Abstract
Black hole spontaneous scalarization has been attracting more and more attention as it circumvents the well-known no-hair theorems. In this work, we study the scalarization in Einstein–scalar-Gauss–Bonnet theory with a probe scalar field in a black hole background with different curvatures. We first probe the signal of black hole scalarization with positive curvature in different spacetimes. The scalar field in AdS spacetime could be formed easier than that in flat case. Then, we investigate the scalar field around AdS black holes with negative and zero curvatures. Comparing with negative and zero cases, the scalar field near AdS black hole with positive curvature could be much easier to emerge. And in negative curvature case, the scalar field is the most difficult to be bounded near the horizon.
Highlights
Besides GR solutions with a trivial scalar field configuration, the scalarized hairy solutions for black holes and stars could exist, which evades the no-hair theorems [35,36,37]. It was shown in [38] that below a certain mass the Schwarzschild black hole background may become unstable in regions of strong curvature, and when the scalar field backreacts to the metric, a scalarized hairy black hole emerges and it is physically favorable
The studies of spontaneous scalarization in the existed literatures were focused on the black hole with spherical horizon, but as known in AdS spacetime, one can have AdS black holes with different topologies and it is interesting to study how these topologies of the black hole horizon affect the scalarization procedure
We find that the scalar hair around the black hole with spherical horizon is easier to form in AdS spacetime than in flat case
Summary
Where is the cosmological constant, λ is the coupling constant between Gauss–Bonnet term and scalar field with the dimension of length and ∇μ denotes the covariant derivative. When the scalar field vanishes, i.e., = 0, the gravity theory admits black hole solutions, whose metric takes the general form, ds. In order to explore the (in)stability of the background black holes, we consider a small fluctuation in the form δ = e−iωt φ(r )Y (x1, x2) where Y (x1, x2) satisfies Laplace– Beltrami equation 2Y (x1, x2) = − AY (x1, x2). We shall rewrite = ±3/L2 where L is the curvature AdS radius As it was discussed in [61,72] because the scalar field is coupled to GB invariant, the effective mass of the scalar field receives a contribution from this coupling. For various values of the coupling, the effective mass of the scalar field becomes tachyonic outside the background black hole and the fluctuations are unstable destabilizing in this way the background metric
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