Covariant Quantum-Corrected RN Black Hole

• The topic involves covariant quantum corrections to Reissner–Nordström black holes using frameworks like effective field theory, RG improvement, and constraint-preserving Hamiltonian methods.
• Different quantization schemes yield distinct predictions, including varied horizon shifts, singularity treatments, and corrections to thermodynamic properties.
• Key analyses span perturbative corrections, geodesic studies, and phenomenological extensions that highlight model-dependent observational and conceptual implications.

The expression covariant quantum-corrected Reissner–Nordström black hole denotes a family of charged, spherically symmetric black-hole geometries in which quantum effects are incorporated without abandoning diffeomorphism covariance. In the literature surveyed here, this label covers several technically distinct constructions: effective-field-theory corrections built from local $R^2$ terms and non-local logarithms, covariant effective spacetimes derived from a constraint-algebra-preserving Hamiltonian framework, RG-improved geometries in asymptotic safety, coherent-state mean-field geometries, and minisuperspace quantizations with a reparametrization-covariant radial Schrödinger evolution. These models agree on the classical limit $f(r)\to 1-2M/r+Q^2/r^2$, but they differ on whether the outer horizon shifts, whether the center is regularized, which thermodynamic quantities are corrected at leading order, and how covariance is implemented at the quantum level (Delgado, 2022, Ahmed et al., 17 Feb 2026, Jalalzadeh et al., 26 Feb 2026, Ishibashi et al., 2021, Pailas, 2020).

1. Covariance and the range of meanings attached to the term

In this context, “covariant” does not refer to a single formalism. In effective-field-theory treatments, covariance means that the action is written from diffeomorphism-invariant local and non-local curvature scalars, such as

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$

with

$$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$

and

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$

supplemented by the Maxwell action $$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$ (Delgado, 2022). In this usage, the geometry is obtained from covariant equations of motion, and renormalization-group running of the local couplings cancels the explicit $\mu$-dependence of observables (Delgado, 2022).

In another usage, “covariant” means that the effective spacetime is derived from a generally covariant effective theory whose constraint algebra is preserved. The $\zeta$-deformed RN model used in recent phenomenological work is explicitly presented as the electrovacuum, $\Lambda=0$, sector of covariant effective spacetimes obtained from an effective Hamiltonian constraint that preserves diffeomorphism covariance, rather than as an ad hoc coordinate-dependent modification (Ahmed et al., 17 Feb 2026). Its lapse function is

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$

with $f(r)\to 1-2M/r+Q^2/r^2$0 (Ahmed et al., 17 Feb 2026).

A third meaning appears in asymptotic-safety constructions, where covariance is tied to a diffeomorphism-invariant scale identification for RG-improved couplings. There the classical RN metric is “improved” by promoting $f(r)\to 1-2M/r+Q^2/r^2$1 and the gauge coupling to running quantities $f(r)\to 1-2M/r+Q^2/r^2$2 and $f(r)\to 1-2M/r+Q^2/r^2$3, with the RG scale fixed covariantly by the classical Kretschmann scalar through

$f(r)\to 1-2M/r+Q^2/r^2$4

or by proper-distance prescriptions (Ishibashi et al., 2021, González et al., 2015).

A recurrent misconception is that “covariant quantum-corrected RN black hole” refers to one universal metric. The literature instead contains several inequivalent covariant implementations, and their predictions differ precisely because they encode different quantum sectors, truncations, and approximation schemes. This suggests that the phrase is best understood as a class label rather than a unique solution.

