Non-Minimally Coupled Horndeski Black Hole

• Non-minimally coupled Horndeski black holes are solutions in scalar-tensor gravity where the scalar field couples to curvature via derivative interactions, preserving second-order field equations.
• They exhibit a rich structure with both static and rotating configurations, where scalar hair arises from a vanishing radial shift current and mimics an effective cosmological constant.
• Key investigations reveal modified thermodynamics, stability properties, and observational signatures that set these black holes apart from standard general-relativistic solutions.

Searching arXiv for recent and foundational papers on non-minimally coupled Horndeski black holes. A non-minimally coupled Horndeski black hole is a black-hole solution in a Horndeski scalar-tensor theory in which the scalar field is coupled to curvature through derivative interactions rather than only through a canonical kinetic term. In the literature most directly associated with this designation, the defining interaction is the Einstein-tensor kinetic coupling, schematically $G^{\mu\nu}\nabla_\mu\phi\nabla_\nu\phi$, often combined with the ordinary kinetic term and, in some cases, a cosmological constant or Maxwell field. This sector is shift symmetric and retains second-order field equations. It supports several exact black-hole branches, including static and rotating BTZ geometries with scalar hair in three dimensions, asymptotically AdS and asymptotically flat four-dimensional solutions, topological charged black holes, and related planar higher-dimensional geometries. Across these constructions, the scalar hair is typically sustained either by the vanishing of the radial component of the conserved scalar current or by closely related constraints, and the resulting thermodynamics, perturbative stability, and observational signatures differ in model-dependent ways from their general-relativistic counterparts (Bravo-Gaete et al., 2014).

1. Horndeski sector and defining non-minimal coupling

The class most commonly discussed under this heading is a shift-symmetric Horndeski subsector in which Einstein gravity is coupled to a scalar through both the standard kinetic term and a derivative coupling to the Einstein tensor. In three dimensions one representative action is

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$

with scalar Lagrangian

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$

The same structure is generalized to arbitrary dimension by

$$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$

This is a truncation of full Horndeski because one keeps only the Einstein-Hilbert term, the cosmological constant, the canonical kinetic term, and the particular derivative coupling $G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$, while discarding the rest of the general Horndeski functions. The defining property is that the field equations remain second order (Bravo-Gaete et al., 2014).

A closely related four-dimensional realization is

$$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$

which was one of the earliest exact black-hole constructions in this sector. There is no explicit cosmological constant in that action, yet the solution behaves asymptotically like Schwarzschild–AdS for $z>0$, so the derivative coupling generates an effective negative cosmological constant scale (Rinaldi, 2012).

The same derivative-coupling structure also appears in Einstein–Maxwell–Horndeski models,

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$

and in the topological charged solutions

$$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$

In all of these cases, “non-minimal coupling” refers specifically to the derivative coupling of the scalar kinetic structure to curvature, rather than to a Brans–Dicke-type coupling of the scalar itself (Cisterna et al., 2014).

2. Field equations, conserved current, and the current-vanishing mechanism

Varying the Einstein-tensor-coupled action yields Einstein equations of the form

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$

together with the scalar equation

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$0

Because of shift symmetry, the scalar equation is a current conservation law,

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$1

For a static radial scalar, the key component is

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$2

The central simplifying condition imposed in several exact constructions is

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$3

equivalent to $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$4 without forcing $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$5. Physically, it means the scalar hair is supported in such a way that there is no radial flux of the shift current through the geometry (Bravo-Gaete et al., 2014).

This current-based mechanism is also central to the older four-dimensional solution with coupling $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$6. There the reduced equations admit a radial shift current

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$7

and regularity at a black-hole horizon requires the current to vanish there, hence everywhere, so black-hole solutions exist only in the $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$8 branch. This is one of the clearest ways the derivative-coupled Horndeski sector navigates around standard no-hair arguments: the theory has shift symmetry and a conserved current, but regularity kills the corresponding charge (Rinaldi, 2012).

In shift-symmetric, reflection-symmetric Horndeski models with linearly time-dependent scalar hair,

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$9

the same general structure persists, but the stability properties are more subtle. For the Einstein-tensor coupling model

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$0

the near-horizon odd-parity stability coefficients satisfy

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$1

when $$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$2 approaches a finite nonzero value at the horizon. In that sector, linearly time-dependent scalar hair is therefore generically unstable near the horizon, whereas static scalar-hair branches can satisfy nontrivial perturbative stability inequalities in restricted parameter regions (Tretyakova et al., 2017).

