---
title: Non-Minimally Coupled Horndeski Black Hole
url: https://www.emergentmind.com/topics/non-minimally-coupled-horndeski-black-hole
type: topic
---

# Non-Minimally Coupled Horndeski Black Hole

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 [1405.4935].

## 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 [1405.4935].

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 [1208.0103].

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 [1401.4479].

## 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
\[
\nabla_\mu\!\left[\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\nu\phi\right]=0.
\]
Because of shift symmetry, the scalar equation is a current conservation law,
\[
\nabla_\mu J^\mu=0, \qquad J^\mu=\left(\alpha g^{\mu\nu}-\eta G^{\mu\nu}\right)\nabla_\nu\phi.
\]
For a static radial scalar, the key component is
\[
J^r=\left(\alpha g^{rr}-\eta G^{rr}\right)\phi'(r).
\]
The central simplifying condition imposed in several exact constructions is
\[
\alpha g^{rr}-\eta G^{rr}=0,
\]
equivalent to \(J^r=0\) without forcing \(\phi'(r)=0\). 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 [1405.4935].

This current-based mechanism is also central to the older four-dimensional solution with coupling \(z\). There the reduced equations admit a radial shift current
\[
J_r=\frac{\psi \sqrt{FG}\,\big(m_p^2r^2G+zG-z\big)}{m_p^2G^2}=-K,
\]
and regularity at a black-hole horizon requires the current to vanish there, hence everywhere, so black-hole solutions exist only in the \(K=0\) 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 [1208.0103].

In shift-symmetric, reflection-symmetric Horndeski models with linearly time-dependent scalar hair,
\[
\phi(t,r)=qt+\psi(r),
\]
the same general structure persists, but the stability properties are more subtle. For the Einstein-tensor coupling model
\[
G_2(X)=-2\Lambda+2\eta X, \qquad G_4(X)=\zeta+\beta X,
\]
the near-horizon odd-parity stability coefficients satisfy
\[
\mathcal F\mathcal G \approx -\left(\frac{2q^2}{h}G_{4X}\right)^2<0
\]
when \(G_{4X}\) 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 [1702.03502].

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

In three dimensions, imposing \(J^r=0\) on the static ansatz
\[
ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\varphi^2, \qquad \phi=\phi(r),
\]
forces
\[
f(r)=\frac{2\alpha r\,h(r)}{\eta h'(r)},
\]
and, away from the degenerate sector \(\alpha=\eta\Lambda\), yields
\[
h(r)=Cr^2-M, \qquad f(r)=\frac{\alpha}{\eta C}\,(Cr^2-M).
\]
With
\[
C=l^{-2}, \qquad \frac{\alpha}{\eta}=l^{-2},
\]
the metric becomes exactly the static BTZ geometry,
\[
ds^2=-\left(\frac{r^2}{l^2}-M\right)dt^2 +\frac{dr^2}{\frac{r^2}{l^2}-M} +r^2 d\varphi^2.
\]
The scalar profile is nontrivial,
\[
\bigl(\phi'(r)\bigr)^2 = -\,\frac{2(\Lambda l^2+1)}{\eta\left(\frac{r^2}{l^2}-M\right)},
\]
and
\[
\phi(r)=\pm\sqrt{-\frac{2l^2(\Lambda l^2+1)}{\eta}\; \ln\!\left(\frac{r}{l}+\sqrt{\frac{r^2}{l^2}-M}\right).
\]
Reality requires
\[
\Lambda\le -\frac1{l^2},
\]
with the endpoint \(\Lambda=-1/l^2\) reducing to the scalarless BTZ solution [1405.4935].

A notable feature of that branch is that although \(\phi'(r)\) diverges like \(1/F(r)\) 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:
\[
\frac12\left[\alpha T^{(1)}_{\mu\nu}+\eta T^{(2)}_{\mu\nu}\right]_{\rm on\ shell} = \left(\Lambda+\frac1{l^2}\right)g_{\mu\nu}.
\]
Consequently the metric solves the ordinary BTZ Einstein equations with AdS radius \(l\) even though the scalar is nontrivial [1405.4935].

