Probing the relationship between the null geodesics and thermodynamic phase transition for rotating Kerr-AdS black holes

In this paper, we aim to examine the relationship between the unstable circular photon orbit and the thermodynamic phase transition for a rotating Kerr-AdS black hole. On one side, we give a brief review of the phase transition for the Kerr-AdS black hole. The coexistence curve and the metastable curve corresponding to the phase transition are clearly shown. On the other side, we calculate the radius and the angular momentum of the unstable circular orbits for a photon by analyzing the effective potential. Then combining these two sides, we find the following results. i) The radius and the angular momentum of the unstable circular photon orbits demonstrate the nonmonotonic behaviors when the thermodynamic phase transition takes place. So from the behavior of the circular orbit, one can determine whether there exists a thermodynamic phase transition. ii) The difference of the radius or the angular momentum for the coexistence small and large black holes can be treated as an order parameter to describe the phase transition. And near the critical point, it has a critical exponent of $\frac{1}{2}$. iii) The temperature and pressure corresponding to the extremal points of the radius or the angular momentum of the unstable circular photon orbit completely agree with that of the metastable curves from the thermodynamic side. Thus, these results confirm the relationship between the geodesics and thermodynamic phase transition for the Kerr-AdS black hole. Therefore, on one hand, we are allowed to probe the thermodynamic phase transition from the gravity side. On the other hand, the signature of the strong gravitational effect can also be revealed from the black hole thermodynamics.

Figure 1

(a) The horizon radii as a function of the phase transition temperature. The top and bottom curves are for the coexistence large and small black holes, respectively. (b) The difference of the horizon radii as a function of the phase transition temperature.

Figure 2

(a) $\tilde{T}\text{−}{\tilde{r}}_{\mathrm{h}}$ phase diagram. (b) $\tilde{P}\text{−}\tilde{T}$ phase diagram.

Figure 3

Behavior of the effective potential with fixed $S=1$ and $J=1$. (a) $L=2$ and $P=0.02$, 0.04, 0.066, 0.09 from bottom to top. (b) $L=5$ and $P=0.001$, 0.002, 0.0033, 0.005 from bottom to top.

Figure 4

Phase transition temperature as a function of the radius and the angular momentum of the circular orbit. (a) $\tilde{T}\text{−}{\tilde{r}}_{\mathrm{co}}$. (b) $\tilde{T}\text{−}{\tilde{L}}_{\mathrm{co}}$. The pressure $\tilde{P}=0.88$, 0.92, 0.96, 1, and 1.04 from bottom to top. And the black dashed curves correspond to the extremal points.

Figure 5

Phase transition pressure as a function of the radius and the angular momentum of the circular orbit. (a) $\tilde{P}\text{−}{\tilde{r}}_{\mathrm{co}}$. (b) $\tilde{P}\text{−}{\tilde{L}}_{\mathrm{co}}$. The temperature $\tilde{T}=0.88$, 0.92, 0.96, 1, and 1.04 from bottom to top. And the black dashed curves correspond to the extremal points.

Figure 6

(a) The radius of the circular orbit for the coexistence small and large black holes. (b) The difference of the radii of the circular orbit.

Figure 7

(a) The angular momentum of the circular orbit for coexistence small and large black holes. (b) The difference of the angular momenta of the circular orbit.

Figure 8

Behaviors of the difference of the radius and angular momentum of the circular orbit near the critical point of the small-large black hole phase transition. Blue thick dashed lines are for the numerical results and red thin lines are our fitting results. (a) $\ln (\mathrm{\Delta }{r}_{\mathrm{co}}/r_{\mathrm{coc}})$ vs $\ln (1-T/T_{\mathrm{c}})$. (b) $\ln (\mathrm{\Delta }{L}_{\mathrm{co}}/L_{\mathrm{coc}})$ vs $\ln (1-T/T_{\mathrm{c}})$.

Figure 9

Extremal points in the $\tilde{P}\text{−}\tilde{T}$ diagram. Blue dashed curve and red dotted curve are the metastable large and small black hole curves from the thermodynamics, respectively. The blue square markers and red circle markers are the extremal points for the large and small black holes calculated from the circular orbit.