---
title: Hairy Kiselev Black Hole
url: https://www.emergentmind.com/topics/hairy-kiselev-black-hole
type: topic
---

# Hairy Kiselev Black Hole

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 [2307.04556]. 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 [2605.25245]. 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 [2404.08026].

## 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
$$
P(r)=\omega \rho(r),\qquad
\rho(r)=-\frac{3\omega N}{r^{3(\omega+1)}}.
$$
The hair sector is anisotropic,
$$
\Theta^0_0=\Theta^1_1=-\bar\rho,\qquad
\Theta^2_2=\Theta^3_3=\bar P_t,
$$
and the primary hairs are encoded by \((\alpha,l)\) through the mass shift \(M=\mathcal M+\alpha l/2\) and the exponential deformation [2307.04556].

A precursor of this construction appears in the broader gravitational-decoupling literature. Under the horizon-preserving condition
$$
e^\nu=e^{-\lambda},
$$
the effective radial equation of state becomes
$$
\tilde p_r=-\tilde\rho.
$$
Within that framework, one explicit branch is identified as a Kiselev black hole, with metric of power-law type
$$
e^\nu=e^{-\lambda}=1-\frac{2\mathcal M}{r}+\frac{\ell^n}{r^n},
$$
source terms
$$
\tilde\rho=\alpha\frac{(n-1)\ell^n}{k^2 r^{n+2}},\qquad
\tilde p_r=-\tilde\rho,\qquad
\tilde p_t=\frac n2\,\tilde\rho,
$$
and the mapping \(n=3\omega+1\) 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 [2006.06735].

## 2. Horizon equation and causal structure

For the direct hairy Kiselev solution, horizons are defined by the roots of
$$
f(r)=1-\frac{2M}{r}-\frac{N}{r^{3\omega+1}}+\alpha e^{-\frac{2r}{2M-\alpha l}}=0.
$$
The direct geodesic study emphasizes that not every parameter choice yields a black hole: some parameter sets lead instead to a naked singularity, because \(r=0\) is singular and whether it is hidden depends on the existence of roots of \(f(r)=0\) [2307.04556].

The most explicit horizon analysis has been carried out for equatorial light bending. For \(\omega=-2/3\), the metric reduces to
$$
f(r)=1-\frac{2M}{r}-Nr+\alpha \exp\!\left(-\frac{2r}{2M-\alpha \ell}\right),
$$
and the paper reports that, in the parameter region examined, varying \(N\) and \(\alpha\) does not change the number of horizons: the spacetime still has a single event horizon, while the horizon radius increases with both \(N\) and \(\alpha\). In the same study, \(\omega=-4/3\) and \(\omega=-1\) yield two positive roots, whereas \(\omega=-2/3\) and \(\omega=0\) yield one positive root [2507.15298].

When a cosmological constant is included, the horizon condition becomes
$$
1-\frac{2M}{r_h}-N\,r_h^{-(3\omega_q+1)}+\alpha e^{-r_h/\sigma}-\frac{\Lambda}{3}r_h^2=0,
\qquad
\sigma=M-\frac{\alpha\ell}{2}.
$$
This relation is implicit in \(M\), because \(\sigma\) depends on \(M\). 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; \(\Lambda>0\) can introduce a cosmological horizon, whereas \(\Lambda<0\) supports large black holes [2605.25245].

## 3. Geodesic motion, photon sphere, and light bending

The timelike geodesic analysis is developed in Eddington-Finkelstein form on the equatorial plane. With action
$$
S=\frac12\int\left(-f\dot v^2+2\varepsilon \dot v\dot r+r^2\dot\varphi^2\right)d\tau,
$$
one has
$$
L=r^2\dot\varphi,
$$
and the radial equation
$$
\ddot r=-\frac12\left(1+\frac{L^2}{r^2}\right)f'+f\frac{L^2}{r^3}.
$$
For the hairy Kiselev lapse, the acceleration is decomposed into Schwarzschild, surrounding-field, and hair contributions. The direct study states that for \(\omega\in(-1/3,0)\) the Newtonian force is strengthened by the Kiselev sector, whereas for other \(\omega\) it is weakened; for \(\omega\in(-1,0)\) the relativistic correction is strengthened, otherwise it is weakened. The same analysis concludes that for \(\omega>0\) the modifications can become important near the event horizon, while for \(\omega<-\frac13\) they become relevant far from the black hole [2307.04556].

