Charged Kiselev Black Holes

• Charged Kiselev black holes are exact solutions to the Einstein–Maxwell equations that incorporate electric charge and an anisotropic fluid via a quintessence-like term.
• Their thermodynamic behavior reveals mass-dependent horizon products, phase transitions, and stability conditions with practical implications for gravitational lensing and orbital dynamics.
• Recent extensions include analyses in modified gravity, black string configurations, and chaos bounds, offering new insights into exotic matter effects and energy condition violations.

A charged Kiselev black hole is a family of exact solutions to the Einstein–Maxwell equations representing a spherically symmetric black hole with electric (or magnetic) charge, embedded in a generic anisotropic quintessential fluid or more generally an anisotropic perfect fluid. The defining feature is the presence of a term in the metric proportional to $1/r^{3\omega+1}$, with equation-of-state parameter $\omega$ controlling the surrounding fluid’s pressure-to-density ratio. These solutions generalize the Reissner–Nordström metric by the inclusion of dark-energy–like or exotic matter surrounding the black hole, and admit a broad parameter space encompassing radiation, dust, cosmological constant, quintessence, or phantom fields. Recent developments have extended these solutions to include charged anisotropic fluids, power–Maxwell electrodynamics, higher derivative corrections, and various modified gravity scenarios.

1. Metric Structure and Physical Interpretation

The line element for a charged Kiselev black hole is typically parametric in ADM mass $M$, electric charge $Q$, and a fluid normalization parameter $\sigma$: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\, dr^2 + r^2 \left(d\theta^2 + \sin^2\theta\, d\phi^2\right)$$ with

$$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{\sigma}{r^{3\omega + 1}}$$

The $-\sigma/r^{3\omega+1}$ term models the back-reaction of the surrounding anisotropic fluid, typically generalized quintessence with $-1 < \omega < -1/3$, but can encode dust ($\omega=0$), radiation ($\omega=1/3$), cosmological constant ($\omega=-1$), or phantom energy ($\omega<-1$).

Charge may be sourced by a nonlinear electric or magnetic field. For the magnetic case, the presence of a magnetic monopole endows the spacetime with residual SO(3) symmetry and has marked consequences for test particle dynamics (Dariescu et al., 2023, Dariescu et al., 6 Aug 2025). The charged anisotropic fluid may itself carry current, thereby modifying both the gravitational and electromagnetic field equations (Dariescu et al., 6 Aug 2025).

2. Thermodynamics, Horizon Products, and Phase Structure

Charged Kiselev black holes exhibit a rich thermodynamic structure (Majeed et al., 2015). The event ($r_+$) and Cauchy ($r_-$) horizons are determined by $f(r)=0$. Key thermodynamic quantities at each horizon include:

• Surface gravity: $$\kappa_\pm = \frac{1}{2} \left. \frac{df}{dr} \right|_{r_\pm}$$
• Hawking temperature: $T_\pm = \kappa_\pm/(2\pi)$
• Horizon area: $\mathcal{A}_\pm = 4\pi r_\pm^2$
• Bekenstein–Hawking entropy: $S_\pm = \mathcal{A}_\pm/4$
• Komar energy: $E_\pm = 2S_\pm T_\pm$

Products of surface gravities, temperatures, and Komar energies across the two horizons depend explicitly on the ADM mass and are thus not universal. In contrast, in RN-type cases (surrounded by radiation or dust), the product of areas and entropies is mass-independent and thus universal; $$\mathcal{A}_+ \mathcal{A}_- = 16\pi^2(Q^2-\sigma_r)^2$$.

The first law of black hole thermodynamics is verified: $$dM = \mathcal{T}_\pm d\mathcal{A}_\pm + \Phi_\pm dQ$$ with an effective surface tension $\mathcal{T}_\pm = \kappa_\pm/(8\pi)$ and electromagnetic potential $\Phi_\pm$.

