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.Comment: Final version. 7 pages, 2 figures, Keywords: Supplee's submarine
paradox, theory of relativity, gravitomagnetism, Archimedes principle,
Lorentz forc
