Phenomenological Charged Extensions of the Quantum Oppenheimer–Snyder Collapse Model

Abstract

This work presents a semi-classical, quantum-corrected model of gravitational collapse for a charged, spherically symmetric dust cloud, extending the classical Oppenheimer–Snyder (OS) framework through loop quantum gravity effects. Our goal is to study phenomenological quantum modifications to geometry, without necessarily embedding them within full loop quantum gravity (LQG). Building upon the quantum Oppenheimer–Snyder (qOS) model, which replaces the classical singularity with a nonsingular bounce via a modified Friedmann equation, we introduce electric and magnetic charges concentrated on a massive thin shell at the boundary of the dust ball. The resulting exterior spacetime generalizes the Schwarzschild solution to a charged, regular black hole geometry akin to a quantum-corrected Reissner–Nordström metric. The Israel junction conditions are applied to match the interior APS (Ashtekar–Pawlowski–Singh) cosmological solution to the charged exterior, yielding constraints on the shell’s mass, pressure, and energy. Stability conditions are derived, including a minimum radius preventing full collapse and ensuring positivity of energy density. This study also examines the geodesic structure around the black hole, focusing on null circular orbits and effective potentials, with implications for the observational signatures of such quantum-corrected compact objects.

1. Introduction

The term dust ball in cosmology refers to a spherically symmetric object, such as a static star or a massive neutron star, whose internal pressures, both radial and angular, are assumed to be zero. In their groundbreaking study, Oppenheimer and his student Snyder analyzed the gravitational collapse of such an object under its own self-gravity [1]. This paper was the third in a series of seminal works published by Oppenheimer between 1938 and 1939 [1,2,3], which fundamentally reshaped our understanding of black hole formation and singularities in spacetime. The first of these, a brief Letter to the Editor titled On the Stability of Stellar Neutron Cores [2], was co-authored by Oppenheimer and Robert Serber. In it, they examined the conditions for stability of neutron cores in stars, challenging an earlier estimate by Landau [4], who had proposed a minimum stable core mass of just 0.001 solar masses. Taking into account both gravitational and kinetic energy, Oppenheimer and Serber argued that the actual minimum mass for stability is closer to one-quarter of a solar mass. They also investigated how nuclear interactions between neutrons could lower this threshold. While certain assumptions, especially those involving spin-exchange nuclear forces, could allow stable neutron cores with masses as low as a few percent of the Sun’s, their analysis suggested that such cores are unlikely to form in stars with solar-like masses.

The second paper, On Massive Neutron Cores [3], co-authored with George Volkoff, explored the gravitational equilibrium of neutron cores using the equation of state for a cold Fermi gas in the framework of general relativity. They found that for neutron cores with masses less than about one-third of a solar mass, only one stable equilibrium solution exists, which can be described using nonrelativistic Fermi statistics and Newtonian gravity. For intermediate masses (between one-third and three-quarters of a solar mass), two solutions emerge, one stable and one unstable. However, when the mass exceeds roughly 0.75 solar masses, no static equilibrium solutions are possible, implying that the object must undergo unbounded gravitational collapse. They compared their findings to analytic solutions developed earlier by Tolman [5] and discussed broader implications for stellar evolution. They concluded that neutron cores are unlikely to form in solar-mass stars unless nuclear fuel is nearly depleted and nuclear interactions allow a lower mass threshold for stability.

The third and most influential paper in the series, On Continued Gravitational Contraction [1], co-authored with Snyder, addressed the ultimate fate of massive stars after their thermonuclear energy sources are exhausted. Within the framework of general relativity, they demonstrated that if such a star has a mass exceeding about 0.7 solar masses and cannot lose mass or fragment, it will undergo unstoppable gravitational collapse. They derived an exact solution to Einstein’s field equations for a collapsing sphere of dust, matter with zero pressure, showing that the star’s radius asymptotically approaches its Schwarzschild radius. As this happens, the star becomes causally disconnected from the external universe, meaning that no signal from within can escape to distant observers. To a distant observer, the collapse appears to slow indefinitely as the star nears the Schwarzschild radius, while an infalling comoving observer experiences the collapse in finite time. This study was among the first to rigorously demonstrate the inevitability of complete gravitational collapse in sufficiently massive stars, a phenomenon now recognized as the formation of a black hole. It is also noteworthy that the well-known Tolman–Oppenheimer–Volkoff (TOV) equation, which describes the structure of a spherically symmetric object in static gravitational equilibrium under general relativity, originated from this third paper [1], building on analytic work previously performed by Tolman [5].

