Wightman Function and Casimir Densities for Robin Plates in the Fulling–rindler Vacuum

Abstract

Positive frequency Wightman function and vacuum expectation value of the energymomentum 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 there 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.

Casimir effect for parallel plates in a Friedmann-Robertson-Walker universe

“…for u z 0 m and Z(η, u) = 0 for u < z 0 m. Substituting the expression (5.2) into the general formulas given above, we obtain the VEVs for the Robin plates in Minkowski spacetime (see [14] for the VEVs in the massless case and [20] for a massive scalar field. Note that in [20] the VEVs in the Minkowski bulk are obtained as a limiting case of the corresponding problem with two uniformly accelerated plates moving through the Fulling-Rindler vacuum state).…”

Casimir energy-momentum tensor for a quantized bulk scalar field in the geometry of two curved branes on Friedmann-Robertson-Walker background

“…For a fixed value of k, this equation has an infinite set of real solutions [53]. Here, it is assumed that they are arranged in the ascending order ω n < ω n+1 .…”

“…In the region between the plates, and with respect to the standard Klein-Gordon orthonormality condition (18), the solutions induced by boundaries are [53]…”

Casimir densities for two spherical branes in Rindler-like spacetimes

“…In addition to applications in braneworld models on the AdS black hole bulk, the problem under consideration is also of separate interest as an example with gravitational and boundary-induced polarizations of the vacuum, where all calculations can be performed in a closed form. Note that the vacuum densities induced by a single and two parallel flat branes in the bulk geometry Ri × R D−1 for both scalar and electromagnetic fields are investigated in [10,11].…”

“…From the boundary condition on the brane ξ = a we find that the possible values for ω are roots to the equation 11) with the notation…”