Holographic Einstein rings of an AdS black hole in massive gravity

In the context of holography, the Einstein ring of an AdS black hole (BH) in massive gravity (MG) is depicted. An oscillating Gaussian source on one side of the AdS boundary propagates in bulk, and we impose a response function to explain it. Using a wave optics imaging system, we obtain the optical appearance of the Einstein ring. Our research reveals that the ring can change into a luminosity-deformed ring or light spots depending on the variation of parameters and observational positions. When observers are positioned at the north pole, the holographic profiles always appear as a ring with concentric stripe surroundings, and a bright ring appears at the location of the photon sphere of the BH. To investigate the distinct features of the MG AdS BH from previous studies, we discussed the relation between the temperature T and the event horizon ue, which shows the decaying behavior with the increasing values of the event horizon ue, when the graviton parameter m is fixed. These in turn influence the behavior of the response function and the Einstein ring. For example, the amplitude of the lensed response function |(O)| increases with the increasing values of the graviton parameter m, for the fixed value of the horizon ue. On the other hand, the amplitude of the lensed response function |(O)| decreases with the increasing values of the horizon ue, for the fixed value of graviton parameter m. These differences are also reflected in the Einstein ring, where the intensities and the locations of the Einstein ring significantly vary according to the numerical values of the involved parameters. These findings are also observed in the brightness profiles and the best fit comparison between the results obtained by wave optics and geometric optics for different values of graviton parameter m. In this perspective, our study contributes to a better understanding of the analytical studies of holographic theory, which can be used to evaluate different types of BHs for a fixed wave source and optical system.