Charged Kiselev Black Hole Dynamics

• Charged Kiselev black hole is a solution to Einstein's equations that incorporates electric/magnetic charge and quintessence fluid, modifying horizon structures and geodesic motion.
• It enables exploration of gravitational lensing, modified thermodynamics, and phase transitions through extensions in nonlinear electrodynamics and alternative gravity theories.
• Its rich dynamics offer insights into chaotic motion, photon spheres, and periapsis shifts, illuminating astrophysical signatures in anisotropic fluid environments.

A charged Kiselev black hole is a solution to the Einstein field equations representing a static, spherically or axisymmetrically symmetric black hole embedded in an anisotropic fluid (“quintessence-like”) environment and endowed with electric and/or magnetic charge. The metric generalizes both Reissner–Nordström and Kiselev’s original quintessence black hole by incorporating a barotropic fluid of equation-of-state parameter $w_q$ and a charge-dependent term. Extensions to nonlinear electrodynamics, magnetized backgrounds, and alternative gravity have been developed, enabling a broad exploration of horizon structure, geodesic motion, thermodynamics, chaos, and astrophysical signatures.

1. Geometric Structure and Charged Kiselev Metric

The canonical spherically symmetric charged Kiselev black hole metric is given by

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

where the lapse function is

$$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{c}{r^{3w_q+1}}\,.$$

Here %%%%1%%%% is the ADM mass, $Q$ is the electric charge, $w_q$ the equation-of-state parameter ($p = w_q\rho$), and $c>0$ a normalization linked to the fluid density $[c]=[L]^{3w_q+1}$ (Majeed et al., 2015, Gao et al., 2022, Azreg-Aïnou et al., 2017).

In more general constructions, e.g., in power–Maxwell electrodynamics (Dariescu et al., 2023), the lapse function takes the form

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

with $p = - (3w_q + 1)$ and $k$ a charge-dependent constant, and exact forms differ depending on whether the electromagnetic source is electric or magnetic.

The magnetized Kiselev black hole, incorporating an external poloidal magnetic field, is described by the axisymmetric line element

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

with $\Lambda(r,\theta) = 1 + B^2 r^2 \sin^2\theta$ (Lungu, 7 Apr 2025).

Extensions to alternative gravity (Rastall theory) modify the fall-off exponent and effective fluid properties, while retaining the generic form

$$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - N_s r^{-\delta}\,,$$

where $\delta$ is a function of Rastall parameters and the fluid equation-of-state (Heydarzade et al., 2017).

2. Horizons, Singularities, and Causal Structure

The event horizon(s) are located at the real, positive roots of $f(r) = 0$. For $w_q = -2/3$, $c > 0$,

$f(r) = 1 - \frac{2M}{r} - c r\,,$

leading to a quadratic equation with roots

$r_\pm = \frac{1 \pm \sqrt{1-8cM}}{2c}\,,$

where $r_-$ is the black hole (event) horizon and $r_+$ a cosmological/quintessence horizon (Lungu, 7 Apr 2025, Majeed et al., 2015). In the general charged case,

$$1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{c}{r^{3w_q+1}} = 0$$

is solved numerically. CQBH may possess two, three, or one positive real root depending on ($M,Q,c,w_q$), with possible merging at extremality. The central curvatural singularity is always present at $r = 0$.

For magnetized metrics, the field $B$ appears only in multiplicative factors and does not affect the locations of the Killing horizons (Lungu, 7 Apr 2025).

3. Particle and Photon Dynamics

Null Geodesics and Photon Spheres

For null geodesics, the equation for motion in equatorial symmetry $\theta = \pi/2$ reduces to

$$\dot{r}^2 + V_{\rm eff}(r) = E^2\,,\qquad V_{\rm eff}(r) = \frac{L^2}{r^2} f(r)\,,$$

with standard conserved energy $E$ and angular momentum $L$ (Azreg-Aïnou et al., 2017, Lungu, 7 Apr 2025). The photon sphere is determined by

$\frac{dV_{\rm eff}}{dr}\bigg|_{r_{\rm ph}} = 0\,,$

which yields a single unstable circular orbit between the horizons. For the magnetized Kiselev black hole, an algebraic quartic for $r_{\rm ph}$ incorporates both $B$ and $c$ (Lungu, 7 Apr 2025).

Charged Particle Motion

Test particles of mass $m$ and charge $q$ obey

$m^2 \dot{r}^2 + V_{\rm eff}(r) = E^2$

with

$$V_{\rm eff}(r) = f(r)\left(m^2 + \frac{L^2}{r^2}\right) - \left(E - \frac{qQ}{r}\right)^2\,.$$

Circular orbits satisfy $V_{\rm eff}(r_0) = 0$ and $V'_{\rm eff}(r_0) = 0$ (Gao et al., 2022). In the presence of nonlinear or magnetic charge (Power–Maxwell or magnetized spacetimes), the effective potentials generalize accordingly; for magnetic backgrounds, charged trajectories are confined to Poincaré cones, with the cone angle determined by $\varepsilon Q_m$ (Dariescu et al., 2023).

