Flat limit of three dimensional asymptotically anti–de Sitter spacetimes

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. * gbarnich@ulb.ac.be; Research Director of the Fund for Scientific Research Belgium † agomberoff@unab.cl ‡ hdgonzal@uc.cl arXiv:1204.3288v1 [gr-qc] 15 Apr 2012 [1] A. Maloney and E. Witten, JHEP 1002, 029 (2010), arXiv:0712.0155 [hep-th].

A 2D Field Theory Equivalent to 3D Gravity with No Cosmological Constant

“…We are going to skip the derivation in this note. It is a well known result, and a derivation using the notation and normalizations used here may be found in [8].…”

“…Note that here there is a subtlety. The generators of BMS 3 must be properly rescaled before taking the limit, so that the central extension becomes proportional to c/l, which is finite in the limit as one may see from (3) (see also [8] for details).…”

“…The action (8), with β = √ 32πG and arbitrary ν is equivalent to Einstein gravity with no cosmological constant in the same way Liouville theory is when the cosmological constant is turned on 1 . The theory is invariant under the BMS 3 group, as it must be, because it is the asymptotic symmetry group of asymptotically flat 3-dimensional gravity.…”

“…The theory is invariant under the BMS 3 group, as it must be, because it is the asymptotic symmetry group of asymptotically flat 3-dimensional gravity. The particular form the fields are transformed by the group may be found in [8].…”

“…In the present talk, which is based on the work originally published in [8], we are going to show how we may take the limit of vanishing cosmological constant so that a field theory of asymptotically flat gravity is obtained. Although this procedure may appear to be trivial, it turns out that the limit is not well defined in the Lagrangian action when keeping a finite value of Newton's constant G. We will show, however, that in the Hamiltonian formulation a well defined limit may be taken for any value of G.…”