Beyond Extremality: Weak Gravity Conjecture Constraints on Gravitational Lensing in Gravity's Rainbow

Abstract

We investigate the constraints imposed by the Weak Gravity Conjecture (WGC) on gravitational lensing in gravity's rainbow, focusing in particular on scenarios beyond extremality and on the interplay between the WGC and the Weak Cosmic Censorship Conjecture (WCCC) in the context of Reissner-Nordström-Anti-de Sitter black holes modified by rainbow gravity. Using topological methods, we first analyze the configuration of photon spheres and confirm that unstable circular photon spheres with topological charge $(ω= -1)$ exist outside the event horizon throughout the parameter space, thereby verifying the simultaneous validity of both the WGC and the WCCC. The rainbow functions $f(\varepsilon)$ and $g(\varepsilon)$, which encode Planck-scale corrections through the energy ratio $(\varepsilon =E/E_P)$, modify both the spacetime metric and the extremality bound. We derive the corresponding modified extremal charge-to-mass ratio, $(q^2/m^2)>(Q^2/M^2)_{\text{ext}}$, and show that gravity's rainbow offers a natural mechanism for reconciling these two fundamental conjectures. By applying the Gauss-Bonnet theorem in conjunction with Jacobi-Maupertuis optical geometry, we compute the weak deflection angles for both photons and massive particles to second order. The rainbow function $g(\varepsilon)$ appears with powers $(g^{-2})$ and $(g^{-4})$, enhancing the deflection angle when $g(\varepsilon)<1$, while $f(\varepsilon)$ influences only the charge-dependent contributions. At extremality, the deflection angle becomes independent of $f(\varepsilon)$, yielding a universal prediction that can be tested without specifying the form of the rainbow functions. We further find that super-extremal configurations exhibit stronger lensing effects than extremal black holes, suggesting a potential observational discriminator between WGC-satisfying naked singularities and WCCC-preserving black holes.