Dynamical Formation of Kerr Black Holes with Synchronized Hair: An Analytic Model

East and Pretorius have successfully evolved, using fully nonlinear numerical simulations, the superradiant instability of the Kerr black hole (BH) triggered by a massive, complex vector field. Evolutions terminate in stationary states of a vector field condensate synchronized with a rotating BH horizon. We show that these end points are fundamental states of Kerr BHs with synchronized Proca hair. Motivated by the “experimental data” from these simulations, we suggest a universal (i.e., field-spin independent), analytic model for the subset of BHs with synchronized hair that possess a quasi-Kerr horizon, applicable in the weak hair regime. Comparing this model with fully nonlinear numerical solutions of BHs with a synchronized scalar or Proca hair, we show that the model is accurate for hairy BHs that may emerge dynamically from superradiance, whose domain we identify.

Figure 1

Part of the domain of existence of Kerr BHs with scalar (main panel) or Proca (inset) hair (shaded blue region) in a $M$ vs ${\mathrm{\Omega }}_H$ diagram, both in units of $\mu$. The Kerr limit—existence line ($p=0$, blue-dotted line)—and the solitonic limit ($p=1$, red-solid line) are shown, together with the line $p=0.3$ and the lines ${\omega }_H=0.3;0.5$, in both panels.

Figure 2

The relative errors are shown in the allowed strip on the ($p$, $w_H$)-plane for the reduced angular momentum $j$ (left panels), area $a_H$ (middle panels), and temperature $t_H$ (right panels) for both BHs with scalar ($\mathbf{S}$—top panels) and Proca ($\mathbf{P}$—bottom panels) hair. The red lines are the set of solutions with $j=1$; solutions to the left of these lines have $j<1$.

Figure 3

$j$ vs $a_H$ diagram. The black stars are Proca numerical solutions; the red points are for scalar hairy BHs. The curves correspond to the analytical model for fixed ${\omega }_H$ values.

Figure 4

Normalized irreducible mass, horizon mass, and angular momentum: numerical solutions (points) vs the analytic model (curves). Here we have taken $j=0.9$ (in [1] the authors took $j=0.99$) to show the universality of the agreement.

Figure 5

(Main panel) Domain of existence of fundamental states of KBHSPH (shaded-blue region). Vacuum Kerr BHs exist below the black-solid line (corresponding to extremal Kerr). The horizontal-dotted lines show the migration trajectories from vacuum Kerr BHs (black dots) to hairy BHs (red triangles) of the evolutions in [1]. The inset shows constant $j$ lines for both vacuum Kerr (dashed) and hairy BHs (solid), which always meet at the existence line, for any $j<1$. Migration of a Kerr BH with spin $j$ terminates when the horizontal line (constant $M$) meets a hairy BH with that $j$ value. See [9] for an earlier discussion of adiabatic evolutions, in the scalar case, based on linear theory.