TOPOLOGICAL REVERBERATIONS IN FLAT SPACE–TIMES

Abstract

We study the role played by multiply-connectedness in the time evolution of the energy E(t) of a radiating system that lies in static flat space–time manifolds ℳ4 whose t= const spacelike sections ℳ3 are compact in at least one spatial direction. The radiation reaction equation of the radiating source is derived for the case where ℳ3 has any nontrivial flat topology, and an exact solution is obtained. We show that the behavior of the radiating energy E(t) changes remarkably from exponential damping, when the system lies in ℛ3, to a reverberation pattern (with discontinuities in the derivative Ė(t) and a set of relative minima and maxima) followed by a growth of E(t), when ℳ3 is endowed with any one of the 17 multiply-connected flat topologies. It emerges from this result that the compactness in at least one spatial direction of Minkowski space–time is sufficient to induce this type of topological reverberation, making clear that topological fragilities can arise not only in the usual cosmological modelling, but also in ordinary flat space–time manifolds. An explicit solution of the radiation reaction equation for the case where [Formula: see text] is discussed in detail, and graphs which reveal how the energy varies with the time are presented and analyzed.