Supplee's paradox

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.

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 ${\displaystyle g(\gamma ^{2}-1)}$, where ${\displaystyle g}$ is the gravitational acceleration and ${\displaystyle \gamma }$ 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 from 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 both 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 (${\displaystyle F=\gamma mg}$), 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.

• Bell's spaceship paradox
• Ehrenfest paradox
• Ladder paradox
• Twin paradox

• Light Speed Submarine - article about the paradox in Physical Review Focus