Classics Illustrated: Limits of Spacetimes

TL;DR

This paper investigates the limits of Reissner-Nordström spacetimes as charge and mass parameters vary, using Geroch's framework and anti-de Sitter embeddings to produce accurate Penrose diagrams.

Contribution

It introduces a method to analyze spacetime limits with Geroch's definition, providing precise Penrose diagrams for different parameter limits.

Findings

Successfully derived metrically accurate Penrose diagrams for the limits

Identified distinct limit behaviors as charge approaches mass and charge approaches zero

Enhanced understanding of spacetime structure in charged black hole models

Abstract

We carefully study the $e \rightarrow m$ and $e \rightarrow 0$ limits of the Reissner-Nordstr\"om spacetime using Geroch's definition of limits of spacetimes. This is implemented by embedding the one-parameter family of spacetimes in anti-de Sitter space, and as a result we obtain metrically correct Penrose diagrams. For $e \rightarrow m$ two distinct limits are studied.