Explicit Global Coordinates for Schwarzschild and Reissner-Nordstroem

TL;DR

This paper develops explicit global coordinate systems for the Schwarzschild and Reissner-Nordström solutions using elementary functions, providing simpler alternatives to Kruskal coordinates and leveraging two-dimensional action descriptions.

Contribution

It introduces explicit, globally valid coordinate charts for these solutions, simplifying the representation and analysis of their spacetime geometries.

Findings

Constructed global coordinates covering all Reissner-Nordström solutions with m>|q| and m=|q|.

Provided a simpler alternative to Kruskal coordinates for the chargeless limit.

Utilized two-dimensional actions for describing highly symmetrical metrics.

Abstract

We construct coordinate systems that cover all of the Reissner-Nordstroem solution with m>|q| and m=|q|, respectively. This is possible by means of elementary analytical functions. The limit of vanishing charge q provides an alternative to Kruskal which, to our mind, is more explicit and simpler. The main tool for finding these global charts is the description of highly symmetrical metrics by two-dimensional actions. Careful gauge fixing yields global representatives of the two-dimensional theory that can be rewritten easily as the corresponding four-dimensional line elements.