Kiselev Black Holes Overview

• Kiselev black holes are static, spherically symmetric spacetimes defined by an anisotropic fluid that mimics quintessence behavior in its stress-energy profile.
• The metric structure allows for diverse parameter regimes, revealing unique horizon classifications and lensing corrections compared to standard black holes.
• Research on Kiselev black holes spans thermodynamic analysis, wave dynamics, and modified gravity, providing insights into evaporation rates and AdS stability.

Searching arXiv for recent and foundational work on Kiselev black holes to ground the article in the current literature. Kiselev black holes are static, spherically symmetric black-hole spacetimes sourced by an anisotropic fluid that is often described as “quintessence-like,” but whose stress tensor is not that of a perfect fluid. In its simplest asymptotically flat form, the metric can be written as $ds^2=-f(r)\,dt^2+f(r)^{-1}dr^2+r^2d\Omega_2^2$ with $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$, while equivalent conventions write $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$ (Xia et al., 1 Dec 2025, Shchigolev et al., 2016). The subject has developed into a broad research program spanning geodesics and lensing, thermodynamics, quasinormal spectra, modified gravity, rotating and regular extensions, and analogue-gravity realizations; at the same time, later work has emphasized that the Kiselev source is intrinsically anisotropic and should not be identified with cosmological quintessence in the strict perfect-fluid sense (Xia et al., 1 Dec 2025, Boonserm et al., 2019).

1. Definition and matter source

The defining matter sector of a Kiselev black hole is an anisotropic fluid with nonzero components $T^t_{\ t}=-\rho(r)$, $T^r_{\ r}=p_r(r)$, and $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$. In the asymptotically flat form discussed in later analyses, the source satisfies

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$

with averaged pressure $\bar p=(p_r+2p_t)/3=w\rho$ (Xia et al., 1 Dec 2025). This is the origin of the persistent but potentially misleading shorthand “quintessence”: the averaged pressure obeys a barotropic relation, but the local fluid is anisotropic.

That anisotropy is not incidental. One later decomposition shows that the Kiselev stress tensor can be mimicked by a perfect fluid plus either an electromagnetic component or a scalar-field component, depending on the sign of $K\,w(1+w)$. In the one-component model, $K\,w(1+w)<0$ corresponds to a perfect fluid plus electromagnetic field, $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$0 reduces to Schwarzschild or Kottler, and $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$1 corresponds to a perfect fluid plus scalar field; in the generalized $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$2-component case the same trichotomy is controlled by the sign of $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$3 (Boonserm et al., 2019). This decomposition quantifies the precise sense in which the Kiselev geometry fails to be a perfect-fluid spacetime.

A further reinterpretation replaces the anisotropic fluid by nonlinear electrodynamics. In Einstein–power-Maxwell theory, the geometry

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$4

is an exact solution sourced either by an electric or by a magnetic power-Maxwell field, with different powers $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$5 for the two realizations. In that formulation, the Kiselev amplitude is fixed by the electric or magnetic charge, while the source still satisfies the characteristic anisotropic relation $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$6 (Dariescu et al., 2022).

2. Metric structure, parameter regimes, and horizons

In the standard spherical form, the Kiselev metric interpolates among several familiar geometries through the equation-of-state parameter. The cases $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$7, $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$8, $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$9, $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$0, and $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$1 respectively generate Schwarzschild–de Sitter/anti–de Sitter, a linear-in-$f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$2 quintessential term, a constant shift, a dust-like $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$3 correction, and a radiation-like $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$4 term (Dariescu et al., 2022, Qu et al., 2023). In the convention used for null-geodesic calculations,

$f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$5

with $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$6; in the limit $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$7, the metric reduces to a cosmological-constant term with $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$8 in the conventions of that treatment (Shchigolev et al., 2016).

The case $f(r)=1-\frac{2m}{r}-\sigma r^{-(3w_q+1)}$9 is especially important because the metric simplifies to

$T^t_{\ t}=-\rho(r)$0

The horizons are then

$T^t_{\ t}=-\rho(r)$1

This yields a standard classification into nonextreme black holes for $T^t_{\ t}=-\rho(r)$2, an extreme Kiselev black hole at $T^t_{\ t}=-\rho(r)$3 with degenerate horizon $T^t_{\ t}=-\rho(r)$4, and a naked singularity for $T^t_{\ t}=-\rho(r)$5 (Younas et al., 2015).

A distinct regime arises in the “reduced Kiselev black hole,” obtained by setting the Schwarzschild mass term to zero and working in $T^t_{\ t}=-\rho(r)$6. Writing $T^t_{\ t}=-\rho(r)$7, the metric becomes

$T^t_{\ t}=-\rho(r)$8

with a single Killing horizon at $T^t_{\ t}=-\rho(r)$9, asymptotic flatness, and a spacelike curvature singularity at $T^r_{\ r}=p_r(r)$0. Its causal structure is the same as Schwarzschild, but the geometry is supported entirely by the anisotropic fluid sector rather than by an explicit mass term (Qu et al., 2023).

