Charged Non-Rotating Black Hole

• Charged non-rotating black hole is a solution of the Einstein–Maxwell equations featuring mass, electric charge, and zero angular momentum, described by the Reissner–Nordström metric.
• It reveals complex interactions between gravitational and electromagnetic forces that affect particle dynamics, ISCO stability, and the formation of accretion structures.
• Research on these black holes probes alternative gravity theories and quantum corrections, offering insights into high-energy astrophysical phenomena and particle acceleration.

A charged non-rotating black hole is a solution of the Einstein–Maxwell equations characterized by nonzero electric charge and vanishing angular momentum. Such solutions, most notably the Reissner–Nordström (RN) metric, serve as fundamental models in the paper of classical and quantum gravity, relativistic astrophysics, horizon and singularity structure, as well as the dynamics of fields and matter in strong gravitational and electromagnetic backgrounds. Charged non-rotating black holes are central to analyzing the interplay of gravitational and electromagnetic forces, particle acceleration, stability of accretion flows, and a variety of phenomena in both standard and modified gravity theories.

1. Spacetime Geometry and Electromagnetic Structure

The prototypical charged, non-rotating black hole in four dimensions is described by the RN metric: $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta\, d\phi^2),$$ with

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

where $M$ and $Q$ are the mass and electric charge, respectively. Horizons are located at $r_{\pm} = M \pm \sqrt{M^2 - Q^2}$. The electromagnetic field comprises only a radial component: $A_t = -Q/r$, with field tensor $F_{tr} = Q/r^2$. The singularity at $r=0$ is spacelike and stronger than for Schwarzschild.

In extensions involving external fields or new physics, the local structure is altered:

• Immersion in an external, asymptotically uniform magnetic field leads to an axially symmetric “magnetized Reissner–Nordström” metric, featuring an additional field parameter $B$ and metric functions $H(r, \theta)$, $\omega(r, \theta)$ that encode the interplay between $Q$, $B$, and geometry (Shaymatov et al., 2021).
• In Lorentz-violating bumblebee gravity, the charged spherically symmetric solution admits a modified metric function,

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

with $\ell = \xi b^2$, providing an effective modification to both the asymptotic geometry and 1/r² charge-tail, while preserving the principal horizon structure (Liu et al., 11 Jul 2024).

• The Bertotti–Robinson (AdS₂ × S²) solution demonstrates how a RN hole can be globally matched to a universe with a homogeneous electric field, modifying asymptotics to AdS (Alekseev, 8 Nov 2025).

2. Test Particle Dynamics and Effective Potentials

The joint gravitational and EM field decisively alters geodesics, especially for charged test particles. The Hamiltonian formalism yields the equations of motion: $$p^\mu = g^{\mu\nu}(\pi_\nu - q A_\nu), \quad H = \frac{1}{2}g^{\mu\nu}(\pi_\mu - qA_\mu)(\pi_\nu - qA_\nu),$$ where $\pi_\mu$ is the canonical momentum, and $q$ is the test particle charge. Stationarity and axial symmetry ensure conservation of energy $E$ and angular momentum $L$.

The resulting radial equation can be expressed as: $\frac{1}{2}\dot{r}^2 + R(r) = 0$ or, equivalently,

$\dot{r}^2 = [E - V_{\rm eff}(r)][E - V_{-}(r)],$

where the physically relevant effective potential for charged particles—especially in the magnetized case—reads

$$V_{\rm eff}(r) = \omega(L - q A_\phi) - qA_t + \sqrt{\frac{H\,r^2[H(L - q A_\phi)^2 + r^2]}{r^2 (\omega-1)\omega + f H^2}}$$

(Shaymatov et al., 2021).

Circular orbits at $r = r_c$ are characterized by

$$V_{\rm eff}(r_c) = E, \quad \left. \frac{dV_{\rm eff}}{dr}\right|_{r_c} = 0,$$

while the innermost stable circular orbit (ISCO) is at the inflection point: $$\left. \frac{d^2V_{\rm eff}}{dr^2}\right|_{r_c} = 0.$$ The ISCO structure is highly sensitive to the interplay of $Q$, $q$, and $B$: positive (negative) test charge leads to an outward (inward) shift, and a nonzero magnetic field further compresses the ISCO (Shaymatov et al., 2021).

3. Accretion Structures and Matter Configurations

In the presence of external magnetic fields and/or test charge, charged-fluid configurations—such as tori and polar clouds—can stably exist around non-rotating charged black holes.

• The construction of these structures utilizes analytic solutions for the Euler equations in the test-field approximation, allowing for both toroidal (equatorial) and cloud (axial) geometries stabilized by the balance of gravitational, electromagnetic, and centrifugal forces (Kovář et al., 2014).
• The pressure and density distribution are encoded in a scalar field $h(r,\theta)$, with level sets determining equipotential surfaces. Rigid rotation ($\omega = \text{const}$), polytropic equations of state, and integrability conditions yield explicit parameter constraints for the existence and local stability of these fluid structures.
• Parameter ranges enabling stability are highly model dependent. Pure magnetic or pure charge cases require small rotation rates and suitable values for $Q$, $B$, and the rotation parameter. Mixed cases (both $Q$ and $B$ nonzero) are necessary to support polar clouds.

Such analytic equilibrium regions may have astrophysical implications for the trapping of charged plasma or dust in black hole magnetospheres.

4. High-Energy Processes and Particle Acceleration

A key dynamical consequence of nonzero black hole charge is the possibility of efficient energy extraction and particle acceleration in the black hole vicinity.