Heat capacity $C_\pm$ is computed as $$C_\pm = [2\pi r_\pm^2(r_\pm^2 - Q^2 + \sigma_r)]/[3Q^2 - 3\sigma_r - r_\pm^2]$$. Its sign signals (in)stability, and divergences identify the locus of phase transitions.

Charged dilaton black holes differ sharply: the inner horizon area and entropy vanish, annihilating all horizon products (Majeed et al., 2015).

3. Test Particle Dynamics, Precession, and Orbital Structure

Charged test particles in the electric Kiselev black hole background move in planar orbits, with radial motion described by: $$\dot r^2 = [E - \epsilon A_t]^2 - f(r)[1 + L^2/r^2]$$ and effective potential

$V_+(r) = \epsilon A_t + \sqrt{f(r)[1 + L^2/r^2]}$

Magnetically charged Kiselev black holes fundamentally alter orbital geometry: charged particles are forced onto Poincaré cones due to the conserved total angular momentum $\vec J = \vec L + \vec S$, with half-opening angle $\cos\alpha = |\epsilon Q_m|/|\vec J|$ (Dariescu et al., 2023, Dariescu et al., 6 Aug 2025).

Periapsis (perihelion) shift: $$\Delta\phi_P = 2\pi \left( \frac{\omega_\phi - \omega_r}{\omega_r} \right)$$ is prograde for uncharged particles, but may become retrograde for charged particles, whether in the electric or magnetic Kiselev black hole backgrounds, depending on the interplay between charge, fluid parameters, and electromagnetic field strength (Dariescu et al., 6 Aug 2025).

4. Gravitational Lensing and Photon Spheres

Strong gravitational lensing computations for a charged Kiselev black hole employ the null geodesic equation in the metric above (Azreg-Aïnou et al., 2017): $$\left( \frac{du}{d\phi} \right)^2 = \frac{1}{b^2} - u^2 f(u)$$ and expand via perturbation series in mass, charge, and quintessence parameters. The bending angle $\alpha$ includes corrections both from finite observer/source distance and quintessence: $$\alpha = 4M u_n + \frac{(15\pi - 16)M^2 - 3\pi Q^2}{4}u_n^2 - \frac{M(1 + M u_n)(u_o^2 + u_s^2)}{u_n} + \ldots$$ For all $\omega_q$, only the photon sphere (the unstable circular orbit for null rays) exists, and there are no stable closed null orbits. The charge $Q$ impacts corrections only at higher order in $1/\text{distance}^2$.

In modified gravity extensions, photon sphere existence and location are further altered by $f(R,T)$ corrections and the Kiselev parameter $\omega$, allowing for a larger domain of black hole behavior (Gashti et al., 3 Oct 2024).

5. Energy Conditions, Stress–Energy Decomposition, and Fluid Mimicry

Generic Kiselev black holes possess stress–energy tensors with intrinsic anisotropy. These can be decomposed into a perfect fluid and either electromagnetic ($p_t - p_r > 0$) or scalar field ($p_r - p_t > 0$) contributions (Boonserm et al., 2019): $$T^{ab}_\text{total} = T^{ab}_\text{fluid} + T^{ab}_\text{EM} + T^{ab}_\text{scalar}$$ The decomposition quantifies exactly how the matter content diverges from a perfect fluid, with the electromagnetic mimic as the physically acceptable regime (satisfying the null energy condition, NEC), while the scalar field mimic typically violates the NEC and may be unstable.

Generalized N-component Kiselev black holes can possess “onion-like” structures, wherein regions of electromagnetic or scalar mimicry alternate with radius, determined by the sign of $\rho'(r)$ (Boonserm et al., 2019).

6. Extensions: Modified Gravity, Black Strings, Critical Bounds, and Topological Analysis

Modified Gravity

Charged Kiselev black holes arise in Rastall theory with a modified energy–momentum conservation law. The exponent of the fluid term in the metric becomes: $$\gamma = \frac{1 + 3w_s - 6\kappa\lambda(1+w_s)}{1 - 3\kappa\lambda(1+w_s)}$$ This alters the effective equation of state felt by the black hole, allowing regular matter in Rastall theory to mimic exotic fluids in GR and vice versa (Heydarzade et al., 2017).

