Asymptotically anti-de Sitter spacetimes in three dimensions

Abstract

In this thesis, we reviewed several aspects of asymptotically anti-de Sitter (AAdS) spacetimes in three dimensional Einstein gravity by following some important historical work. Starting with a brief introduction to anti-de Sitter (AdS) spacetimes where also the BTZ black hole solution is given we de ned Noether-Wald charges using Noether theorems. Next, we compared di erent de nitions of AAdS spacetimes. Here, we adopted the Fefferman-Graham coordinates and solved Einstein equations order by order to prove that the Fefferman-Graham expansion of AAdS spacetimes terminates at second order in three dimensions, as first shown by Skenderis and Solodukhin. Lastly, we considered two sets of boundary conditions and presented their asymptotic symmetry algebras and charge algebras. Imposing Brown-Henneaux boundary conditions we arrived at Banados metric, which is the most general metric for AAdS spacetimes under these conditions. Then we showed that the asymptotic symmetry algebra is two copies of the Virasoro algebra. Under the Compere-Song-Strominger boundary conditions, we calculated the most general metric and showed the charge algebra is a semidirect sum of Virasoro and Kac-Moody algebras. We concluded with some comments and future research directions.