Hairy Kiselev Black Hole

• Hairy Kiselev black holes are defined as static, spherically symmetric solutions combining a Kiselev fluid term with an exponential deformation that introduces primary hair.
• The construction uses extended gravitational decoupling to derive modified metric functions, anisotropic source splitting, and explicit horizon conditions.
• Key studies of these black holes address geodesic motion, light deflection, thermodynamics, and gravitational-wave memory effects.

A hairy Kiselev black hole is a static, spherically symmetric black-hole geometry in which the Schwarzschild mass term is supplemented by both a Kiselev surrounding-fluid term and an additional exponential deformation interpreted as primary hair. In the direct construction developed through extended gravitational decoupling, a representative lapse function is

$$f(r)=1-\frac{2M}{r}-\frac{N}{r^{3\omega+1}}+\alpha e^{-\frac{2r}{2M-\alpha l}}, \qquad M=\mathcal M+\frac{\alpha l}{2},$$

so that the geometry interpolates among Schwarzschild, ordinary Kiselev, and hairy Schwarzschild limits (Heydarzade et al., 2023). A later thermodynamic extension adds a cosmological constant,

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$

and treats the surrounding field as quintessential matter (Ahmed et al., 24 May 2026). The direct literature is not fully uniform in sign conventions for the Kiselev sector: one treatment of gravitational-wave memory uses $+N/r^{3\omega+1}$ rather than $-N/r^{3\omega+1}$, which suggests that the surrounding-field parameter is handled with different conventions across related formulations (Hadi et al., 2024).

1. Definition, derivation, and source structure

The direct construction is formulated within extended gravitational decoupling. The total source is split as

$\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$

where $T_{ik}$ is the Kiselev-type surrounding fluid and $\Theta_{ik}$ is the anisotropic hair sector. The seed metric is taken in static spherical form,

$$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$

and the deformation acts on both metric potentials,

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$

For the Kiselev sector, the symmetry conditions and the additivity and linearity principle give

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$

with

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$0

The hair sector is anisotropic,

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$1

and the primary hairs are encoded by $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$2 through the mass shift $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$3 and the exponential deformation (Heydarzade et al., 2023).

A precursor of this construction appears in the broader gravitational-decoupling literature. Under the horizon-preserving condition

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$4

the effective radial equation of state becomes

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$5

Within that framework, one explicit branch is identified as a Kiselev black hole, with metric of power-law type

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$6

source terms

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$7

and the mapping $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$8 up to sign convention for the Kiselev coefficient. In that sense, the later hairy Kiselev construction is best understood as a Kiselev surrounding field plus an extra exponential primary-hair sector generated by the same decoupling logic (Ovalle et al., 2020).

2. Horizon equation and causal structure

For the direct hairy Kiselev solution, horizons are defined by the roots of

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$9

The direct geodesic study emphasizes that not every parameter choice yields a black hole: some parameter sets lead instead to a naked singularity, because $+N/r^{3\omega+1}$0 is singular and whether it is hidden depends on the existence of roots of $+N/r^{3\omega+1}$1 (Heydarzade et al., 2023).

The most explicit horizon analysis has been carried out for equatorial light bending. For $+N/r^{3\omega+1}$2, the metric reduces to

$+N/r^{3\omega+1}$3

and the paper reports that, in the parameter region examined, varying $+N/r^{3\omega+1}$4 and $+N/r^{3\omega+1}$5 does not change the number of horizons: the spacetime still has a single event horizon, while the horizon radius increases with both $+N/r^{3\omega+1}$6 and $+N/r^{3\omega+1}$7. In the same study, $+N/r^{3\omega+1}$8 and $+N/r^{3\omega+1}$9 yield two positive roots, whereas $-N/r^{3\omega+1}$0 and $-N/r^{3\omega+1}$1 yield one positive root (Kukreti et al., 21 Jul 2025).

When a cosmological constant is included, the horizon condition becomes

$-N/r^{3\omega+1}$2

This relation is implicit in $-N/r^{3\omega+1}$3, because $-N/r^{3\omega+1}$4 depends on $-N/r^{3\omega+1}$5. The thermodynamic analysis emphasizes that the number of horizons depends on the competition among the Schwarzschild, quintessential, exponential-hair, and cosmological terms. In that treatment, the exponential hair mainly affects the small-horizon regime, while the quintessential sector and the cosmological constant produce significant changes in the large-scale behavior; $-N/r^{3\omega+1}$6 can introduce a cosmological horizon, whereas $-N/r^{3\omega+1}$7 supports large black holes (Ahmed et al., 24 May 2026).