The OS model was the first general relativistic treatment of gravitational collapse, describing how a homogeneous, pressureless (dust) spherical cloud collapses to form a black hole. While groundbreaking, it has several limitations and idealizations that make it an incomplete description of realistic black hole formation. For instance, (i) the OS model assumes pressureless dust, but real stars have thermal pressure, radiation pressure, and degeneracy pressure that resist collapse (see [6] that discusses realistic equations of state in stellar collapse). (ii) Most astrophysical objects rotate, leading to accretion disks or Kerr black holes rather than OS’s non-rotating case (see [7,8]). (iii) Real stars have density gradients and shell structure, which can lead to inhomogeneous collapse or fragmentation [9]. (iv) The OS model predicts direct collapse, but real massive stars often undergo core bounce and supernova explosions before forming black holes [10,11]. (v) The OS model predicts a spacetime singularity, but quantum gravity (e.g., loop quantum cosmology) may prevent it [12,13]. (vi) The OS collapse forms a singularity before the event horizon fully develops, potentially violating the Cosmic Censorship Hypothesis [14,15].

Recently, in a significant development, Lewandowski et al. revisited the Oppenheimer–Snyder (OS) model by incorporating loop quantum gravity corrections, resulting in what is known as the Quantum Oppenheimer–Snyder (QOS) model [16]. In this quantum-corrected Schwarzschild black hole framework, the interior region is modeled as a dust-filled spacetime, described by the semi-classical Ashtekar–Pawlowski-Singh (APS) line element [17]

$$d{s}_{int}^2=-d{\tau }^2+a{\left(\tau \right)}^2\left\{d{\tilde{r}}^2+{\tilde{r}}^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)\right\}.$$

(1)

This interior solution is matched to the exterior metric

$$d{s}^2=-\left(1-{\displaystyle \frac{2GM}{r}}+{\displaystyle \frac{\alpha G^2{M}^2}{r^4}}\right)d{t}^2+{\displaystyle \frac{d{r}^2}{1-\frac{2GM}{r}+\frac{\alpha G^2{M}^2}{r^4}}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$$

(2)

at the interface hypersurface $r={\tilde{r}}_{0}a\left(\tau \right).$ Here, M is the constant mass of the dust ball, equal to the black hole’s mass. The radial coordinate r is defined such that the interior region spans $0<r\le R,$ while the exterior region corresponds to $R\le r,$ with $R={\tilde{r}}_{0}a\left(\tau \right).$ The interface is a timelike hypersurface given by $\Sigma =r-R=0$. The energy–momentum tensor for the dust ball inside the interior spacetime is expressed as $T_{\mu }^{\nu }=diag\left[-\rho ,0,0,0\right],$ where the proper energy density is given by $\rho ={\displaystyle \frac{M}{\frac{4}{3}\pi R^3}}.$ This expression is derived from the energy conservation equation $T_{;\nu }^{\mu \nu }=0,$ which explicitly takes the form

$${\displaystyle \frac{d\rho }{d\tau }}+{\displaystyle \frac{3\dot{a}}{a}}\rho =0.$$

(3)

Furthermore, the expansion function $a\left(\tau \right)$ satisfies a quantum-corrected Friedmann equation

$${\left({\displaystyle \frac{\dot{a}}{a}}\right)}^2={\displaystyle \frac{8\pi G}{3}}\rho \left(1-{\displaystyle \frac{\rho }{{\rho }_c}}\right).$$

(4)

Here, ${\rho }_c$ is the critical energy density, chosen such that in the classical limit, ${\displaystyle \frac{\rho }{{\rho }_c}}\to 0,$ the equation reduces to the standard Friedmann equation. Conversly, when ${\displaystyle \frac{\rho }{{\rho }_c}}\sim 1,$ the modified Friedmann equation allows for a solution where $a\left(\tau \right)$ does not shrink to zero, ensuring that the energy density $\rho$ remains finite and preventing the formation of a singularity in spacetime. To ensure a smooth matching of the interior and exterior spacetimes at the hypersurface $\Sigma ,$ the Israel junction conditions [18,19] are imposed. These conditions require the continuity of both the first and second fundamental forms across $\Sigma ,$ leading to a determination of the parameter $\alpha$ in the exterior metric as $\alpha ={\displaystyle \frac{3}{2\pi G{\rho }_c}}.$ Recent studies have investigated the physical properties of the quantum-corrected Oppenheimer–Snyder (qOS) black hole (see, for instance, [20,21,22,23,24]). The qOS regular black hole essentially corresponds to a Schwarzschild metric modified by quantum mechanical corrections and does not possess an electric charge. In this letter, we introduce a charged generalization of this quantum-corrected Schwarzschild black hole, which we refer to as the quantum-mechanically corrected Reissner–Nordström spacetime.

