 / N02 .

↑ Peregoudov, DV: Comment on 'Note on Dewan-Beran-Bell's spaceship trouble' . In: European Journal of Physics . 30, No. 1, 2009, pp. L3-L5. bibcode : 2009EJPh ... 30L ... 3P . doi : 10.1088 / 0143-0807 / 30/1 / L02 .

↑ Redžić, Dragan V .: Reply to 'Comment on "Note on Dewan-Beran-Bell's spaceship problem"' . In: European Journal of Physics . 30, No. 1, 2009, pp. L7-L9. bibcode : 2009EJPh ... 30L ... 7R . doi : 10.1088 / 0143-0807 / 30/1 / L03 .

^ Gu, Ying-Qiu: Some Paradoxes in Special Relativity and the Resolutions . In: Advances in Applied Clifford Algebras . 21, No. 1, 2009, pp. 103-119. arxiv : 0902.2032 . doi : 10.1007 / s00006-010-0244-6 .

↑ Miller, DJ: A constructive approach to the special theory of relativity . In: American Journal of Physics . 78, No. 6, 2010, pp. 633-638. arxiv : 0907.0902 . doi : 10.1119 / 1.3298908 .

↑ remote Flores, Francisco: Bell's Spaceships problem and the Foundations of Special Relativity . In: International Studies in the Philosophy of Science . 25, No. 4, 2011, pp. 351-370. doi : 10.1080 / 02698595.2011.623364 .

^ Kassner, Klaus: Spatial geometry of the rotating disk and its non-rotating counterpart . In: American Journal of Physics . 80, No. 9, 2011, pp. 772-781. arxiv : 1109.2488 . bibcode : 2012AmJPh..80..772K . doi : 10.1119 / 1.4730925 .

↑ Grøn, Ø .: Relativistic description of a rotating disk with angular acceleration . In: Foundations of Physics . 9, No. 5-6, 1979, pp. 353-369. doi : 10.1007 / BF00708527 .

↑ MacGregor, MH: Do Dewan-Beran relativistic stresses actually exist? . In: Lettere al Nuovo Cimento . 30, No. 14, 1981, pp. 417-420. doi : 10.1007 / BF02817127 .

↑ Grøn, Ø .: Energy considerations in connection with a relativistic rotating ring . In: American Journal of Physics . 50, No. 12, 1982, pp. 1144-1145. doi : 10.1119 / 1.12918 .

↑ Øyvind Grøn: Space Geometry in a Rotating Reference Frame: A Historical Appraisal . In: G. Rizzi and M. Ruggiero (Eds.): Relativity in Rotating Frames . Springer, 2004, ISBN 1-4020-1805-3 . Archived from the original on October 16, 2013 Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (Accessed September 8, 2013). @1@ 2

literature

Hrvoje Nikolić: Relativistic contraction of an accelerated rod . In: American Association of Physics Teachers (Ed.): American Journal of Physics . 67, No. 11, April 6, 1999, pp. 1007-1012. arxiv : physics / 9810017 . bibcode : 1999AmJPh..67.1007N . doi : 10.1119 / 1.19161 .
Misner, Charles; Thorne, Kip S .; and Wheeler, John Archibald: Gravitation . WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 , p. 165.

Web links

Michael Weiss: Bell's Spaceship Paradox. The Original Usenet Physics FAQ, 1995, accessed June 18, 2016 .
Norbert Dragon, Geometry of Relativity (PDF; 2.5 MB) and FAQ

[dewanberan-1] A ^b ^c ^d Edmond M. Dewan, Beran, Michael J .: Note on stress effects due to relativistic contraction . In: American Association of Physics Teachers (Ed.): American Journal of Physics . 27, No. 7, March 20, 1959, pp. 517-518. bibcode : 1959AmJPh..27..517D . doi : 10.1119 / 1.1996214 .

[bell1-2] J. S. Bell, How to teach special relativity , Progress in Scientific Culture, 1 (1976)

[bell2-3] J. S. Bell, Speakable and unspeakable in quantum mechanics , Cambridge University Press (1987), ISBN 0-521-52338-9 (contains the above article by Bell from 1976)

[dewan2-4] A ^b ^c ^d Edmond M. Dewan: Stress Effects due to Lorentz Contraction . In: American Journal of Physics . 31, No. 5, May 1963, pp. 383-386. bibcode : 1963AmJPh..31..383D . doi : 10.1119 / 1.1969514 .

[flores-5] Flores, Francisco J .: Bell's spaceships: a useful relativistic paradox . In: Physics Education . 40, No. 6, 2005, pp. 500-503. bibcode : 2005PhyEd..40..500F . doi : 10.1088 / 0031-9120 / 40/6 / F03 .

[franklin-6] Franklin, Jerrold: Lorentz contraction, Bell's spaceships, and rigid body motion in special relativity . In: European Journal of Physics . 31, No. 2, 2010, pp. 291-298. arxiv : 0906.1919 . bibcode : 2010EJPh ... 31..291F . doi : 10.1088 / 0143-0807 / 31/2/006 .

[petkov-7] Vesselin Petkov (2009): Accelerating spaceships paradox and physical meaning of length contraction, arxiv : 0903.5128 , published in: Veselin Petkov: Relativity and the Nature of Spacetime . Springer, 2009, ISBN 3-642-01962-5 .

[8] Feynman, RP: The Feynman Lectures on Physics . tape 2 . Addison-Wesley Longman, 1970, ISBN 0-201-02115-3 , 21-6: The potentials for a charge moving with constant velocity; the Lorentz formula .

