Photon orbits and thermodynamic phase transition of $d$-dimensional charged AdS black holes

We study the relationship between the null geodesics and thermodynamic phase transition for the charged AdS black hole. In the reduced parameter space, we find that there exist nonmonotonic behaviors of the photon sphere radius and the minimum impact parameter for the pressure below its critical value. The study also shows that the changes of the photon sphere radius and the minimum impact parameter can serve as order parameters for the small-large black hole phase transition. In particular, these changes have an universal exponent of $\frac{1}{2}$ near the critical point for any dimension $d$ of spacetime. These results imply that there may exist universal critical behavior of gravity near the thermodynamic critical point of the black hole system.

Figure 1

The effective potential for the four-dimensional charged AdS black hole. The parameters are set to $S=5$, $Q=1$, and $P=0.003$. The angular momentum $L/E$ of the photon varies from 0.8 to 4.4 from bottom to top (in the right side). The red thick line has the critical angular momentum $L_c/E\approx 3.6$.

Figure 2

(a) $\tilde{T}$ vs. ${\tilde{r}}_{ps}$ for fixed reduced pressure $\tilde{P}=0.9$, 0.94, 0.98, 1.0, and 1.02 from bottom to top. (b) $\tilde{T}$ vs. ${\tilde{u}}_{ps}$ for fixed $\tilde{P}=0.7$, 0.8, 0.9, 1.0, and 1.1 from bottom to top.

Figure 3

$\tilde{T}$ vs. ${\tilde{r}}_{ps}$ for fixed $\tilde{P}=0.9$. The horizontal line BD corresponds to the phase transition temperature $\tilde{T}=0.9608$.

Figure 4

Behaviors of $\mathrm{\Delta }{\tilde{r}}_{ps}$ and $\mathrm{\Delta }{\tilde{u}}_{ps}$ as a function of the phase transition temperature $\tilde{T}$. (a) $\mathrm{\Delta }{\tilde{r}}_{ps}$ vs. $\tilde{T}$. (b) $\mathrm{\Delta }{\tilde{u}}_{ps}$ vs. $\tilde{T}$.

Figure 5

(a) $\tilde{T}$ vs. ${\tilde{r}}_{ps}$ and (b) $\tilde{T}$ vs. ${\tilde{u}}_{ps}$ for fixed reduced pressure $\tilde{P}=0.8$, and the dimension of spacetime $d=6–10$. The arrows indicate the increase of $d$.

Figure 6

Behaviors of $\mathrm{\Delta }{\tilde{r}}_{ps}$ and $\mathrm{\Delta }{\tilde{u}}_{ps}$ as functions of the phase transition temperature $\tilde{T}$. (a) $\mathrm{\Delta }{\tilde{r}}_{ps}$ vs. $\tilde{T}$. (b) $\mathrm{\Delta }{\tilde{u}}_{ps}$ vs. $\tilde{T}$. The dimension of spacetime is set to $d=5–10$ from top to bottom.