2. The Model

For the interior metric, we continue using the APS model described by (1); however, for the exterior region, we adopt the following line element,

$$\begin{array}{c}d{s}^2=-\left(1-{\displaystyle \frac{2G\left(M+\mathcal{E}\right)}{r}}+{\displaystyle \frac{G\left(Q^2+P^2\right)}{r^2}}+{\displaystyle \frac{\alpha G^2{M}^2}{r^4}}\right)d{t}^2+\hfill \\ \hfill {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{d{r}^2}{1-\frac{2G\left(M+\mathcal{E}\right)}{r}+\frac{G\left(Q^2+P^2\right)}{r^2}+\frac{\alpha G^2{M}^2}{r^4}}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).\end{array}$$

(5)

It is worth noting that in LQG, the metric (or, more precisely, the triad variables) is not a freely chosen background, but is dynamically derived from the modified Hamiltonian constraint. Therefore, our model is not a derivation from full LQG, but rather a phenomenological extension inspired by expected quantum gravitational features. In other words, the exterior quantum-corrected metric is a toy model or effective background, similar to how Hayward, Dymnikova, or other authors treat non-singular black holes with heuristic corrections. Q and P represent the electric and magnetic monopole charges of the spacetime, while M is the mass of the dust ball, and $\mathcal{E}$ is a additional asymptotic energy introduced due to the presence of a charged massive thin shell [25,26,27]. Let us add that, by analogy with the Reissner–Nordström solution in Einstein’s theory, which in its general form includes both electric and magnetic monopole charges, we have also assumed the presence of both types of charges in our model. The inclusion of the magnetic monopole is primarily for completeness, and it can be omitted without affecting the model by simply setting $P=0$. We should also note that, in a complete loop quantum gravity treatment, the electromagnetic field contributes to the Ashtekar connection and should, in principle, receive quantum corrections. However, in the present model, we have restricted quantum corrections to the geometric sector via the modified Friedmann equation. This approach is consistent with the standard treatment in the existing semiclassical loop quantum cosmology literature.

Furthermore, in our model, the interior region remains neutral and is described by the APS metric. All electric and magnetic charges are confined to a thin shell at the boundary. This setup is intentionally chosen to avoid the well-known difficulties associated with constructing a charged, homogeneous cosmology.

Since the APS model describes a neutral dust ball, the electric and magnetic charges must reside on the interface thin shell $\Sigma .$ Unlike the qOS model, where $\Sigma$ was both massless and chargeless, in this extended model, the interface hypersurface carries charge and contributes a nonzero mass to the shell. To analyze this system further, we apply the Israel junction conditions to the new charged configuration. According to the first Israel junction condition, the induced metric must remain continuous across $\Sigma$, leading to the condition

$$f\left(R\right){\dot{t}}^2-{\displaystyle \frac{{\dot{R}}^2}{f\left(R\right)}}=1.$$

(6)

Here, the dot denotes differentiation with respect to the proper time on the shell (and within the interior). As a result, the induced line element across $\Sigma$ takes the form

$$d{s}^2=-d{\tau }^2+R^{2}d{\Omega }^2.$$

(7)

Recalling that $R=a\left(\tau \right){\tilde{r}}_0$ defines the hypersurface $\Sigma ,$ we now apply the second Israel junction condition, which is given by

$$\left[K_i^j\right]-\left[K\right]{\delta }_i^j=-8\pi G{S}_i^j.$$

(8)

Here, $\left[K_i^j\right]$ represents the discontinuity in the extrinsic curvature across $\Sigma$, while $\left[K\right]$ is its trace. The quantity $S_i^j=diag\left[-\sigma ,p,p\right]$ corresponds to the energy–momentum tensor of the surface fluid on $\Sigma$. Applying this condition, we obtain the surface energy density

$$\sigma ={\displaystyle \frac{1}{4\pi GR}}\left(1-\sqrt{1-{\displaystyle \frac{2G\mathcal{E}}{R}}+{\displaystyle \frac{G\left(Q^2+P^2\right)}{R^2}}}\right).$$

