Horizon geometry for Kerr black holes with synchronised hair

Abstract

We study the horizon geometry of Kerr black holes (BHs) with scalar synchronised hair, a family of solutions of the Einstein-Klein-Gordon system that continuously connects to vacuum Kerr BHs. We identify the region in parameter space wherein a global isometric embedding in Euclidean 3-space, ${\mathbb{E}}^3$, is possible for the horizon geometry of the hairy BHs. For the Kerr case, such embedding is possible iff the horizon dimensionless spin $j_H$ (which equals the total dimensionless spin, $j$), the sphericity $\mathfrak{s}$ and the horizon linear velocity $v_H$ are smaller than critical values, $j^{(\mathrm{S})},{\mathfrak{s}}^{(\mathrm{S})},v_H^{(\mathrm{S})}$, respectively. For the hairy BHs, we find that $j_H<j^{(\mathrm{S})}$ is a sufficient, but not necessary, condition for being embeddable; $v<v_H^{(\mathrm{S})}$ is a necessary, but not sufficient, condition for being embeddable; whereas $\mathfrak{s}<{\mathfrak{s}}^{(\mathrm{S})}$ is a necessary and sufficient condition for being embeddable in ${\mathbb{E}}^3$. Thus the latter quantity provides the most faithful diagnosis for the existence of an ${\mathbb{E}}^3$ embedding within the whole family of solutions. We also observe that sufficiently hairy BHs are always embeddable, even if $j$ -- which for hairy BHs (unlike Kerr BHs) differs from $j_H$ --, is larger than unity.

arXiv comment12 pages, 6 figures

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