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

TL;DR

This paper analyzes the possible boundary conditions for fields in anti-de Sitter spacetime, showing that besides Dirichlet and Neumann, other conditions may be valid depending on the field type, mass, and dimensions.

Contribution

It extends the analysis of boundary conditions in AdS spacetime to electromagnetic and gravitational perturbations, identifying all possible self-adjoint extensions and boundary conditions.

Findings

Boundary conditions depend on effective mass and spacetime dimensions.

Other boundary conditions beyond Dirichlet and Neumann are possible.

The analysis applies to scalar, electromagnetic, and gravitational perturbations.

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.