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

TL;DR

The paper addresses how gravity's rainbow modifies extremality bounds and gravitational lensing for Reissner–Nordström–AdS black holes, testing compatibility between the Weak Gravity Conjecture and the Weak Cosmic Censorship Conjecture. It combines topological analysis of photon spheres with Gauss–Bonnet/Jacobi–Maupertuis methods to derive rainbow-corrected deflection angles, highlighting a universal extremal angle independent of the rainbow function $f(\varepsilon)$ and a dominant $g(\varepsilon)$-driven lensing enhancement. A key result is the rainbow-induced shift of the extremality bound to $(Q/M)_{\text{ext}} = 1/f(\varepsilon)$, which can reconcile WGC with WCCC by allowing larger charge-to-mass ratios before extremality while preserving horizons. The findings yield concrete, testable predictions for lensing observables in the rainbow framework, offering observational handles to probe Planck-scale modifications to spacetime geometry.

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.

Figures (8)

• Figure 1: The metric function $V(r)$ as a function of the radial coordinate for different parameter configurations. Four cases are shown: $(M=1, Q=2, f(\varepsilon)=0.1, g(\varepsilon)=0.5)$ depicted by the solid blue curve; $(M=1, Q=2, f(\varepsilon)=0.1, g(\varepsilon)=1.2)$ shown as the dashed red curve; $(M=0.1, Q=0.3, f(\varepsilon)=0.1, g(\varepsilon)=0.5)$ represented by the dash-dotted green curve; and $(M=0.1, Q=0.3, f(\varepsilon)=0.1, g(\varepsilon)=1.1)$ indicated by the dotted purple curve. All cases use $l=1$. The zero crossings mark the event horizon locations, demonstrating how rainbow parameters shift horizon radii relative to standard R-N-AdS BHs.
• Figure 2: Normalized vector field $\mathbf{n}$ in the $(r, \theta)$ plane showing PS locations for $l=1$ and $f(\epsilon)=0.1$. Panel (a): $Q=2$, $M=1$ with $g(\epsilon)=0.5$ (left) and $g(\epsilon)=1.2$ (right). Panel (b): $Q=0.3$, $M=0.1$ with $g(\epsilon)=0.5$ (left) and $g(\epsilon)=1.1$ (right). The zero points of the vector field (marked by circular patterns) indicate unstable PSs with topological charge $\omega=-1$. In all configurations, the PS resides outside the event horizon, validating both WGC and WCCC compatibility.
• Figure 3: Gaussian curvature $\mathcal{K}(r)$ of the Jacobi optical geometry for gravity's rainbow R-N-AdS BHs with AdS radius $l=1$. Four parameter sets are shown: Set 1 (blue solid): $M=1$, $Q=2$, $f=0.1$, $g=0.5$; Set 2 (red dashed): $M=1$, $Q=2$, $f=0.1$, $g=1.2$; Set 3 (green dash-dot): $M=0.1$, $Q=0.3$, $f=0.1$, $g=0.5$; Set 4 (purple solid): $M=0.1$, $Q=0.3$, $f=0.1$, $g=1.1$. The inset panels magnify the curvature structure near $r \sim 3$--$4$: the cyan inset (upper right) highlights Sets 1 and 3 with $g=0.5$, while the red inset (lower left) compares Sets 2 and 4 with $g > 1$. Sets 1 and 3 exhibit strongly negative curvature (reaching $\mathcal{K} \sim -35000$ at $r=20$), whereas Sets 2 and 4 show near-zero values. This dramatic difference quantifies the amplification induced by $g(\varepsilon) < 1$. The persistent negative curvature reflects the defocusing character of the optical geometry in Lorentzian spacetimes, with the magnitude directly controlling deflection angle strength.
• Figure 4: Weak deflection angle $\hat{\alpha}$ versus impact parameter $b$ for gravity's rainbow R-N-AdS BHs with $l=1$. Five configurations are displayed: Set 1 photon (blue solid): $M=1$, $Q=2$, $f=0.1$, $g=0.5$; Set 2 photon (red dashed): $M=1$, $Q=2$, $f=0.1$, $g=1.2$; Set 3 photon (green dash-dot): $M=0.1$, $Q=0.3$, $f=0.1$, $g=0.5$; Set 4 photon (purple long-dash): $M=0.1$, $Q=0.3$, $f=0.1$, $g=1.1$; Set 1 massive $v=0.9c$ (orange solid): $M=1$, $Q=2$, $f=0.1$, $g=0.5$. All curves follow the characteristic $\hat{\alpha} \propto 1/b$ scaling at large $b$, confirming first-order dominance. The marked enhancement for $g=0.5$ (Sets 1, 3) relative to $g > 1$ (Sets 2, 4) demonstrates the $g^{-2}$ amplification. The massive particle curve (orange) exceeds its photon counterpart due to the velocity factor $(1+v^2)/v^2 \approx 2.23$ for $v=0.9c$.
• Figure 5: Extended deflection angle analysis for gravity's rainbow parameters. (a) Deflection versus $g(\varepsilon)$ at fixed $M$, $Q$, $f$, showing the $g^{-2}$ enhancement as $g$ decreases below unity. (b) Photon ($v=c$) versus massive particle ($v=0.5c, 0.7c, 0.9c, 0.99c$) deflection, revealing the velocity-dependent amplification $(1+v^2)/v^2$ that enhances bending for slower particles. (c) Charge variation effects, illustrating the negative (repulsive) $Q^2$ contribution that reduces net deflection for highly charged BHs. (d) Extended parameter space mapping combining variations in $M$, $Q$, and rainbow functions across the theory parameter space.