Infinitely many strings in de Sitter spacetime: Expanding and oscillating elliptic function solutions

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 infinitely many (different and independent) strings. The world-sheet time τ 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 (Ṡ > 0): E = 0 even at S = 0, E decreases for small S and increases ∝ S for large S. For an oscillating string (0 ≤ S ≤ 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.For each b, the two independent solutions S + and S − are analyzed. For b < 1/4, all the strings of the S − solution are unstable (S max = ∞) and never collapse to a point (S min = 0). S + describes one stable (S max is bounded) oscillating string and < E > is an increasing function of b for 0 ≤ b ≤ 1/4. For b > 1/4, all strings (for both S + and S − ) are unstable and have a collapse during their evolution. For b = 1/4, S − describes two strings (one stable and one unstable for large de Sitter radius), while S + describes one stable nonoscillating string.2

Back Reaction of Strings in Self Consistent String Cosmology

“…In addition, exact circular string solutions were found in terms of elliptic functions [4]. All these solutions describe one string, several strings or even an infinite number of different and independent strings.…”

New classes of exact multistring solutions in curved spacetimes

“…[1][2][3]. In curved spacetimes, it is generally impossible to obtain the exact and complete solution to the string equations of motion and constraints.…”

Dynamics of toroidal spiral strings around five-dimensional black holes

“…Signature of F h µν is Euclidean for a timelike ξ, and is Lorentzian for a spacelike ξ. There are a lot of works dealing with the cohomogeneity-one strings in a variety of context [21,22,23,24,25].…”

“…The metric (24) provides the scheme of the Hopf fibration of S 3 , where the projection along the integral curves of the Killing vector field ∂ ψ E defines a map from S 3 to S 2 . The first term in the metric (24) is the metric on the S 2 base space, and the second term is the metric on the S 1 fiber of the Hopf fibration of S 3 .…”