The same paper extends the mechanism to arbitrary dimension with planar horizon,
\[
ds^2=-N(r)^2F(r)\,dt^2+\frac{dr^2}{F(r)}+r^2 dx_{D-2}^2, \qquad \phi=\phi(r),
\]
and again imposing \(J^r=0\) gives the exact planar Schwarzschild–AdS black hole
\[
ds^2=-F(r)\,dt^2+\frac{dr^2}{F(r)}+r^2 dx_{D-2}^2,
\qquad
F(r)=\frac{r^2}{l^2}-\frac{M}{r^{D-3}},
\]
with coupling relation
\[
\frac{\alpha}{\eta}=\frac{(D-1)(D-2)}{2l^2}.
\]
The scalar remains nontrivial and again sources the geometry as an effective cosmological constant on shell [1405.4935].

In four dimensions, the solution
\[
F(r) = \frac34+\frac{r^2}{l^2}-\frac{2M}{m_p^2 r} +\frac{\sqrt{z}}{4m_p r}\arctan\!\left(\frac{m_pr}{\sqrt z}\right),
\]
\[
G(r) = \frac{(m_p^2r^2+2z)^2}{4(m_p^2r^2+z)^2\,F(r)},
\]
with
\[
l^2=\frac{12z}{m_p^2},
\]
is asymptotically Schwarzschild–AdS-like even though no bare cosmological constant is present. For \(M>0\), \(F(r)\) has one zero only, so there is a single event horizon [1208.0103].

Charged topological and spherical solutions in Einstein–Maxwell–Horndeski theory add Coulombic and higher-order charge terms, as well as characteristic \(\arctan\) contributions to the metric and gauge potential. The four-dimensional spherical branch has
\[
ds^2=-F(r)\,dt^2+G(r)\,dr^2+r^2d\Omega^2,
\]
with asymptotically locally AdS behavior set by
\[
l^{-2}:=\frac{\alpha}{3\eta},
\]
and a physically relevant branch obeying \(\alpha\eta>0\) and a scalar-reality bound involving \(\Lambda\), \(\alpha\), \(\eta\), \(q\), and the horizon radius [1401.4479].

## 4. Thermodynamics and horizon mechanics

The three-dimensional BTZ-based solution admits a reduced Euclidean action
\[
I_E[N,F,\xi] = 2\pi\beta\int_{r_+}^{\infty} dr\, N\left[ F' +2\Lambda r +\frac{\alpha}{2}rF\xi +\frac{3\eta}{4}FF'\xi +\frac{\eta}{2}F^2\xi' \right] +B_E,
\]
whose last Euler–Lagrange equation is precisely the constraint implementing \(J^r=0\). On shell,
\[
I_E = \beta\,\mathcal M-\mathcal S = \pi\beta M\,(1-\Lambda l^2)-4\pi^2 r_+\,(1-\Lambda l^2),
\]
so that
\[
\mathcal M=\pi M(1-\Lambda l^2),\qquad \mathcal S=4\pi^2(1-\Lambda l^2)\,r_+.
\]
Both mass and entropy are rescaled relative to ordinary BTZ by the same overall factor, while the Hawking temperature remains the BTZ one,
\[
T=\frac{r_+}{2\pi l^2}.
\]
The first law
\[
d\mathcal M=T\,d\mathcal S
\]
and the usual three-dimensional Smarr formula
\[
\mathcal M=\frac{T}{2}\,\mathcal S
\]
follow from the reduced action and its scaling symmetry [1405.4935].

In higher dimensions, the planar Schwarzschild–AdS branch has
\[
\mathcal M = \frac{(D-1)(D-2)-2l^2\Lambda}{2(D-1)}\,M\,\mathrm{Vol}(\Sigma_{D-2}),
\]
\[
\mathcal S = \frac{(D-1)(D-2)-2l^2\Lambda}{2(D-1)}\, 4\pi r_+^{D-2}\,\mathrm{Vol}(\Sigma_{D-2}),
\]
again differing from the general-relativistic values by a universal overall factor and obeying
\[
d\mathcal M=T\,d\mathcal S, \qquad \mathcal M=\frac{D-2}{D-1}\,T\mathcal S
\]
[1405.4935].

For the four-dimensional Einstein-tensor-coupled black hole with coupling \(z\), the Euclidean temperature is
\[
\beta=\frac{8\pi z\,r_h}{m_p^2r_h^2+2z},
\qquad
T=\frac{m_p^2r_h^2+2z}{8\pi z\,r_h}.
\]
The normalized Euclidean volume, energy, entropy, and heat capacity are modified in a nontrivial way,
\[
V= \frac{\pi z\,x\big(-2x^3+3x+3\arctan x\big)}{3(x^2+2)},
\]
\[
E = M+\frac{m_p\sqrt z\,x^3(x^2+2)^2}{8(x^2-2)(x^2+1)},
\]
\[
S = \frac{\pi z\,x^2(2x^4+x^2-2)}{(x^2+1)(x^2-2)},
\]
\[
C = \frac{2\pi z\,x^2(x^2+2)(2x^8-4x^6-11x^4-4x^2+4)} {(x^2+1)^2(x^2-2)^3},
\]
with \(x=m_pr_h/\sqrt z\). The paper emphasizes that entropy is not generically the area law and that there is a Hawking–Page-like phase structure [1208.0103].

