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

Hideki Maeda; Cristián Martínez

doi:10.1140/epjc/s10052-018-6334-7

EPJ

a
b
c
d
e
ap
st
h
plus
ds
pv
ti
qt
am
n

2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2018) 78: 860
https://doi.org/10.1140/epjc/s10052-018-6334-7

Regular Article - Theoretical Physics

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

Hideki Maeda¹ and Cristián Martínez²^*

¹ Department of Electronics and Information Engineering, Hokkai-Gakuen University, Sapporo, 062-8605, Japan
² Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia, Chile

^* e-mail: martinez@cecs.cl

Received: 6 August 2018
Accepted: 12 October 2018
Published online: 25 October 2018

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 Ansätze 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.