New look at AdS black holes with conformal scalar hair

Abstract

We revisit static, spherically symmetric solutions to AdS-Einstein gravity with a conformally coupled scalar field (and no self-interaction potential) in four dimensions. We first observe that a convenient choice of coordinates leads to a significant simplification of the field equations, which enables one to identify various roots of the indicial equations and thus distinct branches of solutions. Next, we construct an explicit 2-parameter hairy black hole solution in terms of an infinite power series around the event horizon. The black hole is non-extremal with a regular scalar field on and outside the event horizon, and it reduces to the Schwarzschild-AdS metric in the limit of vanishing hair. Its properties are illustrated for various values of the parameters and compared with previous numerical results by other authors. In addition, the analysis reveals the presence of a photon sphere and how the scalar field affects its size and the angular radius of the corresponding shadow. The thermodynamics of the solution is also briefly discussed.

24 pages, 8 figures, 2 tables. v2: a few new comments on the black hole shadow, table 2 expanded, minor changes in the text, typo in eq. (13) fixed, refs. added (main results unchanged). v3: title slightly changed to conform to the journal version, new footnote 10, energy-momentum tensor added in new appendix A, refs. added

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Cited by in corpus (2)

• Charged black holes in quadratic gravity
• Revisiting the static spherically symmetric solutions of gravity with a conformally coupled scalar field