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 [2308.01082].

## 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
\[
P(\pm)=1\pm \frac{\eta\kappa}{2B}\left(\frac{d\phi_0}{dr}\right)^2.
\]
With a suitable \(S\)-deformation, the deformed potential becomes
\[
V_S = \gamma\,\frac{A}{C\,P(+)},
\]
and for the relevant exact hairy solutions with real scalar profile one has \(V_S\ge0\). This establishes mode stability under linear odd-parity perturbations for that class of black holes. In the same setup, slowly rotating solutions satisfy
\[
\omega(r)=c_1+\frac{c_2}{r},
\]
so the exterior frame-dragging profile coincides with the general-relativistic one in the slow-rotation limit [1508.06413].

A different exact spherically symmetric derivative-coupled solution, often associated with Rinaldi, has
\[
g_{tt}(r)=-\frac14 F(r),\qquad
g_{rr}(r)=\frac{(r^2+2\ell_\eta^2)^2}{(r^2+\ell_\eta^2)^2\,F(r)},
\]
\[
F(r)=3+\frac{r^2}{3\ell_\eta^2}-\frac{8m}{r}+\frac{\ell_\eta}{r}\arctan\!\left(\frac{r}{\ell_\eta}\right).
\]
Its asymptotics are effectively AdS-like with
\[
R_{\rm eff}=\sqrt{3}\,\ell_\eta,
\]
even though there is no cosmological constant in the action. Axial gravitational perturbations satisfy a Regge–Wheeler-like equation with a modified propagation factor \(1/(AB)\), 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 \(m/\ell_\eta\) and transitions between a photon-sphere-dominated regime, an intermediate echoing regime, and a rapidly depleted ringing regime with an exponential tail [2109.02678].

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
\[
ds^2=-F(r)dt^2+\frac{h^2(r)}{F(r)}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).
\]
Away from the special point \(z=1/3\) in units \(m_p=l=1\), the geometry differs from Schwarzschild–AdS through both \(h(r)\neq 1\) and the extra \(\arctan\) term in \(F(r)\). For minimally coupled test scalars, increasing the derivative coupling \(z\) 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 \(\omega\propto z^{-1/2}\) [1709.01641].

## 6. Phenomenology, observational properties, and open issues

In the derivative-coupling Horndeski–Galileon sector
\[
S = \int dx^4 \sqrt{-g} \left( \zeta R - \eta \left( \partial \phi \right)^2 + \beta G^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - 2\Lambda \right),
\]
the scalar is often taken as
\[
\phi(r)=qt+\psi(r).
\]
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,
\[
h(r)\approx (B+C)-\frac{\mu}{r},
\]
but the geodesic analysis shows that bound circular orbits require
\[
0<(B+C)<\frac{9}{8}.
\]
If \((B+C)<0\) or \((B+C)>9/8\), bound orbits may not exist. Light deflection constrains the offset tightly,
\[
|1-(C+B)|<3\times 10^{-4},
\]
and the de Sitter-type term is also required to be tiny on astrophysical scales [1606.08569].

A different non-minimally coupled Horndeski branch, the quartic square-root Horndeski black hole, has
\[
f(r)=1-\frac{2M}{r}-\frac{\beta^2}{2\eta r^2}.
\]
This is asymptotically flat and formally resembles a Reissner–Nordström-type deformation of Schwarzschild, but the \(1/r^2\) 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 \(\beta\), \(\eta\), 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 [2507.17280].

A closely related quartic square-root Horndeski black hole immersed in a perfect-fluid dark-matter halo has
\[
f(r) = 1 - \frac{2M}{r} - \frac{\beta^2}{2 \eta r^2} + \frac{b}{r} \ln \left( \frac{r}{b}\right).
\]
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 \(\beta\), \(\eta\), and \(b\) to remain small if the geometry is to be consistent with Sgr A* shadow data [2506.15763].

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 [1702.03502]. 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 [2308.01082]. 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 [2109.02678].

Source: https://www.emergentmind.com/topics/non-minimally-coupled-horndeski-black-hole