Conserved Charges in Asymptotically (Locally) AdS Spacetimes

TL;DR

This paper reviews the construction and properties of conserved charges in asymptotically AdS spacetimes, discussing boundary conditions, symmetries, and the AdS/CFT correspondence from a gravitational physics perspective.

Contribution

It provides a comprehensive review of conserved charges, boundary conditions, and symmetries in asymptotically AdS spacetimes, emphasizing gravitational methods and the AdS/CFT framework.

Findings

Conserved charges in AlAdS spacetimes can be constructed using boundary stress tensors.

The boundary symmetry algebra includes a central extension in AdS3.

The boundary observables form an algebra consistent with gravitational physics principles.

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.