2. Covariant effective-action geometries and horizon shifts

The most explicit action-based construction for the asymptotically flat RN case is the perturbative EFT analysis of a four-dimensional non-extremal black hole in the regime $f(r)\to 1-2M/r+Q^2/r^2$5, expanded consistently to $f(r)\to 1-2M/r+Q^2/r^2$6 and $f(r)\to 1-2M/r+Q^2/r^2$7 (Delgado, 2022). Starting from

$f(r)\to 1-2M/r+Q^2/r^2$8

the quantum-corrected metric is obtained in the form

$f(r)\to 1-2M/r+Q^2/r^2$9

with

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$0

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$1

and a corrected electric field

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$2

(Delgado, 2022). In this model, unlike the Schwarzschild case at the same EFT order, the RN geometry is corrected already at second order in curvature because the classical background has $\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$3 but nonvanishing $\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$4 (Delgado, 2022).

The outer horizon is defined by $\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$5, and its perturbative shift is

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$6

so the horizon position is no longer the classical RN value (Delgado, 2022). By contrast, in the $\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$7-deformed covariant effective spacetime the horizon equation factorizes as

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$8

and the outer horizon remains the classical

$\Gamma=\Gamma_L+\Gamma_{NL}+\Gamma_M,$9

while any extra quantum roots lie inside $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$0 (Ahmed et al., 17 Feb 2026). The same $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$1-independence of the outer event horizon is emphasized in the electric Penrose-process analysis built on the same metric (Chen et al., 4 Jan 2026).

The existence of both behaviors—shifted outer horizon in the EFT curvature expansion and unchanged outer horizon in multiplicative $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$2-deformations—shows that horizon invariance is model dependent, not a universal signature of “covariant” quantization.

3. Thermodynamics: Wald entropy, temperature, pressure, and horizon-by-horizon quantization

In the EFT approach, thermodynamics is computed directly from the corrected geometry and the full action. The entropy is obtained from the Wald formula

$$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$3

with the full $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$4 and $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$5 dependence retained in the entropy calculation even if Gauss–Bonnet identities are used to simplify the equations of motion (Delgado, 2022). The result in the small-charge expansion is

$$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$6

where

$$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$7

The $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$8 limit reproduces the Schwarzschild result of Calmet and Kuipers, and the full entropy is RG invariant once the running of $$\Gamma_L=\int d^4x\sqrt{-g}\left[\frac{R}{16\pi G_N}+c_1(\mu)R^2+c_2(\mu)R_{\mu\nu}R^{\mu\nu}+c_3(\mu)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right]$$9 is included (Delgado, 2022).

The corresponding Hawking temperature is

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$0

and the electrostatic potential is

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$1

with no $$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$2 correction in that setup (Delgado, 2022). To maintain the first law with fixed charge, a nonzero pressure term is introduced,

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$3

leading to

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$4

where the first term is the classical RN pressure on the outer horizon and the second is the pure Schwarzschild quantum contribution (Delgado, 2022).

A distinct thermodynamic formulation appears in the semiclassical horizon-quantization program based on the Misner–Sharp–Hernandez mass. There each horizon carries a quasi-local energy

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$5

and obeys the horizon-by-horizon first law and Smarr relation

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$6

Reduced phase-space quantization yields the discrete spectrum

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$7

so the minimal entropy spacing is $$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$8 (Jalalzadeh et al., 26 Feb 2026). Quantum transitions between adjacent levels produce corrected temperatures

$$\Gamma_{NL}=-\int d^4x\sqrt{-g}\left[\alpha R\ln\!\left(\frac{\Box}{\mu^2}\right)R+\beta R_{\mu\nu}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}\ln\!\left(\frac{\Box}{\mu^2}\right)R^{\mu\nu\rho\sigma}\right],$$9

and a logarithmic entropy correction

$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$0

(Jalalzadeh et al., 26 Feb 2026). These corrections are encoded geometrically through the multiplicative deformation

$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$1

which preserves the classical horizon radii while lowering both horizon temperatures (Jalalzadeh et al., 26 Feb 2026).

The coexistence of Wald-entropy calculations, horizon-by-horizon thermodynamics, and phenomenological thermal-fluctuation corrections illustrates that “quantum-corrected thermodynamics” of RN black holes is framework dependent even when each construction remains internally covariant.

