Kiselev Black Hole Solution

• The Kiselev black hole solution is a static, spherically symmetric model where the Schwarzschild metric is modified by the inclusion of an anisotropic fluid characterized by a free parameter w.
• It generalizes classical black hole models by introducing a power-law matter term in the metric function, linking to nonlinear electrodynamics, modified gravity, and various state equations.
• This framework influences horizon structure, thermodynamic behavior, and gravitational lensing, making it pivotal for understanding complex astrophysical and cosmological phenomena.

The Kiselev black hole solution describes a static, spherically symmetric black hole surrounded by an anisotropic fluid whose equation of state is characterized by a free parameter $w$. It generalizes the Schwarzschild geometry by introducing a power-law matter term in the metric function, providing a flexible framework for modeling black holes embedded in quintessential, radiation, dust, and other fluid environments. Recent developments have connected the Kiselev metric to nonlinear electrodynamics, modified gravity, black hole thermodynamics including Rényi-type entropies, regular black hole models, and extensions to rotation and cosmological backgrounds.

1. Metric Structure and Anisotropic Fluid Source

The canonical Kiselev metric in Schwarzschild-like coordinates $(t,r,\theta,\varphi)$ is

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta\, d\varphi^2)$$

with

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

where %%%%2%%%% is the ADM mass, $c$ the fluid density normalization, and $w$ is a constant barotropic index $(p_r=w\rho)$ (Dariescu et al., 2022). The stress–energy tensor of the anisotropic fluid is

$$T^\mu{}_\nu = \mathrm{diag}(-\rho, p_r, p_t, p_t),\qquad p_r = w\rho, \quad p_t = \frac{1}{2}(1+3w)\rho$$

(Abbas et al., 2019).

For $w$ within $(-1, -1/3)$ and $c>0$, the energy density is positive and the spacetime admits two horizons—a black hole and a cosmological horizon—analogous to Schwarzschild–de Sitter or quintessence black holes.

2. Horizon Structure, Energy Conditions, and Limiting Cases

Horizons are roots of $f(r) = 0$,

$r^{3w+1} - 2M r^{3w} - c = 0$

with explicit solutions for special $w$, e.g., $w=-2/3$ yields a quadratic

$$c\,r^2 - r + 2M = 0 \implies r_\pm = \frac{1 \pm \sqrt{1 - 8Mc}}{2c}$$

with a two-horizon structure for $c < 1/(8M)$ (Boonserm et al., 2019, Abbas et al., 2019).

Energy conditions depend critically on the fluid parameters:

• The null energy condition (NEC) requires $\rho + p_t \geq 0$, and is violated for $Kw(1+w) > 0$, corresponding to exotic fluids (Boonserm et al., 2019).
• The stress–energy tensor can be decomposed into a perfect fluid plus either an electromagnetic or scalar field for $N$-component extensions (Boonserm et al., 2019).

Limiting cases include:

• $w \to -1$: Cosmological constant, $f(r) = 1 - 2M/r - c r^2$ (Schwarzschild–(A)dS).
• $w = 0$: Dust, $f(r) = 1 - 2M/r - c/r$.
• $w = 1/3$: Radiation, $f(r) = 1 - 2M/r - c/r^2$.
• $w = -1/3$: Border of acceleration, resulting in a constant shift (Abbas et al., 2019, Santos, 27 Nov 2024).

3. Kiselev Solution in Nonlinear Electrodynamics and Power–Maxwell Theory

The Kiselev geometry can be derived as an exact solution of Einstein equations coupled to nonlinear electrodynamics, specifically "power–Maxwell" theory: $\mathcal{L}(F) = -\alpha (F_{\mu\nu}F^{\mu\nu})^q$ where $q$ is a real power. Both electric and magnetic ansatzes can generate the Kiselev form, with distinct relations between the power $q$, the equation-of-state parameter, and the metric exponent (Dariescu et al., 2022). For example, the metric retains

$f(r) = 1 - \frac{2M}{r} - k r^{p}$

with $p$ related to $w$ and the electrodynamic parameters. This demonstrates that the Kiselev metric naturally emerges in the context of nonlinear sources, not merely as a phenomenological fluid.

Thermodynamic properties such as the Hawking temperature,

$$T_H = \frac{1}{4\pi} \Big( \frac{2M}{r_+^2} - k p r_+^{p-1} \Big)$$

and the distinctive heat capacity peak (Schottky anomaly), signal the potential for cyclic heat engine behavior (Dariescu et al., 2022).

4. Extensions: Magnetized Solutions, Rotation, and Modified Gravity

Magnetized Kiselev Black Holes

A Harrison–Ernst transformation applied to the seed Kiselev metric yields a family with asymptotically Melvin–type magnetic fields: $$ds^2 = - f(r)\Lambda(r,\theta)^2 dt^2 + \frac{\Lambda(r,\theta)^2}{f(r)} dr^2 + \Lambda(r,\theta)^2 r^2 d\theta^2 + \frac{r^2 \sin^2\theta}{\Lambda(r,\theta)^2} d\varphi^2$$ with $f(r) = 1 - 2M/r - k r$ and $\Lambda(r,\theta) = 1 + B^2 r^2 \sin^2\theta$ (Lungu, 7 Apr 2025). The magnetic field parameter $B$ modifies photon spheres and increases lensing effects.

