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A Global Isometric Embedding of the Reissner-Nordström Metric into Pseudo-Euclidean Spacetime

A. T. Eberlein, C. N. Pope

TL;DR

The paper tackles the problem of globally embedding the maximally extended Reissner-Nordström spacetime into a higher-dimensional pseudo-Euclidean space. It advances the method by constructing a $\,9$-dimensional embedding with explicit local expressions in Boyer-Lindquist coordinates and a complementary Kruskal-Szekeres formulation, augmented by coordinates $R_+$ and $R_-$ to handle the dual horizons. A key contribution is the four-coordinate-free level-set representation that encodes the RN geometry, yielding finiteness across both horizons and across all regions of the maximal extension. The work broadens the arsenal of geometric tools for black hole spacetimes and suggests pathways to global embeddings for other multi-horizon metrics, while noting the extremal case remains outside its scope.

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-Nordström 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-Nordström 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-Nordström 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.

A Global Isometric Embedding of the Reissner-Nordström Metric into Pseudo-Euclidean Spacetime

TL;DR

The paper tackles the problem of globally embedding the maximally extended Reissner-Nordström spacetime into a higher-dimensional pseudo-Euclidean space. It advances the method by constructing a -dimensional embedding with explicit local expressions in Boyer-Lindquist coordinates and a complementary Kruskal-Szekeres formulation, augmented by coordinates and to handle the dual horizons. A key contribution is the four-coordinate-free level-set representation that encodes the RN geometry, yielding finiteness across both horizons and across all regions of the maximal extension. The work broadens the arsenal of geometric tools for black hole spacetimes and suggests pathways to global embeddings for other multi-horizon metrics, while noting the extremal case remains outside its scope.

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-Nordström 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-Nordström 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-Nordström 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.

Paper Structure

This paper contains 7 sections, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Current State of Global Isometric Embeddings of the Reissner-Nordström Metric
  3. Definitions
  4. Global Isometric Embedding of the Schwarzschild Metric
  5. Global Isometric Embedding of the Reissner-Nordström Metric
  6. Discussion and Conclusions
  7. Embedding Coordinates in Local Kruskal-Szekeres Coordinates

Figures (2)

  • Figure 1: The Kruskal diagram for the maximal analytic extension of the Schwarzschild metric. The $45\degree$ horizon corresponds to $U^{KS}=0$ and the $-45\degree$ horizon corresponds to $V^{KS}=0.$
  • Figure 2: The Kruskal diagram for the maximally extended Reissner-Nordström manifold.