The same formalism also admits regularizations. One construction matches a Kiselev exterior to a de Sitter interior across a timelike dust shell, producing a globally regular black hole in which the $T^r_{\ r}=p_r(r)$1 singularity is replaced by a de Sitter core. Another allows the equation-of-state parameter to vary radially, $T^r_{\ r}=p_r(r)$2, and derives regular solutions with finite curvature invariants at the origin and de Sitter-like cores (Saadati et al., 2020, Santos, 2024).

3. Null geodesics, photon spheres, and lensing

For static spherical Kiselev geometries, null geodesics are conveniently written in terms of $T^r_{\ r}=p_r(r)$3. In the notation of the homotopy-perturbation treatment, the general geodesic equation becomes

$T^r_{\ r}=p_r(r)$4

with the quintessence contribution disappearing from this equation in the limit $T^r_{\ r}=p_r(r)$5 because of the prefactor $T^r_{\ r}=p_r(r)$6 (Shchigolev et al., 2016). For $T^r_{\ r}=p_r(r)$7, the added term is proportional to $T^r_{\ r}=p_r(r)$8; for $T^r_{\ r}=p_r(r)$9, it is constant, and this difference drives qualitatively different bending-angle corrections.

The homotopy-perturbation method yields analytic weak-field series for both the orbit and the deflection angle. At lowest order, the Kiselev corrections are

$T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$0

and

$T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$1

with higher-order terms involving $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$2, $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$3, and mixed $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$4 contributions. The appearance of the $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$5 term for $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$6 is the distinctive feature of that case. The same work also derives a homotopy-based root-finding series for the deflection angle that does not require assuming $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$7 and is extendable to arbitrary order in the weak-field regime (Shchigolev et al., 2016).

In the strong-lensing analysis of the $T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$8 metric, the photon effective potential is

$T^\theta_{\ \theta}=T^\varphi_{\ \varphi}=p_t(r)$9

Circular null orbits satisfy $$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$0, giving

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$1

with the unstable photon sphere identified as $$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$2. The critical impact parameter is

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$3

The exact bending angle can then be written in terms of incomplete and complete elliptic integrals, and the comparative ordering of deflections is

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$4

for the parameter ranges studied (Younas et al., 2015).

These geodesic structures persist in rotating and modified-gravity extensions, where they reappear in shadow radii, relativistic images, and photon-sphere topology. A plausible implication is that Kiselev deformations are best viewed not as a single universal lensing correction, but as a family of distinct optical signatures controlled by the exponent $$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$5 and by the sign convention chosen for the surrounding medium.

4. Thermodynamics, evaporation, and wave dynamics

In the standard Kiselev metric written as

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$6

the horizon equation is $$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$7, and the Hawking temperature is

$$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$8

Since $$\rho(r)=-p_r(r)=-\frac{3K\,w}{8\pi\,r^{3(1+w)}},\qquad p_t(r)=-\frac{3K\,w(1+3w)}{8\pi\,r^{3(1+w)}},$$9 in the quintessential range, the second term is negative, so the surrounding medium lowers the temperature relative to Schwarzschild at fixed $\bar p=(p_r+2p_t)/3=w\rho$0 (Wu et al., 17 Jun 2026). In the evaporation study based on Page’s geometrical-optics approximation, decreasing $\bar p=(p_r+2p_t)/3=w\rho$1 lowers the non-final-stage temperature and markedly prolongs the evaporation lifetime. For $\bar p=(p_r+2p_t)/3=w\rho$2 and $\bar p=(p_r+2p_t)/3=w\rho$3, the reported lifetimes are $\bar p=(p_r+2p_t)/3=w\rho$4, $\bar p=(p_r+2p_t)/3=w\rho$5, while $\bar p=(p_r+2p_t)/3=w\rho$6, essentially Schwarzschild-like (Wu et al., 17 Jun 2026).

Rainbow-gravity deformations preserve the Kiselev horizon equation but rescale the temperature by the rainbow function. In that framework, no generic remnant appears: for $\bar p=(p_r+2p_t)/3=w\rho$7 the evaporation does not halt, and only a peculiar $\bar p=(p_r+2p_t)/3=w\rho$8, $\bar p=(p_r+2p_t)/3=w\rho$9 configuration with a cosmological horizon yields a $K\,w(1+w)$0 endpoint at the degenerate horizon (Morais et al., 2021). In Einstein–power-Maxwell realizations, the event-horizon temperature is

$K\,w(1+w)$1

and the heat capacity at fixed charge is negative throughout the physical range, but it exhibits a Schottky peak characteristic of multi-horizon systems (Dariescu et al., 2022).

Wave dynamics around Kiselev backgrounds has likewise become a substantial subfield. In quantum fluctuation modified gravity, the metric is written as

$K\,w(1+w)$2

and scalar and vector perturbations obey Schrödinger-type equations with effective potentials

$K\,w(1+w)$3

Using 6th-order WKB, the real part of the quasinormal frequency decreases as the quantum-fluctuation parameter $K\,w(1+w)$4 rises, while the greybody factors are significantly affected by $K\,w(1+w)$5, $K\,w(1+w)$6, and the multipole number $K\,w(1+w)$7 (Sajjad et al., 2 Oct 2025).

