Supplee's paradox

In relativistic physics, Supplee's paradox (also called the submarine paradox) arises when considering the buoyant force exerted on a relativistic bullet immersed in a fluid subject to an ambient gravitational field. If a bullet with neutral buoyancy passes through a fluid at a relativistic speed, will its increased mass in the fluid's frame of reference cause it to sink, or will the fluid's increased mass in the bullet's frame of reference cause it to rise? The paradox was apparently first discussed by James M. Supplee.

A bit about buoyancy

To simplify the analysis, it is customary to neglect drag and viscosity, and even to assume that the fluid has constant density.

Consider a small object immersed in a container of fluid subject to a uniform gravitational field. Then the object will be subject to a net downward gravitational force. Compare this 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, we say that the object has neutral buoyancy and it will neither rise nor sink.

Statement of the paradox

It is simplest to assume that at rest, the bullet has neutral buoyancy.

In the rest frame of the fluid, if the bullet moves at speed $v$ , then according to the kinematic laws of special relativity, its density (as measured in the frame of the fluid) increases by the square of the Lorentz Factor $\gamma^2 = \frac{1}{1-v^2/c^2}$ . Therefore, a rapidly moving bullet should sink.

In the rest frame of the bullet, the density of fluid, as measured in the frame of the bullet, increases by the same factor, so the moving bullet should rise.

Resolution of the paradox

The resolution comes down to observing that the gravitational force lies outside the domain of kinematics; when it is treated properly, the paradox disappears.

Supplee himself concluded that the paradox can be resolved by noting that in the frame of the bullet, the shape of the container of fluid is altered (viz. the sea floor is curved upwards). Given certain assumptions about how to treat the gravitational force, he argued that the bullet sinks with acceleration $g (\gamma^2-1)$ , where g is the acceleration due to gravity (assumed to be uniform over the scale of the thought experiment) and $\gamma^2$ is the factor mentioned above.

The paradox has also been studied by George Matsas, who used mathematical methods from general relativity to remove Supplee's assumptions. In particular, he modeled the situation using a Rindler chart. 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.

References

Matsas, George E. A. (2003). "Relativistic Arquimedes law for fast moving bodies and the general-relativistic resolution of the "submarine paradox"". Phys. Rev. D 68 (2): 027701. arXiv:gr-qc/0305106. Bibcode:2003PhRvD..68b7701M. doi:10.1103/PhysRevD.68.027701. See also the eprint version.
Supplee, J. M. (1989). "Relativistic buoyancy". Am. J. Phys. 57: 75–7. Bibcode:1989AmJPh..57...75S. doi:10.1119/1.15875.

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