The null-geodesic and lensing analysis is most explicit for \(\omega=-2/3\). In that case,
$$
V_{\rm eff}(r)=\left(1-\frac{2M}{r}-Nr+\alpha e^{-2r/(2M-\alpha \ell)}\right)\frac{L^2}{r^2}.
$$
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
$$
\frac{6M}{r}+Nr+2\alpha e^{-2r/(2M-\alpha \ell)}
\left(\frac{r}{2M-\alpha \ell}-1\right)=2,
$$
with an approximate solution
$$
r_{c\pm}=
\frac{(\alpha+1)\pm\sqrt{(\alpha+1)^2-6M(\alpha\beta+N)}}{\alpha\beta+N},
\qquad
\beta=\frac{6}{2M-\alpha \ell}.
$$
The physically relevant photon-sphere radius is the smaller root \(r_{c-}=r_{\rm ps}\), and the paper reports that \(r_{\rm ps}\) increases with both \(\alpha\) and \(N\) [2507.15298].

The same lensing study derives the turning-point condition
$$
\frac{1}{b^2}=\frac{f(r_0)}{r_0^2},
$$
the critical impact parameter
$$
b_{sc}= \sqrt{ \frac{r_{\rm ps}^3}
{r_{\rm ps}-2M-Nr_{\rm ps}^2+\alpha r_{\rm ps}e^{-2r_{\rm ps}/(2M-\alpha \ell)} } },
$$
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 [2507.15298].

## 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
$$
ds^{2} = -\left( 1-\frac{2m}{r} +\frac{N}{r^{3\omega+1}} +\alpha e^{-\frac{r}{m-\alpha l/2}} \right)du^{2} -2\,du\,dr +r^{2}d\Omega^{2},
\qquad
m=\mathcal M+\frac{\alpha l}{2}.
$$
For the Bondi-type analysis, the treatment specializes to \(\omega=0\), so that the large-\(r\) geometry becomes
$$
ds^{2} = -\left(1-\frac{2m}{r}+\frac{N}{r}\right)du^{2}-2\,du\,dr+r^{2}d\Omega^{2}.
$$
This is then matched to a Bondi-Sachs expansion in which the News tensor is chosen as
$$
N_{AB}=A_0\,\mathrm{sech}^{2}(u)\,Y
\begin{pmatrix}
1 & 0\\
0 & -\sin^2\theta
\end{pmatrix},
$$
and the Bondi mass balance equation is
$$
\dot M = \frac{1}{4}D_A D_B N^{AB} -\frac{1}{8}N_{AB}N^{AB} -4\pi r^2 \bar T_{uu}.
$$
By comparison with the asymptotic metric, the Bondi mass is identified as
$$
M_B(u,\theta)=M-m+\frac N2,
$$
and on the equatorial plane the explicit expression includes the pulse-dependent \(\tanh u\) term together with the background shift
$$
-\mathcal M-\frac{\alpha l}{2}+\frac N2.
$$
In that formulation, both the surrounding-field parameter \(N\) and the hair parameters \((\alpha,l)\) leave an imprint on asymptotic mass measurements [2404.08026].

The same study treats memory through neighboring geodesics in the pulse-deformed background
$$
ds^{2} = -f(r)du^{2}-2dudr +(r^{2}+rH(u))d\theta^{2}
 +(r^{2}-rH(u))\sin^{2}\theta d\phi^{2},
$$
with
$$
f(r)=1-\frac{2m}{r}+\frac{N}{r^{3\omega+1}},
\qquad
H(u)=A\,\mathrm{sech}^{2}(u-u_0).
$$
Displacement memory is measured through \(\Delta r\) and \(\Delta u\); velocity memory through \(\Delta V^r\) and \(\Delta V^u\). The reported trends are that increasing \(N\) decreases \(\Delta r\) and \(\Delta V^r\) but increases \(\Delta u\) and \(\Delta V^u\), while increasing \(\alpha\) produces the opposite pattern in the \(r\)- and \(u\)-channels. The treatment imposes \(\omega>-\frac13\) 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 [2404.08026].

## 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,
$$
f(r)=1-\frac{2M}{r}-N\,r^{-(3\omega_q+1)}+\alpha e^{-r/\sigma}-\frac{\Lambda}{3}r^2,
\qquad
\sigma=M-\frac{\alpha\ell}{2}.
$$
The horizon equation \(f(r_h)=0\) yields an implicit mass-radius relation because \(\sigma\) itself depends on \(M\). The Hawking temperature is obtained from
$$
T_H=\frac{f'(r_h)}{4\pi},
$$
with
$$
f'(r)=\frac{2M}{r^2}+(3\omega_q+1)N\,r^{-(3\omega_q+2)}
-\frac{\alpha}{\sigma}e^{-r/\sigma}-\frac{2\Lambda}{3}r,
$$
so that
$$
T_H=
\frac{1}{4\pi}\left[
\frac{2M}{r_h^2}
+(3\omega_q+1)N\,r_h^{-(3\omega_q+2)}
-\frac{\alpha}{\sigma}e^{-r_h/\sigma}
-\frac{2\Lambda}{3}r_h
\right].
$$
Entropy is taken to obey the usual area law,
$$
S=\pi r_h^2,
$$
the heat capacity is
$$
C=\frac{\partial M}{\partial T_H}
=\frac{\partial M/\partial r_h}{\partial T_H/\partial r_h},
$$
and the Gibbs free energy is
$$
G=M-T_HS.
$$
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 [2605.25245].