5. Modified-gravity, rotating, regular, topological, and analogue extensions

Kiselev black holes have been generalized extensively in modified gravity. In $K\,w(1+w)$8 gravity, the static spherical solution takes the form

$K\,w(1+w)$9

Only the $K\,w(1+w)<0$0 case reproduces the GR exponent exactly; the other sectors acquire $K\,w(1+w)<0$1-dependent falloffs, modified energy-condition bounds, and altered Hawking temperatures (Santos et al., 2023). In quantum fluctuation modified gravity, the corresponding exponent becomes

$K\,w(1+w)<0$2

so that the GR Kiselev exponent is recovered only in the limit $K\,w(1+w)<0$3 (Hua et al., 2024).

Rotating solutions have also been constructed. In $K\,w(1+w)<0$4 gravity, the rotating Kiselev metric generated from a spherical seed has

$K\,w(1+w)<0$5

and includes Kerr for $K\,w(1+w)<0$6 and Kerr–Newman for $K\,w(1+w)<0$7, $K\,w(1+w)<0$8. The horizon structure exhibits one or two critical values of the coupling $K\,w(1+w)<0$9, depending on the spin $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$00, and the black-hole domain in $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$01 is correspondingly segmented by extremal branches (Ghosh et al., 2023). A different rotating construction, applied to the reduced Kiselev geometry, yields

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$02

with allowed spin values smaller than in the corresponding Kerr family, an approximate shadow-shape degeneracy at fixed $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$03, and elliptic shadows for certain parameter ranges (Benali et al., 2024).

Regularization has proceeded in two complementary ways. One approach lets $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$04 vary radially and solves

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$05

obtaining regular black holes with finite $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$06, $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$07, and Kretschmann scalar at the origin (Santos, 2024). Another glues a Kiselev exterior to a de Sitter interior across a timelike dust shell, producing a stationary globally regular black hole with a stable shell for specific parameter ranges (Saadati et al., 2020).

Thermodynamic-topology methods have introduced a different classification layer. For Kiselev–AdS black holes in $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$08 gravity, the Duan $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$09-mapping framework gives total topological charge $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$10 in the temperature-based method and total topological number $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$11 in the generalized Helmholtz free-energy method, while the coupled parameters $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$12 and $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$13 alter the number and arrangement of individual defects and the photon-sphere structure (Gashti et al., 2024). A related study of quantum-corrected AdS–Reissner–Nordström black holes in Kiselev spacetime finds that most quintessence-like cases remain in the $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$14 class, while a dust case can give $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$15 and a phantom case $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$16 (Sadeghi et al., 2024).

The subject has even entered analogue gravity. Starting from Gross–Pitaevskii theory, one acoustic construction reproduces

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$17

so that Schwarzschild, dust, radiation-like, quintessence, and $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$18-type Kiselev geometries all arise from a unified acoustic metric. In that setting, the quasibound-state spectra are analytically accessible, while quasinormal modes are computed by 6th-order WKB (Santos et al., 25 Jun 2025).

6. AdS viability, controversies, and open directions

The sharpest recent critique concerns anti–de Sitter embeddings. For the Kiselev–AdS metric

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$19

asymptotic AdS behavior requires $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$20, but the Seiberg–Witten brane-nucleation analysis shows that avoiding instability requires

$f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$21

This is the opposite of the dark-energy range $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$22. Accordingly, for $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$23 and $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$24, the Euclidean brane action becomes negative outside the horizon and is unbounded below, so Kiselev black holes in AdS are declared nonviable in that parameter range (Xia et al., 1 Dec 2025). Spherical horizons are the least unstable; planar and hyperbolic horizons worsen the instability.

This result is deliberately specific. It does not show that no black hole can coexist with an anisotropic dark-energy fluid in AdS; it rules out the particular Kiselev embedding with that stress tensor and metric function. The same paper explicitly leaves open the endpoint of the instability after backreaction from copious brane production (Xia et al., 1 Dec 2025).

A broader conceptual controversy concerns nomenclature. Across the later literature, the “Kiselev quintessence” is repeatedly reinterpreted as an anisotropic medium rather than a genuine cosmological quintessence field, whether through perfect-fluid-plus-field decompositions, nonlinear-electrodynamics sourcing, or modified-gravity effective stresses (Boonserm et al., 2019, Dariescu et al., 2022). This does not invalidate the geometry as a mathematical or phenomenological model, but it changes the physical reading of lensing, thermodynamic, and observational inferences drawn from it.

Several open directions remain explicit in the current literature: alternative homotopies that remove weak-field dependence in analytic lensing constructions (Shchigolev et al., 2016); alternative anisotropic-fluid black holes consistent with AdS holography (Xia et al., 1 Dec 2025); and further classes generated by tailored radial profiles $f(r)=1-\frac{2M}{r}-\frac{K}{r^{1+3w}}$25, including regular cores and nonstandard asymptotics (Santos, 2024). Taken together, these lines of work place Kiselev black holes at the intersection of black-hole optics, anisotropic matter modeling, modified gravity, and analogue spacetime engineering, while keeping the status of the source interpretation and AdS consistency under active scrutiny.