Topological Reverberations in Flat Space-times

TL;DR

This paper investigates how multiply-connected flat space-times influence the energy evolution of radiating systems, revealing a topological reverberation effect characterized by energy oscillations and growth due to space-time topology.

Contribution

It derives an exact radiation reaction equation for systems in multiply-connected flat space-times and demonstrates the topological reverberation effect on energy evolution.

Findings

Energy exhibits reverberation patterns with minima and maxima.

Multiply-connected topology induces energy oscillations and growth.

Explicit solutions for R^2 x S^1 topology are provided.

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 M_4 whose t=const spacelike sections M_3 are compact in at least one spatial direction. The radiation reaction equation of the radiating source is derived for the case where M_3 has any non-trivial flat topology, and an exact solution is obtained. We also show that when the spacelike sections are multiply-connected flat 3-manifolds the energy E(t) exhibits a reverberation pattern with discontinuities in the derivative of E(t) and a set of relative minima and maxima, followed by a growth of E(t). 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 our radiating system is topologically fragile. An explicit solution of the radiation reaction equation for the case where M_3 = R^2 x S^1 is discussed, and graphs which reveal how the energy varies with the time are presented and analyzed.