Charge-Induced Pole Cancellation and Horizon Transitions in Scale-Dependent Gravitational Collapse

Abstract: We construct a charged Oppenheimer-Snyder-like collapse model in scale-dependent gravity by matching a spatially flat FLRW interior to a charged scale-dependent exterior across a timelike thin shell. The electric charge is confined to the stellar surface, preserving interior homogeneity and isotropy. The exterior geometry is supported by a phenomenological Bianchi-consistent effective source, while the shell dynamics follow from the Israel-Maxwell junction conditions. A barotropic surface equation of state closes the shell system, with a charged-dust shell as the minimal realization. For a negative scale-dependent parameter, $\tildeω&lt;0$, the exterior contains a finite-radius boundary $x_s$ defined by $D(x_s)=0$. Charge separates the solutions into three regimes. For $0\le q<sup>2&lt;x_s$, the lapse develops a negative pole at a curvature singularity, the physical exterior contains one outer horizon, and a representative monotonic collapse crosses this horizon before reaching $x_s$; no future-directed locally outgoing radial null branch emerges from the singular boundary. At $q<sup>2=x_s$, simultaneous zeros of the numerator and denominator cancel the curvature pole, although the prescribed running coupling remains singular. For $q<sup>2&gt;x_s$, the curvature singularity persists with a positive pole and locally outgoing radial null branches exist. Depending on the physical extremality condition $x_e&gt;x_s$, the exterior may contain two simple horizons, one degenerate horizon, or no horizon. These results show that charge qualitatively changes the singular and horizon structure of scale-dependent collapse and provide model-level evidence for horizon shielding in the negative-pole regime.

• The paper constructs a Bianchi-consistent charged thin-shell collapse model with a neutral FLRW interior, a scale-dependent charged exterior, and closed Israel–Maxwell dynamics governed by a finite parameter set.
• Charge cancellation at q²=x_s removes the finite-radius curvature singularity from the metric through pole-factor cancellation, although the running gravitational coupling G(x) remains divergent there.
• The paper shows that physical extremality requires the degenerate horizon to lie outside the singular boundary, while the pole sign controls local null emission and supports horizon-shielding evidence only for 0≤q²<x_s.

Model construction and effective exterior geometry

The paper develops a charged extension of the scale-dependent Oppenheimer–Snyder collapse model, in which a spatially flat FLRW interior is matched across a timelike thin shell to a charged, scale-dependent exterior (2608.16341). The key structural choice is that electric charge resides entirely on the stellar surface: a volume electric field in the interior would break FLRW homogeneity and isotropy, so the neutral interior is paired with a charged exterior carrying a Maxwell surface current at the boundary.

The exterior is not derived from an action-based scale-dependent Einstein–Maxwell theory. Instead, the authors posit an effective field equation $$G^\mu{}_\nu = 8\pi G(r) T^{\mathrm{EM}}{}^\mu{}_\nu + \Theta^\mu{}_\nu$$, where the polarization tensor $\Theta^\mu{}_\nu$ is reconstructed so that (i) the Schwarzschild gauge condition $g_{tt}g_{rr}=-1$ is admissible, (ii) the classical Einstein–Maxwell limit is recovered as $G'(r)\to 0$, and (iii) the modified conservation relation $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$ required by the contracted Bianchi identity is satisfied. The resulting lapse is

$$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$

with running coupling $g(x)=x^3/D(x)$, $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$, and $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$. The model correctly reduces to the uncharged scale-dependent case at $q=0$ and to Reissner–Nordström at $\Theta^\mu{}_\nu$0. This phenomenological, equation-level construction is a deliberate concession: the lapse is not claimed to follow from the standard action formulation of scale-dependent gravity, and the authors state this plainly.

Closed thin-shell dynamics

The shell system is closed by combining the Maxwell junction condition, the Israel equations, and shell energy conservation. Charge conservation on the shell gives $\Theta^\mu{}_\nu$1 with $\Theta^\mu{}_\nu$2 constant. The angular Israel equation yields $\Theta^\mu{}_\nu$3, which must be retained as an unsquared branch condition requiring $\Theta^\mu{}_\nu$4. Imposing the barotropic surface equation of state $\Theta^\mu{}_\nu$5 integrates the conservation law to $\Theta^\mu{}_\nu$6, producing a first-order equation of motion $\Theta^\mu{}_\nu$7 governed by the finite parameter set $\Theta^\mu{}_\nu$8.

