Infinitely Many Strings in De Sitter Spacetime: Expanding and Oscillating Elliptic Function Solutions

TL;DR

This paper provides a detailed analysis of circular string evolution in 2+1 dimensional de Sitter spacetime using elliptic functions, revealing that a single world-sheet can describe infinitely many distinct strings with unique energies and motions.

Contribution

It introduces a novel insight that one world-sheet can represent infinitely many strings in de Sitter spacetime, a feature absent in flat spacetime, and classifies string behaviors based on energy parameters.

Findings

Strings exhibit oscillatory, hyperbolic, or unbounded motion depending on energy parameter.

Energy behavior differs for expanding and oscillating strings, with explicit formulas derived.

A single world-sheet encodes infinitely many strings with distinct physical properties.

Abstract

The exact general evolution of circular strings in $2+1$ dimensional de Sitter spacetime is described closely and completely in terms of elliptic functions. The evolution depends on a constant parameter $b$, related to the string energy, and falls into three classes depending on whether $b<1/4$ (oscillatory motion), $b=1/4$ (degenerated, hyperbolic motion) or $b>1/4$ (unbounded motion). The novel feature here is that one single world-sheet generically describes {\it infinitely many} (different and independent) strings. The world-sheet time $\tau$ is an infinite-valued function of the string physical time, each branch yields a different string. This has no analogue in flat spacetime. We compute the string energy $E$ as a function of the string proper size $S$, and analyze it for the expanding and oscillating strings. For expanding strings $(\dot{S}>0)$: $E\neq 0$ even at $S=0$, $E$ decreases for small $S$ and increases $\propto\hspace*{-1mm}S$ for large $S$. For an oscillating string $(0\leq S\leq S_{max})$, the average energy $<E>$ over one oscillation period is expressed as a function of $S_{max}$ as a complete elliptic integral of the third kind.