4. Gravitational Lensing, Deflection, and Periapsis Shifts

The lensing of photons by charged Kiselev black holes has been analyzed both via perturbative and exact approaches. The photon bending angle for sources and observers at finite distances is given by

$$\alpha = 2 \int_0^{u_0} \frac{du}{\sqrt{u_0^2 f(u_0) - u^2 f(u)}} - \pi - \dots + \arcsin(\cdots)\,,$$

where $u_0 = 1/r_0$ is the inverse-radius of closest approach (Azreg-Aïnou et al., 2017). In strong-field expansions around the photon sphere, leading corrections due to mass ($M$), electric charge ($Q$), and quintessence are obtained in explicit series (Azreg-Aïnou et al., 2017, Lungu, 7 Apr 2025).

Periapsis advance of bounded orbits for both electric and magnetic Kiselev spacetimes exhibits that, while uncharged particles experience always-prograde shifts, charged particles can have retrograde precession. For nearly circular orbits, the analytic shift is

$$\Delta\varphi_P = 2\pi\left[\frac{1}{\sqrt{A}} - 1\right]$$

with $A = (\omega_r/\omega_\varphi)^2$ and $\omega_r,\omega_\varphi$ the epicyclic frequencies (Dariescu et al., 6 Aug 2025). The sign of $\Delta\varphi_P$ depends sensitively on the Lorentz force term and can be reversed by sufficiently large test charge or external field.

5. Thermodynamics and Phase Structure

The thermodynamics of charged Kiselev black holes closely parallels Reissner–Nordström, but acquires modifications from the fluid term. The surface gravities at the horizons are

$\kappa_\pm = \tfrac{1}{2} f'(r_\pm),$

yielding Hawking temperatures $T_\pm = \kappa_\pm/(2\pi)$. Entropies obey the area law $S_\pm = \pi r_\pm^2$; Komar energies and irreducible masses are also modified by the additional term (Majeed et al., 2015).

Heat capacities read

$$C_{Q,\pm} = \frac{2\pi r_\pm^2 (r_\pm^2 - Q^2 + c r_\pm^{1-3w})}{(3Q^2 - r_\pm^2) - c(3w+1) r_\pm^{-3w}}$$

and diverge when the denominator vanishes, signaling a second-order phase transition (Majeed et al., 2015). The universality of thermodynamic product relations (e.g., between areas/entropies at different horizons) holds only in special cases ($w\geq -1/3$), otherwise all products depend on the mass (Majeed et al., 2015, Heydarzade et al., 2017).

6. Nonlinear and Magnetized Extensions

In “power-Maxwell” electrodynamics, the electromagnetic Lagrangian is generalized to $L(F) = -\alpha F^q$, with metric function $f(r) = 1 - 2M/r - k r^p$, and parameters set by the charge and exponent $p = -(3w_q+1)$ (Dariescu et al., 2023). The resulting charged Kiselev black hole displays both electric and magnetic configurations, with the allowed horizon structure and stable orbits depending on the nonlinearity.

The “magnetized Kiselev” black hole incorporates a nontrivial $\theta$-dependence via $\Lambda(r,\theta)$, leading to axisymmetry and modified photon dynamics. The effect of the magnetic field is to increase the photon sphere radius and the lensing deflection angle, while quintessence shrinks the photon sphere and reduces the splitting between lensed images (Lungu, 7 Apr 2025).

7. Chaotic Dynamics and Maldacena–Shenker–Stanford Bound

The Lyapunov exponent $\lambda$ for radial perturbations of circular orbits in charged Kiselev black holes can be computed using

$$\lambda^2 = \frac{f(r_0)}{2(E - qQ/r_0)^2} \left[ f''(r_0)(m^2 + L^2/r_0^2) + \frac{6L^2}{r_0^4} f(r_0) - \frac{2qQ}{r_0^2} f'(r_0) \right]\,.$$

The chaos bound $\lambda \leq 2\pi T_H$ is saturated at the horizon ($r=r_+$), but can be violated at finite radii outside the horizon for specific ranges of black hole charge, particle charge-to-mass ratio, and small quintessence normalization $c$ (Gao et al., 2022). Increasing $c$ suppresses such violations.

8. Stress-Energy Structure and Energy Conditions

The Kiselev stress-energy can be decomposed as a sum of a perfect fluid (potentially dark-energy-like) and a (possibly position-dependent) electromagnetic field. For parameter choices satisfying $K w (1+w) < 0$, this decomposition is both mathematically and physically well-defined, and the null energy condition (NEC) is satisfied (Boonserm et al., 2019). In power–Maxwell and Rastall generalizations, the effective equation of state and fall-off exponent for the surrounding fluid are further modified, driving a range of possible causal and thermodynamic behaviors (Heydarzade et al., 2017).

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References:

• (Lungu, 7 Apr 2025): Null geodesics around a magnetized Kiselev black hole
• (Majeed et al., 2015): Thermodynamic Relations for Kiselev and Dilaton Black Hole
• (Dariescu et al., 2023): Charged particles in the background of the Kiselev solution in power-Maxwell electrodynamics
• (Boonserm et al., 2019): Decomposition of total stress-energy for the generalised Kiselev black hole
• (Gao et al., 2022): Chaos bound and its violation in charged Kiselev black hole
• (Dariescu et al., 6 Aug 2025): Periapsis shifts in the electric and magnetic Kiselev black hole spacetimes
• (Azreg-Aïnou et al., 2017): Strong Gravitational Lensing by a Charged Kiselev Black Hole
• (Heydarzade et al., 2017): Black Hole Solutions Surrounded by Perfect Fluid in Rastall Theory