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), where a non-rigorous explanation was presented. George Matsas 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 Ricardo Soares Vieira.
Buoyancy
To simplify the analysis, it is customary to neglect drag and viscosity, and even to assume that the fluid has constant density.
An 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 weight of the displaced fluid is greater than the object’s weight. The difference between the forces results in an upward pointing vector, the buoyant force, and the object will rise. If the object is more dense than the fluid, it will sink. If the object and the fluid have equal density, the object is said to have neutral buoyancy and it will not experience any net force.
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