On the meaning of Painlev\'e--Gullstrand synchronization

TL;DR

This paper explores the connection between Newtonian gravity and general relativity, introducing a scalar potential linked to proper time and analyzing spacetimes with Painlevé-Gullstrand synchronization.

Contribution

It defines a scalar potential related to proper time in the context of Painlevé-Gullstrand synchronized spacetimes, deepening the understanding of their relationship.

Findings

Scalar potential equals proper time of the vector potential.

Relationship established between Newtonian gravitation and general relativity.

Analysis of spacetimes with Painlevé-Gullstrand synchronization.

Abstract

Following on from two recent papers, here we examine the relationship between Newtonian gravitation and general relativity in more depth. This allows us to define a scalar potential which is just the proper time of the vector potential when the latter is interpreted as the geodesic velocity field. The results are closely related to spacetimes that admit Painlev\'{e}--Gullstrand synchronization.