Probing correlation between photon orbits and phase structure of charged AdS black hole in massive gravity background

The phase structure of charged Anti-de Sitter black hole in massive gravity is investigated using the unstable circular photon orbits formalism, concretely we establish a direct link between the null geodesics and the critical behavior thermodynamic of such black hole solution. Our analysis reveals that the radius and the impact parameter corresponding to the unstable circular orbits can be used to probe the thermodynamic phase structure. We also show that the latter are key quantities to characterize the order of Van der Waals-like phase transition. Namely, we found a critical exponent around $\delta = 1/2$. All these results support further that the photon trajectories can be used as a useful and crucial tool to probe the thermodynamic black holes criticality.