Parallel waves in Einstein-non linear sigma models

TL;DR

This paper explores Einstein-nonlinear sigma models with different target manifolds, revealing smooth cosmic string solutions and novel parallel wave spacetimes with interesting topological and causal properties.

Contribution

It introduces new smooth cosmic string solutions with topological charges and extends stationary solutions to parallel waves with non-planar wavefronts in Einstein-nonlinear sigma models.

Findings

Cosmic string solutions are smooth and free of singularities.

Wavefronts are classified as sub-quadratic, ensuring causal well-behaved spacetimes.

Models exhibit non-vanishing baryon current and distinctive geometric features.

Abstract

We study a family of solutions of Einstein-non linear sigma models with $S^2$ and $SU(2) \sim S^3$ target manifolds. In the $S^2$ case, the solutions are smooth everywhere, free of conical singularities, and approach asymptotically the metric of a cosmic string, with a mass per length that is proportional to the absolute value of the winding number from topological spheres onto the target $S^2$. This gives an interesting example of a relation between a mass and a topological charge. The case with target $SU(2)$ generalizes the stationary solution found in Eur. Phys. J. C (2021) 81:55 to parallel waves with a non-planar wavefront $\mathcal{W}$. We prove that these $\mathcal{W}$-fronted parallel waves are sub-quadratic in the classification in Class. Quant. Grav. \textbf{20} (2003) 2275, and thus causally well behaved. These spacetimes have a non-vanishing baryon current and their geometry has many striking features.