Excited Kerr black holes with scalar hair

In the context of a complex scalar field coupled to Einstein gravity theory, we present a novel family of solutions of Kerr black holes with excited state scalar hair inspired by the work of Herdeiro and Radu in [Phys. Rev. Lett. 112, 221101 (2014)], which can be regarded as numerical solutions of rotating compact objects with excited scalar hair, including boson stars and black holes. In contrast to Kerr black holes with ground state scalar hair, we find that the first excited Kerr black holes with scalar hair have two types of nodes, including radial $n_r=1$ and angular $n_{\theta }=1$ nodes. Moreover, in the case of both nodes the curves of the mass versus the frequency form nontrivial loops. Furthermore, we also show the numerical results of the second excited states with even parity, and find that the curves can be divided into two kinds: closed and open loops. We also study the dependence of the horizon area on angular momentum and Hawking temperature in these excited states.

Figure 1

Two numerical solutions of the scalar field ${\varphi }_1$ as a function of $x$ and $\theta$ for azimuthal harmonic index $m=1$ and frequency $\omega =0.9$. The red dashed lines represent zero value.

Figure 2

The physical property of the rotating $n_r=1$ excited state of boson star with $m=1$, 2, 3. Left graph: the BS mass $M$ as a function of the frequency $\omega$. Right graph: the BS mass $M$ as a function of the angular momentum $J$.

Figure 3

The physical property of the $n_r=1$ excited state of Kerr black holes with $m=1$ scalar hair. Left graph: the BH mass $M$ as a function of the frequency $\omega$ with the event horizon $r_H=0.01$ (green), 0.05 (red) and 0.1 (blue), respectively. Right graph: the BH mass $M$ as a function of the angular momentum $J$ with the event horizon $r_H=0.01$.

Figure 4

The surface area $A_H$ of the $n_r=1$ excited state of Kerr black holes with $m=1$ scalar hair. Left graph: the surface area as a function of the angular momentum $J$ with the event horizon $r_H=0.01$ (black), 0.05 (red) and 0.1 (green), respectively. Right graph: the surface area as a function of the Hawking temperature $T_H$, the parameters of curves from top to bottom are $r_H=0.28$, $r_H=0.1$, $r_H=0.05$ and $r_H=0.01$.

Figure 5

The numerical solutions of the scalar field ${\varphi }_1$ and $W$ as a function of $x$ and $\theta$ with the frequency $\omega =0.805$ for boson star solution with azimuthal harmonic index $m=1$.

Figure 6

The physical properties of the $n_{\theta }=1$ excited state of rotating boson stars and KBHsSHs. Left graph: the BS mass $M$ as a function of the frequency $\omega$ with the azimuthal harmonic index $m=1$, 2, 3, respectively. Right graph: the BH mass $M$ as a function of the frequency $\omega$ with the event horizons $r_H=0.02$ (red), 0.05 (green), 0.1 (purple) and 0.19 (blue), respectively.

Figure 7

The spatial profiles of three typical numerical results for the scalar field ${\varphi }_2$ of rotating BSs with $m=1$ and $\omega =0.77$. In the bottom right panel, we show the distribution of the scalar field as a function of the $x$ coordinate in the equatorial plane at $\theta =\pi /2$.

Figure 8

The physical property of the rotating $n_{\theta }=1$ excited state of boson stars with $m=1$, 2, 3. Left panel: the BS mass $M$ as a function of the frequency $\omega$. Right panel: the BS mass $M$ as a function of the angular momentum $J$.

Figure 9

The physical property of the $n_{\theta }=1$ excited state of Kerr black holes with $m=1$ scalar hair. Left graph: the BH mass $M$ as a function of the frequency $\omega$ with the event horizon $r_H=0.001$, 0.003, 0.005, 0.05, 0.07 and 0.1, respectively. Right graph: the BH mass $M$ as a function of the angular momentum $J$ with the event horizon $r_H=0.05$.

Figure 10

The surface area $A_H$ of the second excited state of Kerr black holes with $m=1$ scalar hair. Left panel: the surface area as a function of the angular momentum $J$ with the event horizon $r_H=0.05$ (black), 0.07 (blue), and 0.1 (red), respectively. Right panel: the surface area as a function of the Hawking temperature $T_H$ with the event horizon $r_H=0.01$, 0.05, 0.07, and 0.1, respectively.