Photon sphere and reentrant phase transition of charged Born-Infeld-AdS black holes
In this paper, we extend the study of the relationship between the photon sphere and the thermodynamic phase transition, especially the reentrant phase transition, to this black hole background. According to the number of the thermodynamic critical points, the black hole systems are divided into four cases with different values of Born-Infeld parameter $b$, where the black hole systems can have no phase transition, reentrant phase transition, or Van der Waals-like phase transition. For these different cases, we obtain the corresponding phase structures in pressure-temperature diagram and temperature-specific volume diagram. The tiny differences between these cases are clearly displayed. On the other hand, the radius $r_{\rm ps}$ and the minimal impact parameter $u_{\rm ps}$ of the photon sphere are calculated via the effective potential of the radial motion of photons. For different cases, $r_{\rm ps}$ and $u_{\rm ps}$ are found to have different behaviors. In particular, with the increase of $r_{\rm ps}$ or $u_{\rm ps}$, the temperature possesses a decrease-increase-decrease-increase behavior for fixed pressure if there exists the reentrant phase transition. While for fixed temperature, the pressure will show 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 changes of $r_{\rm ps}$ and $u_{\rm ps}$ among the black hole phase transition confirm an universal critical exponent $\frac{1}{2}$. Therefore, all the results indicate that, for the charged Born-Infeld-AdS black holes, not only the Van der Waals-like phase transition, but also the reentrant phase transition can be reflected through the photon sphere.