Orbits of photon in Bardeen-boson stars and their frozen states

In a recent study [1], the Bardeen-boson star (BBS) model involving a scalar field minimally coupled to Einstein gravity and a Bardeen's nonlinear electromagnetic field was investigated. It was found that when the magnetic charge q of the electromagnetic field exceeds a certain critical value qc, a frozen Bardeen-boson star (FBBS) can be obtained with the frequency approaching zero. In this paper, we study the null orbits in the background of the general BBS and FBBS. We find that similar to the boson star (BS), all BBSs do not have the event horizon and possess complete null geodesics, allowing photons to move throughout the entire spacetime of BBS. Among these BBSs, the FBBSs whose spacetime is very similar to that of black holes are particularly special. The null orbits around the FBBSs exhibit sharp deflections near the critical horizon while becoming nearly straight inside the critical horizon. Furthermore, the photon in the background of FBBSs moves for a very long time inside the critical horizon from the perspective of an infinity viewer.