Abstract
We construct electrically charged Kerr black holes (BHs) with scalar hair. Firstly, we take an uncharged scalar field, interacting with the electromagnetic field only indirectly, via the background metric. The corresponding family of solutions, dubbed Kerr-Newman BHs with ungauged scalar hair, reduces to (a sub-family of) Kerr-Newman BHs in the limit of vanishing scalar hair and to uncharged rotating boson stars in the limit of vanishing horizon. It adds one extra parameter to the uncharged solutions: the total electric charge. This leading electromagnetic multipole moment is unaffected by the scalar hair and can be computed by using Gauss's law on any closed 2-surface surrounding (a spatial section of) the event horizon. By contrast, the first sub-leading electromagnetic multipole -- the magnetic dipole moment --, gets suppressed by the scalar hair, such that the gyromagnetic ratio is always smaller than the Kerr-Newman value ($g=2$). Secondly, we consider a gauged scalar field and obtain a family of Kerr-Newman BHs with gauged scalar hair. The electrically charged scalar field now stores a part of the total electric charge, which can only be computed by applying Gauss' law at spatial infinity and introduces a new solitonic limit -- electrically charged rotating boson stars. In both cases, we analyse some physical properties of the solutions.
Highlights
The Kerr metric [1] is the fundamental black holes (BHs) solution in General Relativity, believed to describe an untold number of BHs in equilibrium in the Cosmos
The family of solutions – Kerr-Newman BHs with ungauged scalar hair (KNBHsUSH) – is described by four continuous parameters: (1) the ADM mass, M, which can be split into the horizon and exterior matter/energy contribution, M = MH + M Ψ + M EM; (2) the total angular momentum, J, which can be split in a similar fashion, J = JH + JΨ + JEM; (3) the Noether charge, Q, associated to the global U (1) invariance of the complex scalar field, which obeys Q = JΨ/m, where m is the azimuthal winding number; (4) and the total electric charge, QE
KNBHsUSH are obtained using the metric, scalar field and electromagnetic potential ansatz given by ds2 = −e2F0 N dt2 + e2F1 dr2 + r2dθ2 + e2F2 r2 sin2 θ2, (4)
Summary
The Kerr metric [1] is the fundamental BH solution in General Relativity, believed to describe an untold number of BHs in (or near) equilibrium in the Cosmos. We start by considering an ungauged ( electrically uncharged) scalar field In this case, the family of solutions – Kerr-Newman BHs with ungauged scalar hair (KNBHsUSH) – is described by four continuous parameters (with one non-trivial constraint between them): (1) the ADM mass, M , which can be split into the horizon and exterior matter/energy contribution (composed of the scalar Ψ plus electromagnetic fields), M = MH + M Ψ + M EM; (2) the total angular momentum, J, which can be split in a similar fashion, J = JH + JΨ + JEM; (3) the Noether charge, Q, associated to the global U (1) invariance of the complex scalar field, which obeys Q = JΨ/m, where m is the azimuthal winding number; (4) and the total electric charge, QE.
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