Quantum-Corrected Reissner-Nordström Black Hole

• The quantum-corrected Reissner-Nordström black hole is a static, charged black hole model modified by quantum gravity corrections that alter its spacetime geometry and energy landscapes.
• Quantum corrections introduce a dimensionful parameter ζ that contracts the event horizon and ergoregion, fundamentally changing the dynamics of charged particles around the black hole.
• These modifications lower the maximal electric Penrose process efficiency to below 10% and provide distinct kinematic signatures that may be observable in high-energy astrophysical phenomena.

A quantum-corrected Reissner-Nordström (RN) black hole is a static, spherically symmetric solution to the Einstein-Maxwell equations modified by covariant quantum-gravity corrections. Such corrections introduce a parameter $\zeta$ that encapsulates leading-order quantum effects and alters both the spacetime metric and the effective energy landscape for test particles. This has direct consequences for energy extraction mechanisms, particularly the electric Penrose process, which relies on negative-energy states in the vicinity of the black hole.

1. Spacetime Structure and Quantum Modification

The standard RN line element in geometric units $(G = c = 1)$ is

$$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$

with

PRESERVED_PLACEHOLDER_^^^^^^^^3^^^^^^^^

where $M$ and $Q$ are the mass and charge of the black hole.

The quantum-corrected RN metric modifies the lapse function as (Chen et al., 4 Jan 2026): $$f(r) = \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right) \left[1 + \frac{\zeta^2}{r^2} \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)\right]$$ Here, $\zeta$ is a quantum gravity parameter, regarded as small and dimensionful. The electromagnetic four-potential remains $A_a = -\frac{Q}{r}\,(\mathrm{d}t)_a$. The outer event horizon is the largest root of $f(r) = 0$.

Quantum corrections ($(G = c = 1)$0) contract the event horizon, raise the effective potential, and affect the kinematics of charged particles in the black hole exterior. These modifications are formally of order $(G = c = 1)$1.

2. Dynamics of Charged Particles and Effective Potential

For a test particle of mass $(G = c = 1)$2, charge $(G = c = 1)$^^^^^^^^3 and specific charge $(G = c = 1)$4, the Lagrangian density is

$(G = c = 1)$5

Conserved quantities, due to $(G = c = 1)$6- and $(G = c = 1)$7-cyclicity: $(G = c = 1)$8 Radial equation of motion, restricting to the equatorial plane: $(G = c = 1)$9 or, equivalently, defining the effective potential $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$0 (Chen et al., 4 Jan 2026),

$$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$1

^^^^^^^^3 Generalized Electro-Ergoregion and Negative-Energy States

Negative-energy orbits, essential for the electric Penrose process, occur where the test particle's energy $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$2 satisfies $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$^^^^^^^^3 The boundary (electro-ergosurface) is determined by $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$4 (Chen et al., 4 Jan 2026): $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$5 Quantum corrections shrink the ergoregion: as $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$6 increases, $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$7 moves inward, reducing the spatial extent where negative-energy states are permitted. For fixed $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$8, typical numerical results show $$\mathrm{d}s^2 = -f_{\rm RN}(r)\,\mathrm{d}t^2 + \frac{1}{f_{\rm RN}(r)}\,\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$$9, PRESERVED_PLACEHOLDER_^^^^^^^^3 PRESERVED_PLACEHOLDER_^^^^^^^^3 (Chen et al., 4 Jan 2026).

