The flat limit of three dimensional asymptotically anti-de Sitter spacetimes

TL;DR

This work develops a rigorous procedure to relate three-dimensional asymptotically AdS spacetimes to asymptotically flat spacetimes by a carefully constructed flat limit. By reformulating AdS$_3$ results in Robinson–Trautman coordinates and then employing a modified Penrose limit within the BMS gauge, the authors demonstrate a clean contraction of the asymptotic symmetry algebra from two Virasoro algebras to the BMS$_3$ algebra, with explicit central charges $c^{\pm}=\frac{3l}{2G}$ and $c_1=0$, $c_2=\frac{3}{G}$. They provide detailed mappings of the solution spaces, charges, and zero-mode content between AdS$_3$ (including BTZ) and flat spacetimes, clarifying the geometric and holographic interpretation of the flat limit in three dimensions. The results offer a concrete bridge between AdS/CFT-like structures and flat-space holography in 3D, with explicit procedures to translate between $ ext{Fefferman–Graham}$-based data and $ ext{BMS}_3$ data.

Abstract

In order to get a better understanding of holographic properties of gravitational theories with a vanishing cosmological constant, we analyze in detail the relation between asymptotically anti-de Sitter and asymptotically flat spacetimes in three dimensions. This relation is somewhat subtle because the limit of vanishing cosmological constant cannot be naively taken in standard Fefferman-Graham coordinates. After reformulating the standard anti-de Sitter results in Robinson-Trautman coordinates, a suitably modified Penrose limit is shown to connect both asymptotic regimes.

Figures (2)

• Figure 1: Penrose diagrams of anti-de Sitter and Minkowski spacetimes. Arrows in the diagrams represent outgoing null rays.
• Figure 2: Zero mode solutions of 2+1 gravity. Figure (a) depicts the case of non-vanishing cosmological constant. The slope $\beta$ is given by $\tan \beta=\frac{1}{l}$. Figure (b) shows the limit $l \to \infty$, when $\beta$ vanishes. No solutions are lost in the limit, but the horizon of the BTZ black holes gets pushed to infinity, hence the time coordinate becomes spatial everywhere and the line element describes the non-static, cosmological solution (\ref{['cosmology']}).