The spinning Q-lump cosmic string — a straight, stationary string of the O(3) sigma model with a uniaxial anisotropy term, whose internal target-space rotation carries a Noether charge and generates an active gravitational mass — has been characterized gravitationally only in the weak-field limit of Einstein's equations and only for winding number N=2N=2 (E. Šimánek, Phys. Rev. D 78, 045014 (2008)). That leading-order analysis found a long-range Newtonian acceleration resembling a rotating massive cylindrical shell, Lense-Thirring-type frame dragging, a light-deflection angle 4πGϵ4\pi G\epsilon plus a frame-dragging correction δΦd\delta\Phi_d evaluated only by replacing the exact integrand with a simplified form, and evidence that g00g_{00} is not asymptotically flat (logarithmic growth at large distance). The open problem is the self-consistent, fully nonlinear solution of the coupled Einstein-sigma-model equations for the stationary Q-lump string at arbitrary admissible winding number N≥2N \ge 2 (finite-energy Q-lumps require N≥2N \ge 2; the anisotropy contribution to the energy per unit length diverges for N=1N=1); the resulting metric asymptotics; the exact frame-dragging contribution to light deflection; and the energy-loss budget and evolution of closed loops of spinning string, including the possible electromagnetic radiation channel if the sigma model is gauged with a Chern-Simons term.

Background

Cosmic strings are line-like topological defects that may have formed during early-universe symmetry-breaking phase transitions. In the simplest case, the Abelian Higgs model (a complex scalar field coupled to a gauge field, whose vacuum manifold is a circle), a straight static string has no Newtonian gravitational field outside its core: spacetime is locally flat but conical, with a deficit angle 4πGϵ4\pi G\epsilon set by the energy per unit length ϵ\epsilon, an effect that nonetheless double-images background light. A pure O(3) sigma-model string — built from a three-component field ϕ⃗\vec{\phi} mapping the transverse plane onto a two-sphere, classified by an integer winding number NN counting how many times the target sphere is wrapped — behaves similarly: its tension and energy density balance exactly, so its active (Newtonian) mass density vanishes (Comtet and Gibbons, Nucl. Phys. B 299, 719 (1988)).

Q-lumps, introduced by Leese (Nucl. Phys. B 366, 283 (1991)), are time-dependent solitons of the O(3) sigma model with an added potential term: the field precesses in internal space at a frequency α\alpha while its position and energy stay static, carrying a conserved Noether charge QQ associated with the global U(1) phase symmetry. Their core size λ\lambda is not fixed by the charge — for each charge there is a family of solutions — so λ\lambda is an intrinsic free parameter of the model. A Q-lump string is obtained by extending such a soliton trivially along a spatial axis, giving a straight, stationary string whose matter fields rotate in internal space.

Šimánek (Phys. Rev. D 78, 045014 (2008)) added a uniaxial anisotropy term (favoring one internal direction, analogous to the anisotropy energy of a ferromagnet) to the O(3) sigma-model string and studied the resulting spinning Q-lump string. Because the internal rotation makes the fields time dependent, the Lorentz-boost invariance along the string axis that had enforced the equality of energy density and tension disappears, and a finite active gravitational mass density appears. Working in the weak-field limit — linearizing Einstein's equations as gμν=ημν+hμνg_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} — and evaluating all integrals at winding N=2N=2, that work obtained: a linear active mass density m=ϵ0π(αλ)2/8m=\epsilon_0\pi(\alpha\lambda)^2/8, where ϵ0=4πη2\epsilon_0=4\pi\eta^2 is the pure sigma-model string tension and η\eta the symmetry-breaking scale; a frame-dragging (Lense-Thirring) angular velocity ω(r)≃4JG/r2\omega(r)\simeq 4JG/r^2 with angular momentum per unit length J3≃π2αη2λ2J_3\simeq\pi^2\alpha\eta^2\lambda^2, so that the string gravitates like a rotating cylindrical mass shell of radius of order the string width λ\lambda; a light-deflection angle δΦ≃4πGϵ\delta\Phi\simeq 4\pi G\epsilon (enhanced by roughly 40% for αλ∼1\alpha\lambda\sim 1) plus a drag-induced correction δΦd\delta\Phi_d that was estimated only by replacing the exact integrand with the simplified form (π/2)(r2+πλ2/2)−1(\pi/2)(r^2+\pi\lambda^2/2)^{-1}; and a weak-field metric perturbation h00h_{00} growing like log⁡(r/λ)\log(r/\lambda) at large distance, meaning g00g_{00} is not asymptotically flat. The cosmological motivation is the Vilenkin scenario, in which strings seed density perturbations: galaxy formation requires Gm∼10−6Gm\sim 10^{-6}, which for the Q-lump string amounts to αλ∼1\alpha\lambda\sim 1, and closed oscillating loops must lose energy dominantly by gravitational radiation for the seeding scenario to work.

