Regular coordinate systems for Schwarzschild and other spherical spacetimes

TL;DR

This paper advocates for using simpler, pedagogically effective coordinate systems like Painleve-Gullstrand for Schwarzschild and spherical spacetimes, highlighting their advantages over traditional Kruskal-Szekeres coordinates.

Contribution

It introduces generalizations of Painleve-Gullstrand coordinates for Schwarzschild and other spherical spacetimes, promoting their pedagogical and practical benefits.

Findings

Painleve-Gullstrand coordinates are simpler and more pedagogically accessible.

Generalizations of these coordinates are developed for broader spherical spacetimes.

The paper demonstrates the effectiveness of these coordinates in various contexts.

Abstract

The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a pedagogical point of view, using these coordinates comes with several drawbacks, and we advocate the use of simpler, but equally effective, coordinate systems. One such system, introduced by Painleve and Gullstrand in the 1920's, is especially simple and pedagogically powerful; it is, however, still poorly known today. One of our purposes here is therefore to popularize these coordinates. Our other purpose is to provide generalizations to the Painleve-Gullstrand coordinates, first within the specific context of Schwarzschild spacetime, and then in the context of more general spherical spacetimes.