Surface Casimir densities on branes orthogonal to the boundary of anti-de Sitter spacetime

TL;DR

This paper analyzes the vacuum expectation value of the surface energy-momentum tensor for a scalar field in anti-de Sitter spacetime with two orthogonal branes, revealing how boundary conditions and separation affect the energy densities and their physical implications.

Contribution

It provides a detailed calculation of the surface energy-momentum tensor in AdS with two branes, including renormalization and the effects of boundary conditions and brane separation.

Findings

Surface energy densities can be positive or negative depending on boundary conditions.

Induced SEMT is finite and free from renormalization ambiguities.

Energy density vanishes for Dirichlet and Neumann boundary conditions on the brane.

Abstract

We investigate the vacuum expectation value of the surface energy-momentum tensor (SEMT) for a scalar field with general curvature coupling in the geometry of two branes orthogonal to the boundary of anti-de Sitter (AdS) spacetime. For Robin boundary conditions on the branes, the SEMT is decomposed into the contributions corresponding to the self-energies of the branes and the parts induced by the presence of the second brane. The renormalization is required for the first parts only and for the corresponding regularization the generalized zeta function method is employed. The induced SEMT is finite and is free from renormalization umbiguities. For an observer living on the brane, the corresponding equation of state is of the cosmological constant type. Depending on the boundary conditions and on the separation between the branes, the surface energy densities can be either positive or negative. The energy density induced on the brane vanishes in special cases of Dirichlet and Neumann boundary conditions on that brane. The effect of gravity on the induced SEMT is essential at separations between the branes of the order or larger than the curvature radius for AdS spacetime. In the large separation limit the decay of the SEMT, as a function of the proper separation, follows a power law for both massless and massive fields. For parallel plates in Minkowski bulk and for massive fields the fall-off of the corresponding expectation value is exponential.