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

We construct exact, regular and topologically nontrivial 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 nontrivial winding number. We reduce the full coupled four-dimensional Einstein nonlinear sigma model system to a single second order ordinary differential equation. When the cosmological constant vanishes, such a master equation can be further reduced to an Abel equation. Two interesting regular solutions correspond to a stationary traversable wormhole (whose only “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 nontrivial quantization conditions on the parameters of the theory.