All static and electrically charged solutions with Einstein base manifold in the arbitrary-dimensional Einstein-Maxwell system with a massless scalar field

TL;DR

This paper classifies all static, electrically charged solutions with a massless scalar field in higher-dimensional Einstein-Maxwell systems, revealing multiple asymptotically flat solutions and their geometric properties.

Contribution

It provides a complete classification of static solutions with a scalar field in arbitrary dimensions, including explicit forms and geometric analysis.

Findings

Nine solutions for dimensions n≥4 with non-constant scalar fields.

Three solutions for three-dimensional cases.

Asymptotically flat solutions are not unique for n≥4.

Abstract

We present a simple and complete classification of static solutions in the Einstein-Maxwell system with a massless scalar field in arbitrary $n(\ge 3)$ dimensions. We consider spacetimes which correspond to a warped product $M^2 \times K^{n-2}$, where $K^{n-2}$ is a $(n-2)$-dimensional Einstein space. The scalar field is assumed to depend only on the radial coordinate and the electromagnetic field is purely electric. Suitable Ans\"atze enable us to integrate the field equations in a general form and express the solutions in terms of elementary functions. The classification with a non-constant real scalar field consists of nine solutions for $n\ge 4$ and three solutions for $n=3$. A complete geometric analysis of the solutions is presented and the global mass and electric charge are determined for asymptotically flat configurations. There are two remarkable features for the solutions with $n\ge 4$: (i) Unlike the case with a vanishing electromagnetic field or constant scalar field, asymptotically flat solution is not unique, and (ii) The solutions can asymptotically approach the Bertotti-Robinson spacetime depending on the integrations constants. In accordance with the no-hair theorem, none of the solutions are endowed of a Killing horizon.