On the existence of conformally coupled scalar field hair for black holes in (anti-)de Sitter space

TL;DR

This work investigates whether black holes can support hair from a conformally coupled scalar field in spacetimes with a cosmological constant, i.e., asymptotically (A)dS. By applying a conformal transformation to map the non-minimally coupled system to a minimally coupled one with potential U(Φ), the authors establish no-hair in de Sitter space (Λ>0) and construct stable hairy black holes in anti-de Sitter space (Λ<0) when the scalar mass satisfies μ^2 ≤ -2Λ/3. Numerical AdS solutions exhibit monotone scalar profiles decaying as φ ∼ r^{-k} with k = 3/2 − sqrt(1/4 − 3 μ^2/Λ), and a linear-stability analysis against spherical perturbations confirms stability via a positive perturbation potential outside the horizon. The results highlight that a negative cosmological constant enables stable hair for a conformally coupled scalar, in contrast to the dS no-hair result, and connect hairy AdS black holes to the minimally coupled system through the conformal map.

Abstract

The Einstein-conformally coupled scalar field system is studied in the presence of a cosmological constant. We consider a massless or massive scalar field with no additional self-interaction, and spherically symmetric black hole geometries. When the cosmological constant is positive, no scalar hair can exist and the only solution is the Schwarzschild-de Sitter black hole. When the cosmological constant is negative, stable scalar field hair exists provided the mass of the scalar field is not too large.

Figures (11)

• Figure 1: (a) Graph of $U(\Phi )/\Lambda$ (vertical axis) against $\Phi$ (horizontal axis) for positive $\Lambda$ and vanishing mass. For a massive conformally coupled scalar field and $\Lambda >0$, the potential has the same overall shape. (b) Graph of $U(\Phi )/|\Lambda |$ (vertical axis) against $\Phi$ (horizontal axis) for negative $\Lambda$ and vanishing mass.
• Figure 2: Graphs of $U(\Phi )/|\Lambda |$ (vertical axis in each case) against $\Phi$ (horizontal axis in each case) for negative $\Lambda$ and various values of $q=-1+\frac{3\mu ^{2}}{|\Lambda |}$. For $q\le 0$, the potential is negative everywhere as in the massless case (graph (a)). For $0< q < 1$ (graphs (b) and (c)), the potential is negative for small $\Phi$ but becomes positive for sufficiently large $\Phi$. If $q\ge 1$ (graph (d)), then the potential is positive for all $\Phi$.
• Figure 3: Graph of $\Phi$ (vertical axis) against ${\bar{r}}$ (horizontal axis) for $\mu ^{2}=0$, $-\Lambda /3$ and $-\Lambda /2$, with $\Lambda =-0.1$, ${\bar{r}}_{h}=1$ and $\Phi ({\bar{r}}_{h})=1$. For all values of the mass, $\Phi$ has no zeros and tends to zero at infinity.
• Figure 4: Graph of ${\@fontswitch{}{\mathcal{}} {A}}$ (given by (\ref{['eq:bonuscond']})) (vertical axis) against ${\bar{r}}$ (horizontal axis) for $\mu ^{2}=0$, $-\Lambda /3$ and $-\Lambda /2$, with the other parameters as in figure \ref{['fig:Phi']}. The graph shows clearly that this quantity never vanishes, so that condition (\ref{['eq:bonuscond']}) holds and the conformal transformation remains valid for these solutions. Similar results are found for other values of the parameters.
• Figure 5: Graph of $\phi$ (vertical axis) against $r$ (horizontal axis), for the parameter values given previously. The scalar field function is monotonic for all values of the mass.