On the generalization of the Kruskal-Szekeres coordinates: a global conformal charting of the Reissner-Nordstrom spacetime

TL;DR

This paper extends Kruskal-Szekeres coordinates to Reissner-Nordstrom spacetime, introducing two new global conformal coordinate systems that handle multiple horizons and improve metric smoothness.

Contribution

It develops a novel method for constructing Kruskal-like coordinates and introduces two distinct classes of global conformal charts for Reissner-Nordstrom spacetime.

Findings

Constructed two global conformal coordinate systems for Reissner-Nordstrom spacetime.

Achieved $C^ ext{infinity}$ differentiability of the metric in these coordinates.

Explicitly expressed the conformal metric factor in terms of original coordinates.

Abstract

The Kruskal-Szekeres coordinates construction for the Schwarzschild spacetime could be viewed geometrically as a squeezing of the $t$-line associated with the asymptotic observer into a single point, at the event horizon $r=2M$. Starting from this point, we extend the Kruskal charting to spacetimes with two horizons, in particular the Reissner-Nordstr\"om manifold, $\mathcal{M}_{RN}$. We develop a new method for constructing Kruskal-like coordinates and find two algebraically distinct classes charting $\mathcal{M}_{RN}$. We pedagogically illustrate our method by constructing two compact, conformal, and global coordinate systems labeled $\mathcal{GK_{I}}$ and $\mathcal{GK_{II}}$ for each class respectively. In both coordinates, the metric differentiability can be promoted to $C^\infty$. The conformal metric factor can be explicitly written in terms of the original $t$ and $r$ coordinates for both charts.