Asymptotically Flat Horndeski Black Holes

• Asymptotically flat Horndeski black holes are scalar-tensor solutions whose metrics approach Minkowski space while featuring nontrivial, often stealth, scalar profiles.
• They are derived using both static spherically symmetric and rotating ansatzes, combining analytical and numerical methods to produce Schwarzschild-, Kerr-, or Kerr–Newman-like geometries.
• These solutions exhibit distinct thermodynamic, lensing, and shadow features that offer practical observational signatures and test the limits of stability and quantum corrections.

Asymptotically flat Horndeski black holes are black-hole solutions of Horndeski or beyond-Horndeski scalar-tensor theories whose metric approaches Minkowski space at large radius while the scalar sector is nontrivial, or is nontrivial but stealthy so that the metric remains exactly a vacuum General Relativity geometry. The subject includes static, spherically symmetric hairy solutions, stealth Schwarzschild and Kerr configurations, numerically constructed stationary and axisymmetric spinning black holes in shift-symmetric Einstein–scalar–Gauss–Bonnet theory, and phenomenological rotating metrics used in shadow studies. Across these constructions, the recurrent technical issues are the realization of scalar hair, the precise meaning of asymptotic flatness, the evasion of no-hair theorems, and the corresponding thermodynamic and observational signatures (Babichev et al., 2017, Delgado et al., 2020, Bakopoulos et al., 2022).

1. Framework and asymptotic conditions

A large part of the literature is formulated in terms of the scalar kinetic density

$X \equiv -\frac12\,(\nabla\phi)^2,$

with the shift-symmetric Horndeski or beyond-Horndeski action written as

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$

For static, spherically symmetric configurations one typically adopts

$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$

with asymptotic flatness imposed through

$$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$

In the shift-symmetric beyond-Horndeski formulation, a compact rewriting in terms of auxiliary functions $Z(X)$, $Y(X)$, $\mathcal{A}$, and $\mathcal{B}$ is especially useful; under parity symmetry, the field equations become directly integrable and generate multiple asymptotically flat black-hole families (Babichev et al., 2017, Bakopoulos et al., 2022).

A distinct but closely related asymptotically flat sector is the shift-symmetric Einstein–scalar–Gauss–Bonnet model with linear scalar coupling to the Gauss–Bonnet invariant,

$$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$

Its spinning black holes are constructed with the stationary, axisymmetric ansatz

$$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$

and asymptotic flatness is encoded by

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$0

with

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$1

In this branch the ADM mass $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$2, angular momentum $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$3, and scalar charge $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$4 are defined by the large-$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$5 fall-offs, and the scalar hair is secondary rather than independent (Delgado et al., 2020).

2. Static asymptotically flat solution classes

The static landscape contains non-stealth hairy metrics, stealth geometries with a hidden scalar, and branches in which the scalar depends linearly on time while the metric remains static.

Class	Representative metric or condition	Hair type
Quartic Horndeski	$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$6	Secondary
Quartic beyond Horndeski	$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$7	Secondary
Pure quartic stealth	$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$8 with constant $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$9	Stealth
Cubic Galileon branch	$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$0, $$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$1	Primary $$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$2
Bi-scalar extension	$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$3	Primary $$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$4

In the shift-symmetric quartic Horndeski model with

$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$5

imposing $$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$6 yields

$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$7

and the metric becomes

$$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$8

The scalar fall-off amplitude is fixed by $$ds^2=-h(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,d\Omega^2,\qquad \phi=\phi(r),$$9, so there is no independent scalar charge and the hair is secondary. In the quartic beyond-Horndeski model with

$$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$0

the asymptotics are again Reissner–Nordström-like,

$$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$1

with secondary hair fixed by the couplings. By contrast, pure quartic theories with

$$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$2

can satisfy the algebraic conditions

$$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$3

so that the metric is exactly Schwarzschild while the scalar remains nontrivial with constant $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$4 (Babichev et al., 2017).

