Dynamics in non-globally-hyperbolic static spacetimes: III. Anti-de Sitter spacetime

Abstract

In recent years, there has been considerable interest in theories formulated in anti-de Sitter (AdS) spacetime. However, AdS spacetime fails to be globally hyperbolic, so a classical field satisfying a hyperbolic wave equation on AdS spacetime need not have a well defined dynamics. Nevertheless, AdS spacetime is static, so the possible rules of dynamics for a field satisfying a linear wave equation are constrained by our previous general analysis-given in paper II-where it was shown that the possible choices of dynamics correspond to choices of positive, self-adjoint extensions of a certain differential operator, A. In the present paper, we reduce the analysis of electromagnetic, and gravitational perturbations in AdS spacetime to scalar wave equations. We then apply our general results to analyse the possible dynamics of scalar, electromagnetic, and gravitational perturbations in AdS spacetime. In AdS spacetime, the freedom (if any) in choosing self-adjoint extensions of A corresponds to the freedom (if any) in choosing suitable boundary conditions at infinity, so our analysis determines all of the possible boundary conditions that can be imposed at infinity. In particular, we show that other boundary conditions besides the Dirichlet and Neumann conditions may be possible, depending on the value of the effective mass for scalar field perturbations, and depending on the number of spacetime dimensions and type of mode for electromagnetic and gravitational perturbations.

Holographic renormalization of Einstein-Maxwell-Dilaton theories

“…Holographic renormalization is a central part of extracting physical quantities of the dual field theory from the gravity side [19]- [23]. See also a previous work in the context of EMD theories [17] that considered the scalar and vector variations separately.…”

Erratum: Energy bounds in designer gravity [Phys. Rev. D74, 064006 (2006)]

“…When these scalars have mass at or slightly above the Breitenlohner-Freedman (BF) bound, the analysis of [5] suggests that the energy is bounded below under a range of boundary conditions on the scalar field. It was further found in [6] that requiring the wave equation to have a sufficiently well-posed deterministic evolution law under general boundary conditions restricts the scalar field mass to be in this same range.…”

Kaluza–Klein reductions and AdS/Ricci-flat correspondence

“…This leads to the decomposition (2.9) for the metric perturbation: scalars can be decomposed into scalar harmonics, vectors are decomposed into a vector harmonics that is divergent-free and the gradient of scalars as a consequence of the Hodge decomposition theorem, and symmetric tensors are decomposed in traceless symmetric conserved tensor (the tensor harmonic), and derivatives of vector and scalar harmonics (see e.g. [49] or [50]). …”