Exact \textit{pp}-waves, (A)dS waves and Kundt spaces in the Abelian-Higgs model

TL;DR

This paper presents new exact solutions in the Abelian-Higgs model coupled with General Relativity, describing various wave and Kundt space geometries with superconducting currents, and analyzes their electromagnetic confinement properties.

Contribution

It introduces novel exact solutions involving extit{pp}-waves, AdS waves, and Kundt spaces with superconducting currents, linking geometry to scalar and electromagnetic fields.

Findings

Solutions include extit{pp}-waves, AdS waves, and Kundt spaces with aligned fields.

The geometry of orthogonal surfaces in Kundt spaces is governed by the Liouville equation.

Electromagnetic fields can be confined within bounded regions due to gravitational and scalar interactions.

Abstract

We find new exact solutions of the Abelian-Higgs model coupled to General Relativity, characterized by a non-vanishing superconducting current. The solutions correspond to \textit{pp}-waves, AdS waves, and Kundt spaces, for which both the Maxwell field and the gradient of the phase of the scalar are aligned with the null direction defining these spaces. In the Kundt family, the geometry of the two-dimensional surfaces orthogonal to the superconducting current is determined by the solutions of the two-dimensional Liouville equation, and in consequence, these surfaces are of constant curvature, as it occurs in a vacuum. The solution to the Liouville equation also acts as a potential for the Maxwell field, which we integrate into a closed-form. Using these results, we show that the combined effects of the gravitational and scalar interactions can confine the electromagnetic field within a bounded region in the surfaces transverse to the current.