Testing the Weak Gravity Conjecture via Gravitational Lensing, Black Hole Shadows, and Barrow Thermodynamics in F(R)-Euler-Heisenberg (A)dS Black Holes

Abstract

We investigate the interplay of the Weak Gravity Conjecture (WGC) and the Weak Cosmic Censorship Conjecture (WCCC) in $F(R)$-Euler-Heisenberg black holes in Anti-de Sitter and de Sitter backgrounds. The solution is characterized by the electric charge $q$, the $F(R)$ deviation $f_{R_0}$, the Euler--Heisenberg coupling $λ$, and the constant scalar curvature $R_0$. We establish a universal entropy--extremality relation that provides thermodynamic evidence for the WGC independently of $f_{R_0}$ and $R_0$. Photon sphere analysis from both geodesic and topological perspectives confirms the simultaneous compatibility of the WGC and WCCC, with the Euler--Heisenberg coupling restoring photon spheres in the naked singularity regime. Gravitational lensing in the strong- and weak-deflection limits reveals that the photon sphere radius is independent of the cosmological background while the critical impact parameter nearly doubles in de Sitter. Black hole shadow images under isotropic accretion are constructed. Within the Barrow entropy framework, we uncover van der Waals-type phase transitions and analyze Joule-Thomson expansion, identifying the small black hole phase as the WGC-compatible thermodynamic regime accessible via isenthalpic cooling.

Figures (19)

• Figure 1: Blackening function $h(r)$ for the $F(R)$--EH BH in the AdS background ($R_0 = -1$, $M=1$). The solid black curve corresponds to the Schwarzschild-AdS limit ($q=0$) with a single horizon at $r_h \simeq 0.932$. The blue dashed curve ($q=0.3$, $f_{R_0}=-0.1$) and the red solid curve ($q=0.4$, $f_{R_0}=0$) represent non-extremal configurations with two horizons. The purple dotted curve ($q=0.8$, $f_{R_0}=0.5$) stays entirely above the $h=0$ axis, indicating a naked singularity.
• Figure 2: Blackening function $h(r)$ for the $F(R)$--EH BH in the dS background ($R_0 = +1$, $M=1$). The solid black curve is the Schwarzschild-dS solution with a BH horizon at $r_h \simeq 1.116$ and a cosmological horizon at $r_c \simeq 2.769$. The blue dashed curve ($q=0.3$, $f_{R_0}=-0.1$) exhibits three horizons: an inner horizon, a BH horizon, and a cosmological horizon. The red solid curve ($q=0.4$, $\lambda=0.5$, $f_{R_0}=0$) shows a two-horizon configuration. All curves turn over and become negative at large $r$, reflecting the dS asymptotic structure.
• Figure 3: Three-dimensional diagram of the $F(R)$--EH-AdS BH spatial geometry for $f_{R_0}=-0.1$, $\lambda=0.05$, $q=1$, and $R_0=-1$. The red ring marks the event horizon. The contour lines trace the monotonic growth of $h(r)$ toward the AdS boundary, and the color gradient encodes the magnitude of the metric function across the radial extent of the spacetime.
• Figure 4: Three-dimensional surface of the Hawking temperature $T_H(r_h, q)$ for the $F(R)$--EH-AdS BH with $R_0=-1$, $f_{R_0}=-0.1$, and $\lambda=0.05$. The temperature is non-monotonic in $r_h$: it decreases to a local minimum before rising due to the AdS curvature contribution. The red region corresponds to $T_H \leq 0$ (clipped), which marks configurations beyond the extremal limit.
• Figure 5: Three-dimensional surface of the Hawking temperature $T_H(r_h, q)$ for the $F(R)$--EH-dS BH with $R_0=+1$, $f_{R_0}=0.1$, and $\lambda=0.05$. The temperature decreases monotonically with $r_h$ and vanishes at the Nariai limit. The extensive red plateau at $T_H=0$ reflects the large domain in which the BH horizon approaches the cosmological horizon.