Solution of Supplee's submarine paradox through special and general relativity

TL;DR

This paper resolves Supplee's submarine paradox by demonstrating the importance of relativistic gravitomagnetic effects and speed-dependent gravity in explaining the apparent sinking or floating of a submarine.

Contribution

It provides a relativistic formulation of Archimedes principle and incorporates curved spacetime effects to clarify the paradox.

Findings

Relativistic force fields can be decomposed into static and dynamic parts.

Gravitational force near Earth increases with the body's speed.

The paradox is explained through gravitomagnetic effects and speed-dependent gravity.

Abstract

In 1989 Supplee described an apparent relativistic paradox on which a submarine seems to sink to observers at rest within the ocean, but it rather seems to float in the submarine proper frame. In this letter, we show that the paradox arises from a misuse of the Archimedes principle in the relativistic case. Considering first the special relativity, we show that any relativistic force field can be written in the Lorentz form, so that it can always be decomposed into a \emph{static} (electric-like) and a \emph{dynamic} (magnetic-like) part. These gravitomagnetic effects provide a relativistic formulation of Archimedes principle, from which the paradox is explained. Besides, if the curved spacetime on the vicinity of the Earth is taken into account, we show that the gravitational force exerted by Earth on a moving body must increase with the speed of the body. The submarine paradox is then analyzed again with this speed-dependent gravitational force.