Kerr black holes with scalar hair

TL;DR

A family of solutions of Einstein's gravity minimally coupled to a complex, massive scalar field, describing asymptotically flat, spinning black holes with scalar hair and a regular horizon is presented.

Abstract

We present a family of solutions of Einstein's gravity minimally coupled to a complex, massive scalar field, describing asymptotically flat, spinning black holes with scalar hair and a regular horizon. These hairy black holes (HBHs) are supported by rotation and have no static limit. Besides mass M and angular momentum J, they carry a conserved, continuous Noether charge Q measuring the scalar hair. HBHs branch off from the Kerr metric at the threshold of the superradiant instability and reduce to spinning boson stars in the limit of vanishing horizon area. They overlap with Kerr black holes for a set of (M,J) values. A single Killing vector field preserves the solutions, tangent to the null geodesic generators of the event horizon. HBHs can exhibit sharp physical differences when compared to the Kerr solution, such as J/M^2>1, quadrupole moment larger than J^2/M and larger orbital angular velocity at the innermost stable circular orbit. Families of HBHs connected to the Kerr geometry should exist in scalar (and other) models with more general self interactions.

Figures (5)

• Figure 1: $M$ vs $\Omega_H$ for Kerr BHs. The black solid curve corresponds to extremal BHs, which obey $M=\frac{1}{2\Omega_H}$; Kerr BHs exist below it (shaded region). A scalar cloud with parameters $(n,\ell,m)$ exists along a line. Five such (dotted blue) lines are shown, for $n=0$, $\ell=m$ and different $m$'s. (Inset) $R_{11}(r)$, from $r=r_H$, and normalized such that $R_{11}(r_H)=1$ for two clouds with $m=1$. The corresponding points in the $m=1$ existence (blue) line are shown with the same colors.
• Figure 2: Domain of existence of HBHs for $m=1$ in $M$-$w$ space (shaded blue region). The black solid line and the dotted blue lines are the same as in Fig. \ref{['linear']}. (Inset) The boson star lines for $m=1,2$.
• Figure 3: Domain of existence of HBHs for $m=1$ in $M$-$J$ space. The color code is the same as in Figs. \ref{['linear']}-\ref{['non-linear']}. (Inset) Area as a function of $J$ along constant $M$ curves. Solid green (dashed black) curves correspond to Kerr (HBHs). For the same $M$ they bifurcate in the (dotted blue) Kerr line.
• Figure 4: Ratio of the quadrupole moment of HBHs to that of a Kerr solution with the same $M,J$. Several lines of constant $\Omega_H$ (dashed black) and $q$ (c.f. caption) are displayed.
• Figure 5: Orbital frequency for counter- (bottom curves) and co-rotating (top curves) orbits at the ISCO for Kerr ($q=0$) and HBHs ($q>0$). The color code is the same as in Fig. \ref{['quadrupole']}.