Hairy extension of the Bertotti–Robinson spacetime in the Einstein–Maxwell-scalar theory is a black hole in closed spatial geometries

Abstract

Abstract In analogy with the Einstein–Maxwell–Dilaton black hole which is a hairy extension of the Reissner–Nordström black hole, we introduce a hairy extension for the Bertotti–Robinson spacetime in the context of the Einstein–Maxwell-scalar theory. The new spacetime is a closed S 3 black hole powered by a pure magnetic field. The energy–momentum tensor is an anisotropic fluid-like form such that p r/ρ and p ⊥/ρ are u (the radial coordinate)-dependent and at the horizon becomes − 1 3 and 1 3 , respectively. We use the concept of quasilocal mass and find the well-known Misner–Sharp mass to show that the new black hole satisfies the unified first law of the black hole.