BTZ black hole from the structure of the algebra so(2,n)

TL;DR

This paper explores the algebraic structures of so(2,n) to understand the formation of BTZ black holes in anti-de Sitter space, focusing on Lie algebra decompositions, singularity definitions, and horizon geometry.

Contribution

It provides a detailed algebraic framework linking Lie algebra structures to black hole geometry in various dimensions, including a new coherent structure for AdS_2.

Findings

Singularity characterized by closed orbits of the Iwasawa group

Equivalence of singularity definitions via orbits and vector field norms

Derived the shape of the black hole horizon in anti-de Sitter space

Abstract

We study the relevant structure of so(2,n) which makes the BTZ black hole possible in the anti de Sitter space. We pay a particular attention of the reductive Lie algebra structures and Iwasawa decompositions and the way these structures evolves when one increases the dimension. The singularity is defined as the closed orbits of the Iwasawa group and we show that this definitionn is equivalent (in every dimension) to the definition by means of vanishing norm of a fundamental vector field. Then we derive the shape of the horizon and, as a small bonus, we define a coherent black hole structure on AdS_2. This paper contains a "short" version (15 pages) in which only the main steps are given in order to clarify the construction and a "long" version (50 pages) in which all the intermediate steps and computation are given.