In higher derivative gravity, new charged solutions deviate from Kiselev metrics by deriving modified asymptotic behaviors and horizon structures without adding quintessence-like matter, instead attributing modifications to curvature corrections. “Hair” parameters and negative mass states may appear (Lin et al., 2016).

Black Strings

Charged Kiselev black holes generalize to charged rotating black strings in AdS (Barbosa et al., 3 Mar 2025), with

$$f(r) = \frac{r^2}{l^2} - \frac{2ml}{r} + \frac{N_q}{r^{3w_q+1}} + \frac{l^2 Q^2}{r^2}$$

Event horizon location, closed timelike curve conditions, and conserved charges (mass, angular momentum, electric charge) are governed by $N_q$, $w_q$, $Q$, $a$, and $l$. Thermodynamic quantities, Hawking temperature from quantum tunneling, and heat capacity reflect the impact of quintessence and charge on stability and phase transitions.

Maximum Force and Chaos Bounds

Charged Kiselev black holes surrounded by quintessential matter ($w=-2/3$) possess a maximum force bound at the horizon that is independent of mass and generally less than in pure Schwarzschild (Atazadeh, 2021). For certain fluid parameter choices, gravitational attraction is perfectly balanced by fluid repulsion, so the net force vanishes.

The Lyapunov exponent for chaos around charged Kiselev black holes can violate the chaos bound ($\lambda\leq \kappa$) at finite distance from the horizon when charge and angular momentum are large enough, especially for small normalization factor $\alpha$. However, the bound is always saturated in the near-horizon region for fixed charge-to-mass ratio (Gao et al., 2022).

Topological Thermodynamics

Quantum corrections regularize singularities and alter thermodynamic topology. Topological charges (winding numbers) calculated from Duan’s $\phi$-mapping theory via off-shell Helmholtz free energy typically stabilize at $W=+1$, but exotic choices of $\omega=-4/3$ or parameter configurations can yield $W=0$ or $W=-1$ (Sadeghi et al., 19 Aug 2024). In $f(R,T)$ gravity, similar methodology reveals that increasing the gravity parameter $\gamma$ expands the admissible parameter space and can increase the number of thermodynamic topological charges, especially in phantom field cases (Gashti et al., 3 Oct 2024).

Stability, WGC, and WCCC

Perturbative analysis of quantum-corrected RN–AdS black holes in Kiselev spacetime verifies universal relationships for extremality. Both the Weak Gravity Conjecture (WGC) and Weak Cosmic Censorship Conjecture (WCCC) hold: the black hole can never be overcharged by perturbations or absorption of charged scalar fields, so singularities are always shielded behind horizons and lighter particles with higher $q/m$ exist in the spectrum (Anand et al., 28 Oct 2024).

7. Observational and Astrophysical Implications

The range of possible periapsis shifts (both prograde and retrograde), departures from the pure perfect fluid case, non-universal thermodynamic products, and the influence of topological defects, all provide testable deviations from pure RN or Schwarzschild black holes. This suggests new avenues for indirect detection via orbital dynamics, gravitational lensing signatures, phase transition structure, or horizon stability in the strong field regime. The inclusion of charged anisotropic fluids, modified gravity, and higher derivative corrections provides a broad context for interpreting electromagnetic, gravitational wave, and thermodynamic signals from environments where dark energy–like matter or exotic fields surround astrophysical black holes.

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In summary, charged Kiselev black holes constitute a unifying framework for investigating black hole solutions interacting with generic anisotropic matter or fields. Their key features—modified horizon and thermodynamic structures, the emergence of nontrivial stress–energy decompositions, unique orbital and lensing phenomena, and intricate topological and stability properties—position them as a principal model for studying the impact of non-vacuum environments, fundamental bounds, and exotic matter on black hole physics.