Self-gravitating black hole scalar wigs

It has long been known that no static, spherically symmetric, asymptotically flat Klein-Gordon scalar field configuration surrounding a nonrotating black hole can exist in general relativity. In a series of previous papers, we proved that, at the effective level, this no-hair theorem can be circumvented by relaxing the staticity assumption: for appropriate model parameters, there are quasibound scalar field configurations living on a fixed Schwarzschild background which, although not being strictly static, have a larger lifetime than the age of the universe. This situation arises when the mass of the scalar field distribution is much smaller than the black hole mass, and following the analogies with the hair in the literature we dubbed these long-lived field configurations wigs. Here we extend our previous work to include the gravitational backreaction produced by the scalar wigs. We derive new approximate solutions of the spherically symmetric Einstein-Klein-Gordon system which represent self-gravitating scalar wigs surrounding black holes. These configurations interpolate between boson star configurations and Schwarzschild black holes dressed with the long-lived scalar test field distributions discussed in previous papers. Nonlinear numerical evolutions of initial data sets extracted from our approximate solutions support the validity of our approach. Arbitrarily large lifetimes are still possible, although for the parameter space that we analyze in this paper they seem to decay faster than the quasibound states. Finally, we speculate about the possibility that these configurations could describe the innermost regions of dark matter halos.

Figure 1

The wave function $\psi (r)$ of a static boson star in the ground state configuration, with $\mu =0.1$, ${\psi }_c=0.0113$ and $\omega =0.0990$.

Figure 2

Left column: The real and imaginary parts of the wave function $\psi (r)$ describing a quasibound state in the ground state configuration, with $M_{\mathrm{BH}}=1$, $\mu =0.1$ and $s=-0.0000153092+0.09945i$, in Schwarzschild coordinates. Right column: Same as in left column but in iEF coordinates. Note that the use of iEF coordinates moves the divergence of the wave function from the horizon to the spacetime singularity.

Figure 3

The real and imaginary parts of the wave function $\psi (r)$, the Misner-Sharp mass $m(r)$, and the lapse function $\alpha (r)$, for a black hole scalar wig with $A=0.0354$, $M_{\mathrm{BH}}=1$, and $\mu =0.1$. Here the inner boundary has been chosen at $r_1=2.1M_{\mathrm{BH}}$. See the fourth line in Table 1 for more details about this self-gravitating object.

Figure 4

The total to black hole mass ratio, $M_{\mathrm{T}}/M_{\mathrm{BH}}$, of a black hole scalar wig of $M_{\mathrm{BH}}=1$ and $\mu =0.1$ as function of the field amplitude $A$.

Figure 5

The energy density $\rho$ as defined in Eq. (26) for different configurations of same field amplitude $A$. Note that the black hole scalar wig appears more compact than the quasibound state. At large radii the presence of the central black hole is not relevant and the black hole scalar wig and the static boson star approach each other.

Figure 6

The different limits of a black hole scalar wig. If the mass of the black hole tends to zero, static boson stars are recovered. As long as the field amplitude remains small, the quasibound states are a good approximation.

Figure 7

Real and imaginary parts of $s=\sigma +i\omega$, as well as of the wave function $\psi (r)$, as a function of the ratio, $M_{\mathrm{T}}/M_{\mathrm{BH}}$, for a family of black hole scalar wigs characterized by $\mu =0.1$, $A=5.66\times {10}^{-3}$. Notice that a quasibound state and a static boson star are recovered in the limits $M_{\mathrm{T}}/M_{\mathrm{BH}}\to 1$ and $M_{\mathrm{T}}/M_{\mathrm{BH}}\to \infty$, respectively.

Figure 8

Real and imaginary parts of the scalar field $\mathrm{\Phi }$ as a function of $r$ at fixed iEF time $t_{\mathrm{EF}}=0$.

Figure 9

Metric coefficients for $t_{\mathrm{f}}\sim 1800$ as a function of $r$. The inset is a zoom onto the region in which most of the scalar field density is concentrated.

Figure 10

Maximum (minimum) value of the metric coefficient $a$ ($b$). The inset shows the behavior at earlier times. At $t=0$ the spacetime is conformally flat and then it evolves towards a quasistationary state in the $1+\log$ slice.

Figure 11

Evolution of the scalar field mass for models $A_1$ and $A_2$.

Figure 12

Same plot as in Fig. 7 for the magnitude of the decay rate $|\sigma |$ as a function of the total to black hole mass ratio, now in a logarithmic scale. This plot makes the power-law behavior for $M_{\mathrm{T}}/M_{\mathrm{BH}}\gtrsim 3$ of this function evident.