3. Exact black-hole geometries and scalar-hair structure

In three dimensions, imposing $$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$3 on the static ansatz

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$4

forces

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$5

and, away from the degenerate sector $$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$6, yields

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$7

With

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$8

the metric becomes exactly the static BTZ geometry,

$$\mathcal L_\phi=-\frac12\alpha\,(\nabla\phi)^2+\frac12\eta\,G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi.$$9

The scalar profile is nontrivial,

$$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$0

and

$$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$1

Reality requires

$$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$2

with the endpoint $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$3 reducing to the scalarless BTZ solution (Bravo-Gaete et al., 2014).

A notable feature of that branch is that although $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$4 diverges like $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$5 near the horizon, the integrated scalar field is well-defined there, and on shell the scalar stress tensor behaves as an effective cosmological constant shift: $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$6 Consequently the metric solves the ordinary BTZ Einstein equations with AdS radius $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$7 even though the scalar is nontrivial (Bravo-Gaete et al., 2014).

The same paper extends the mechanism to arbitrary dimension with planar horizon,

$$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$8

and again imposing $$S=\int d^Dx\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\nabla^\nu\phi\right].$$9 gives the exact planar Schwarzschild–AdS black hole

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$0

with coupling relation

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$1

The scalar remains nontrivial and again sources the geometry as an effective cosmological constant on shell (Bravo-Gaete et al., 2014).

In four dimensions, the solution

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$2

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$3

with

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$4

is asymptotically Schwarzschild–AdS-like even though no bare cosmological constant is present. For $G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$5, $G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$6 has one zero only, so there is a single event horizon (Rinaldi, 2012).

Charged topological and spherical solutions in Einstein–Maxwell–Horndeski theory add Coulombic and higher-order charge terms, as well as characteristic $G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$7 contributions to the metric and gauge potential. The four-dimensional spherical branch has

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$8

with asymptotically locally AdS behavior set by

$G_{\mu\nu}\nabla^\mu\phi\nabla^\nu\phi$9

and a physically relevant branch obeying $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$0 and a scalar-reality bound involving $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$1, $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$2, $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$3, $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$4, and the horizon radius (Cisterna et al., 2014).

4. Thermodynamics and horizon mechanics

The three-dimensional BTZ-based solution admits a reduced Euclidean action

$$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$5

whose last Euler–Lagrange equation is precisely the constraint implementing $$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$6. On shell,

$$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$7

so that

$$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$8

Both mass and entropy are rescaled relative to ordinary BTZ by the same overall factor, while the Hawking temperature remains the BTZ one,

$$L=\frac{m_p^2}{2}R-\frac12\left(g^{\mu\nu}-\frac{z}{m_p^2}G^{\mu\nu}\right)\partial_\mu\varphi\,\partial_\nu\varphi,$$9

The first law

$z>0$0

and the usual three-dimensional Smarr formula

$z>0$1

follow from the reduced action and its scaling symmetry (Bravo-Gaete et al., 2014).

In higher dimensions, the planar Schwarzschild–AdS branch has

$z>0$2

$z>0$3

again differing from the general-relativistic values by a universal overall factor and obeying

$z>0$4

(Bravo-Gaete et al., 2014).

For the four-dimensional Einstein-tensor-coupled black hole with coupling $z>0$5, the Euclidean temperature is

$z>0$6

The normalized Euclidean volume, energy, entropy, and heat capacity are modified in a nontrivial way,

$z>0$7

$z>0$8

$z>0$9

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$0

with $$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$1. The paper emphasizes that entropy is not generically the area law and that there is a Hawking–Page-like phase structure (Rinaldi, 2012).

More generally, black-hole thermodynamics in Horndeski theories requires care because the standard Wald entropy formula may not be directly applicable in the presence of higher-derivative interactions and nonminimal derivative couplings. Using the original Iyer–Wald formulation, one obtains the entropy differential and total mass variation directly from the conservation of the Hamiltonian. In shift-symmetric theories, the paper shows that for static scalar hair the black-hole entropy follows the ordinary area law even in the presence of a nontrivial scalar profile, whereas for linearly time-dependent scalar hair the entropy also depends on the profile of the scalar field (Minamitsuji et al., 2023).

5. Stability, perturbations, and dynamical behavior

Odd-parity perturbations of black holes in the non-minimal derivative coupling sector can be reduced to a master equation with background-dependent functions

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$2

With a suitable $$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$3-deformation, the deformed potential becomes

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$4

and for the relevant exact hairy solutions with real scalar profile one has $$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$5. This establishes mode stability under linear odd-parity perturbations for that class of black holes. In the same setup, slowly rotating solutions satisfy

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$6

so the exterior frame-dragging profile coincides with the general-relativistic one in the slow-rotation limit (Cisterna et al., 2015).

