Particle emission spectrum and thermodynamics phase transition of RN-AdS black hole in massive gravity via a new prescription

In this paper, we investigate the shadow and energy emission rate of a charged AdS black hole in massive gravity. We develop a new prescription based on maximum frequencies of emission spectrum to probe the black holes thermodynamics and shed light into its phase transition criticality. We establish a link between the thermodynamic behavior and the maximum emission frequency of the black hole for both photons and massive bosonic modes. Remarkably, we show that the frequency maximizing the spectrum combined to the shadow radius can serve as a powerful physical tool to delve into the thermodynamics and phase structure of RN-AdS black holes in massive gravity.