A Global Isometric Embedding of the Reissner-Nordstr\"om Metric into Pseudo-Euclidean Spacetime

TL;DR

This paper constructs a comprehensive nine-dimensional pseudo-Euclidean embedding of the entire Reissner-Nordstr"om spacetime, including multiple horizons, extending previous embeddings limited to external regions or single patches.

Contribution

It provides the first global isometric embedding of the maximally extended Reissner-Nordstr"om spacetime into nine dimensions, with explicit local coordinates and level-set descriptions.

Findings

Embedding is finite and well-defined on all horizons.

Includes additional coordinates to handle multiple horizons.

Extends previous embeddings limited to external regions.

Abstract

The event horizon of the Schwarzschild black hole has been well studied and the singular behavior of the Schwarzschild metric on horizon is understood as a coordinate singularity rather than an essential singularity. One demonstration of this non-singular behavior on horizon was provided by Fronsdal in 1959, by finding a global isometric embedding of the Schwarzschild metric into a six-dimensional pseudo-Euclidean spacetime. Isometric embeddings for the Reissner-Nordstr\"om metric have also been constructed, but they only embed the region external to the inner horizon or in a single Eddington-Finkelstein patch. This paper presents a global isometric embedding for the maximally extended Reissner-Nordstr\"om spacetime into a nine-dimensional pseudo-Euclidean spacetime. We present the solution in terms of explicit local four-dimensional coordinates, and also as a level-set of functions of the higher-dimensional embedding spacetime. While the Reissner-Nordstr\"om embedding presented has several similarities to the Fronsdal embedding of the Schwarzschild metric, the presence of the second horizon requires additional embedding coordinates and terms not found in the Fronsdal embedding, in order that the embedding is defined and finite on each horizon.