The beyond-Horndeski construction based on the auxiliary functions $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$5, $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$6, $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$7, and $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$8 makes it possible to classify asymptotically flat families more systematically. In the parity-symmetric sector, direct integrability gives $$h(r)\to 1-\frac{2M}{r}+\cdots,\qquad f(r)\to 1-\frac{2M}{r}+\cdots,\qquad \phi(r)\to \phi_\infty+\frac{Q_\phi}{r}+\cdots.$$9. For the asymptotically flat branch one sets

$Z(X)$0

and obtains

$Z(X)$1

so that a canonical kinetic term in $Z(X)$2 robustly generates RN-like $Z(X)$3 and $Z(X)$4 tails. This branch is explicitly interpreted as scalar-hairy and asymptotically flat (Bakopoulos et al., 2022).

A different evasion mechanism appears in the cubic Galileon subclass

$Z(X)$5

for which the four-dimensional asymptotically flat sector is obtained by setting

$Z(X)$6

Then the scalar ansatz

$Z(X)$7

supports a family of static metrics with

$Z(X)$8

and the parameter $Z(X)$9 acts as primary hair (Babichev et al., 2016).

Not every asymptotically flat Horndeski branch yields a regular black hole. In the shift-symmetric Gauss–Bonnet model represented by

$Y(X)$0

imposing $Y(X)$1 produces asymptotically flat solutions with ultra-suppressed $Y(X)$2 corrections,

$Y(X)$3

but numerical integration reaches a naked curvature singularity rather than a regular horizon. In this specific $Y(X)$4 construction, asymptotic flatness does not imply black-hole regularity (Babichev et al., 2017).

3. Rotating and stationary asymptotically flat black holes

The main exact rotating solutions currently available in Horndeski theory are the stationary, axisymmetric, asymptotically flat black holes of shift-symmetric Einstein–scalar–Gauss–Bonnet theory. They are obtained numerically from five coupled nonlinear elliptic equations for $Y(X)$5, with input parameters $Y(X)$6. These black holes possess a nontrivial scalar field outside a regular event horizon, a single Kerr-like $Y(X)$7 ergosurface defined by $Y(X)$8, and a scalar charge fixed by horizon data through

$Y(X)$9

The scalar hair is therefore secondary. The domain of existence in the $\mathcal{A}$0 plane is bounded by the GR limit, the static branch, an extremal line, and a critical line where horizon regularity breaks down. Small violations of the Kerr bound occur,

$\mathcal{A}$1

while representative deviations from Kerr in ISCO and light-ring frequencies remain at the level of a few to several percent (Delgado et al., 2020).

Exact stealth rotation is realized in pure quartic Horndeski theories. When the two algebraic conditions at $\mathcal{A}$2,

$\mathcal{A}$3

are satisfied, any Ricci-flat vacuum GR solution is also a solution of the quartic theory with nontrivial scalar field. In particular, Kerr in Boyer–Lindquist coordinates remains exactly Kerr while the scalar profile is nontrivial and regular outside the horizon. This is a genuinely stealth configuration: the metric is exactly that of GR, but the scalar sector is not trivial (Babichev et al., 2017).

A separate rotating branch, used extensively in shadow calculations, is the phenomenological Gürses–Gürsey class obtained by applying the Newman–Janis algorithm to a static seed metric with mass function $\mathcal{A}$4. For the Horndeski-inspired Kerr–Newman-like specialization one takes

$\mathcal{A}$5

Asymptotic flatness is ensured by $\mathcal{A}$6, and the horizon condition is

$\mathcal{A}$7

If $\mathcal{A}$8 the solution reduces to Kerr–Newman, whereas in the Horndeski interpretation the effective charge-like parameter can have either sign. The Newman–Janis rotation is explicitly described as a procedure that should be applied with care beyond GR, but in this context it serves as a phenomenological generator of rotating asymptotically flat metrics for shadow studies (Badía et al., 2021).