3. Geodesic motion, photon sphere, and light bending

The timelike geodesic analysis is developed in Eddington-Finkelstein form on the equatorial plane. With action

$-N/r^{3\omega+1}$8

one has

$-N/r^{3\omega+1}$9

and the radial equation

$\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$0

For the hairy Kiselev lapse, the acceleration is decomposed into Schwarzschild, surrounding-field, and hair contributions. The direct study states that for $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$1 the Newtonian force is strengthened by the Kiselev sector, whereas for other $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$2 it is weakened; for $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$3 the relativistic correction is strengthened, otherwise it is weakened. The same analysis concludes that for $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$4 the modifications can become important near the event horizon, while for $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$5 they become relevant far from the black hole (Heydarzade et al., 2023).

The null-geodesic and lensing analysis is most explicit for $\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$6. In that case,

$\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$7

The effective potential has one maximum and no minimum, so there are no stable circular null orbits; the unique maximum corresponds to the unstable circular photon orbit. The photon-sphere condition is written as

$\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$8

with an approximate solution

$\tilde T_{ik}=\alpha \Theta_{ik}+T_{ik},$9

The physically relevant photon-sphere radius is the smaller root $T_{ik}$0, and the paper reports that $T_{ik}$1 increases with both $T_{ik}$2 and $T_{ik}$3 (Kukreti et al., 21 Jul 2025).

The same lensing study derives the turning-point condition

$T_{ik}$4

the critical impact parameter

$T_{ik}$5

and an exact equatorial deflection angle in terms of elliptic integrals. Its main comparative conclusion is that the deflection angle decreases monotonically with impact parameter, ordinary Kiselev black holes produce the largest deflection, Schwarzschild is intermediate, and hairy Kiselev black holes give the smallest deflection. In the language of that paper, the quintessential field enhances deflection, whereas the scalar-hair sector suppresses it in the strong-field regime (Kukreti et al., 21 Jul 2025).

4. Gravitational-wave memory and Bondi-type observables

A distinct line of work studies the hairy Kiselev black hole through gravitational-wave pulse and memory effects. In outgoing-null coordinates, the geometry is written as

$T_{ik}$6

For the Bondi-type analysis, the treatment specializes to $T_{ik}$7, so that the large-$T_{ik}$8 geometry becomes

$T_{ik}$9

This is then matched to a Bondi-Sachs expansion in which the News tensor is chosen as

$\Theta_{ik}$0

and the Bondi mass balance equation is

$\Theta_{ik}$1

By comparison with the asymptotic metric, the Bondi mass is identified as

$\Theta_{ik}$2

and on the equatorial plane the explicit expression includes the pulse-dependent $\Theta_{ik}$3 term together with the background shift

$\Theta_{ik}$4

In that formulation, both the surrounding-field parameter $\Theta_{ik}$5 and the hair parameters $\Theta_{ik}$6 leave an imprint on asymptotic mass measurements (Hadi et al., 2024).

The same study treats memory through neighboring geodesics in the pulse-deformed background

$\Theta_{ik}$7

with

$\Theta_{ik}$8

Displacement memory is measured through $\Theta_{ik}$9 and $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$0; velocity memory through $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$1 and $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$2. The reported trends are that increasing $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$3 decreases $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$4 and $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$5 but increases $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$6 and $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$7, while increasing $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$8 produces the opposite pattern in the $$ds^2=-e^{\nu(r)}dt^2+e^{\lambda(r)}dr^2+r^2d\Omega^2,$$9- and $$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$0-channels. The treatment imposes $$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$1 for asymptotic flatness in the geodesic-memory section, and the Bondi-type part is explicitly presented as an analogy with Bondi-Sachs formalism rather than a fully exact radiative solution (Hadi et al., 2024).

