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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.

european physical journal. c, particles and fields

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