Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime with dynamical boundary conditions of Wentzell type

TL;DR

This paper analyzes solutions for a Klein-Gordon field in anti-de Sitter spacetime with generalized Wentzell boundary conditions, solving the equations explicitly and exploring bound states and symmetry properties.

Contribution

It provides a complete solution to the bulk and boundary equations for a Klein-Gordon field with Wentzell boundary conditions in AdS, extending previous Robin boundary condition results.

Findings

Explicit solutions for bulk and boundary fields

Existence of bound state solutions

Wentzell boundary conditions preserve AdS symmetries

Abstract

We study a real, massive Klein-Gordon field in the Poincar\'e fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of dynamical boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincar\'e fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.