Global embedding of BTZ spacetime using generalized method of symmetric embeddings construction

TL;DR

This paper introduces a novel algorithm for explicitly embedding pseudo-Riemannian manifolds with symmetries into higher-dimensional spaces, simplifying the process and enabling new embeddings such as that of the BTZ black hole.

Contribution

It presents a symmetry-based algorithm that avoids traditional equations, making the construction of embeddings more straightforward and applicable to complex spacetimes like BTZ black holes.

Findings

Explicit global embedding of spinning BTZ black hole in 7D space

Simplified construction of known embeddings using the new algorithm

Development of a group-theoretical method for embedding construction

Abstract

It is often easier to study pseudo-Riemannian manifolds by presenting them as surfaces in some ambient space. We propose an algorithm for construction of explicit isometric embeddings of pseudo-Riemannian manifolds with symmetries into an ambient space of higher dimension. While most of the existing methods are based on Gauss-Codazzi-Mainardi-Peterson equations, we do not use them and instead concentrate on a system of equations which connects the metric on the manifold and the embedding function of the surface. Our algorithm is based on the group theoretical method of separation of variables that we developed earlier (arXiv:1202.1204). The algorithm makes this method more convenient and simple to use. It allowed us to simplify the construction of many known embeddings as well as obtain some new ones. In particular, we obtain explicit global embeddings of spinning BTZ black hole in 7-dimensional flat space.