Casimir Energy Densities for Parallel Plate on Background of Conformally Flat Brane-World Geometries and Cosmological Constant Problem

In this paper, we calculate the stress-energy tensor for a quantized massless conformally coupled scalar field in the background of a conformally flat brane-world geometries, where the scalar field satisfying Robin boundary conditions on two parallel plates. In the general case of Robin boundary conditions formula are derived for the vacuum expectation values of the energy-momentum tensor. Further the surface energy per unit area are obtained . As an application of the general formula we have considered the important special case of the AdS$_{4+1}$ bulk, moreover application to the Randall-Sundrum scenario is discused. In this specific example for a certain choice of Robin coefficients, one could make the effective cosmological constant vanish.