(9)

Similarly, the angular pressure is found to be

$$p={\displaystyle \frac{1}{8\pi G}}\left({\displaystyle \frac{1}{R}}\left(1-\sqrt{1-{\displaystyle \frac{2G\mathcal{E}}{R}}+{\displaystyle \frac{G\left(Q^2+P^2\right)}{R^2}}}\right)+{\displaystyle \frac{G\left(Q^2+P^2\right)}{R^3\sqrt{1-\frac{2G\mathcal{E}}{R}+\frac{G\left(Q^2+P^2\right)}{R^2}}}}\right).$$

(10)

In deriving Equations (9) and (10), we utilized the modified Friedmann Equation (4). Since the presence of electric and magnetic charges leads to a nonzero surface energy density $\sigma ,$ the mass of the shell is given by

$$m=4\pi R^2\sigma ={\displaystyle \frac{R}{G}}\left(1-\sqrt{1-{\displaystyle \frac{2G\mathcal{E}}{R}}+{\displaystyle \frac{G\left(Q^2+P^2\right)}{R^2}}}\right).$$

(11)

By setting $P=Q=0$, the additional asymptotic energy term $\mathcal{E}$ must vanish to revert the dqOS model back to the qOS black hole. Additionally, since the charge distribution is spherically symmetric, the electric and magnetic fields vanish inside the shell, while in the exterior region, they are purely radial and given by

$$\mathbf{E}={\displaystyle \frac{Q}{r^2}}\widehat{r},$$

(12)

and

$$\mathbf{B}={\displaystyle \frac{P}{r^2}}\widehat{r}.$$

(13)

Given the known mass and charges of the thin shell, the energy term $\mathcal{E}$ is determined as

$$\mathcal{E}=m-{\displaystyle \frac{G{m}^2}{2R}}+{\displaystyle \frac{Q^2+P^2}{2R}},$$

(14)

subject to the condition

$$0\le {\displaystyle \frac{Gm}{R}}\le 1.$$

(15)

This condition ensures that the asymptotic energy remains positive, which is necessary for the energy density to be physically meaningful. The inequality also establishes a minimum radius $R_{min},$ given by $R_{min}=Gm,$ since for any $R<R_{min},$ the surface energy density $\sigma$ becomes negative, which is unphysical for normal matter. Next, we require the asymptotic energy to be minimized at $R=R_{min}.$ To achieve this, we express it as

$$E\left(R_{min}\right)={\displaystyle \frac{R_{min}}{2G}}+{\displaystyle \frac{Q^2+P^2}{2R_{min}}},$$

(16)

and solve the equation ${\displaystyle \frac{d\mathcal{E}\left(R_{min}\right)}{d{R}_{min}}}=0$, which leads to the result

$$R_{min}=\sqrt{G\left(Q^2+P^2\right)},$$

(17)

or equivalently,

$$Q^2+P^2=G{m}^2.$$

(18)

Furthermore, we derive

$${\displaystyle \frac{d^2\mathcal{E}\left(R_{min}\right)}{d{R}_{min}^2}}={\displaystyle \frac{Q^2+P^2}{R_{min}^3}}>0,$$

(19)

which confirms that the asymptotic energy at $R=R_{min}$ is indeed a minimum, provided that $Q^2+P^2=G{m}^2.$ As a result, the minimum asymptotic energy at the shell’s minimum radius is

$$\mathcal{E}\left(R_{min}\right)=m.$$

(20)

Under this configuration, we obtain

$${\sigma }_{min}={\displaystyle \frac{1}{4\pi G{R}_{min}}}$$

(21)

while the pressure p in (10) diverges. This is a physically reasonable outcome, as at the minimum radius, a strong repulsive force would act to expand or rebound the shell, thereby preventing the complete collapse of spacetime.

