Conserved Charges in Asymptotically (Locally) AdS Spacetimes

Abstract

We review issues related to conservation laws for gravity with a negative cosmological constant subject to asymptotically (locally) anti-de Sitter boundary conditions. Beginning with the empty AdS spacetime, we introduce asymptotically (locally) AdS (AlAdS) boundary conditions, important properties of the boundary metric, the notion of conformal frames, and the Fefferman-Graham expansion. These tools are used to construct variational principles for AlAdS gravity, to more properly define the notion of asymptotic symmetry, and to construct the associated boundary stress tensor. The resulting conserved charges are shown to agree (up to possible choices of zero-point) with those built using Hamiltonian methods. Brief comments are included on AdS positive energy theorems and the appearance of a central extension of the AdS$_3$ asymptotic symmetry algebra. We also describe the algebra of boundary observables and introduce the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence using only tools from gravitational physics (and without other input from string theory). Our review focuses on motivations, current status, and open issues as opposed to calculational details. We emphasize the relativist (as opposed to particle physics) perspective and assume as background a standard graduate course in general relativity.

Figures (4)

• Figure 1: The hyperboloid \ref{['eq:AdShyperboloid']} embedded in $\mathbb{M}^{2,d}$, defining anti-de Sitter space.
• Figure 2: Conformal diagrams of AdS$_{d+1}$, showing both the global spacetime and the region covered by the Poincaré patch. In both figures, the $\tau$ direction extends infinitely to the future and to the past. In (a), a full $S^{d-1}$ of symmetry has been suppressed, leaving only the $\tau$, $r_*$ coordinates of \ref{['eq:AdSEinstein']}. The dotted line corresponds to $r_* = 0$. In (b), one of the angular directions has been shown explicitly to guide the reader's intuition; the axis of the cylinder corresponds to the dotted line in (a). The Poincaré patch covers a wedge-shaped region of the interior of the cylinder which meets the boundary at the lines marked $\mathscr{I}^\pm$ and the points marked $i^\pm, i^0$. These loci form the null, timelike, and spacelike infinities of the associated region (conformal to Minkowski space) on the AdS boundary.
• Figure 3: A sketch of the spacetime ${\cal M}$. The codimension two surface $C$ is a Cauchy surface of the boundary $\partial M$.
• Figure 4: An illustration of the definition of $B^-_\epsilon$. A source term $J=\epsilon A$ is added to the action and the gauge invariant function $B$ is calculated for the deformed solution $s_\epsilon$ subject to the boundary conditions that $s$ and $s_\epsilon$ coincide in the far past. Dashed lines indicate the boundary of the causal future of $J$. Only functions $B$ which have support in this region can have $B(s_\epsilon)\ne B(s)$. For visual clarity we have chosen our gauge invariant function $A$ and $B$ to have compact support though this is not required.