Photon sphere and reentrant phase transition of charged Born-Infeld-AdS black holes

Comparing to a charged anti-de Sitter (AdS) black hole in general relativity, a new interesting phase transition-the reentrant phase transition-is observed in a charged Born-Infeld-AdS black hole system. It is worth extending the study of the relationship between the photon sphere and thermodynamic phase transition (especially the reentrant phase transition) to this black hole background. Black hole systems are divided into four cases according to the number of thermodynamic critical points, with different values of the Born-Infeld parameter b, where the black hole systems can have no phase transition, a reentrant phase transition, or a van der Waals-like phase transition. For these different cases, we obtain the corresponding pressure-temperature phase structures and temperature-specific volume diagrams. The tiny differences between these cases are clearly displayed. Then, we calculate the radius r(ps) and the minimal impact parameter u(ps) of the photon sphere via the effective potential of the radial motion of photons. r(ps) and u(ps) are found to have different behaviors for the different cases. In particular, with the increase of r(ps) or u(ps) the temperature exhibits a decrease-increase-decrease-increase behavior for fixed pressure if there is a reentrant phase transition, while for fixed temperature the pressure exhibits an increase-decrease-increase-decrease behavior instead. These behaviors are quite different from that of the van der Waals-like phase transition. Near the critical point, the behavior of r(ps )and u(ps) for the black hole phase transitions confirms a universal critical exponent of 1/2. We also find that the temperature and pressure corresponding to the extremal points of r(ps) and u(ps) are highly consistent with the thermodynamic metastable curve for black hole systems with different values of b. Furthermore, we also extend the corresponding study to higher-dimensional black holes cases. The results show that the photon sphere behaves quite differently for the van der Waals-like and reentrant phase transitions, and both phase transitions can be identified via the photon sphere.