[9] Paul J. Nawrocki: Stress Effects due to Relativistic Contraction . In: American Journal of Physics . 30, No. 10, October 1962, pp. 771-772. bibcode : 1962AmJPh..30..771N . doi : 10.1119 / 1.1941785 .

[10] Bell Speakable and unspeakable in quantum mechanics , p. 68. Bell added: Of course many people who gave this wrong answer at first get the right answer on further reflection.

[Matsuda-11] Matsuda, Takuya; and Kinoshita, Atsuya: A Paradox of Two Space Ships in Special Relativity . In: AAPPS Bulletin . 14, No. 1, 2004, pp. 3-7.

[12] Evett, Arthur A .; Wangsness, Roald K .: Note on the Separation of Relativistically Moving Rockets . In: American Journal of Physics . 28, No. 6, 1960, pp. 566-566. bibcode : 1960AmJPh..28..566E . doi : 10.1119 / 1.1935893 .

[13] Romain, Jacques E .: A Geometrical Approach to Relativistic Paradoxes . In: American Journal of Physics . 31, No. 8, 1963, pp. 576-585. bibcode : 1963AmJPh..31..576R . doi : 10.1119 / 1.1969686 .

[14] Evett, Arthur A .: A Relativistic Rocket Discussion problem . In: American Journal of Physics . 40, No. 8, 1972, pp. 1170-1171. bibcode : 1972AmJPh..40.1170E . doi : 10.1119 / 1.1986781 .

[15] Gershtein, SS; Logunov, AA: JS Bell's problem . In: Physics of Particles and Nuclei . 29, No. 5, 1998, pp. 463-468. doi : 10.1134 / 1.953086 .

[tartaglia-16] Tartaglia, A .; Ruggiero, ML: Lorentz contraction and accelerated systems . In: European Journal of Physics . 24, No. 2, 2003, pp. 215-220. arxiv : gr-qc / 0301050 . doi : 10.1088 / 0143-0807 / 24/2/361 .

[17] Cornwell, DT: Forces due to contraction on a cord spanning between two spaceships . In: EPL (Europhysics Letters) . 71, No. 5, 2005, pp. 699-704. bibcode : 2005EL ..... 71..699C . doi : 10.1209 / epl / i2005-10143-x .

[18] Semay, Claude: Observer with a constant proper acceleration . In: European Journal of Physics . 27, No. 5, 2006, pp. 1157-1167. arxiv : physics / 0601179 . doi : 10.1088 / 0143-0807 / 27/5/015 .

[styer-19] Styer, Daniel F .: How do two moving clocks fall out of sync? A tale of trucks, threads, and twins . In: American Journal of Physics . 75, No. 9, 2007, pp. 805-814. bibcode : 2007AmJPh..75..805S . doi : 10.1119 / 1.2733691 .

[20] Jürgen Freund: The Rocket-Rope Paradox (Bell's Paradox) . In: Special Relativity for Beginners: A Textbook for Undergraduates . World Scientific, 2008, ISBN 981-277-159-X , pp. 109-116.

[21] Redžić, Dragan V .: Note on Dewan Beran Bell's spaceship problem . In: European Journal of Physics . 29, No. 3, 2008, pp. N11-N19. bibcode : 2008EJPh ... 29 ... 11R . doi : 10.1088 / 0143-0807 / 29/3 / N02 .

[22] Peregoudov, DV: Comment on 'Note on Dewan-Beran-Bell's spaceship trouble' . In: European Journal of Physics . 30, No. 1, 2009, pp. L3-L5. bibcode : 2009EJPh ... 30L ... 3P . doi : 10.1088 / 0143-0807 / 30/1 / L02 .

[23] Redžić, Dragan V .: Reply to 'Comment on "Note on Dewan-Beran-Bell's spaceship problem"' . In: European Journal of Physics . 30, No. 1, 2009, pp. L7-L9. bibcode : 2009EJPh ... 30L ... 7R . doi : 10.1088 / 0143-0807 / 30/1 / L03 .

[24] Gu, Ying-Qiu: Some Paradoxes in Special Relativity and the Resolutions . In: Advances in Applied Clifford Algebras . 21, No. 1, 2009, pp. 103-119. arxiv : 0902.2032 . doi : 10.1007 / s00006-010-0244-6 .

[25] Miller, DJ: A constructive approach to the special theory of relativity . In: American Journal of Physics . 78, No. 6, 2010, pp. 633-638. arxiv : 0907.0902 . doi : 10.1119 / 1.3298908 .

[26] remote Flores, Francisco: Bell's Spaceships problem and the Foundations of Special Relativity . In: International Studies in the Philosophy of Science . 25, No. 4, 2011, pp. 351-370. doi : 10.1080 / 02698595.2011.623364 .

[kassner-27] Kassner, Klaus: Spatial geometry of the rotating disk and its non-rotating counterpart . In: American Journal of Physics . 80, No. 9, 2011, pp. 772-781. arxiv : 1109.2488 . bibcode : 2012AmJPh..80..772K . doi : 10.1119 / 1.4730925 .

[28] Grøn, Ø .: Relativistic description of a rotating disk with angular acceleration . In: Foundations of Physics . 9, No. 5-6, 1979, pp. 353-369. doi : 10.1007 / BF00708527 .

[29] MacGregor, MH: Do Dewan-Beran relativistic stresses actually exist? . In: Lettere al Nuovo Cimento . 30, No. 14, 1981, pp. 417-420. doi : 10.1007 / BF02817127 .

[30] Grøn, Ø .: Energy considerations in connection with a relativistic rotating ring . In: American Journal of Physics . 50, No. 12, 1982, pp. 1144-1145. doi : 10.1119 / 1.12918 .