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

TL;DR

This paper constructs exact, topologically non-trivial solutions in the 4D Einstein-nonlinear sigma model, revealing new regular configurations like traversable wormholes and cylindrical spacetimes with quantized parameters.

Contribution

It introduces novel exact solutions in the Einstein-nonlinear sigma model with non-trivial topology, including a traversable wormhole and a cylindrical spacetime, reducing the system to solvable differential equations.

Findings

Existence of a traversable wormhole solution with negative cosmological constant.

A cylindrical solution with a Lorentzian squashed sphere section.

Quantization conditions derived from the Poschl-Teller solvable potential.

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.