Scale-invariant Casimir cavities: exact constancy of the proximity-force parameter for equiangular spirals

Abstract

We identify a class of Casimir cavities in which the dimensionless parameter controlling the proximity-force approximation (PFA) is not merely small but exactly constant along the entire boundary. For a cavity bounded by two equiangular (logarithmic) spirals of common pitch b, related by a rotation, the ratio of local perpendicular gap to local radius of curvature is independent of position, and we prove that the equiangular spiral is the only plane curve with this property. Integrating the PFA pressure yields a closed-form Casimir force per transverse height, apparently the first such result for a continuously self-similar gap geometry. Because the geometry is scale-free, the accuracy of the PFA cannot be improved by shrinking a gap, so we calibrate the leading correction directly by a boundary-element computation. Exploiting translational invariance, the electromagnetic mode sum separates into TE and TM sectors; in the TE sector we obtain Q_TE = 0.682, giving Q_EM in [0.68, 0.84]. This settles a discrepancy between two incompatible estimates obtained by extrapolating small-gap expansions beyond their validity. The golden pitch is treated as a worked example and shown explicitly NOT to be a physical optimum.