In this particular setup, where $Q^2+P^2=G{m}^2$, the metric function for the exterior region is given by

$$f\left(r\right)=1-{\displaystyle \frac{2G\left(M+m\right)}{r}}+{\displaystyle \frac{G^2{m}^2}{r^2}}+{\displaystyle \frac{\alpha G^2{M}^2}{r^4}}.$$

(22)

Enforcing the condition $Q^2+P^2=G{m}^2$ ensures that when the shell radius reaches its minimum, the asymptotic energy reaches its lowest value at that radius, while the angular pressure diverges to positive infinity, preventing the collapse of the thin shell. Next, we introduce the dimensionless parameters $\mu ={\displaystyle \frac{Gm}{R}}$ with $0\le \mu \le 1,$ $\Xi ={\displaystyle \frac{GM}{R}}$ with $0\le \Xi ,$ and $x={\displaystyle \frac{r}{R}}$ with $x>0,$ which transforms the metric function into

$$f\left(x\right)=1-{\displaystyle \frac{2\left(\mu +\Xi \right)}{x}}+{\displaystyle \frac{{\mu }^2}{x^2}}+{\displaystyle \frac{\tilde{\alpha }{\Xi }^2}{x^4}},$$

(23)

where $\tilde{\alpha }={\displaystyle \frac{\alpha }{R^2}}.$ In Figure 1, we plot $f\left(x\right)$ against x for different values of $\mu \in \left[0,1\right],$ with fixed parameters $\Xi =0.1,$ and $\tilde{\alpha }=0.5.$ The plot shows that for $\mu <0.5584,$ the shell radius is greater than the event horizon radius. More generally, for arbitrary values of $\Xi$ and $\tilde{\alpha },$ the condition $0<\mu <1-\sqrt{2\Xi -\tilde{\alpha }{\Xi }^2},$ ensures that the thin-shell radius at $x=1$ is greater than the event horizon radius $x=x_{+},$ where $f\left(x_{+}\right)=0.$ Although $\Xi$ and $\tilde{\alpha }$ are free parameters, they must satisfy the constraint $0<2\Xi -\tilde{\alpha }{\Xi }^2<1$. We also note that to prevent the event horizon from exceeding the shell radius for a given $\tilde{\alpha },$ the parameter $\Xi$ must be fine-tuned. Based on Figure 1, such configurations do indeed exist.

3. Null Circular Geodesics Around a Quantum-Corrected Charged Black Hole

In this section, we investigate null circular geodesics confined to the equatorial plane near a quantum-mechanically corrected charged black hole. The spacetime is described by the line element

$$d{s}^2=-f\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{f\left(r\right)}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right),$$

(24)

where $f\left(r\right)$ is given in Equation (22). The Lagrangian for a massless (null) particle moving in this spacetime is given by

$$\mathcal{L}={\displaystyle \frac{1}{2}}\left(-f\left(r\right){\dot{t}}^2+{\displaystyle \frac{{\dot{r}}^2}{f\left(r\right)}}+r^2\left({\dot{\theta }}^2+{sin}^2\theta {\dot{\varphi }}^2\right)\right),$$

(25)

where dots denote derivatives with respect to the affine parameter. Owing to the spacetime symmetries associated with the Killing vectors ${\partial }_t$ and ${\partial }_{\varphi },$ there are two conserved quantities: the energy e and angular momentum $\ell ,$ given by

$$e=-{\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{t}}}=f\left(r\right)\dot{t},$$

(26)

and

$$\ell ={\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{\varphi }}}=r^2{sin}^2\theta \dot{\varphi }.$$

(27)

On the equatorial plane $\theta ={\displaystyle \frac{\pi }{2}}$, we impose the null condition $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0,$ which yields

$${\dot{r}}^2+V_{eff}=e^2,$$

(28)

where the effective potential is defined as

$$V_{eff}={\displaystyle \frac{{\ell }^2}{r^2}}f.$$

(29)

To analyze circular motion, we apply the conditions $\dot{r}=0$, and $\ddot{r}=0.$ Setting $\dot{r}=0$ in Equation (28) gives

$${\displaystyle \frac{e^2}{{\ell }^2}}={\displaystyle \frac{1}{r_c^2}}f\left(r_c\right),$$

(30)

and $\ddot{r}=0$ leads to

$$r_c{f}^{\prime }\left(r_c\right)-2f\left(r_c\right)=0.$$

(31)

Equation (31) can, in principle, be solved to express the radius of the circular orbit $r_c$ in terms of the other parameters $G,$ $m,$ $M,$ and $\alpha .$ For physical relevance, $r_c$ must be positive and lie outside any event horizon to ensure a meaningful (positive) value of ${\displaystyle \frac{e^2}{{\ell }^2}}.$ Explicitly, Equation (31) becomes

$$r_c^4-3G\left(M+m\right)r_c^3+2G^2{m}^2{r}_c^2+3\alpha G^2{M}^2=0.$$

(32)

