AdS spacetimes and isometric embeddings

TL;DR

This paper constructs explicit isometric embeddings of BTZ black holes and AdS spacetimes into higher-dimensional flat spacetimes, providing new geometric insights and generalizations to higher dimensions.

Contribution

It presents explicit algebraic embeddings of BTZ black holes and cosmological AdS spacetimes into flat ambient spaces, including higher-dimensional generalizations.

Findings

Embedded BTZ black hole in 3+2 dimensional flat spacetime.

Constructed a family of embeddings for cosmological AdS spacetimes.

Generalized embeddings to higher-dimensional AdS spaces.

Abstract

An algebraic global isometric embedding of the nonrotating BTZ black hole is presented. The ambient spacetime is $\mathbb{M}^{2,3}$, the 3+2 dimensional flat spacetime. We also present the analogous embedding for the Euclidean BTZ spacetime and by performing a kind of double analytic continuation construct a 1-parameter family of embeddings of cosmological AdS spacetime into $\mathbb{M}^{2,3}$ which coincide asymptotically with the embedded BTZ manifold of the appropriate mass. Finally we note that the family of embeddings of cosmological AdS$_{n}$ into $\mathbb{M}^{2,n}$ generalises to higher dimensions.