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Exact black hole solutions in the Einstein–Maxwell-scalar theory are constructed. They are the extensions of dilaton black holes in de Sitter or anti de Sitter universe. As a result, except for a scalar potential, a coupling function between the scalar field and the Maxwell invariant is present. Then the corresponding Smarr formula and the first law of thermodynamics are investigated.

We construct higher-dimensional and exact black holes in Einstein-Maxwell-scalar theory. The strategy we adopted is to extend the known, static and spherically symmetric black holes in the Einstein-Maxwell dilaton gravity and Einstein-Maxwell-scalar theory. Then we investigate the black hole thermodynamics. Concretely, the generalized Smarr formula and the first law of thermodynamics are derived.

We investigate a slowly rotating black hole solution in a novel Einstein–Maxwell-scalar theory, which is prompted by the classification of general Einstein–Maxwell-scalar theory. The gyromagnetic ratio of this black hole is calculated, and it increases as the second free parameter $$\beta $$ β increases, but decreases with the increasing parameter $$\gamma \equiv \frac{2 \alpha ^{2}}{1+\alpha ^2}$$ γ ≡ 2 α 2 1 + α 2 . In the Einstein–Maxwell-dilaton (EMD) theory, the parameter $$\beta $$ β vanishes but the free parameter $$\alpha $$ α governing the strength of the coupling between the dilaton and the Maxwell field remains. The gyromagnetic ratio is always less than 2, the well-known value for a Kerr–Newman (KN) black hole as well as for a Dirac electron. Scalar hairs reduce the magnetic dipole moment in dilaton theory, resulting in a drop in the gyromagnetic ratio. However, we find that the gyromagnetic ratio of two can be realized in this Einstein–Maxwell-scalar theory by increasing $$\beta $$ β and the charge-to-mass ratio Q/M simultaneously (recall that the gyromagnetic ratio of KN black holes is independent of Q/M). The same situation also applies to the angular velocity of a locally non-rotating observer. Moreover, we analyze the period correction for circular orbits in terms of charge-to-mass ratio, as well as the correction of the radius of the innermost stable circular orbits. It is found the correction increases with $$\beta $$ β but decreases with Q/M. Finally, the total radiative efficiency is investigated, and it can vanish once the effect of rotation is considered.

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By using the Taylor series method and the solution-generating method, we construct exact black hole solutions with minimally coupled scalar field. We find that the black hole solutions can have many hairs except for the physical mass. These hairs come from the scalar potential. Unlike the mass, there is no symmetry corresponding to these hairs, thus they are not conserved and one cannot understand them as Noether charges. They arise as coupling constants. Although there are many hairs, the black hole has only one horizon. The scalar potential becomes negative for sufficient large φ (or in the vicinity of black hole singularity). Therefore, the no-scalar-hair theorem does not apply to our solutions since the latter does not obey the dominant energy condition. Although the scalar potential becomes negative for sufficient large φ, the black holes are stable to both odd parity perturbations and scalar perturbations. As for even parity perturbations, we find there remains parameter space for the stability of the black holes. Finally, the black hole thermodynamics are developed.

The Bekenstein’s theorem allows us to generate an Einstein-conformal scalar solution from a single Einstein-ordinary scalar solution. In this article, we extend this theorem to the Einstein–Maxwell-scalar (EMS) theory with a non-minimal coupling between the scalar and Maxwell field when a scalar potential is also included. As applications of this extended theorem, the well-known static dilaton solution and rotating solution with a specific coupling between dilaton and Maxwell field are considered, and new conformal dilaton black hole solutions are found. The Noether charges, such as mass, electric charge, and angular momentum, are compared between the old and new black hole solutions connected by conformal transformations, and they are found conformally invariant. We speculate that the theorem may be useful in the computations of metric perturbations and spontaneous scalarization of black holes in the Einstein–Maxwell-conformal-scalar theory since they can be mapped to the corresponding EMS theories, which have been investigated in detail.

The Bekenstein's theorem allows us to generate a Einstein-conformal scalar solution from a single Einstein-ordinary scalar solution. In this article, we extend this theorem to Einstein-Maxwell-scalar (EMS) theory with a non-minimal coupling between the scalar and Maxwell field. As applications of this extended theorem, the well-known static dilaton solution and rotating solution with a specific coupling between dilaton and Maxwell field are considered, and new conformal dilaton black hole solutions are found. The Noether charges such as the mass, electric charge, angular momentum are compared between the old and new black hole solutions connected by conformal transformations, and they are found conformally invariant. We speculate that the theorem may be helpful in the computations of metric perturbations and spontaneous scalarization of black holes in the Einstein-Maxwell-conformal-scalar theory since they can be mapped to the corresponding EMS theories, which have been investigated in detail.