Wightman function and Casimir densities on AdS bulk with application to the Randall-Sundrum braneworld

TL;DR

This paper calculates the Wightman function and vacuum energy densities for a scalar field in AdS space with boundaries, analyzing boundary effects, forces, and applications to the Randall-Sundrum model.

Contribution

It provides a general analysis of vacuum densities with Robin boundary conditions in AdS, including different coefficients, and applies results to the Randall-Sundrum braneworld.

Findings

Interaction forces are attractive for Dirichlet boundary conditions.

There exists a parameter region with repulsive small-distance forces and attractive large-distance forces.

Vacuum densities exhibit specific asymptotic behaviors near boundaries and at large distances.

Abstract

Positive frequency Wightman function and vacuum expectation value of the energy-momentum tensor are computed for a massive scalar field with general curvature coupling parameter subject to Robin boundary conditions on two parallel plates located on $D+1$ - dimensional AdS background. The general case of different Robin coefficients on separate plates is considered. The mode summation method is used with a combination of a variant of the generalized Abel-Plana formula for the series over zeros of combinations of cylinder functions. This allows us to extract manifestly the parts due to the AdS spacetime without boundaries and boundary induced parts. The asymptotic behavior of the vacuum densities near the plates and at large distances is investigated. The vacuum forces acting on the boundaries are presented as a sum of the self-action and interaction forces. The first one contains well-known surface divergences and needs further regularization. The interaction forces between the plates are attractive for Dirichlet scalar. We show that threre is a region in the space of parameters defining the boundary conditions in which the interaction forces are repulsive for small distances and attractive for large distances. An application to the Randall-Sundrum braneworld with arbitrary mass terms on the branes is discussed.