General relativity and background Lorentz transformations solve Supplee's submarine paradox

TL;DR

This paper resolves Supplee's submarine paradox by applying background Lorentz transformations and spacetime geometry, providing a simpler, more accurate explanation within general relativity that corrects previous misconceptions.

Contribution

It introduces a novel approach using background Lorentz transformations to clarify the paradox, avoiding assumptions of force transformation and correcting prior errors.

Findings

The paradox is resolved through metric transformation analysis.

The method simplifies previous complex treatments in general relativity.

A correction is made to the gravitational force expression used in earlier studies.

Abstract

A submarine moving at relativistic horizontal velocity sinks in Earth's rest frame due to length contraction while appearing to float in its own frame. Using spacetime geometry and the Lorentz transformations, we show that the resolution lies in how metric components transform between reference frames in relative motion. This solution frees us from assumptions made in previous studies on how a Newtonian gravitational force should transform. The method of background Lorentz transformations is technically simpler than previous treatments in the framework of general relativity. Moreover, we find a novel and intuitive understanding of the paradox, and correct an erroneous expression for the gravitational force obtained by Supplee and used again in the literature.