Reissner-Nordström black holes surrounded by perfect fluid dark matter: testing the viability of weak gravity conjecture and weak cosmic censorship conjecture simultaneously

Abstract: A possible violation of the weak gravity conjecture (WGC) by cosmic censorship is one of the major challenges in the field of general relativity. However, in this paper, we explore the possibility of reconciling the WGC and the WCCC by considering Reissner-Nordstr\"om (R-N) black holes embedded in perfect fluid dark matter (PFDM) in asymptotically flat spacetimes. These two conjectures are seemingly unrelated, but a recent proposal suggested that they are connected surprisingly. In particular, We argue a promising class of valid counterexamples to the WCCC in the four-dimensional Einstein-Maxwell theory, considering a charged black hole when WGC is present. We demonstrate that by imposing certain constraints on the parameters of the metric, the WGC and the WCCC can be compatible. Furthermore, we investigate the properties of the charged black hole in the presence of PFDM for $Q > M$ and present some intriguing figures to test the validity of the WGC and the WCCC simultaneously. When PFDM is absent ($\gamma=0$), the RN black hole either has two event horizons if $Q^2/M^2\leq 1$ or none if $Q^2/M^2> 1$. The second scenario results in a naked singularity, which contradicts the WCCC. But when PFDM is present ($\gamma\neq 0$), the RN black hole has event horizons with regard to Q and M. This implies that the singularity is always covered, and the WGC and the WCCC are fulfilled. Furthermore, we demonstrate that there is a critical value of $\gamma$, called $\gamma_{ext}$, that makes the RN black hole extremal when $\gamma=\gamma_{ext}$. In this situation, the black hole has an event horizon, and the WGC and the WCCC are still fulfilled. We infer that PFDM can make the WGC and the WCCC compatible with the RN black hole and that the WGC and the WCCC agree with each other when PFDM is present.