Horizon geometry for Kerr black holes with synchronized hair

We study the horizon geometry of Kerr black holes (BHs) with scalar synchronized hair [1], 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; $v_H<v_H^{(\mathrm{S})}$ 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.

Figure 1

The sphericity $\mathfrak{s}$ (top solid blue curve) and the horizon linear velocity, $v_H$ (bottom dashed green curve) as a function of the dimensionless angular momentum, $j$, for the Kerr horizon. The vertical line gives the Smarr point dimensionless spin value, $j^{(\mathrm{S})}$.

Figure 2

(Left panel) Domain of existence of KBHsSH solutions with $m=1$ (dark blue and upper light blue shaded regions) in an $M$ versus ${\mathrm{\Omega }}_H$ diagram, both in units of the scalar field mass $\mu$. Three lines of constant ${\mathrm{\Omega }}_H$ solutions are displayed; these sequences will be used below. Also the line of solutions with $j=1$ is shown, which starts at the extremal vacuum Kerr. The bottom light blue shaded region corresponds to vacuum Kerr BHs which is limited by a dashed-dotted brown line corresponding to extremal Kerr BHs. (Right panel) Same plot showing the embeddable (dark blue) and not embeddable (intermediate blue) regions of KBHsSH and Kerr BHs delimited by the Smarr line (solid black line) and the Smarr point (pink cross—extended as the pink dotted line). The lightest blue region of the domain has not been examined in detail (being a numerically challenging region). Six specific solutions are highlighted, whose embedding diagrams are shown in Fig. 3. The remaining lines are the same lines as in the left panel.

Figure 3

Embedding diagrams, in ${\mathbb{E}}^3$, for the horizon of the six highlighted solutions in the right panel of Fig. 2. First (second) row, left to right: solutions 1,2,3 (4,5,6). Solutions 1-6 have parameters $(j;j_H;\mathfrak{s};v_H)$, respectively: (0.92; 6.27; 1.04; 0.09); (0.88; 5.53; 1.24; 0.29); (0.88; 4.92; 1.27; 0.33); (0.86; 1.69; 1.46; 0.65); (0.89; 1.16; 1.23; 0.48); (1.08; 1.64; 1.01; 0.08).

Figure 4

Domain of existence of KBHsSH solutions in a $\mathfrak{s}$ versus $r_H$ diagram. As before, the dark blue region corresponds to embeddable solutions, the intermediate blue region corresponds to nonembeddable solutions and the light blue region to the region we have not analyzed. Three lines of constant horizon angular velocity ${\mathrm{\Omega }}_H/\mu =\{0.75;0.80;0.90\}$ (dark green dot-dashed lines) are also exhibited. The remaining lines/points are the same as in Fig. 2.

Figure 5

Domain of existence of KBHsSH solutions in a $v_H$ versus $r_H$ diagram. All color coding is as in Fig. 4.

Figure 6

Horizon dimensionless angular momentum, $j_H$ versus the hairiness parameters $p$ (left panel) and $q$ (right panel). The color coding is the same as in previous figures.