4. Singularities, core geometry, and inner-horizon structure

The central singularity is treated very differently across models. In the EFT small-charge construction, the analysis is explicitly perturbative around the classical RN geometry and does not address singularity resolution; the non-extremal, small-$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$2 regime is the domain of validity (Delgado, 2022). In the $$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$3-deformed covariant effective spacetime, the focus is on horizon structure, orbital dynamics, perturbations, and thermodynamics, and the outer horizon is unchanged while extra quantum roots stay inside $$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$4; regularity of the center is not presented as the main result (Ahmed et al., 17 Feb 2026).

By contrast, asymptotic-safety RG improvement can replace the classical singularity with a regular core. With the covariant Kretschmann-based identification

$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$5

the improved lapse

$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$6

behaves near the center as

$$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$7

so all curvature components vanish at $$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$8, yielding a Minkowski core (Ishibashi et al., 2021). The same paper shows that more general scale identifications $$\Gamma_M=-\frac14\int d^4x\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}$$9 divide the space of improved geometries into regions with Minkowski cores, de Sitter cores, anti-de Sitter cores, or weak singularities, depending on $\mu$0 (Ishibashi et al., 2021).

A related asymptotic-safety construction for RN-(A)dS promotes $\mu$1, $\mu$2, and $\mu$3 to running couplings and uses a covariant scale setting $\mu$4. There the improved geometry develops a new internal horizon, and the paper argues that no minimal mass is required to avoid weak cosmic censorship because the new internal horizon shields the $\mu$5 singularity (González et al., 2015). It also concludes that there is no stable remnant (González et al., 2015).

The coherent-state mean-field construction introduces yet another possibility. There the corrected potential is

$\mu$6

with

$\mu$7

where $\mu$8 is the Dawson function (Antonelli et al., 2 Jun 2025). The center is fully regular only for the tuned value

$\mu$9

while for generic $\zeta$0 the geometry has an integrable singularity: curvature scalars diverge, but the effective energy density behaves as $\zeta$1, which is locally integrable with the spherical volume element (Antonelli et al., 2 Jun 2025). This model can also remove the inner Cauchy horizon in part of parameter space (Antonelli et al., 2 Jun 2025).

These results rule out a common simplification: covariant quantum corrections do not universally regularize the RN center. Depending on the framework, the center may remain singular, become weakly singular, become integrably singular, or be replaced by a regular Minkowski core.

5. Geodesics, photon spheres, shadows, perturbations, and energy extraction

Once a covariant quantum correction is specified, the exterior geometry can be probed through null and timelike geodesics. In the $\zeta$2-deformed covariant effective spacetime, the photon sphere satisfies

$\zeta$3

with $\zeta$4 and $\zeta$5 (Ahmed et al., 17 Feb 2026). The critical impact parameter is

$\zeta$6

and the paper reports that $\zeta$7 shifts both $\zeta$8 and $\zeta$9 (Ahmed et al., 17 Feb 2026). For neutral timelike circular orbits,

$\Lambda=0$0

and explicitly

$\Lambda=0$1

so $\Lambda=0$2 lowers $\Lambda=0$3, softens $\Lambda=0$4, and pushes the ISCO outward (Ahmed et al., 17 Feb 2026).

The same metric underlies the charged-particle Penrose analysis. There the effective potentials are

$\Lambda=0$5

and the generalized ergoregion boundary $\Lambda=0$6 is set by $\Lambda=0$7 with $\Lambda=0$8 (Chen et al., 4 Jan 2026). Because $\Lambda=0$9 increases monotonically with $$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$0 for $$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$1, the generalized ergoregion shrinks as $$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$2 grows, and the efficiency

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$3

decreases monotonically with $$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$4 for fixed kinematics (Chen et al., 4 Jan 2026). The paper therefore describes $$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$5 as having an obstructive effect on the electric Penrose process (Chen et al., 4 Jan 2026).