4. Electric Penrose Process and Extraction Efficiency

Consider an incident particle ("1") split at radial location PRESERVED_PLACEHOLDER_^^^^^^^^3 into two fragments ("2" and "^^^^^^^^3 with masses PRESERVED_PLACEHOLDER_^^^^^^^^3 PRESERVED_PLACEHOLDER_^^^^^^^^3 and charges PRESERVED_PLACEHOLDER_^^^^^^^^3 PRESERVED_PLACEHOLDER_^^^^^^^^3 Conservation laws (Chen et al., 4 Jan 2026):

• Charge: PRESERVED_PLACEHOLDER_^^^^^^^^3
• Energy: PRESERVED_PLACEHOLDER_^^^^^^^^3
• Angular momentum: PRESERVED_PLACEHOLDER_^^^^^^^^3

At the turning point $M$0,

$M$1

for $M$2. Defining the extraction efficiency,

$M$^^^^^^^^3^^^^^^^^

The analytic expressions for the distribution of angular momentum and energies are provided in (Chen et al., 4 Jan 2026), with all conservation constraints enforced.

Quantum corrections raise the potential barrier and suppress efficiency. Numerically, $M$4 decreases with increasing $M$5: $M$6, $M$7 (see Fig. 5 (Chen et al., 4 Jan 2026)). For small $M$8, the decay is quadratic: $M$9

5. Rigorous Conditions for Escape: Kinematic Constraints

Under simplified assumptions (planar motion, fragment "2" with $Q$0, split at turning point $Q$1), it is shown that the fragment "^^^^^^^^3 can always escape to infinity with $Q$2 (Chen et al., 4 Jan 2026). The derivative of the effective potential $Q$^^^^^^^^3^^^^^^^^ ensures outward motion. The rigorous inequalities hold for arbitrary $Q$4, provided $Q$5, making the result universal across a broad class of charged static black holes.

6. Quantum Obstruction and Kinematic Thresholds

Quantum corrections obstruct the Penrose mechanism in two principal ways:

• The ergoregion contracts (smaller $Q$6), so there is less spatial room for negative-energy processes.
• The effective potential barrier increases, narrowing the set of escaping trajectories and further reducing efficiency.

Special cases are observed: a particle that escapes in classical RN may become trapped in the quantum-corrected case for sufficiently large $Q$7 or inappropriate choice of $Q$8 (Chen et al., 4 Jan 2026). There exists a window in $Q$9 parameter space where energy extraction is possible.

7. Kinematic and Observational Signatures

Quantum corrections imprint several distinctive kinematic signatures:

• Reduced size of the electro-ergoregion: high-energy emission reliant on negative-energy orbits is suppressed for sizable $$f(r) = \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right) \left[1 + \frac{\zeta^2}{r^2} \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)\right]$$0.
• Shifted turning radii and altered ejection trajectories: escape becomes more difficult and the angular momentum distribution of outgoing particles shifts.
• Lower maximal extraction efficiency, with typically $$f(r) = \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right) \left[1 + \frac{\zeta^2}{r^2} \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)\right]$$1, compared to $$f(r) = \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right) \left[1 + \frac{\zeta^2}{r^2} \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)\right]$$2 in classical RN.
• Discriminable particle trajectories and efficiency decrements, offering potential observational handles to distinguish quantum-corrected black holes from classical ones.

8. Comparison with Other Charged Quantum-Corrected Spacetimes

The obstructive effects observed here contrast with higher efficiency ratios in nonlinear electrodynamics solutions, such as the ABG and EGB black holes, often reaching efficiency enhancements by factors up to $$f(r) = \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right) \left[1 + \frac{\zeta^2}{r^2} \left(1 - \frac{2M}{r} + \frac{Q^2}{r^2}\right)\right]$$^^^^^^^^3^^^^^^^^ over standard RN (Chen et al., 18 Aug 2025, Alloqulov et al., 2024). In the quantum-corrected RN case, however, the principal effect is suppression, not enhancement, of the Penrose process (Chen et al., 4 Jan 2026).

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In summary, quantum-corrected Reissner-Nordström black holes exhibit contracted electro-ergoregions and elevated effective potentials, suppressing the maximal efficiency and available phase space for electric Penrose energy extraction. These quantum modifications provide distinct kinematic thresholds and may manifest in astrophysical and observational contexts sensitive to jet formation and high-energy accretion phenomena (Chen et al., 4 Jan 2026).