A notable feature is the explicit consistency check: substituting the second-order equation $\Theta^\mu{}_\nu$9 back into the temporal junction equation reproduces $g_{tt}g_{rr}=-1$0 exactly, confirming mutual consistency of the angular and temporal junction equations, the conservation law, and the equation of state. The charged-dust case ($g_{tt}g_{rr}=-1$1) gives constant shell material energy $g_{tt}g_{rr}=-1$2 and the effective potential $g_{tt}g_{rr}=-1$3.

Two clarifications guard against common errors. First, the electromagnetic force per unit area $g_{tt}g_{rr}=-1$4 is already encoded in $g_{tt}g_{rr}=-1$5 and the normal Israel equation; adding it separately to $g_{tt}g_{rr}=-1$6 would double-count. Second, the initial data are constrained: $g_{tt}g_{rr}=-1$7, so prescribing $g_{tt}g_{rr}=-1$8 independently overdetermines the shell. The paper also distinguishes a genuine massive charged shell from a Maxwell-current-only configuration ($g_{tt}g_{rr}=-1$9), which has no gravitational thin shell and should not be labeled as one.

Geometry at the finite-radius boundary and the cancellation condition

For $G'(r)\to 0$0 with $G'(r)\to 0$1, $G'(r)\to 0$2 has a unique positive root $G'(r)\to 0$3. Since $G'(r)\to 0$4 and $G'(r)\to 0$5, the lapse behaves as

$G'(r)\to 0$6

so the sign of $G'(r)\to 0$7 determines the sign of the pole. For $G'(r)\to 0$8, the Kretschmann scalar diverges as $G'(r)\to 0$9, establishing $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$0 as a curvature singularity in both the negative- and positive-pole sectors. Charge above $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$1 reverses the pole sign but does not remove the singularity.

The exceptional value $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$2 produces simultaneous zeros of numerator and denominator; l'Hôpital's rule gives the finite value $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$3, and the common factor cancels so the metric admits an analytic extension through $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$4. This is the only charge for which the finite-radius curvature singularity disappears. An important caveat is stated explicitly: the running coupling $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$5 remains divergent at $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$6, so the geometry is regular while the chosen parametrization of the scale-dependent sector is not.

Horizon classification and physical extremality

The paper's central technical correction concerns the extremality condition. Horizons satisfy $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$7; a degenerate horizon obeys the pair $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$8, giving the algebraic cubic $$\nabla_\mu\Theta^\mu{}_\nu = -8\pi T^{\mathrm{EM}}{}_\mu{}^\nu\,\partial_\nu G$$9. However, because the physical exterior is restricted to $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$0, an algebraic double root with $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$1 is unphysical. Using $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$2, a physical extremal root exists only when $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$3, equivalently $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$4 with

$$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$5

For $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$6, $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$7 and $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$8; the extremal curve is physically relevant only for $$f_q(x) = 1 - \frac{g(x)}{x} + \frac{q^2 g(x)}{x^2} = \frac{H_q(x)}{D(x)},$$9. Portions of a phase diagram extending to, say, $g(x)=x^3/D(x)$0 must not be read as containing a physical degenerate horizon. This restriction materially changes the classification:

Regime	Condition	Behavior at $g(x)=x^3/D(x)$1	Horizons in $g(x)=x^3/D(x)$2
Negative-pole	$g(x)=x^3/D(x)$3	Curvature singularity, $g(x)=x^3/D(x)$4	One outer horizon
Cancellation	$g(x)=x^3/D(x)$5	Pole removed	Depends on $g(x)=x^3/D(x)$6
Two-horizon	$g(x)=x^3/D(x)$7, $g(x)=x^3/D(x)$8	Singularity, $g(x)=x^3/D(x)$9	Two simple horizons
Extremal	$D(x)=x^3+\tilde{\omega}(x+\gamma/2)$0, $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$1	Singularity	One degenerate horizon
Horizonless	$D(x)=x^3+\tilde{\omega}(x+\gamma/2)$2, or $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$3 if $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$4	Singularity	None

Here $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$5 is the physical horizon-loss threshold. The implication is that charge does not merely displace the outer horizon: it can change the horizon count, produce a degenerate configuration, or eliminate horizons entirely, leaving a finite-radius curvature singularity with no shielding horizon.