4. Scalar hair, stealth structure, and no-hair evasion

The asymptotically flat Horndeski literature distinguishes sharply between secondary hair, primary hair, and stealth scalar structure. In the quartic Horndeski and quartic beyond-Horndeski RN-like branches, the scalar fall-off amplitude is fixed by the couplings and there is no independent scalar charge, so the hair is secondary. In the cubic Galileon asymptotically flat branch, the parameter $\mathcal{A}$9 labels a continuous family of solutions at fixed mass and is primary hair. In the exact bi-scalar extensions of Horndeski gravity, the conformally coupled scalar can carry an independent primary charge $\mathcal{B}$0, with

$\mathcal{B}$1

while the time-dependent Galileon

$\mathcal{B}$2

regularizes the horizon. In the shift-symmetric Einstein–scalar–Gauss–Bonnet rotating branch, by contrast, the scalar charge satisfies $\mathcal{B}$3 and is again secondary (Babichev et al., 2017, Babichev et al., 2016, Charmousis et al., 2014, Delgado et al., 2020).

The no-hair theorems are evaded by violating specific assumptions rather than by abandoning asymptotic flatness. In the shift-symmetric analysis, the standard theorem assumes analyticity near $\mathcal{B}$4, the presence of a canonical kinetic term, asymptotic flatness, and finite $\mathcal{B}$5 down to the horizon. Two explicit evasion mechanisms are identified. The first uses non-analytic terms such as

$\mathcal{B}$6

which inject $\mathcal{B}$7-independent pieces into the radial current $\mathcal{B}$8, allowing $\mathcal{B}$9 with $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$0. The second removes the canonical $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$1 term in a pure quartic theory and imposes the algebraic conditions that make the scalar stress tensor cancel on Ricci-flat metrics, yielding stealth Schwarzschild and Kerr solutions (Babichev et al., 2017).

Time dependence is another recurrent mechanism. In the cubic Galileon branch, the ansatz $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$2 keeps the metric static because the field equations depend on $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$3 only through invariant combinations. In the bi-scalar extension, the same strategy cures the classic BBMB horizon singularity while preserving primary scalar hair $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$4. This suggests that, in asymptotically flat Horndeski sectors, regular scalar hair is often tied to a controlled departure from strictly static scalar profiles even when the geometry itself remains static (Babichev et al., 2016, Charmousis et al., 2014).

Spontaneous scalarization provides a different route to asymptotically flat hairy black holes. On a Schwarzschild background with $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$5, regular shift-symmetric Horndeski couplings analytic at $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$6 lead to

$$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$7

so no scalarization occurs. By contrast, a generalized quartic coupling with

$$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$8

can generate a compact negative well in the effective potential while preserving $$\mathcal{S}=\int d^4x\,\sqrt{-g}\left[\,R-\frac{1}{2}\,\partial_\mu\phi\,\partial^\mu\phi+\alpha\,\phi\,R_{\rm GB}^2\right].$$9, thereby realizing a near-horizon tachyonic instability. Numerically, the negative region appears once $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$0 is sufficiently negative, with threshold approximately

$$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$1

A combined quartic-plus-quintic model interpolates continuously to Einstein–scalar–Gauss–Bonnet theory and reproduces the scalarization threshold

$$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$2

at the ESGB point $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$3 (Minamitsuji et al., 2019).

5. Geodesics, lensing, shadows, and related observables

A representative static asymptotically flat Horndeski black hole used in strong-deflection lensing is

$$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$4

with $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$5. The outer horizon exists for $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$6, and for $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$7 the metric is isometric to Reissner–Nordström with $$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$8. The photon sphere satisfies

$$ds^2=-e^{2F_0}N\,dt^2+e^{2F_1}\left(\frac{dr^2}{N}+r^2\,d\theta^2\right)+e^{2F_2}\,r^2\sin^2\theta\,(d\varphi-W\,dt)^2, \qquad N\equiv 1-\frac{r_H}{r},$$9

and the critical impact parameter is

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$00

In the strong-deflection limit the bending angle takes the Bozza form

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$01

with

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$02

Negative $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$03 strengthens strong-field lensing, whereas positive $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$04 weakens it (Badía et al., 2017).

Null and timelike geodesics have also been analyzed in the kinetic-coupling branch with

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$05

where the physical horizon is

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$06

For photons, the unstable circular orbit is at

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$07

and the weak-field bending angle is

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$08

In the same class, the gravitational redshift and Shapiro delay lead to meter-level constraints on the Horndeski parameter,

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$09

For massive particles, the effective potential is

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$10

the perihelion shift becomes

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$11

and Mercury yields the bound $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$12 in that notation (Carvajal et al., 3 Mar 2025, Carvajal et al., 13 Jul 2025).

