Bounds on length scales of classical spacetime foam models

Abstract

Simple models of a classical spacetime foam are considered, which consist of identical static defects embedded in Minkowski spacetime. Plane-wave solutions of the vacuum Maxwell equations with appropriate boundary conditions at the defect surfaces are obtained in the long-wavelength limit. The corresponding dispersion relations \omega^2=\omega^2(\vec{k}) are calculated, in particular, the coefficients of the quadratic and quartic terms in \vec{k}. Astronomical observations of gamma-ray bursts and ultra-high-energy cosmic rays then place bounds on the coefficients of the dispersion relations and, thereby, on particular combinations of the fundamental length scales of the static spacetime-foam models considered. Spacetime foam models with a single length scale are excluded, even models with a length scale close to the Planck length (as long as a classical spacetime remains relevant).