• The "electric Penrose process" enables neutral infalling particles, upon ionization near the horizon, to acquire substantial energy via the black hole's electric field. The energy gain is determined by the jump in Coulomb potential:

$E_{\max} \simeq m + \frac{qQ}{r_0},$

with Lorentz factors as high as $\gamma_{\max} \simeq 10^6$ for maximal astrophysical charges (Tursunov et al., 2021).

• This process can accelerate protons to PeV ("PeVatron") energies, relevant for cosmic ray astrophysics, even when $Q \ll M$. The mechanism is isotropic and does not rely on spin-driven processes.
• External homogeneous EM fields (Bertotti–Robinson classes) provide a means of black hole acceleration by balancing conical singularities, connecting the black hole's global acceleration to its charge-to-mass ratio (Alekseev, 8 Nov 2025).
• Near-horizon particle collisions in the magnetized RN geometry can achieve arbitrarily large center-of-mass energies, with both black hole charge and external field amplifying $E_{\rm CM}$:

$$E_{\rm CM}^2 = -g_{\mu\nu}(p_1^\mu + p_2^\mu)(p_1^\nu + p_2^\nu) = 2m^2[1 - g_{\mu\nu}u_1^\mu u_2^\nu],$$

with ratios $E_{\rm CM}/2m > 10$ for moderate parameter choices (Shaymatov et al., 2021).

These mechanisms have significant implications for UHECR generation, radiative phenomena, and the energetic budget of accreting plasmas.

5. Modifications from Alternate Gravity and Quantum Effects

Charged non-rotating black holes are sensitive probes of deviations from general relativity and semiclassical effects.

• In Einstein–bumblebee gravity, the Lorentz-violating parameter $\ell$ alters the metric function, renormalizing the effective charge and affecting the horizon structure, asymptotics, and curvature invariants (Liu et al., 11 Jul 2024).
• The presence of a cosmological constant introduces an additional cubic term, modifying the global horizon structure and yielding up to three real roots.
• Quantum tunneling calculations incorporating a Generalized Uncertainty Principle (GUP) lead to robust corrections to both temperature and entropy for lower-dimensional (BTZ) charged black holes:

$$T_{\text{GUP}} = T_H \Big[1 + \frac{\lambda l_p}{2 r_+} - \frac{\lambda^2 l_p^2}{2 r_+^2}\Big],$$

$$S_{\text{GUP}} = S_0 - 2\pi \lambda l_p \ln(r_+) - \frac{2\pi \lambda^2 l_p^2}{r_+} + \text{const},$$

mirroring the expected "log + inverse area" corrections from string and loop quantum gravity, and implying a Planck-scale remnant for the evaporation endpoint (Sadeghi et al., 2016).

This suggests that charged non-rotating black holes are informative laboratories for testing both infrared (cosmological, Lorentz-violating) and ultraviolet (quantum gravity) modifications of black hole thermodynamics and causal structure.

6. Observational and Astrophysical Consequences

The macroscopic electric charge of astrophysical black holes is expected to be extremely small due to efficient neutralization, but even tiny values $Q/M \ll 1$ may produce observable effects:

Phenomenon	Effect of Q and B	Reference
ISCO radius $r_{\rm ISCO}$	Inward shift, strong Q/B	(Shaymatov et al., 2021)
Black hole shadow radius (bumblebee)	Decreases with Q and ℓ	(Liu et al., 11 Jul 2024)
Center-of-mass energy $E_{\rm CM}$	Grows rapidly with Q, B	(Shaymatov et al., 2021)
High-energy cosmic rays from Sgr A*	PeVatron activity possible	(Tursunov et al., 2021)
Polarization and Faraday rotation	Upper limits on Q	(Tursunov et al., 2021)
Trapping of charged matter	Existence of tori, clouds	(Kovář et al., 2014)

The degeneracy of ISCO radii between magnetized RN and rotating Kerr geometries—capable of mimicking spins up to $a/M \sim 0.8$—demonstrates challenges in disentangling charge/magnetic effects from spin-based observational inferences based on orbital dynamics. Furthermore, the presence of equipotential surfaces supporting tori or clouds may influence spectral signatures, polarization, and variability in accreting compact objects (Kovář et al., 2014).

7. Horizon and Causal Structures

Charged, non-rotating black holes generically exhibit an inner (Cauchy) and an outer (event) horizon for $Q^2 < M^2$. As $Q \rightarrow M$, the horizons coalesce in the extremal limit, with distinct modifications under alternative gravity or cosmological constant. The spacetime may asymptote to flatness (GR), AdS (Bertotti–Robinson), or non-Minkowskian behaviors (bumblebee gravity).

The global causal structure is adjusted by the asymptotic properties—AdS asymptotia lead to "box-like" Penrose diagrams, while external fields or Lorentz violation rescale the radial distance or introduce singularities on antipodal axes at cosmological distances (Alekseev, 8 Nov 2025, Liu et al., 11 Jul 2024). In all cases, the curvature singularity at $r = 0$ persists, except for lower-dimensional models where topology may play a role.

In summary, charged non-rotating black holes serve as a central theoretical construct elucidating the interplay between gravity and electromagnetism, revealing complex effects in particle dynamics, energy extraction, equilibrium matter configurations, and sensitivity to both high- and low-energy modifications of GR. Their paper remains critical for advancing both fundamental theory and the interpretation of astrophysical data.