Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime

TL;DR

This work addresses the problem of defining deterministic dynamics for linear fields in anti-de Sitter spacetime, which lacks a global Cauchy surface. By reducing scalar, electromagnetic, and gravitational perturbations to scalar wave equations on a two-dimensional AdS space via spherical harmonic decomposition, the authors show that admissible dynamics are in one-to-one correspondence with positive self-adjoint extensions of a central operator $A$. They classify all such extensions across dimensions, identifying a unique extension when $\nu^2\ge 1$ and a one-parameter family $A_\alpha$ for $0<\nu^2<1$ or $\nu^2=0$, with explicit positivity conditions that translate into boundary conditions at infinity (Dirichlet, Neumann, or Robin type). The results illuminate how boundary conditions at infinity depend on the effective mass and dimension, revealing when boundary data are needed and how energy positivity can be maintained—relevant for AdS/CFT and bulk/boundary correspondences. Overall, the paper provides a comprehensive framework to enumerate and interpret all consistent bulk dynamics for AdS perturbations and connects the mathematical structure to physical boundary prescriptions, including the Breitenlohner-Freedman-type stability bounds.

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.