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

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.