Global Embedding of the Reissner-Nordstr\"om Metric in the Flat Ambient Space

TL;DR

This paper presents three new global isometric embeddings of the Reissner-Nordström black hole metric into flat space, covering different regions and maintaining smoothness at horizons, including extremal cases.

Contribution

It introduces minimal-dimension, smooth, global embeddings for non-extremal, extremal, and hyperextremal Reissner-Nordström black holes, covering all regions including horizons.

Findings

Three new embeddings in minimal flat space dimension

Embeddings are smooth across horizons for all black hole types

Time lines in embeddings are more complex than simple conic sections

Abstract

We study isometric embeddings of non-extremal Reissner-Nordstr\"om metric describing a charged black hole. We obtain three new embeddings in the flat ambient space with minimal possible dimension. These embeddings are global, i.e. corresponding surfaces are smooth at all values of radius, including horizons. Each of the given embeddings covers one instance of the regions outside the horizon, one instance between the horizons and one instance inside the internal horizon. The lines of time for these embeddings turn out to be more complicated than circles or hyperbolas. The obtained embeddings are also smooth at all values of radius for extremal and hyperextremal black holes.