Topologically non-trivial configurations in the 4d Einstein--nonlinear $\sigma$-model system

Abstract

We construct exact, regular and topologically non-trivial\ configurations of the coupled Einstein-nonlinear sigma model in (3+1) dimensions. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and circumvents Derrick's theorem because it depends explicitly on time, but in such a way that its energy-momentum tensor is compatible with a stationary metric. Moreover, the $SU(2)$ configuration cannot be continuously deformed to the trivial Pion vacuum as it possesses a non-trivial winding number. We reduce the full coupled 4D Einstein nonlinear sigma model system to a single second order ordinary differential equation. When the cosmological constant vanishes, such master equation can be further reduced to an Abel equation. Two interesting regular solutions correspond to a stationary traversable wormhole (whose only \textquotedblleft exotic matter" is a negative cosmological constant) and a (3+1)-dimensional cylinder whose (2+1)-dimensional section is a Lorentzian squashed sphere. The Klein-Gordon equation in these two families of spacetimes can be solved in terms of special functions. The angular equation gives rise to the Jacobi polynomials while the radial equation belongs to the Poschl-Teller family. The solvability of the Poschl-Teller problem implies non-trivial quantization conditions on the parameters of the theory.

arXiv comment14 pages, to appear in Physical Review D

References in corpus (21)

• Misner String Entropy
• Nut Charge, Anti-de Sitter Space and Entropy
• Analytic self-gravitating Skyrmions, cosmological bounces and AdS wormholes
• Entropy of Rotating Misner String Spacetimes
• Rehabilitating space-times with NUTs
• Lie - point and variational symmetries in minisuperspace Einstein's gravity
• Hedgehog ansatz and its generalization for self-gravitating Skyrmions
• General Analytic Solutions of Scalar Field Cosmology with Arbitrary Potential
• NUT wormholes
• Non-linear superposition law and Skyrme crystals
• Exact multi-soliton solutions in the four dimensional Skyrme model
• The geometric origin of Lie point symmetries of the Schrödinger and the Klein Gordon equations
• Generalized Hedgehog ansatz and Gribov copies in regions with non trivial topologies
• SU(N) Multi-Skyrmions at Finite Volume
• Cosmological Einstein-Skyrme solutions with non-vanishing topological charge
• Multi-Skyrmions with orientational moduli
• Causality and the Skyrme Model
• Hypersurface homogeneous locally rotationally symmetric spacetimes admitting conformal symmetries
• Constraining Monopoles by Topology: an Autonomous System
• $SU(N)$ BPS Monopoles in ${\mathcal{M}}^2\times S^2$
• SU(2) Cosmological Solitons

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