On the generalization of the Kruskal–Szekeres coordinates: a global conformal charting of the Reissner–Nordström spacetime

Abstract

Abstract The Kruskal–Szekeres coordinate construction for the Schwarzschild spacetime could be interpreted simply as a squeezing of the t-line into a single point, at the event horizon r = 2 M . Starting from this perspective, we extend the Kruskal charting to spacetimes with two horizons, in particular the Reissner–Nordström manifold, M RN . We develop a new method to construct Kruskal-like coordinates through casting the metric in new null coordinates, and find two algebraically distinct ways to chart M RN , referred to as classes: type-I and type-II within this work. We pedagogically illustrate our method by crafting two compact, conformal, and global coordinate systems labeled GK I and GK II as an example for each class respectively, and plot the corresponding Penrose diagrams. In both coordinates, the metric differentiability can be promoted to C ∞ in a straightforward way. Finally, the conformal metric factor can be written explicitly in terms of the t and r functions for both types of charts. We also argued that the chart recently reported in Soltani (2023 arXiv:2307.11026) could be viewed as another example for the type-II classification, similar to GK II .