Exact classical sphere-plate Casimir interaction in (D+1)-dimensional spacetime

Abstract

We consider the high temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D+1)$-dimensional Minkowski spacetime. The high temperature leading term of the Casimir free interaction energy is known as the classical term since it does not depend on the Planck constant $\hslash$. From the functional representation of the zero temperature Casimir interaction energy, we use Matsubara formalism to derive the classical term. It can be expressed as a weighted sum over logarithms of determinants. Using similarity transforms of matrices, we re-express this classical term as an infinite series. This series is then computed exactly using generalized Abel-Plana summation formula. From this, we deduce the short distance asymptotic expansions of the classical Casimir interaction force. As expected, the leading term agrees with the proximity force approximation. The next two terms in the asymptotic expansion are also computed. It is observed that the ratio of the next-to-leading order term to the leading order term is proportional to the dimension of spacetime. Hence, a larger correction to the proximity force approximation is expected in spacetime with higher dimensions. This is similar to a previous result deduced for the zero temperature case.

14 pages

References in corpus (19)

• Casimir forces between arbitrary compact objects
• The Casimir effect within scattering theory
• Casimir forces in a T operator approach
• The Casimir effect for a sphere and a cylinder in front of plane and corrections to the proximity force theorem
• Casimir Forces between Compact Objects: I. The Scalar Case
• Fluctuation induced quantum interactions between compact objects and a plane mirror
• Multiple Scattering Methods in Casimir Calculations
• Casimir interaction between plane and spherical metallic surfaces
• Casimir force for a sphere in front of a plane beyond Proximity Force Approximation
• Generalized Lifshitz formula for a cylindrical plasma sheet in front of a plane beyond proximity force approximation
• Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique
• Casimir forces between arbitrary compact objects: Scalar and electromagnetic field
• Finite Temperature Casimir Effect in Kaluza-Klein Spacetime
• Finite temperature Casimir effect in spacetime with extra compactified dimensions
• Exact Casimir Interaction Between Semitransparent Spheres and Cylinders
• Finite temperature Casimir effect in piston geometry and its classical limit
• Geothermal Casimir phenomena for the sphere-plate and cylinder-plate configurations
• Mode summation approach to Casimir effect between two objects
• Casimir interaction between spheres in $(\mathbf{D}+1)$-dimensional Minkowski spacetime

Cited by in corpus (3)

• Finite temperature Casimir interaction between spheres in $(D+1)$-dimensional spacetime: Exact computations and asymptotic expansions
• Derivative expansion for the electromagnetic and Neumann Casimir effects in $2+1$ dimensions with imperfect mirrors
• Exact Casimir interaction of perfectly conducting three-spheres in four euclidean dimensions