Introduction

The “no-hair” theorems are powerful tools in studying black hole solutions of Einstein
gravity coupled with matter. These “no-hair” theorems describe the existence and stability of four-dimensional asymptotically flat black holes coupled to an electromagnetic
field or in vacuum. In the case of a minimally coupled scalar field in asymptotically flat
spacetime the “no-hair” theorems were proven imposing conditions on the form of the selfinteraction potential [1]. These theorems were also generalized to non-minimally coupled
scalar fields [2].

For asymptotically flat spacetime, a four-dimensional black hole coupled to a scalar field
with a zero self-interaction potential is known [3]. However, the scalar field diverges on
the event horizon and, furthermore, the solution is unstable [4], so there is no violation of
the “no-hair” theorems. In the case of a positive cosmological constant with a minimally
coupled scalar field with a self-interaction potential, black hole solutions were found in [5]
and also a numerical solution was presented in [6], but it was unstable. If the scalar field
is non-minimally coupled, a solution exists with a quartic self-interaction potential [7], but
it was shown to be unstable [8, 9].

In the case of a negative cosmological constant, stable solutions were found numerically
for spherical geometries [10, 11] and an exact solution in asymptotically AdS space with
hyperbolic geometry was presented in [12] and generalized later to include charge [13] and
further generalized to non-conformal solutions [14]. This solution is perturbatively stable
for negative mass and may develop instabilities for positive mass [15]. The thermodynamics
of this solution was studied in [12] where it was shown that there is a second order phase
transition of the hairy black hole to a pure topological black hole without hair. The
analytical and numerical calculation of the quasi-normal modes of scalar, electromagnetic
and tensor perturbations of these black holes confirmed this behaviour [16]. Recently, a new
exact solution of a charged C-metric conformally coupled to a scalar field was presented
in [17, 18]. A Schwarzschild-AdS black hole in five-dimensions coupled to a scalar field was
discussed in [19], while dilatonic black hole solutions with a Gauss-Bonnet term in various
dimensions were discussed in [20].

Recently scalar-tensor theories with nonminimal couplings between derivatives of a
scalar field and curvature were studied. The most general gravity Lagrangian linear in
the curvature scalar R, quadratic in the scalar field φ, and containing terms with four
derivatives was considered in [21]. It was shown that this theory cannot be recast into
Einsteinian form by a conformal rescaling. It was further shown that without considering
any effective potential, an effective cosmological constant and then an inflationary phase
can be generated.

Subsequently it was found [22] that the equation of motion for the scalar field can be
reduced to a second-order differential equation when the scalar field is kinetically coupled
to the Einstein tensor. Then the cosmological evolution of the scalar field coupled to the
Einstein tensor was considered and it was shown that the universe at early stages has
a quasi-de Sitter behaviour corresponding to a cosmological constant proportional to the
inverse of the coupling of the scalar field to the Einstein tensor. These properties of the
derivative coupling of the scalar field to curvature had triggered the interest of the study of the cosmological implications of this new type of scalar-tensor theory [23, 24, 25, 26].

Also local black hole solutions were discussed in [27].
The dynamical evolution of a scalar field coupled to the Einstein tensor in the background of a Reissner-Nordstr¨om black hole was studied in [28]. By calculating the quasinormal spectrum of scalar perturbations it was found that for weak coupling of the scalar

field to the Einstein tensor and for small angular momentum the effective potential outside
the horizon of the black hole is always positive indicating that the background black hole is
stable for a weaker coupling. However, for higher angular momentum and as the coupling
constant gets larger than a critical value, the effective potential develops a negative gap
near the black hole horizon indicating an instability of the black hole background.
The previous discussion indicates that the presence of the derivative coupling of a scalar
field to the Einstein tensor on cosmological or black hole backgrounds generates an effect
similar to the presence of an effective cosmological constant. In this paper we investigate
this effect further. We consider a spherically symmetric Reissner-Nordstr¨om black hole
and perturb this background by introducing a derivative coupling of a scalar field to the
Einstein tensor. We show that in this gravitating system there exists a critical temperature
in which the system undergoes a second-order phase transition to an anisotropic hairy black
hole configuration and the scalar field is regular on the horizon. This “Einstein hair” is the
result of evading the no-hair theorem thanks to the presence of the derivative coupling of
the scalar field to the Einstein tensor.

The coupled dynamical system of Einstein-Maxwell-Klein-Gordon equations is a highly
non-linear system of equations for which even a numerical solution appears beyond reach.
To solve the field equations, we expand the fields around a Reissner-Nordstr¨om black hole
solution and pertubatively determine the critical temperature. By solving the first-order
equations numerically near the critical temperature, we study the behaviour of the hairy
black hole solution. We calculate the temperature of the new hairy black hole and compare
it with the corresponding temperature of a Reissner-Nordstr¨om black hole of the same
charge. We find that above the critical temperature the Reissner-Nordstr¨om black hole
is unstable and by calculating the free energies we show that the new hairy black hole
configuration is energetically favorable over the corresponding Reissner-Nordstr¨om black
hole.

The paper is organized as follows. In Sec. 2 we set up the field equations and outline our
solution. In Sec. 3 we find the zeroth order solution and calculate the critical temperature
near which the Reissner-Nordstr¨om black hole may become unstable and develop hair. In
Sec. 4 we find the first-order solutions to the system of Einstein-Maxwell-Klein-Gordon
equations which are hairy black holes near the critical temperature. In Sec. 5 we discuss
the thermodynamic stability of our first-order hairy solution. In Sec. 6 we discuss the
validitiy of our perturbative expansion. Finally, in Sec. 7 we conclude.