In relativisticphysics, Supplee's paradox (also called the submarine paradox) is a physical paradox that arises when considering the buoyant force exerted on a relativistic bullet (or in a submarine) immersed in a fluid subject to an ambient gravitational field. If a bullet has neutral buoyancy when it is at rest in a perfect fluid and then it is launched with a relativistic speed, observers at rest within the fluid would conclude that the bullet should sink, since its density will increase due to the length contraction effect. On the other hand, in the bullet's proper frame it is the moving fluid that becomes denser and hence the bullet would float. But the bullet cannot sink in one frame and float in another, so there is a paradox situation.
The paradox was first formulated by James M. Supplee (1989),[1] where a non-rigorous explanation was presented. George Matsas[2] has analysed this paradox in the scope of general relativity and also pointed out that these relativistic buoyancy effects could be important in some questions regarding the thermodynamics of black holes. A comprehensive explanation of Supplee's paradox through both the special and the general theory of relativity was presented by Vieira.[3] Hrvoje Nikolic noticed that rigidity of the submarine is not essential and presented a general relativistic analysis revealing that paradox resolves by the fact that the relevant velocity of the submarine is relative to Earth (which is the source of the gravitational field), not relative to the observer.[4]
To simplify the analysis, it is customary to neglect drag and viscosity, and even to assume that the fluid has constant density.
A small object immersed in a container of fluid subjected to a uniform gravitational field will be subject to a net downward gravitational force, compared with the net downward gravitational force on an equal volume of the fluid. If the object is less dense than the fluid, the difference between these two vectors is an upward pointing vector, the buoyant force, and the object will rise. If things are the other way around, it will sink. If the object and the fluid have equal density, the object is said to have neutral buoyancy and it will neither rise nor sink.
Resolution
The resolution comes down to observing that the usual Archimedes principle cannot be applied in the relativistic case. If the theory of relativity is correctly employed to analyse the forces involved, there will be no true paradox.
Supplee[1] himself concluded that the paradox can be resolved with a more careful analysis of the gravitational buoyancy forces acting on the bullet. Considering the reasonable (but not justified) assumption that the gravitational force depends on the kinetic energy content of the bodies, Supplee showed that the bullet sinks in the frame at rest with the fluid with the acceleration , where is the gravitational acceleration and is the Lorentz factor. In the proper reference frame of the bullet, the same result is obtained by noting that this frame is not inertial, which implies that the shape of the container will no more be flat, on the contrary, the sea floor becomes curved upwards, which results in the bullet getting far away the sea surface, i.e., in the bullet relatively sinking.
The non-justified assumption considered by Supplee that the gravitational force on the bullet should depend on its energy content was eliminated by George Matsas,[2] who used the full mathematical methods of general relativity in order to explain the Supplee paradox and agreed with Supplee's results. In particular, he modelled the situation using a Rindler chart, where a submarine is accelerated from the rest to a given velocity v. Matsas concluded that the paradox can be resolved by noting that in the frame of the fluid, the shape of the bullet is altered, and derived the same result which had been obtained by Supplee. Matsas has applied a similar analysis to shed light on certain questions involving the thermodynamics of black holes.
Finally, Vieira[3] has recently analysed the submarine paradox through the special and general relativity. In the first case, he showed that gravitomagnetic effects should be taken into account in order to describe the forces acting in a moving submarine underwater. When these effects are considered, a relativistic Archimedes principle can be formulated, from which he showed that the submarine must sink in both frames. Vieira also considered the case of a curved space-time in the proximity of the Earth. In this case he assumed that the space-time can be approximately regarded as consisting of a flat space but a curved time. He showed that in this case the gravitational force between the Earth at rest and a moving body increases with the speed of the body in the same way as considered by Supplee (), providing in this way a justification for his assumption. Analysing the paradox again with this speed-dependent gravitational force, the Supplee paradox is explained and the results agree with those obtained by Supplee and Matsas.
^ abSupplee, James M. (1989-01-01). "Relativistic buoyancy". American Journal of Physics. 57 (1): 75–77. Bibcode:1989AmJPh..57...75S. doi:10.1119/1.15875. ISSN 0002-9505.
^ abMatsas, George E. A. (2003-01-01). "Relativistic Archimedes law for fast moving bodies and the general-relativistic resolution of the "submarine paradox"". Physical Review D. 68 (2): 027701. arXiv:gr-qc/0305106. Bibcode:2003PhRvD..68b7701M. doi:10.1103/PhysRevD.68.027701. S2CID 119375791.
^ abVieira, R. S. (2016). "Solution of Supplee's submarine paradox through special and general relativity". EPL. 116 (5): 50007. arXiv:1611.07517. Bibcode:2016EL....11650007V. doi:10.1209/0295-5075/116/50007. S2CID 118904436.
^Nikolic, H. (2022). "Submarine paradox softened". American Journal of Physics. 90 (11): 841–847. arXiv:2112.11162. doi:10.1119/5.0084185. S2CID 245353561.
