Flat limit of three dimensional asymptotically anti–de Sitter spacetimes

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.

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 nonvanishing 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 nonstatic, cosmological solution (62).