Conserved Charges in Asymptotically (Locally) AdS Spacetimes

Abstract

When a physical system is complicated and nonlinear, global symmetries and the associated conserved quantities provide some of the most powerful analytic tools to understand its behavior. This is as true in theories with a dynamical spacetime metric as for systems defined on a fixed spacetime background. Chapter 17 has already discussed the so-called Arnowitt–Deser–Misner (ADM ) conserved quantities for asymptotically flat dynamical spacetimes, exploring in detail certain subtleties related to diffeomorphism invariance. In particular, it showed that the correct notion of global symmetry is given by the so-called asymptotic symmetries; equivalence classes of diffeomorphisms with the same asymptotic behavior at infinity. It was also noted that the notion of asymptotic symmetry depends critically on the choice of boundary conditions. Indeed, it is the imposition of boundary conditions that causes the true gauge symmetries to be only a subset of the full diffeomorphism group and thus allows the existence of nontrivial asymptotic symmetries at all. This chapter will explore the asymptotic symmetries and corresponding conserved charges of asymptotically anti-de Sitter (AdS) spacetimes (and of the more general asymptotically locally AdS spacetimes). There are three excellent reasons for doing so. The first is simply to gain further insight into asymptotic charges in gravity by investigating a new example. Since empty AdS space is a maximally symmetric solution, asymptotically AdS spacetimes are a natural and simple choice. The second is that the structure one finds in the AdS context is actually much richer than that in asymptotically flat space. At the physical level, this point is deeply connected to the fact (see, e. g., [1]) that all multipole moments of a given field in AdS space decay at the same rate at infinity. So while in asymptotically flat space the far field is dominated mostly by monopole terms (with only subleading corrections from dipoles and higher multipoles) all terms contribute equally in AdS. It is therefore useful to describe not just global charges (e. g., the total energy) but also the local densities of these charges along the AdS boundary. In fact, it is natural to discuss an entire so-called boundary stress tensor rather than just the conserved charges it defines. For this reason, we take a somewhat different path to the construction of conserved AdS charges than was followed in Chap. 17 . In particular, we will use covariant as opposed to Hamiltonian methods below, although we will show in Sect. 19.3 that the end results for conserved charges are equivalent. The third reason to study conserved charges in AdS is their fundamental relation to the AdS/CFT correspondence [2] [3] [4], which may well be the most common application of general relativity in twenty-first century physics. While this is not the place for a detailed treatment of either string theory or AdS/CFT, no Handbook of Spacetime would be complete without presenting at least a brief overview of the correspondence. It turns out that this is easy to do once we have become familiar with and its cousins associated with other (nonmetric) fields. So at the end of this chapter (Sect. 19.4) we will take the opportunity to do so. We will introduce AdS/CFT from the gravity side without using tools from either string theory or CFT. We will focus on such modern applications below, along with open questions. We make no effort to be either comprehensive or historical. Nevertheless, the reader should be aware that conserved charges for asymptotically AdS spacetimes were first constructed in [5], where the associated energy was also argued to be positive definite. The plan for this chapter is as follows. After defining and discussing AdS asymptotics in Sect. 19.1, we construct variational principles for asymptotically AdS spacetimes in Sect. 19.2. This allows us to introduce the boundary stress tensor and a similar so-called response function for a bulk scalar field. The conserved charges constructed from are discussed in Sect. 19.24, and we comment briefly on positivity of the energy in Sect. 19.25. Section 19.3 then provides a general proof that the do indeed generate canonical transformations corresponding to the desired asymptotic symmetries. As a result, they agree (up to a possible choice of zero-point) with corresponding ADM-like charges that would be constructed via the AdS-analogs of the Hamiltonian techniques used in Chap. 17 . The interested reader can find such a Hamiltonian treatment in [6] [7] [8]. Below, we generally consider AdS gravity coupled to a simple scalar matter field. More complete treatments allowing more general matter fields can be found in e. g., [9] [10] [11]. Section 19.4 then defines the algebra of boundary observables and provides the above-mentioned brief introduction to AdS/CFT.