Quantum-Corrected Black Holes in Higher Dimensions: From Gravitational Collapse to Remnants, WGC-Like Behavior and Accretion Signatures

Abstract: We study the gravitational collapse of a homogeneous dust sphere in higher-dimensional spacetime, with the loop quantum gravity correction entered through a modified Friedmann equation. Its most striking feature is an endpoint: at a finite horizon radius the temperature vanishes and evaporation stops, leaving a remnant. Reading (α) as an effective charge, we find behavior reminiscent of the Weak Gravity Conjecture (WGC), although no (U(1)) gauge field is present in the model. Solving the degeneracy conditions with the untruncated metric function, rather than with a mass expansion that is not uniform near the endpoint, we obtain closed-form expressions for the remnant radius and mass in (D=4,5,6,7). The unscaled ratio (α/M_{\text{remnant}}^2) is constant only in four dimensions and grows as (α^{4-D}) elsewhere, but it carries dimension (L^{8-2D}), so that growth is an artifact of the units. The dimensionless combination (\tilde{\mathcal{R}} = (α/M^2) r_0^{2D-8}) settles instead on a finite (α)-independent number in every dimension, decreasing from (27/64) in four dimensions to (13824/(15625π^4)) in seven. We also deform the quantum parameter, (α\to α+ \varepsilon), and evaluate (-T(\partial S/\partial \varepsilon)M) in the remnant limit. Because ((\partial M/\partial r)α) vanishes there, the fixed-entropy derivative reduces exactly to a fixed-radius one, and the resulting combination (\mathcal{U}) is finite in each dimension but depends on (\varepsilon) except for (D=5). What is invariant is the product (\mathcal{R}\,\mathcal{U}\,M_{\text{remnant}} = (D-3)/2), which we verify numerically. Lastly, we study Bondi accretion onto the quantum-corrected black hole and obtain the Eddington luminosity for each dimension.

• The paper derives an LQG-corrected black hole from homogeneous dust collapse in dimensions D=4–7, showing that evaporation ends at a finite-radius, zero-temperature remnant controlled by the quantum parameter α.
• The paper finds a properly dimensionless WGC-like ratio that remains finite and α-independent, decreases from about 0.422 in D=4 to 0.009 in D=7, and weakens without vanishing as dimensionality increases.
• The paper links remnant physics to accretion by showing that α ρ(r₀) is dimension-dependent but constant, suggesting that observations of accretion flows could constrain quantum-gravity corrections despite limitations in higher-dimensional radiative modeling.

Overview

This paper constructs a higher-dimensional quantum-corrected black hole geometry by carrying a loop quantum gravity (LQG) correction through an Oppenheimer–Snyder collapse of homogeneous dust, then uses the resulting solution to probe three otherwise separate questions: swampland-type consistency conditions, universal thermodynamic relations, and Bondi accretion phenomenology. The single deformation parameter $\alpha = \gamma^2 \Delta^{2/(d-1)}$, built from the Barbero–Immirzi parameter $\gamma$ and the area gap $\Delta$, controls all quantum effects. The central structural result is that evaporation terminates at a finite horizon radius where the Hawking temperature vanishes, producing a zero-temperature remnant in every dimension studied ($D = 4$ through $7$). The paper is explicit that its Weak Gravity Conjecture (WGC) analysis is an analogy rather than a derivation, since no $U(1)$ gauge field exists in the model.

Quantum-corrected collapse and thermodynamics

The interior FRW dust ball is matched to a static exterior via Darmois–Israel junction conditions. Because the exterior admits a timelike Killing vector and the surface follows a geodesic, the metric functions satisfy $f = F^2 g$ with $g = 1 - \dot{r}^2$, so the entire exterior geometry is encoded in the Hubble parameter. Using the loop-quantum-cosmology-modified Friedmann equation,

$$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$

with critical density $\rho_c = d(d-1)/(2\kappa\gamma^2\Delta^{2/(d-1)})$, the exterior metric function becomes

$\gamma$0

where $\gamma$1. The classical Schwarzschild-(A)dS limit is recovered as $\gamma$2. The temperature profile is non-monotonic: it rises from zero at finite radius, peaks, and approaches the classical curve from above at large radii — the direct signature of the remnant endpoint.

Entropy computed by integrating the first law $\gamma$3 shows an exact cancellation of $\gamma$4 in all dimensions, leaving corrections proportional to $\gamma$5 relative to the Bekenstein–Hawking area law. Notably, four dimensions are exceptional: instead of a power-law correction one obtains $\gamma$6, a logarithmic term consistent with independent microstate countings. Higher-dimensional black holes behave more classically at small scales, as the relative quantum correction diminishes with dimension.

WGC-like scaling of the remnant

Identifying $\gamma$7 and treating the zero-temperature endpoint as an effective extremal state, the degeneracy conditions $\gamma$8 are solved on the untruncated metric function. This methodological choice matters: the first-order mass expansion is not uniform near the endpoint, because there the quantum and classical terms balance rather than one correcting the other; using the truncated mass misestimates $\gamma$9 by a factor of order unity. Since $\Delta$0, the $\Delta$1-dependent terms enter only at relative order $\Delta$2, so the scaling results hold for either sign of $\Delta$3.

