Regular coordinate systems for Schwarzschild and other spherical spacetimes

The continuation of the Schwarzschild metric across the event horizon is a well-understood problem discussed in most textbooks on general relativity. Among the most popular coordinate systems that are regular at the horizon are the Kruskal–Szekeres and Eddington–Finkelstein coordinates. Our first objective in this paper is to popularize another set of coordinates, the Painlevé–Gullstrand coordinates. These were first introduced in the 1920s, and have been periodically rediscovered since; they are especially attractive and pedagogically powerful. Our second objective is to provide generalizations of these coordinates, first within the specific context of Schwarzschild spacetime, and then in the context of more general spherical spacetimes.