Shadow and lensing analyses have been carried out for several quantum-corrected RN families. For the square-root deformation used in shadow studies,

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$6

the corrected horizon radii are

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$7

and the photon sphere radius is

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$8

The corresponding shadow radius admits the expansion

$$f(r)=\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[1+\frac{\zeta^2}{r^2}\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\right],$$9

so $f(r)\to 1-2M/r+Q^2/r^2$00 shrinks the shadow while $f(r)\to 1-2M/r+Q^2/r^2$01 enlarges it slightly (Lobos et al., 16 Jun 2025). A related quintessence model based on the Wu–Liu metric similarly concludes that larger $f(r)\to 1-2M/r+Q^2/r^2$02 slightly enlarges the shadow (Hamil et al., 2023).

Perturbative field propagation has also been studied. In the $f(r)\to 1-2M/r+Q^2/r^2$03-deformed spacetime, a massless scalar satisfies

$f(r)\to 1-2M/r+Q^2/r^2$04

and increasing $f(r)\to 1-2M/r+Q^2/r^2$05 raises and widens the potential barrier, supporting mode stability against massless scalar perturbations and lowering the greybody factor (Ahmed et al., 17 Feb 2026). In the coherent-state quantum RN geometry, scalar quasinormal modes are governed by

$f(r)\to 1-2M/r+Q^2/r^2$06

and the reported trend is that $f(r)\to 1-2M/r+Q^2/r^2$07 decreases with increasing core size $f(r)\to 1-2M/r+Q^2/r^2$08, implying longer-lived ringdown (Antonelli et al., 2 Jun 2025).

6. Quantization-based constructions and conceptual issues

Beyond effective metrics, some RN quantum corrections are obtained from direct quantization of reduced gravitational degrees of freedom. A notable example constructs a reparametrization-covariant radial Schrödinger equation for the spherically symmetric Einstein–Maxwell minisuperspace without fixing the radial gauge (Pailas, 2020). After reduction, the regular Lagrangian becomes

$f(r)\to 1-2M/r+Q^2/r^2$09

leading to the “time”-covariant Schrödinger equation

$f(r)\to 1-2M/r+Q^2/r^2$10

which is form-covariant under radial reparametrizations $f(r)\to 1-2M/r+Q^2/r^2$11 provided $f(r)\to 1-2M/r+Q^2/r^2$12 transforms as a density (Pailas, 2020). In the Bohmian reconstruction of the Gaussian state, the effective quantum-corrected horizons become

$f(r)\to 1-2M/r+Q^2/r^2$13

so both horizons expand relative to classical RN (Pailas, 2020). The same work emphasizes a conceptual tension: DeWitt’s probabilistic criterion indicates singularity avoidance because $f(r)\to 1-2M/r+Q^2/r^2$14 at the classical singularity, while the semiclassical Bohmian geometry can remain curvature singular (Pailas, 2020).

Loop-quantum-gravity corrections in spherically symmetric Einstein–Maxwell theory raise a different covariance issue. With inverse-triad corrections and an unmodified constraint algebra, one can retain standard spacetime covariance and obtain corrected RN-like spacetimes exhibiting three horizons over a finite mass range and a mass threshold beyond which the inner horizon disappears (Tibrewala, 2012). With a modified constraint algebra, however, classical coordinate transformations no longer provide a good symmetry, and covariance is recovered only through a “quantum” notion of mapping from phase space to spacetime (Tibrewala, 2012). Holonomy corrections deform the algebra further, preclude a static solution, and imply signature change in deep quantum regions (Tibrewala, 2012).

These quantization-based models clarify a broader conceptual point. In RN quantum gravity, covariance may survive either because the action remains covariant, because the effective constraint algebra closes in the classical form, or because a new quantum notion of phase-space-to-spacetime mapping replaces the classical one. The term therefore has a precise but model-relative meaning.

