Assessing WGC Compatibility in ModMax Black Holes via Photon Spheres Analysis and WCCC Validation

TL;DR

The paper investigates whether the Weak Gravity Conjecture can be realized without violating the Weak Cosmic Censorship Conjecture for charged black holes within ModMax electrodynamics and its AdS and $F(R)$-gravity extensions. It combines analytic extremality conditions with a topological photon-sphere analysis to map parameter regions where electromagnetism can dominate gravity while preserving a BH and a photon sphere. Across three models—the pure ModMax BH, ModMax-AdS BH, and an $F(R)$-ModMax BH—the study finds that WGC compatibility is generically favored by positive nonlinear parameter $\gamma$ and by gravitational corrections, though the extent of compatibility varies by model. The photon-sphere topological approach provides a diagnostic link between BH structure, extremality, and the feasibility of WGC-compatible regimes, suggesting practical pathways to test quantum-gravity consistency in modified BH theories.

Abstract

It seems that the regime of Hawking radiation and evaporation ultimately drives charged black holes toward super-extremality of the charge parameter and the dominance of extremal conditions. This progression, in turn, lays the groundwork for satisfying the necessary conditions for the Weak Gravity Conjecture (WGC). Preliminary studies indicate that black holes such as the Reissner-Nordstr$ö$m (RN) model, in their initial form, lack the capacity to sustain super-extremality of the charge parameter. If such conditions arise, these black holes transition into naked singularities-a scenario that is highly undesirable due to the loss of causality and the breakdown of space-time geometry. This raises whether the inability to sustain super-extremality is an inherent property of the model or a consequence of the approximations and precision limitations employed in its construction. To address this, we turned to the ModMax model, which represents an extension of the RN model. Our analysis revealed that the ModMax model not only accommodates super-extremality of the charge parameter but also, under certain conditions, emerges as a promising candidate for investigating the WGC. Furthermore, we independently observed how the inclusion of the de Sitter radius ($\ell$) in the AdS model and $f(R)$ gravitational corrections-both of which enhance and complicate the model-can have a direct impact on the range of super-extremal charge tolerance which, in turn, provides the realization of the conditions necessary for the WGC.

Figures (10)

• Figure 1: Combined view of the black hole models and related parameter spaces. (a) The Schwarzschild BH is shown for baseline comparison. (b) The RN BH demonstrates the standard charged black hole solution. (c)--(f) Different snapshots of the ModMax model illustrate various parameter regimes, including the effects of the nonlinear parameter $\gamma$ on screening and extremality conditions.
• Figure 2: The ($Q_{exe}-M_{exe}$) plan for different values of $\gamma$.
• Figure 3: The normal vector in the $(r, \theta)$ plane associated with the photon spheres. (a) $\gamma=0.5$, $Q_{ext}=0.5$, $M_{ext}=0.3894$, $r_{ext}=0.3894$. (b) $\gamma=0.7$, $Q_{ext}=0.5$, $M_{ext}=0.35234$, $r_{ext}=0.35234$. (c) $\gamma=1$, $Q_{ext}=0.5$, $M_{ext}=0.30326$, $r_{ext}=0.30326$.
• Figure 4: The metric function with $\ell=1$ and $M=0.45$. (a) $Q=0.5$ and various $\gamma$, (b) $\gamma=0.5$ and various $Q$.
• Figure 5: The ($Q_{exe}-M_{exe}$) plan for (a) different values of $\gamma$ with $\ell=1$, and (b) different values of $l$ with $\gamma=0.5$.