Relativistic Arquimedes law for fast moving bodies and the general-relativistic resolution of the "submarine paradox"

TL;DR

This paper resolves the submarine paradox in general relativity by extending Archimedes' law, demonstrating that a relativistically moving submarine sinks due to relativistic effects on buoyancy.

Contribution

It provides a general relativistic formulation of Archimedes' law and clarifies the behavior of bodies in relativistic motion within a fluid, resolving the paradox.

Findings

Relativistic effects cause the submarine to sink when moving at high velocity.

The general relativistic extension of Archimedes' law explains the sinking behavior.

The paradox is resolved by considering relativistic density and buoyancy effects.

Abstract

We investigate and solve in the context of General Relativity the apparent paradox which appears when bodies floating in a background fluid are set in relativistic motion. Suppose some macroscopic body, say, a submarine designed to lie just in equilibrium when it rests (totally) immersed in a certain background fluid. The puzzle arises when different observers are asked to describe what is expected to happen when the submarine is given some high velocity parallel to the direction of the fluid surface. On the one hand, according to observers at rest with the fluid, the submarine would contract and, thus, sink as a consequence of the density increase. On the other hand, mariners at rest with the submarine using an analogous reasoning for the fluid elements would reach the opposite conclusion. The general relativistic extension of the Arquimedes law for moving bodies shows that the submarine sinks.