Wightman function and the Casimir effect for a Robin sphere in a constant curvature space
Abstract
We evaluate the Wightman function, the mean field squared and the vacuum expectation value (VEV) of the energymomentum tensor for a scalar field with Robin boundary condition on a spherical shell in the background of a constant negative curvature space. For the coefficient in the boundary condition there is a critical value above which the scalar vacuum becomes unstable. In both interior and exterior regions, the VEVs are decomposed into the boundaryfree and sphereinduced contributions. For the latter, rapidly convergent integral representations are provided. In the region inside the sphere, the eigenvalues are expressed in terms of the zeros of the combination of the associated Legendre function and its derivative and the decomposition is achieved by making use of the AbelPlana type summation formula for the series over these zeros. The sphereinduced contribution to the VEV of the field squared is negative for Dirichlet boundary condition and positive for Neumann one. At distances from the sphere larger than the curvature scale of the background space the suppression of the vacuum fluctuations in the gravitational field corresponding to the negative curvature space is stronger compared with the case of the Minkowskian bulk. In particular, the decay of the VEVs with the distance is exponential for both massive and massless fields. The corresponding results are generalized for spaces with spherical bubbles and for cosmological models with negative curvature spaces.
 Publication:

European Physical Journal C
 Pub Date:
 September 2014
 DOI:
 10.1140/epjc/s1005201430474
 arXiv:
 arXiv:1407.0879
 Bibcode:
 2014EPJC...74.3047B
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Quantum Physics
 EPrint:
 28 pages, 3 figures, LaTeX file