The Effect of Spatial Curvature on the Classical and Quantum Strings

TL;DR

This paper investigates how spatial curvature influences classical and quantum string dynamics, providing exact solutions, analyzing physical properties, and exploring self-consistent spacetime configurations with different curvature signs.

Contribution

It offers the first complete solutions for circular string motion in curved spacetimes and examines their back-reaction, quantization, and spectrum in various curvature scenarios.

Findings

Self-consistent solutions exist only for positively curved (closed) universes.

Mass spectra of strings depend on curvature, with distinct behaviors for positive and negative curvature.

Exact solutions for string fluctuations and stationary states are derived and analyzed.

Abstract

We study the effects of the spatial curvature on the classical and quantum string dynamics. We find the general solution of the circular string motion in static Robertson-Walker spacetimes with closed or open sections. This is given closely and completely in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. We find the {\it back-reaction} effect of these strings on the spacetime: the self-consistent solution to the Einstein equations is a spatially closed $(K>0)$ spacetime with a selected value of the curvature index $K$ (the scale f* is normalized to unity). No self-consistent solutions with $K\leq 0$ exist. We semi-classically quantize the circular strings and find the mass $m$ in each case. For $K>0,$ the very massive strings, oscillating on the full hypersphere, have $m^2\sim K n^2\;\;(n\in N_0)$ {\it independent} of $\alpha'$ and the level spacing {\it grows} with $n,$ while the strings oscillating on one hemisphere (without crossing the equator) have $m^2\alpha'\sim n$ and a {\it finite} number of states $N\sim 1/(K\alpha').$ For $K<0,$ there are infinitely many string states with masses $m\log m\sim n,$ that is, the level spacing grows {\it slower} than $n.$ The stationary string solutions as well as the generic string fluctuations around the center of mass are also found and analyzed in closed form.