Global Embeddings of BTZ and Schwarzschild-ADS Type Black Holes in a Flat Space

Anton Sheykin,Dmitry Solovyev,Sergey Paston

doi:10.3390/sym11070841

Abstract

We study the problem of construction of global isometric embedding for spherically symmetric black holes with negative cosmological constant in various dimensions. Firstly, we show that there is no such embedding for 4D RN-AdS black hole in 6D flat ambient space, completing the classification which we started earlier. Then we construct an explicit embedding of non-spinning BTZ black hole in 6D flat ambient space. Using this embedding as an anzats, we then construct a global explicit embedding of d-dimensional Schwarzschild-AdS black hole in a flat ( d + 3 ) -dimensional ambient space.

Highlights

Black holes are one of the most favorite objects of study in the general relativity
The n-dimensional generalization of the black hole metric was found by Tangherlini [5], who mentioned that the absence of propagating degrees of freedom in (2 + 1)D gravity with point masses is well-known since in (2 + 1) dimensions Gμν = 0 implies Rμναβ = 0
We still able to construct a global embedding for the static case of BTZ with

Summary

Introduction

Black holes are one of the most favorite objects of study in the general relativity. They are simple enough to be described only by a few parameters (for a historical survey of no-hair theorem see, e.g., [1]) but have a very rich structure and many interesting properties. A remarkable example of this approach is the description of gravity as the Regge-Teitelboim embedding theory [17,18,19,20,21,22,23], where dynamics of gravity is formulated in terms of dynamics of curved surface isometrically embedded in flat ambient spacetime In this theory the construction of explicit global embeddings of physically relevant metrics is required for the analysis of their possible non-Einsteinian modification [24,25]. Another one of the main recent applications of explicit embeddings is the analysis of thermodynamic properties of manifolds with horizons. We construct global 7-dimensional embedding for Schwarzschild-AdS black hole, as well as its generalization: the global embedding of d-dimensional Schwarzschild-AdS black hole in (d + 3)D flat spacetime

The Embedding of Static Black Holes

Global Embedding of Static BTZ Spacetime

The Absence of 6-Dimensional Embeddings

Rn l 2

Concluding Remarks