Comparison of spacetime defects which are homeomorphic but not diffeomorphic

Abstract

Certain remnants of a quantum spacetime foam can be modeled by a distribution of defects embedded in a flat classical spacetime. The presence of such spacetime defects affects the propagation of elementary particles. In this article, we show explicitly that both topology and differential structure of the defects are important for the particle motion. Specifically, we consider three types of spacetime defects which are described by the same topological manifold $\mathbb{R}\times\big(\mathbb{R}P^3-\{\text{point}\}\big)$ but which are not diffeomorphic to each other. We investigate the propagation of a massless scalar field over the three different manifolds and find different solutions of the \mbox{Klein--Gordon} equation.