A different exact spherically symmetric derivative-coupled solution, often associated with Rinaldi, has

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$7

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$8

Its asymptotics are effectively AdS-like with

$$I=\int \sqrt{-g} \,\text{d}^4 x \left[(R-2\Lambda)-\frac12(\alpha g_{\mu\nu}-\eta G_{\mu\nu})\nabla^{\mu}\phi \nabla^{\nu}\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$9

even though there is no cosmological constant in the action. Axial gravitational perturbations satisfy a Regge–Wheeler-like equation with a modified propagation factor $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$0, so the theory predicts a modified propagation speed for gravitational waves on this background. The solution is linearly stable under axial gravitational perturbations over the parameter space studied. The ringdown depends primarily on $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$1 and transitions between a photon-sphere-dominated regime, an intermediate echoing regime, and a rapidly depleted ringing regime with an exponential tail (Chatzifotis et al., 2021).

Quasinormal modes of asymptotically AdS black holes in scalar-tensor theories with non-minimal derivative coupling have also been studied on fixed Horndeski black-hole backgrounds of the form

$$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$2

Away from the special point $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$3 in units $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$4, the geometry differs from Schwarzschild–AdS through both $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$5 and the extra $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$6 term in $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$7. For minimally coupled test scalars, increasing the derivative coupling $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$8 at fixed horizon radius lowers both the oscillation frequency and damping rate relative to general relativity, and for large black holes the approximate scaling is $$I[g_{\mu\nu},\phi] =\int d^n x\,\sqrt{-g}\left[ \kappa (R-2\Lambda) -\frac12\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\mu\phi\nabla_\nu\phi -\frac14 F_{\mu\nu}F^{\mu\nu} \right].$$9 (Dong et al., 2017).

6. Phenomenology, observational properties, and open issues

In the derivative-coupling Horndeski–Galileon sector

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$0

the scalar is often taken as

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$1

This linear time dependence is crucial because it avoids singular behavior of the scalar derivative on the horizon and allows one to evade the usual no-hair obstructions while keeping the metric static, thanks to the shift symmetry of the scalar sector. In observationally relevant regimes the metric can reduce to an effectively Schwarzschild-like geometry with a constant offset,

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$2

but the geodesic analysis shows that bound circular orbits require

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$3

If $$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$4 or $$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$5, bound orbits may not exist. Light deflection constrains the offset tightly,

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$6

and the de Sitter-type term is also required to be tiny on astrophysical scales (Tretyakova, 2016).

A different non-minimally coupled Horndeski branch, the quartic square-root Horndeski black hole, has

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$7

This is asymptotically flat and formally resembles a Reissner–Nordström-type deformation of Schwarzschild, but the $$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$8 term is generated by the Horndeski scalar sector rather than by electromagnetism. In a plasma medium, the deflection angle, photon sphere, and shadow radius depend on $$G_{\mu\nu}+\Lambda g_{\mu\nu} =\frac12\left(\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right),$$9, $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$00, and the plasma profile. Shadow-radius comparisons with Event Horizon Telescope bounds on Sgr A* and M87* yield model-dependent allowed regions for these parameters (Kala et al., 23 Jul 2025).

A closely related quartic square-root Horndeski black hole immersed in a perfect-fluid dark-matter halo has

$$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$01

Its specific heat and free energy show that small horizon states are locally stable but are never globally preferred in the parameter ranges studied, while the shadow and scalar quasinormal spectrum constrain $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$02, $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$03, and $$S=\int d^3x\,\sqrt{-g}\left[\,R-2\Lambda-\frac12\left(\alpha g_{\mu\nu}-\eta G_{\mu\nu}\right)\nabla^\mu\phi\,\nabla^\nu\phi\right],$$04 to remain small if the geometry is to be consistent with Sgr A* shadow data (Gohain et al., 18 Jun 2025).

Several broader issues remain open. One concerns the distinction between static scalar hair and linearly time-dependent scalar hair: the latter can evade no-hair assumptions but is generically unstable near the horizon in the pure Einstein-tensor coupling model unless the dangerous coupling effectively vanishes on the background (Tretyakova et al., 2017). Another concerns thermodynamics: for static scalar hair in shift-symmetric Horndeski, the area law can survive, whereas for linearly time-dependent scalar hair the entropy can acquire explicit scalar-profile dependence (Minamitsuji et al., 2023). A further open direction concerns full gravitational perturbations, since axial stability results do not settle the behavior of the polar sector, which generally couples directly to scalar hair (Chatzifotis et al., 2021).