5. Thermodynamics, Hawking-radiation sparsity, and greybody factors

The most comprehensive thermodynamic treatment considers a hairy Kiselev black hole with quintessential matter and cosmological constant,

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$2

The horizon equation $$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$3 yields an implicit mass-radius relation because $$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$4 itself depends on $$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$5. The Hawking temperature is obtained from

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$6

with

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$7

so that

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$8

Entropy is taken to obey the usual area law,

$$e^{\nu(r)}\to e^{\nu(r)+\alpha \xi(r)},\qquad e^{\lambda(r)}\to e^{\lambda(r)}+\alpha \eta(r).$$9

the heat capacity is

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$0

and the Gibbs free energy is

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$1

The analysis states that the heat capacity exhibits divergences associated with second-order phase transitions, while the Gibbs free energy reveals the possibility of competing thermodynamic branches. Its central conclusion is that the exponential hair mainly affects the small-horizon regime, whereas the quintessential sector and the cosmological constant dominate the large-scale behavior (Ahmed et al., 24 May 2026).

The same work studies Hawking-radiation sparsity. For bosons, the number flux is

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$2

with emission rate

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$3

The sparsity parameter is

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$4

and the paper concludes that the Hawking flux is highly intermittent rather than continuous. In the blackbody approximation, the effective emitting area is written as

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$5

where the photon sphere satisfies

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$6

The greybody analysis reduces the Klein-Gordon equation to

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$7

with tortoise coordinate $$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$8 and effective potential

$$T^0_0=T^1_1=-\rho,\qquad T^2_2=T^3_3=\frac12(1+3\omega)\rho,$$9

A lower bound on the transmission coefficient is then given by

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$00

Within that framework, higher angular momentum suppresses transmission, the exponential hair gives only mild deviations from Schwarzschild for the chosen parameters, and positive $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$01 can strongly suppress the bound (Ahmed et al., 24 May 2026).

6. Conceptual status and relation to adjacent hairy-black-hole literature

The expression “hairy Kiselev black hole” should not be extended indiscriminately to every hairy black hole with anisotropic matter. Several nearby constructions are explicitly not Kiselev black holes in the usual sense. Regular hairy black holes built through Minkowski deformation and gravitational decoupling impose $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$02 and produce exponential or de Sitter-core behavior, but they do not introduce a Kiselev quintessence parameter $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$03 or the standard term $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$04 (Ovalle et al., 2023). Static hairy black holes obtained through extended geometric deformation of Schwarzschild likewise produce anisotropic matter and a logarithmic-plus-$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$05 correction,

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$06

but they are not written as Kiselev fluids (Avalos et al., 2023).

The same caution applies to phenomenological descendants of the hairy Schwarzschild solution. Strong and weak lensing analyses based on

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$07

treat the hair as an additional source from the surroundings, “such as dark matter,” yet do not derive a Kiselev equation of state (Jha et al., 2022). Imaging studies of disk and spherical accretion for the same class of geometries focus on single and double photon spheres rather than on Kiselev matter proper (Meng et al., 2023). Geodesic studies of related decoupling-generated black holes with primary hairs $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$08 provide methodology for ISCO, MBO, and photon-sphere analysis, but again without an explicit Kiselev fluid sector (Ramos et al., 2021).

By contrast, gravitational decoupling provides a direct bridge to Kiselev-type structure. One explicit branch in the earlier decoupling literature is identified as a Kiselev black hole, with asymptotic flatness requiring $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$09 and the dominant energy condition selecting $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$10 (Ovalle et al., 2020). A different analogue arises in Lorentz-breaking massive gravity, where the power-law deformation

$$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$11

is thermodynamically rich but sourced by a scalar sector rather than by Kiselev matter (Capela et al., 2012). At a still broader level, the theorem that the nontrivial behavior of hair must extend beyond the photonsphere in static, asymptotically flat Einstein-matter theories supplies a useful conceptual template for analyzing any static hairy Kiselev geometry once its effective $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$12, $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$13, and $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$14 are identified (Hod, 2011).

In this restricted and technically precise sense, the hairy Kiselev black hole is best classified as a Kiselev surrounding-fluid spacetime dressed by a decoupling-generated primary-hair sector. Its defining features are the coexistence of the Kiselev term $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$15 and an exponential hair correction controlled by $$f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/(M-\alpha\ell/2)}-\frac{\Lambda}{3}r^2,$$16, together with a source that is explicitly anisotropic and naturally organized as a background fluid plus a distinct hair sector (Heydarzade et al., 2023).