The same work studies Hawking-radiation sparsity. For bosons, the number flux is
$$
\frac{d^2N}{dt\,d\varpi}=
\frac{1}{2\pi}\sum_{l=0}^{\infty}(2l+1)
\frac{\Gamma_l(\varpi)}{e^{\varpi/T_H}-1},
$$
with emission rate
$$
\dot N=
\frac{1}{2\pi}\sum_{l=0}^{\infty}(2l+1)\int_0^\infty
\frac{\Gamma_l(\varpi)}{e^{\varpi/T_H}-1}\,d\varpi.
$$
The sparsity parameter is
$$
\eta=\frac{\tau_{\rm gap}}{\tau_{\rm loc}}
=\frac{\varpi_\star}{2\pi\dot N},
$$
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
$$
\mathcal A_{\rm eff}=\pi b_c^2,
\qquad
b_c^2=\frac{r_{\rm ph}^2}{f(r_{\rm ph})},
$$
where the photon sphere satisfies
$$
r_{\rm ph}f'(r_{\rm ph})-2f(r_{\rm ph})=0.
$$
The greybody analysis reduces the Klein-Gordon equation to
$$
\frac{d^2u}{dr_*^2}+(\omega^2-V(r))u=0,
$$
with tortoise coordinate \(dr_*/dr=1/f(r)\) and effective potential
$$
V(r)=f(r)\left[\frac{l(l+1)}{r^2}+\frac{f'(r)}{r}\right].
$$
A lower bound on the transmission coefficient is then given by
$$
|T_b|\ge
\mathrm{sech}^{2}\!\left(
\int_{r_h}^{+\infty}\frac{V}{2\omega f(r)}\,dr
\right).
$$
Within that framework, higher angular momentum suppresses transmission, the exponential hair gives only mild deviations from Schwarzschild for the chosen parameters, and positive \(\Lambda\) can strongly suppress the bound [2605.25245].

## 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 \(p_r=-\rho\) and produce exponential or de Sitter-core behavior, but they do not introduce a Kiselev quintessence parameter \(\omega\) or the standard term \(c/r^{3\omega+1}\) [2304.03263]. Static hairy black holes obtained through extended geometric deformation of Schwarzschild likewise produce anisotropic matter and a logarithmic-plus-\(1/r^2\) correction,
$$
F(r)=1-\frac{2M}{r}+\frac54\frac{\alpha M}{r^2}+\frac{\alpha M}{r^2}\ln\!\left(\frac{r}{r_H}\right),
$$
but they are not written as Kiselev fluids [2303.04119].

The same caution applies to phenomenological descendants of the hairy Schwarzschild solution. Strong and weak lensing analyses based on
$$
f(r)=1-\frac{2M}{r}+\alpha e^{-r/(M-\ell_0/2)}
$$
treat the hair as an additional source from the surroundings, “such as dark matter,” yet do not derive a Kiselev equation of state [2205.06052]. 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 [2306.10459]. Geodesic studies of related decoupling-generated black holes with primary hairs \((\alpha,\ell_0)\) provide methodology for ISCO, MBO, and photon-sphere analysis, but again without an explicit Kiselev fluid sector [2107.01146].

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 \(n>1\) and the dominant energy condition selecting \(1\le n\le 2\) [2006.06735]. A different analogue arises in Lorentz-breaking massive gravity, where the power-law deformation
$$
\alpha(r)=1-\frac{2M}{r}-\frac{Q}{r^\lambda}
$$
is thermodynamically rich but sourced by a scalar sector rather than by Kiselev matter [1203.4222]. 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 \(\rho\), \(p\), and \(p_T\) are identified [1112.3286].

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 \(N/r^{3\omega+1}\) and an exponential hair correction controlled by \((\alpha,l)\), together with a source that is explicitly anisotropic and naturally organized as a background fluid plus a distinct hair sector [2307.04556].

Source: https://www.emergentmind.com/topics/hairy-kiselev-black-hole