External links
Light Speed Submarine - article about the paradox in Physical Review Focus
January 20, 2023
supplee, paradox, relativistic, physics, also, called, submarine, paradox, physical, paradox, that, arises, when, considering, buoyant, force, exerted, relativistic, bullet, submarine, immersed, fluid, subject, ambient, gravitational, field, bullet, neutral, b. In relativistic physics Supplee s paradox also called the submarine paradox is a physical paradox that arises when considering the buoyant force exerted on a relativistic bullet or in a submarine immersed in a fluid subject to an ambient gravitational field If a bullet has neutral buoyancy when it is at rest in a perfect fluid and then it is launched with a relativistic speed observers at rest within the fluid would conclude that the bullet should sink since its density will increase due to the length contraction effect On the other hand in the bullet s proper frame it is the moving fluid that becomes denser and hence the bullet would float But the bullet cannot sink in one frame and float in another so there is a paradox situation The paradox was first formulated by James M Supplee 1989 1 where a non rigorous explanation was presented George Matsas 2 has analysed this paradox in the scope of general relativity and also pointed out that these relativistic buoyancy effects could be important in some questions regarding the thermodynamics of black holes A comprehensive explanation of Supplee s paradox through both the special and the general theory of relativity was presented by Vieira 3 Hrvoje Nikolic noticed that rigidity of the submarine is not essential and presented a general relativistic analysis revealing that paradox resolves by the fact that the relevant velocity of the submarine is relative to Earth which is the source of the gravitational field not relative to the observer 4 Contents 1 Buoyancy 2 Resolution 3 See also 4 References 5 External linksBuoyancy EditTo simplify the analysis it is customary to neglect drag and viscosity and even to assume that the fluid has constant density A small object immersed in a container of fluid subjected to a uniform gravitational field will be subject to a net downward gravitational force compared with the net downward gravitational force on an equal volume of the fluid If the object is less dense than the fluid the difference between these two vectors is an upward pointing vector the buoyant force and the object will rise If things are the other way around it will sink If the object and the fluid have equal density the object is said to have neutral buoyancy and it will neither rise nor sink Resolution EditThe resolution comes down to observing that the usual Archimedes principle cannot be applied in the relativistic case If the theory of relativity is correctly employed to analyse the forces involved there will be no true paradox Supplee 1 himself concluded that the paradox can be resolved with a more careful analysis of the gravitational buoyancy forces acting on the bullet Considering the reasonable but not justified assumption that the gravitational force depends on the kinetic energy content of the bodies Supplee showed that the bullet sinks in the frame at rest with the fluid with the acceleration g g 2 1 displaystyle g gamma 2 1 where g displaystyle g is the gravitational acceleration and g displaystyle gamma is the Lorentz factor In the proper reference frame of the bullet the same result is obtained by noting that this frame is not inertial which implies that the shape of the container will no more be flat on the contrary the sea floor becomes curved upwards which results in the bullet getting far away the sea surface i e in the bullet relatively sinking The non justified assumption considered by Supplee that the gravitational force on the bullet should depend on its energy content was eliminated by George Matsas 2 who used the full mathematical methods of general relativity in order to explain the Supplee paradox and agreed with Supplee s results In particular he modelled the situation using a Rindler chart where a submarine is accelerated from the rest to a given velocity v Matsas concluded that the paradox can be resolved by noting that in the frame of the fluid the shape of the bullet is altered and derived the same result which had been obtained by Supplee Matsas has applied a similar analysis to shed light on certain questions involving the thermodynamics of black holes Finally Vieira 3 has recently analysed the submarine paradox through the special and general relativity In the first case he showed that gravitomagnetic effects should be taken into account in order to describe the forces acting in a moving submarine underwater When these effects are considered a relativistic Archimedes principle can be formulated from which he showed that the submarine must sink in both frames Vieira also considered the case of a curved space time in the proximity of the Earth In this case he assumed that the space time can be approximately regarded as consisting of a flat space but a curved time He showed that in this case the gravitational force between the Earth at rest and a moving body increases with the speed of the body in the same way as considered by Supplee F g m g displaystyle F gamma mg providing in this way a justification for his assumption Analysing the paradox again with this speed dependent gravitational force the Supplee paradox is explained and the results agree with those obtained by Supplee and Matsas See also EditBell s spaceship paradox Ehrenfest paradox Ladder paradox Twin paradoxReferences Edit a b Supplee James M 1989 01 01 Relativistic buoyancy American Journal of Physics 57 1 75 77 Bibcode 1989AmJPh 57 75S doi 10 1119 1 15875 ISSN 0002 9505 a b Matsas George E A 2003 01 01 Relativistic Archimedes law for fast moving bodies and the general relativistic resolution of the submarine paradox Physical Review D 68 2 027701 arXiv gr qc 0305106 Bibcode 2003PhRvD 68b7701M doi 10 1103 PhysRevD 68 027701 S2CID 119375791 a b Vieira R S 2016 Solution of Supplee s submarine paradox through special and general relativity EPL 116 5 50007 arXiv 1611 07517 Bibcode 2016EL 11650007V doi 10 1209 0295 5075 116 50007 S2CID 118904436 Nikolic H 2022 Submarine paradox softened American Journal of Physics 90 11 841 847 arXiv 2112 11162 doi 10 1119 5 0084185 S2CID 245353561 External links EditLight Speed Submarine article about the paradox in Physical Review Focus Retrieved from https en wikipedia org w index php title Supplee 27s paradox amp oldid 1133900929, wikipedia, wiki, book, books, library,