Rotating shadow phenomenology is often discussed in the Newman–Janis-generated Horndeski-inspired Kerr–Newman-like geometry. In the presence of a separable, pressureless, nonmagnetized plasma with

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$13

the shadow boundary is obtained from spherical photon orbits through the impact parameters $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$14 and $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$15, and the celestial coordinates are

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$16

For the Shapiro-type profile

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$17

the plasma reduces the shadow area $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$18, reduces the oblateness $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$19, and moves the centroid $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$20 closer to the origin. At sufficiently low $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$21, the condition

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$22

fails in a forbidden region that can eventually envelop the black hole and erase the shadow. In the vacuum or high-frequency limit, the contour approaches the Kerr or Kerr–Newman form (Badía et al., 2021).

6. Thermodynamics, stability, and unresolved issues

The spinning shift-symmetric Einstein–scalar–Gauss–Bonnet black holes have a complete thermodynamic description. Their horizon quantities are

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$23

and the entropy is the Wald entropy

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$24

These solutions satisfy

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$25

At fixed $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$26, increasing $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$27 decreases the reduced area $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$28 but increases the reduced entropy $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$29 and reduced temperature $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$30; at fixed $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$31, increasing $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$32 decreases $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$33, $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$34, and $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$35 (Delgado et al., 2020).

Linear stability is highly subclass-dependent. For static, spherically symmetric black holes with a time-independent scalar field in reflection-symmetric shift-symmetric Horndeski theories with $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$36, a near-horizon analysis shows that hairy solutions are generically unstable: one cannot keep the relevant coefficients $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$37, $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$38, and $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$39 simultaneously positive near the horizon. By contrast, asymptotically flat black-hole solutions with a scalar linearly coupled to the Gauss–Bonnet term, constructed perturbatively for small coupling, are free of ghosts and Laplacian instabilities provided

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$40

This separates unstable reflection-symmetric time-independent hair from the perturbatively stable Gauss–Bonnet branch (Minamitsuji et al., 2022).

Quantum-corrected thermodynamics has been studied for the asymptotically flat shift-symmetric solution

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$41

by applying the generalized uncertainty principle

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$42

With $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$43, the modified temperature is written as

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$44

where $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$45 is the GUP-corrected localization scale, and the entropy and heat capacity acquire explicit $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$46-suppressed corrections. In that treatment the evaporation is argued to halt at a remnant rather than proceeding to complete disappearance (Seifi et al., 2022).

A different asymptotically flat charged Einstein–Horndeski branch exhibits a striking late-time evaporation law. For

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$47

the thermodynamic quantities are

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$48

Because $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$49 as $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$50 and the exterior electric field

$$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$51

suppresses the Schwinger channel near the endpoint, the evaporation becomes ultra slow and the lifetime is much longer than in the Reissner–Nordström case (Liang et al., 2024).

Several limitations remain explicit in the current literature. Exact analytic rotating solutions are unavailable in many Horndeski sectors, and the known nonperturbative spinning EsGB solutions likely have a singular extremal limit. Full linear or nonlinear stability is open in many hairy branches, especially beyond the perturbative Gauss–Bonnet regime. Shadow calculations based on Newman–Janis-rotated metrics are phenomenological rather than exact, and realistic shadow modeling requires emission, absorption, and scattering in addition to the geometric boundary. Finally, higher-dimensional Einstein–Horndeski–Maxwell black holes often belong to asymptotically locally flat rather than strictly asymptotically flat branches; in the ALF family studied in connection with EHT, increasing the spacetime dimension reduces the shadow size, and only the four-dimensional ALF branch is compatible with the M87* $$S=\int d^4x\,\sqrt{-g}\,\Big[L_2+L_3+L_4+L_5+L^{\rm bH}_4+L^{\rm bH}_5\Big].$$52 shadow band (Delgado et al., 2020, Badía et al., 2021, Nozari et al., 2023).