Spherically symmetric black holes and affine-null metric formulation of Einstein's equations

TL;DR

This paper introduces two coordinate approaches for analyzing spherically symmetric black holes, ensuring regularity from the horizon to null infinity, and applies them to derive Reissner-Nordström solutions.

Contribution

It presents two methods for coordinate choices in spherical symmetry that maintain regularity from horizon to infinity, including an affine-null formulation of Einstein's equations.

Findings

Derived Reissner-Nordström black holes in regular coordinates.

Proposed coordinate frameworks avoiding singularities at the horizon.

Enhanced understanding of black hole spacetimes in spherical symmetry.

Abstract

The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g. $r=2M$ in the Schwarzschild black hole) or to an implicit dependence of the chosen coordinate to physical relevant coordinates (e.g. the dependence of the null coordinates in the Kruskal metric). Here we discuss two approaches for coordinate choices in spherical symmetry allowing us to discuss explicitly "solitary" and spherically symmetric black holes from a regular horizon to null infinity. The first approach relies on a construction of a regular null coordinate (where regular is meant as being defined from the horizon to null infinity) given an explicit solution of the Einstein-matter equations. The second approach is based on an affine-null formulation of the Einstein equations and the respective characteristic initial value problem. In particular, we present a derivation of the Reissner-Nordstr\"om black holes expressed in terms of these regular coordinates.