Closely related stationary self-gravitating strings have been solved beyond the weak-field regime numerically: Verbin and Larsen (Phys. Rev. D 70, 085004 (2004)) treated nonsingular stationary global strings — a nonlinear sigma model with a potential for a self-gravitating complex scalar whose oscillatory phase dependence mimics a gauged Kaluza-Klein symmetry — confirming the logarithmic growth of h00h_{00} and the asymptotic flatness of the gravito-magnetic metric component; Brihaye and Delsate (arXiv:1306.4011) and Brihaye and Verbin (Phys. Rev. D 95, 044027 (2017)) constructed static and stationary gravitating Q-tubes (scalar and Proca fields with harmonic time dependence) in full cylindrical gravity. None of these treats the O(3) sigma-model Q-lump string with internal target-space rotation, its frame-dragging and lensing observables, its higher-winding generalizations, or its loops — which is precisely the open territory left by the source paper.

Problem Statement

Construct the gravitational field of the straight, stationary, internally spinning Q-lump cosmic string beyond the weak-field approximation and beyond winding number N=2N=2, by solving the coupled nonlinear Einstein-O(3)-sigma-model equations (with the uniaxial anisotropy term of the source paper) self-consistently for the metric and matter fields. The common objective is a self-consistent account of the gravitational field and astrophysical behavior of internally spinning cosmic strings, of which the source's weak-field, N=2N=2 analysis is the leading approximation; the requested results jointly support this objective as follows. (i) Metric and asymptotics: determine the metric components and their large-distance behavior at general winding number N≥2N \ge 2 (the finite-energy admissibility condition of the source: the anisotropy contribution to the energy per unit length diverges for N=1N=1), internal rotation frequency α\alpha, and core size λ\lambda — in particular whether g00g_{00} is genuinely non-asymptotically flat (logarithmically growing) in the full nonlinear theory, and how the weak-field quantitative results (active mass density, shell-like frame dragging) are modified. (ii) Observables: compute the frame-dragging angular velocity and the light-deflection angle from the self-consistent metric, including an exact evaluation of the drag-induced deflection δΦd\delta\Phi_d whose weak-field estimate replaced the integrand by (π/2)(r2+πλ2/2)−1(\pi/2)(r^2+\pi\lambda^2/2)^{-1}. (iii) Loops: determine the energy-loss budget and evolution of closed loops of spinning Q-lump string — whether gravitational radiation indeed dominates, as argued from the anisotropy-induced boson mass α\alpha suppressing δϕ⃗\delta\vec{\phi} emission — and, as an explicitly secondary branch, whether in the Chern-Simons-gauged extension of the sigma model, where the strings can be electrically charged, electromagnetic radiation competes with gravitational radiation in a way that would alter loop lifetimes and their density-fluctuation signature. Because the Q-lump core size is a free parameter of the model (a family of solutions per Noether charge, known since Leese 1991), answers must be given over the (N,α,λ)(N,\alpha,\lambda) parameter space or with λ\lambda fixed by an explicitly stated additional physical input, not at a single silently chosen point.

References

  1. E. Šimánek, "Gravitational field of a spinning sigma-model cosmic string" (arXiv preprint titled "Gravitational Field of a Spinning Cosmic String"), Phys. Rev. D 78, 045014 (2008). DOI: 10.1103/PhysRevD.78.045014; arXiv:0803.0151. https://arxiv.org/abs/0803.0151
  2. R. A. Leese, "Q-Lumps and their interactions", Nucl. Phys. B 366, 283 (1991). DOI: 10.1016/0550-3213(91)90004-H. https://doi.org/10.1016/0550-3213(91)90004-H
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  4. Y. Verbin and A. L. Larsen, "Non-Singular Stationary Global Strings", Phys. Rev. D 70, 085004 (2004). DOI: 10.1103/PhysRevD.70.085004; arXiv:hep-th/0407034. https://arxiv.org/abs/hep-th/0407034
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  6. Y. Brihaye and T. Delsate, "Gravitating Q-tubes and cylindrical spacetime", arXiv:1306.4011 (2013). https://arxiv.org/abs/1306.4011
  7. Y. Brihaye and Y. Verbin, "Proca Q Tubes and their Coupling to Gravity", Phys. Rev. D 95, 044027 (2017). DOI: 10.1103/PhysRevD.95.044027; arXiv:1611.01803. https://arxiv.org/abs/1611.01803
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