Spacetime Embedding Diagrams for Spherically Symmetric Black Holes

TL;DR

This paper demonstrates how to embed certain spherically symmetric black hole spacetimes into 2+1 Minkowski space, creating dynamic diagrams that illustrate particle motion and relate to horizon properties.

Contribution

It introduces a method for embedding specific black hole spacetimes into Minkowski space, enabling dynamic visualization of their geometry and particle trajectories.

Findings

Embedding is possible for Schwarzschild de-Sitter, anti de-Sitter, and Reissner-Nordstrom near the horizon.

The embedding diagrams are dynamic and illustrate test particle motion.

Conditions for smooth embedding relate to the surface gravity of the horizon.

Abstract

We show that it is possible to embed the 1+1 dimensional reduction of certain spherically symmetric black hole spacetimes into 2+1 Minkowski space. The spacetimes of interest (Schwarzschild de-Sitter, Schwarzschild anti de-Sitter, and Reissner-Nordstrom near the outer horizon) represent a class of metrics whose geometries allow for such embeddings. The embedding diagrams have a dynamic character which allows one to represent the motion of test particles. We also analyze various features of the embedding construction, deriving the general conditions under which our procedure provides a smooth embedding. These conditions also yeild an embedding constant related to the surface gravity of the relevant horizon.