Abstract
We numerically solve the equations of motion (EOM) for two models of circular cosmic string loops with windings in a simply connected internal space. Since the windings cannot be topologically stabilized, stability must be achieved (if at all) dynamically. As toy models for realistic compactifications, we consider windings on a small section of $\mathbb{R}^2$, which is valid as an approximation to any simply connected internal manifold if the winding radius is sufficiently small, and windings on an $S^2$ of constant radius $\mathcal{R}$. We then use Kosambi-Cartan-Chern (KCC) theory to analyze the Jacobi stability of the string equations and determine bounds on the physical parameters that ensure dynamical stability of the windings. We find that, for the same initial conditions, the curvature and topology of the internal space have nontrivial effects on the microscopic behavior of the string in the higher dimensions, but that the macroscopic behavior is remarkably insensitive to the details of the motion in the compact space. This suggests that higher-dimensional signatures may be extremely difficult to detect in the effective $(3+1)$-dimensional dynamics of strings compactified on an internal space, even if configurations with nontrivial windings persist over long time periods.
Highlights
One very powerful approach for studying the properties of systems of differential equations using geometrical methods is Kosambi–Cartan–Chern (KCC) theory, which was initiated in the pioneering works of Kosambi [10], Cartan [11]and Chern [12], respectively
In the geometrical description of dynamical systems proposed by KCC theory, two geometric quantities are associated to each system of second order differential equations: a non-linear connection, and a Berwald type connection, respectively. (These are discussed in detail in Sect. 4.) With the use of these two connections, five geometric invariants of the system are constructed
The KCC theory introduces a geometric description of the time evolution of two-dimensional dynamical systems, described mathematically by second order differential equations, in which the solution curves are described by analogy with the theory of geodesics in a Finsler space
Summary
One very powerful approach for studying the properties of systems of differential equations using geometrical methods is Kosambi–Cartan–Chern (KCC) theory, which was initiated in the pioneering works of Kosambi [10], Cartan [11]. The (3 + 1)-dimensional loop radius exhibits an extremely uniform oscillatory signature, with a single oscillation period which is determined by these parameters It makes very little difference how rapidly the string oscillates in the internal space, or whether its oscillations contain a single or multiple periodicities – in all cases, the effect on the macroscopic string dynamics remains negligible over very large time scales. Perhaps the most important difference between the R2 and S2 examples considered here is that the oscillations in the compact space remain regular (do not increase with time) in the former, but grow increasingly rapidly in the latter They remain small over long time periods, compared to the oscillations of the loop in the Minkowski directions, and have a negligible effect on the macroscopic string dynamics, except via their contribution to. For the models considered in this work, we choose background coordinates, an embedding, and a world-sheet parameterization that ensures this correspondence holds, so that the string EOM are direct expressions of energy-momentum conservation
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