Spacetime Embedding Diagrams for Spherically Symmetric Black Holes
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-Nordström 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 yield an embedding constant related to the surface gravity of the relevant horizon.
Giblin, J.T., Marolf, D. & Garvey, R. General Relativity and Gravitation (2004) 36: 83. doi:10.1023/B:GERG.0000006695.17232.2e
General Relativity and Gravitation