Holographic Einstein rings of AdS black holes in Horndeski theory*

Abstract Using the AdS/CFT correspondence and wave optics techniques, we conduct an extensive study of the imaging properties of holographic Einstein rings in the context of anti-de Sitter (AdS) black holes (BHs) in Horndeski theory. Our results indicate that the optical characteristics of these holographic Einstein rings are significantly influenced by the observer's position, physical parameters of the BH, nature of the wave source, and configuration of the optical system. Specifically, when the observer is positioned at the north pole of the AdS boundary, the holographic image prominently displays a ring structure aligning with the BH's photon sphere. We thoroughly analyze the effect of various physical parameters — including the observation position, event horizon radius, temperature, and the parameter γ in Horndeski theory — on the holographic Einstein rings. These parameters play a crucial role in determining the rings' radius and brightness, with variations potentially causing the ring structures to deform or even transform into bright spots. Furthermore, our comparative analysis between wave and geometric optics reveals a strong agreement in predicting the positions and brightness of both the photon and Einstein rings. This paper offers new insights into the spacetime geometry of BHs in Horndeski theory and proposes a promising framework for exploring the gravitational duals of strongly coupled systems.