The closed-form results are:

$\Delta$4

A key dimensional-analysis point resolves an apparent dichotomy. The raw ratio $\Delta$5 carries dimension $\Delta$6 and is dimensionless only at $\Delta$7; its growth as $\Delta$8 in higher dimensions is an artifact of units, not physics. The properly scaled combination

$\Delta$9

is a finite, $D = 4$0-independent pure number in every dimension, decreasing from $D = 4$1 at $D = 4$2 to $D = 4$3 at $D = 4$4 — a factor of roughly 46. The effective WGC-like bound is therefore strongest in four dimensions and weakens monotonically with added dimensions, but never switches off. Both $D = 4$5 and $D = 4$6 vanish with $D = 4$7: the remnant is a purely quantum object, and removing the area gap eliminates the endpoint entirely rather than leaving a classical relic.

The authors are careful about scope: this is a heuristic analogy. There is no gauge field, the remnant is macroscopic rather than a particle in a spectrum, and nothing constrains the light states the actual conjecture concerns. What the analogy supplies is a sharp consistency target for any model producing a zero-temperature remnant from a minimal-area-gap correction.

Universal extremality relation under deformation

Deforming $D = 4$8 and evaluating $D = 4$9, the first law reduces the combination to $7$0. At the remnant, $7$1 implies $7$2 exactly, so the fixed-entropy derivative collapses onto a fixed-radius derivative and $7$3 can be evaluated from the exact mass function

$7$4

Two findings emerge. First, $7$5 is finite at the extremal point in every dimension, so the universal relation survives the LQG correction. Second — and contrary to what the standard Goon–Penco-type statement would suggest — $7$6 itself is not independent of the deformation strength except at $7$7 ($7$8); elsewhere it inherits $7$9-dependence through $U(1)$0, moving by roughly 26% over the sampled range of $U(1)$1 in four dimensions. What is invariant is the product

$U(1)$2

verified numerically to four significant figures (residuals of a few parts in $U(1)$3 attributable to neglected $U(1)$4 corrections). The authors state plainly that they expected $U(1)$5 alone to be protected and have no general argument forcing the clean coefficients $U(1)$6; the identity is an observation about this family of solutions, checked case by case.

Bondi accretion and observational signatures

Accretion is analyzed for two fluids — a barotropic dark fluid with $U(1)$7 and an exponential density profile $U(1)$8 of galactic-halo type — across dimensions $U(1)$9 through $f = F^2 g$0. Across all cases the radial velocity remains negative (sustained inward flow), the dark-fluid pressure stays negative throughout, and larger $f = F^2 g$1 enhances the barotropic accretion rate and luminosity while affecting the exponential-profile quantities only weakly, making that channel a faint observational signature.

Evaluating the flow at the remnant requires care: expressions carrying inverse powers of $f = F^2 g$2 must be treated as limits since $f = F^2 g$3. Replacing the coordinate expression by the conserved mixed-component flux $f = F^2 g$4 yields a finite, radius-independent rate $f = F^2 g$5, as a steady spherical flow requires. The inner-boundary density satisfies

$f = F^2 g$6

a constant rising weakly from $f = F^2 g$7 at $f = F^2 g$8 toward $f = F^2 g$9 asymptotically. Combined with $g = 1 - \dot{r}^2$0, this gives $g = 1 - \dot{r}^2$1, the accretion counterpart of the WGC-like scaling. This is identified as the most directly observational result: $g = 1 - \dot{r}^2$2 becomes, in principle, boundable by accretion measurements.

The paper concedes two limitations here explicitly. The flux integrals use the four-dimensional equatorial area element and the Eddington luminosity uses the four-dimensional Thomson opacity, so luminosities reported for $g = 1 - \dot{r}^2$3 are four-dimensional radiative estimates on higher-dimensional backgrounds, not genuine $g = 1 - \dot{r}^2$4-dimensional Eddington limits — restoring proper prefactors would change numbers but not the $g = 1 - \dot{r}^2$5-scaling. Additionally, time-dependent accretion episodes onto near-remnant configurations lie outside the steady-flow framework.

Limitations and open questions

Beyond those already noted, several caveats bear on the results. The WGC-like analysis rests on identifying $g = 1 - \dot{r}^2$6 with an effective charge squared, which is a structural analogy only; no dynamical gauge field exists to test the conjecture's actual content. The identity $g = 1 - \dot{r}^2$7 has been verified only within this specific construction, and whether it extends to other quantum-gravity-inspired models (asymptotic safety, string $g = 1 - \dot{r}^2$8 corrections, Lovelock gravity) is left open. The numerical verification retains full $g = 1 - \dot{r}^2$9 dependence only in four dimensions; higher-dimensional numerics with all $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$0 terms are not presented. Rotating analogues, needed for contact with realistic astrophysical environments, are absent. Finally, connections to the broader swampland program — de Sitter, trans-Planckian censorship, and distance conjectures — are flagged but not developed.

Conclusion

The paper delivers a self-contained chain of results: an LQG-corrected Oppenheimer–Snyder collapse yielding closed-form thermodynamics in $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$1–$$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$2; a zero-temperature remnant whose dimensionless charge-to-mass analogue $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$3 is a fixed pure number per dimension; a deformation-tested universal relation surviving in product form $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$4 even though $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$5 alone is not invariant; and an accretion analysis converting the quantum parameter into a potentially observable quantity via $$H^2 = \frac{2\kappa}{d(d-1)}\rho_T\left(1 - \frac{\rho_T}{\rho_c}\right),$$6. The work's discipline in distinguishing analogy from derivation, and in flagging where first-order expansions fail, makes its claims well-scoped. Its most consequential suggestion is that loop quantum gravity parameters could be constrained by astronomical data on black hole systems rather than by internal consistency arguments alone.