General relativity in R: visual representation of Schwarzschild space using different coordinate systems

Abstract

In general relativity, Schwarzschild coordinates for a black hole have desirable properties such as asymptotic matching with flat-space spherical coordinates; but other coordinate systems can be used which have other advantages such as removing the non-physical coordinate singularity at the event horizon. Following Schwarzschild’s original publication in 1916 of his spherically symmetrical solution to the vacuum Einstein field equations, a variety of coordinate transformations have been described that highlight different features of the Schwarzschild metric. These include: Kruskal-Szekeres (Kruskal, 1960; Szekeres, 1960), Eddington-Finkelstein (Eddington, 1924; Finkelstein, 1958), Gullstrand-Painleve (Gullstrand, 1922; Painlevé, 1921), Lemaitre (Lemaître, 1933), and various Penrose transforms with or without a black hole (Hawking & Ellis, 1973). These are described in many undergraduate GR textbooks such as Schutz (2009) and Carroll (2019).