The BTZ spacetime as an algebraic embedding

TL;DR

This paper presents a simple algebraic global isometric embedding of the nonrotating BTZ black hole into higher-dimensional Minkowski space, revealing its geometric structure and embedding class, with extensions to Euclidean and rotating solutions.

Contribution

It introduces a straightforward algebraic embedding for the BTZ black hole and its Euclidean counterpart, and explores embeddings into higher-dimensional spaces, highlighting the spacetime's embedding class.

Findings

Embedding in Minkowski space as intersection of quadrics

BTZ spacetime is of embedding class one in AdS4 or H4

Rotating Euclidean solution admits a quadratic algebraic embedding

Abstract

A simple algebraic global isometric embedding is presented for the nonrotating BTZ black hole and its counterpart of Euclidean signature. The image of the embedding, in Minkowski space of two extra dimensions, is the interection of two quadric hypersurfaces. Furthermore an embedding into $AdS_4$ or $H_4$ is also obtained, showing that the spacetime is of embedding class one with respect to maximally symmetric space of negative curvature. The rotating solution of Euclidean signature is also shown to admit a quadratic algebraic embedding, but seemingly requires more than two extra dimensions.