Since this equation does not admit a closed-form analytic solution for $r_c$, we instead solve it for $\alpha$

$$\alpha ={\displaystyle \frac{r_c^2}{3G^2{M}^2}}\left(3G\left(M+m\right)r_c-2G^2{m}^2-r_c^2\right).$$

(33)

Substituting this into Equation (30) yields

$${\displaystyle \frac{e^2}{{\ell }^2}}={\displaystyle \frac{\left(G^2{m}^2-3G\left(M+m\right)r_c+2r_c^2\right)}{3r_c^4}}.$$

(34)

Introducing the scaled variables $\mu ,$ $\Xi ,$ R, and $x_c={\displaystyle \frac{r_c}{R}},$ we express the previous relations as

$${\displaystyle \frac{e^2{R}^2}{{\ell }^2}}={\displaystyle \frac{{\mu }^2-3\left(\mu +\Xi \right)x_c+2x_c^2}{3x_c^4}},$$

(35)

and

$$\tilde{\alpha }={\displaystyle \frac{\alpha }{R^2}}={\displaystyle \frac{x_c^2}{3{\Xi }^2}}\left(3\left(\mu +\Xi \right)x_c-2{\mu }^2-x_c^2\right).$$

(36)

In Figure 2, we plot $\tilde{\alpha }={\displaystyle \frac{\alpha }{R^2}}$ and ${\displaystyle \frac{e^2{R}^2}{{\ell }^2}}$ as functions of $x_c$ for fixed parameters $\Xi =0.1$ and $\mu =0.4.$ The plot shows that a minimum value $x_c^{min}$ exists where ${\displaystyle \frac{e^2{R}^2}{{\ell }^2}}=0,$ and for $x_c>x_c^{min},$ this quantity becomes positive. Moreover, for each fixed value of $\tilde{\alpha },$ there are two corresponding values of $x_c,$ located on either side of the maximum of the $\tilde{\alpha }$ curve. These correspond to a local minimum (smaller $x_c$) and a local maximum (larger $x_c$) of the effective potential. In Figure 3, we plot the metric function $f\left(x\right)$ for the same values of $\Xi$ and $\mu$ while varying $\tilde{\alpha }$. The figure shows that for $\tilde{\alpha }<3,$ the spacetime admits two horizons; for $\tilde{\alpha }=3,$ there exists a single (degenerate) horizon; and for $\tilde{\alpha }>3,$ the spacetime no longer describes a black hole. Finally, in Figure 4, we plot the scaled effective potential ${\displaystyle \frac{R^2}{{\ell }^2}}{V}_{eff}$ against x for the same parameters. Comparing Figure 3 and Figure 4 reveals that for $\tilde{\alpha }>3,$ the effective potential has two stationary points: a local minimum, corresponding to a stable circular orbit, and a local maximum, corresponding to an unstable circular orbit.

4. Conclusions

In this work, we have presented a phenomenological charged generalization of the quantum-corrected Oppenheimer–Snyder (qOS) black hole model, incorporating electric and magnetic monopole charges localized on a massive thin shell. While the interior region continues to be described by the effective loop quantum gravity-inspired APS metric, the exterior spacetime is modeled as a modified Reissner–Nordström solution that includes higher-order quantum corrections. Applying the Israel junction conditions, we derived expressions for the shell’s surface energy density, pressure, and mass, revealing how these quantities depend on the shell radius and charge content. A key feature of our model is the emergence of a minimal shell radius, below which the surface energy density would become negative, rendering the configuration physically unviable. We demonstrated that when condition $Q^2+P^2=G{m}^2$ is satisfied, the asymptotic energy reaches a minimum at this critical radius, and the diverging shell pressure provides a natural mechanism to halt further collapse. This behavior effectively prevents the formation of a singularity, consistent with expectations from loop quantum gravity. Furthermore, we explored the behavior of null circular geodesics in the quantum-corrected charged exterior spacetime, identifying the influence of the charge and quantum correction parameter $\alpha$ on photon trajectories and effective potentials. Our analysis indicates that the inclusion of charge and quantum effects leads to notable modifications in the geodesic structure, which may have observable implications in astrophysical scenarios involving strong gravitational fields. Overall, the model presented here provides a more realistic and regular framework for gravitational collapse by incorporating both quantum gravitational effects and electromagnetic charge. This construction paves the way for future investigations into the dynamical stability, thermodynamics, and observational signatures of regular black holes arising from loop quantum gravity-inspired models.