How do spherical black holes grow monopole hair?

Black holes in certain modified gravity theories that contain a scalar field coupled to curvature invariants are known to possess (monopole) scalar hair while non-black-hole spacetimes (like neutron stars) do not. Therefore, as a neutron star collapses to a black hole, scalar hair must grow until it settles to the stationary black hole solution with (monopole) hair. In this paper, we study this process in detail and show that the growth of scalar hair is tied to the appearance and growth of the event horizon (before an apparent horizon forms), which forces scalar modes that would otherwise (in the future) become divergent to be radiated away. We prove this result rigorously in general first for a large class of modified theories, and then we exemplify the results by studying the temporal evolution of the scalar field in scalar Gauss-Bonnet gravity in two backgrounds: (i) a collapsing Oppenheimer-Snyder background, and (ii) a collapsing neutron star background. In case (i), we find an exact scalar field solution analytically, while in case (ii) we solve for the temporal evolution of the scalar field numerically, with both cases supporting the conclusion presented above. Our results suggest that the emission of a burst of scalar field radiation is a necessary condition for black hole formation in a large class of modified theories of gravity.

Figure 1

Penrose diagram for static and spherically symmetric BHs for theories whose action is described by Eq. (1). We assume that the conformal structure of BHs in these theories is similar to the conformal structure of static BHs in general relativity. Note that we only show the part of the conformal diagram which is causally connected to the external Universe. In the figure, ${\mathcal{H}}^{\pm }$ denotes the future and the past event horizons, ${\mathcal{I}}^{\pm }$ denotes the future and the past null infinity, $i^{\pm }$ denote the future and past timelike infinities and $i^0$ denotes the point at spatial infinity.

Figure 2

Left: conformal diagram for a spherically symmetric collapse of matter into a Schwarzschild BH in GR. In this figure, the shaded region is the world tube of the stellar interior, ${\mathrm{\Sigma }}_1$ denotes a Cauchy surface on which we prescribe initial data. ${\mathrm{\Sigma }}_2$ denotes another hypersurface to the future of ${\mathrm{\Sigma }}_1$ used in the proof of Theorem 1. Theorem 1 in Sec. 3 shows that solutions to Eq. (21) is regular on the region of spacetime to the future of $\mathrm{\Sigma }$, on and exterior to the event horizon $\mathcal{H}$. Right: shows a noncompactified version of the figure on the left.

Figure 3

Left: conformal diagram of a collapsing spacetime. ${\mathrm{\Sigma }}_{\text{initial}}$ and ${\mathrm{\Sigma }}_{\text{final}}$ are spacelike hypersurfaces draw prior to and after the collapse, respectively. We assume that spacetime is static around ${\mathrm{\Sigma }}_{\text{initial}}$ and ${\mathrm{\Sigma }}_{\text{final}}$, describing the initially static star which collapses and settles to a final static BH. Right: the spacetime volume $\mathcal{V}=[t_0,t_0+T]\times {\mathrm{\Sigma }}_{\text{initial}}$ is shown in the figure. The boundary $\partial \mathcal{V}$ consists of three regions, $\{t_0+T\}\times {\mathrm{\Sigma }}_{\text{initial}}$, $\{t_0\}\times {\mathrm{\Sigma }}_{\text{initial}}$, and $[t_0,t_0+T]\times \{r=\infty \}$. The normals to $\{t_0+T\}\times {\mathrm{\Sigma }}_{\text{initial}}$ and $\{t_0\}\times {\mathrm{\Sigma }}_{\text{initial}}$ point in opposite directions.

Figure 4

Penrose diagram of Oppenheimer-Snyder spacetime.

Figure 5

The surface of the star, EH, and the null surfaces on which we prescribe initial data is shown in OS collapse. The shaded region in the above figure shows the spacetime region to the future of the null surfaces $u=u_0$ and $v=v_0$ up to the EH and inside the star. The EH forms at $t=-9/2M$ and the surface of the star crosses the EH at $t=-4/3M$, after which the star is causally disconnected from external observers.

Figure 6

Simulated spacetime diagram of a collapsing NS in perturbative SGB gravity. Shown are the surface (white), the event horizon (EH, blue) and the apparent horizon (AH, red). The light blue line refers to null geodesic, and the black dot to the time of peak flux. The time $t$ and radii $r$ shown both refer to individual coordinates. The panels show (left) the value of the scalar field $\mathrm{\Phi }$ relative to its final configuration ${\mathrm{\Phi }}_{\text{final}}$ [Eq. (83)] and (right) the scalar energy flux $S_{\mathrm{\Phi }}^r$.