Rotating Kiselev Metrics and Modified Gravity

Rotating reductions generalize the metric via Newman–Janis-type algorithms. In spherical symmetry, the seed metric is extended to

$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2$

and rotation introduces additional parameters (spin $a$), shifting the horizon structure. The effective mass and horizon radii are modified, e.g.,

$$\Delta(r) = r^2 + a^2 - 2 r M(r) + \frac{K}{r^{d-2}}$$

where $K$ and $d$ depend on the specific modified gravity theory, such as $f(R,T)$ gravity (Ghosh et al., 2023, Santos et al., 2023, Benali et al., 2 Jun 2024, Czinner et al., 30 Aug 2025).

5. Thermodynamic Properties: Rényi Entropy and Cosmic Censorship

By demanding the black hole thermodynamic temperature derived from Rényi entropy matches the surface gravity (Hawking temperature), the metric function is uniquely fixed to the Kiselev form: $$\bar f(r,\lambda) = 1 - \frac{2M}{r} - 2\pi\lambda M r$$ with $K = 2\pi\lambda M$ and $w = -2/3$ (Czinner et al., 23 Apr 2025). For such solutions,

$$T_H = \frac{1 - \lambda \pi r^2}{4\pi r (1 + \lambda \pi r^2)}$$

and the Rényi entropy

$$S_R = \frac{1}{\lambda} \ln \left[1 + \lambda S_{BH}\right], \quad S_{BH} = \pi r_h^2$$

satisfies the first law $dM = T_H\,dS_R$. The heat capacity is strictly negative, pointing to local thermodynamic instability. The third law enforces an upper mass bound, $M < 1/(4\sqrt{\pi\lambda})$, prohibiting horizonless solutions (cosmic censorship).

6. Regular Black Hole Extensions and Horizon Stability

Allowing the equation-of-state parameter $w = w(r)$ to vary with radius provides general regular solutions, avoiding singularities at the origin. For specific choices of $w(r)$, one obtains metrics

$f(r) = 1 - \frac{C_3 r^2}{(a^2 + r^2)^{3/2}}$

with finite curvature invariants at $r=0$, satisfying both weak and strong energy conditions over suitable parameter ranges (Santos, 27 Nov 2024).

In asymptotically AdS backgrounds, the Kiselev metric with $w < -1/3$ is unstable against brane nucleation, rendering such solutions nonviable for AdS dark energy environments (Xia et al., 1 Dec 2025).

7. Geodesics, Lensing, and Observational Signatures

The null geodesic structure and gravitational lensing signatures for Kiselev black holes, including charge and magnetization, are rich. The parameter $w$ and the density $c$ impact photon sphere radii, shadow shapes, and deflection angles. In particular, terms proportional to $r$ or higher powers (for $w$ values such as $-2/3$) introduce nontrivial corrections to the bending of light which may be observationally distinguishable (Lungu, 7 Apr 2025, Abbas et al., 2019, Shchigolev et al., 2016).

8. Summary Table of Metric Variants

Solution Type	Metric Function $f(r)$	Source/Context
Static Kiselev	$1-\frac{2M}{r}-\frac{c}{r^{3w+1}}$	Original, anisotropic fluid
Magnetized Kiselev	$1-\frac{2M}{r}-k r$; $\Lambda(r,\theta)$	Harrison–Ernst magnetization
Power–Maxwell Electrodynamics	$1-\frac{2M}{r}-k r^{p}$	Nonlinear electrodynamics
Rotating Kiselev	See $\Delta(r)$, $\Sigma$, etc.	Newman–Janis rotation, $f(R,T)$ or GR
Rényi–Kiselev	$1-2M/r-2\pi\lambda M r$	Thermodynamics, cosmic censorship
Regular (Generalized $w(r)$)	$1 - C_3 r^2 / (a^2+r^2)^{3/2}$, etc.	Radial-varying $w(r)$, regular center

• Null geodesics and magnetized Kiselev: (Lungu, 7 Apr 2025)
• Power–Maxwell electrodynamic realization: (Dariescu et al., 2022)
• Stress–energy decomposition and NEC: (Boonserm et al., 2019)
• Strong lensing and shadow structure: (Abbas et al., 2019, Shchigolev et al., 2016, Czinner et al., 30 Aug 2025, Benali et al., 2 Jun 2024)
• Quantum modified gravity and horizon structure: (Hua et al., 24 Nov 2024)
• Regular solution with $w(r)$: (Santos, 27 Nov 2024)
• Rényi black hole thermodynamics: (Czinner et al., 23 Apr 2025)
• AdS instability: (Xia et al., 1 Dec 2025)
• Bardeen–Kiselev with cosmological constant: (Rodrigues et al., 2022, Wu et al., 2022)
• Rotating extensions: (Ghosh et al., 2023, Santos et al., 2023, Benali et al., 2 Jun 2024)

The Kiselev black hole solution and its generalizations provide a versatile and rigorous framework for analyzing black-hole physics in the presence of general anisotropic fluids, nonlinear electrodynamics, modified gravity, and nonstandard thermodynamics. This continues to have direct implications for lensing, horizon structure, cosmic censorship, and the stability of black holes in complex astrophysical and cosmological settings.