Nonequatorial scalar rings supported by magnetized Schwarzschild-Melvin black holes

Abstract

It has recently been demonstrated that magnetized black holes in composed Einstein-Maxwell-scalar-Gauss-Bonnet field theories with a nonminimal negative coupling of the scalar field to the Gauss-Bonnet curvature invariant may support spatially regular scalar hairy configurations. In particular, it has been revealed that, for Schwarzschild-Melvin black-hole spacetimes, the onset of the near-horizon spontaneous scalarization phenomenon is marked by the numerically computed dimensionless critical relation $(BM{)}_{\mathrm{crit}}\ensuremath{\simeq}0.971$, where ${M,B}$ are respectively the mass and the magnetic field of the spacetime. In the present paper we prove, using analytical techniques, that the boundary between bald Schwarzschild-Melvin black-hole spacetimes and hairy (scalarized) black-hole solutions of the composed Einstein-Maxwell-scalar-Gauss-Bonnet theory is characterized by the exact dimensionless relation $(BM{)}_{\mathrm{crit}}=\sqrt{\frac{\sqrt{6}\ensuremath{-}2}{2\sqrt{6}}+\sqrt{\frac{\sqrt{6}\ensuremath{-}1}{2}}}$ for the critical magnetic strength. Intriguingly, we prove that the critical dimensionless magnetic parameter $(BM{)}_{\mathrm{crit}}$ corresponds to magnetized black holes that support a pair of linearized nonminimally coupled thin scalar rings that are characterized by the nonequatorial polar angular relation $({\mathrm{sin}}^{2}\ensuremath{\theta}{)}_{\mathrm{scalar}\ensuremath{-}\mathrm{ring}}=\frac{690\ensuremath{-}72\sqrt{6}+4\sqrt{3258\sqrt{6}\ensuremath{-}7158}}{789}<1$. It is also proved that the classically allowed angular region for the negative-coupling near-horizon spontaneous scalarization phenomenon of magnetized Schwarzschild-Melvin spacetimes is restricted to the black-hole poles, ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{\text{scalar}}\ensuremath{\rightarrow}0$, in the asymptotic large-strength magnetic regime $BM\ensuremath{\gg}1$.