Outgoing null behavior and horizon shielding

In ingoing Eddington–Finkelstein coordinates, outgoing radial null rays obey $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$6, which near $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$7 integrates to $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$8 with $D(x)=x^3+\tilde{\omega}(x+\gamma/2)$9 and $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$0. The sign of $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$1 is decisive. For $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$2, $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$3 and $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$4: no real outgoing branch exists for $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$5, so the singular boundary is locally non-emitting. For $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$6, real outgoing branches exist for $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$7 and the boundary is locally emitting. Reversing the sign of the lapse pole therefore reverses the local causal character of the singular boundary.

The numerical charged-dust example ($$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$8, $$H_q(x)=x^3-x^2+(\tilde{\omega}+q^2)x+\tilde{\omega}\gamma/2$$9, $q=0$0, $q=0$1, $q=0$2) shows monotonic collapse with no turning point: the shell crosses the outer horizon $q=0$3 at $q=0$4 and reaches $q=0$5 only at $q=0$6. The interior apparent horizon, $q=0$7, intersects the surface at $q=0$8, $q=0$9, after which the trapped region expands through the interior. The authors note that the exterior horizon crossing precedes the apparent-horizon intersection without contradiction, since one is a Killing horizon of the static exterior and the other is a slicing-dependent quasilocal surface.

The paper is careful to scope its claim: the local non-emission property and the ordering $\Theta^\mu{}_\nu$00 along a monotonic, dynamically admissible trajectory together constitute model-level evidence for horizon shielding, not a proof of weak cosmic censorship. In the positive-pole two-horizon case ($\Theta^\mu{}_\nu$01, with $\Theta^\mu{}_\nu$02, $\Theta^\mu{}_\nu$03), local emission does not establish global nakedness, since outgoing rays must still traverse the full dynamical geometry and the outer horizon; moreover, the null-ray analysis rests on a leading-order near-$\Theta^\mu{}_\nu$04 expansion that should not be extrapolated to large radii. The shielding conclusion is explicitly restricted to $\Theta^\mu{}_\nu$05.

Limitations and open questions

Several limitations are acknowledged within the paper itself. The effective source $\Theta^\mu{}_\nu$06 is phenomenological and Bianchi-consistent but is not derived from the action-based scale-dependent Einstein–Maxwell system, so the exterior lapse is a model assumption rather than a prediction of a fundamental formulation. The shell dynamics require a phenomenological barotropic closure; only the dust case is worked out numerically, and the extremal configuration is flagged as requiring a separate collapse analysis. The horizon-shielding argument combines a local null-ray result with a single representative trajectory, and the authors state that neither element alone, nor the pair jointly, constitutes a proof of global censorship. Open questions left by the paper include whether the cancellation surface $\Theta^\mu{}_\nu$07 admits a complete dynamical matching (given the residual divergence of $\Theta^\mu{}_\nu$08), how the positive-pole horizonless sector evolves globally, and whether the local emission result in the positive-pole region survives propagation through the full matched spacetime.

Conclusion

This work constructs a charged thin-shell collapse model in scale-dependent gravity with a Bianchi-consistent effective exterior source and fully closed shell dynamics. Charge partitions the parameter space into a negative-pole regime with a single outer horizon and a locally non-emitting singularity, a tuned cancellation value at which the finite-radius curvature singularity is removed from the metric (though the running coupling remains singular), and a positive-pole regime whose horizon content depends on the physical extremality condition $\Theta^\mu{}_\nu$09. The principal results are the corrected extremality classification, the demonstration that the sign of the lapse pole controls local null emission, and model-level evidence for horizon shielding confined to $\Theta^\mu{}_\nu$10.