On the Casimir interaction between two smoothly deformed cylindrical surfaces

TL;DR

This paper extends the derivative expansion method to analyze the Casimir interaction between smoothly deformed cylindrical surfaces, providing approximate formulas valid for small separations relative to curvature radii.

Contribution

The authors generalize the derivative expansion approach to cylindrical geometries and relate planar Casimir results to cylindrical cases for scalar fields with Dirichlet or Neumann conditions.

Findings

Derived approximate expressions for Casimir energy between cylinders.

Established the applicability limits of the derivative expansion.

Linked planar Casimir results to cylindrical geometries.

Abstract

We generalize the derivative expansion (DE) approach to the interaction between almost-flat smooth surfaces, to the case of surfaces which are optimally described in cylindrical coordinates. As in the original form of the DE, the obtained method does not depend on the nature of the interaction. We apply our results to the study of the static, zero-temperature Casimir effect between two cylindrical surfaces, obtaining approximate expressions which are reliable under the assumption that the distance between those surfaces is always much smaller than their local curvature radii. To obtain the zero-point energy, we apply known results about the thermal Casimir effect for a planar geometry. To that effect, we relate the time coordinate in the latter to the angular variable in the cylindrical case, as well as the temperature to the radius of the cylinders. We study the dependence of the applicability of the DE on the kind of interaction, considering the particular cases where Dirichlet or Neumann conditions are applied to a scalar field.