Gravitational Lensing by Black Holes in Einstein-nonlinear Electrodynamic Theories with Multiple Photon Spheres

Abstract: In this paper, we study the gravitational lensing effects of non-linear electrodynamic black holes. Non-linear electrodynamic black holes serve as typical models for multi-event horizon black holes. Depending on the choice of metric parameters, these black holes can possess more than five event horizons. Consequently, within certain parameter ranges, black holes can have more than three photon spheres of varying sizes outside the event horizon. Specifically, we focus on the strong gravitational lensing effects near the triple photon spheres, particularly the formation of higher-order images of point sources and celestial spheres. The presence of one, two, or three or more photon spheres significantly increases the number of higher-order images of a point source. When a black hole is illuminated by a celestial sphere, the three photon spheres generate three critical curves in the black hole image, with the smallest critical curve coinciding with the shadow's edge. Additionally, since non-linear electrodynamic black holes are models of multi-event horizon black holes, we can infer the gravitational lensing effects and the changes in celestial images for black holes with more than three photon spheres by analyzing the distinctions and patterns between the gravitational lensing effects of one, two, and three photon spheres.