We study the leading long-distance attractive force between two holes in a plate arising from a scalar eld with Dirichlet boundary conditions on the plate. We use a formalism in which the interaction is governed by a non-local eld theory which lives on the two holes. The interaction energy is proportional to Q1Q2=r7 at large separation r, where Q1 and Q2 are certain charges associated with the holes. We compute these charges for round and rectangular holes. We show that the 1=r7 behavior is universal for separations large compared to the linear dimensions of the holes, irrespective of the spin or interactions of the bosonic eld. We also study the interaction between two long thin slits, for which the energy falls o as 1=r 6.
Kabat, D., Karabali, D., & Nair, V. P. (2010). Casimir interaction between holes in a plate. Physical Review D - Particles, Fields, Gravitation and Cosmology, 82(2) DOI: 10.1103/PhysRevD.82.025014