Israel coordinates for all static spherically symmetric spacetimes with vanishing second Ricci invariant

TL;DR

This paper generalizes the Israel coordinate construction to static spherically symmetric spacetimes with zero second Ricci invariant, enabling analysis of completeness and horizon structure without coordinate transformations.

Contribution

It extends the Israel procedure to handle spacetimes with two bifurcation spheres, providing a constructive approach to their completeness.

Findings

Can cover spacetimes with two bifurcation two-spheres

Fails with three bifurcation two-spheres

No coordinate transformations are used

Abstract

Static spherically symmetric spacetimes with vanishing second Ricci invariant constitute an important class of solutions to Einstein's equations and more generally as archetypes of regular black holes. When studying completeness one is most often presented with the Kruskal - Szekeres procedure. However, this procedure only works if the spacetime admits a single non-degenerate Killing horizon (a single bifurcation two-sphere). Here we generalize the Israel procedure to examine a constructive approach to completeness based entirely on the static spherically symmetric nature of spacetimes with a vanishing second Ricci invariant. It is shown by "block gluing" that the Israel procedure can cover two bifurcation two-spheres, but can fail with three. No coordinate transformations are used in this work.