7. Phenomenology, extensions, and current research directions

Recent work has moved from formal construction to phenomenology. The $f(r)\to 1-2M/r+Q^2/r^2$15-deformed covariant effective spacetime has been confronted with quasi-periodic oscillation data from stellar-mass, intermediate-mass, and supermassive black-hole candidates. The analysis uses geodesic QPO models such as the relativistic precession model,

$f(r)\to 1-2M/r+Q^2/r^2$16

and Bayesian parameter estimation with MCMC, reporting nonzero posterior values for $f(r)\to 1-2M/r+Q^2/r^2$17 in all four studied sources within the model assumptions (Ahmed et al., 17 Feb 2026). The same paper also finds that the normalized offset

$f(r)\to 1-2M/r+Q^2/r^2$18

decreases with $f(r)\to 1-2M/r+Q^2/r^2$19, bringing QPO radii closer to the ISCO (Ahmed et al., 17 Feb 2026).

Several extensions combine quantum-corrected RN sectors with environmental matter. A quintessence-surrounded quantum-corrected RN black hole with

$f(r)\to 1-2M/r+Q^2/r^2$20

exhibits a minimal radius $f(r)\to 1-2M/r+Q^2/r^2$21, remnant formation when $f(r)\to 1-2M/r+Q^2/r^2$22 at finite $f(r)\to 1-2M/r+Q^2/r^2$23, and a shadow that remains circular but changes with $f(r)\to 1-2M/r+Q^2/r^2$24, $f(r)\to 1-2M/r+Q^2/r^2$25, and $f(r)\to 1-2M/r+Q^2/r^2$26 (Hamil et al., 2023). A related AdS/Kiselev/string-cloud model uses

$f(r)\to 1-2M/r+Q^2/r^2$27

and reports systematic shifts in photon-sphere radius, ISCO, Hawking temperature, Gibbs free energy, and specific heat (Ahmed et al., 13 Aug 2025). In the AdS/CFT setting, holographic Einstein-ring imaging has been carried out for a quantum-corrected AdS–RN geometry in Kiselev spacetime, with the ring radius decreasing as the quantum parameter $f(r)\to 1-2M/r+Q^2/r^2$28, the equation-of-state parameter $f(r)\to 1-2M/r+Q^2/r^2$29, the temperature $f(r)\to 1-2M/r+Q^2/r^2$30, and the chemical potential $f(r)\to 1-2M/r+Q^2/r^2$31 increase, while increasing with the cosmological-fluid parameter $f(r)\to 1-2M/r+Q^2/r^2$32 (Gui et al., 25 Jan 2025).

An important extension concerns AdS effective-field-theory corrections. For RN–AdS, the one-loop covariant EFT analysis yields corrected metric functions $f(r)\to 1-2M/r+Q^2/r^2$33 and $f(r)\to 1-2M/r+Q^2/r^2$34, an explicit corrected horizon radius, and RG-invariant Wald entropy, temperature, pressure, specific heat, and Helmholtz free energy (Pourhassan et al., 2022). This work concludes that the quantum charged AdS black hole can exist only for a bounded range of masses and can undergo a second-order phase transition as it moves from positive to negative specific heat (Pourhassan et al., 2022).

A final misconception sometimes appears in phenomenological discussions: that current observationally motivated quantum-corrected RN models necessarily imply large astrophysical deviations. Several cited works state the opposite. In the EFT small-charge regime, the corrections are extremely small for astrophysical black holes (Delgado, 2022). In shadow and strong-lensing calculations based on $f(r)\to 1-2M/r+Q^2/r^2$35, the quantum correction is theoretically consistent but negligible at current observational precision (Lobos et al., 16 Jun 2025). Observable effects therefore depend strongly on the chosen quantum parameterization and on whether it is treated as a strictly Planckian scale or as an effective phenomenological parameter.

Taken together, the literature presents the covariant quantum-corrected Reissner–Nordström black hole as a broad research program rather than a settled object: a charged black-hole background in which covariance is preserved while quantum corrections reshape geometry, horizon structure, entropy, radiation, orbital dynamics, and stability in framework-specific ways.