Throttling process of the Kerr–Newman–anti-de Sitter black holes in the extended phase space

The throttling process of the Kerr–Newman–anti-de Sitter (KN–AdS) black holes is systematically studied in the extended phase space. In this framework, the cosmological constant is interpreted as a varying thermodynamic pressure, and the black hole mass is identified with enthalpy. The throttling process is essentially an adiabatic and isenthalpic (i.e., constant-mass) process for the KN–AdS black holes. The Joule–Thomson coefficient, inversion temperature, inversion curve, and isenthalpic curve are investigated in order, with both analytical and numerical methods. It is found that there are no maximum inversion temperatures, but only minimum ones that are around one half of the critical temperatures of the KN–AdS black holes. Two characteristic masses are also discussed to show the detailed features in the throttling behaviors of the KN–AdS black holes.

Figure 1

The JT coefficient $\mu$ as a function of the KN–AdS black hole entropy $S$, with $p=1$, $J=1$, and $Q=1$. $\mu$ diverges at $S=0.9973$, where the Hawking temperature of the KN–AdS black hole is zero. Then, it monotonically increases and reaches 0 at $S=2.371$. Hence, there are only minimum inversion temperatures for the KN–AdS black holes, and the KN–AdS black holes always cool at large entropies.

Figure 2

The inversion curves of the KN–AdS black holes with different values of $J$ and $Q$. The figures in the left column show the inversion curves with fixed $J$ and increasing $Q$, and in the right column with fixed $Q$ and increasing $J$. The detailed values of $J$ and $Q$ are listed in each panel. The cooling and heating regions lie above and below the inversion curves respectively. The shapes of the inversion curves are similar, and different inversion curves tend to coincide with each other with $J$ and $Q$ increasing.

Figure 3

The inversion curves of the KN–AdS black holes with different values of $J$ and $Q$ at high pressures ($100<p<10000$). The detailed values of $J$ and $Q$ are listed in the figure. Both inversion curves are concave functions, meaning that there are no maximum inversion temperatures.

Figure 4

The isenthalpic curves and the inversion curves of the KN–AdS black holes with different values of $J$ and $Q$. The detailed values of $J$ and $Q$ are listed in each panel. The shapes of the isenthalpic curves are similar to those of the RN–AdS or Kerr–AdS black holes, with large $Q$ or $J$. In each panel, the masses of the KN–AdS black holes must satisfy the condition $2M^2>\sqrt{4J^2+Q^4}+Q^2$ to avoid naked singularity.

Figure 5

The isenthalpic curves and the inversion curves of the RN–AdS (a) and KN–AdS (b) black holes, with the black hole masses chosen around their characteristic values $M_{*}$. The detailed values of $J$, $Q$, and $M$ are listed in each panel. If $M>M_{*}$, there is an inversion curve; if $M<M_{*}$, there is no inversion curve.

Figure 6

The intersections of the isenthalpic curves with different enthalpies (i.e., masses) for the RN–AdS (a) and KN–AdS (b) black holes. The detailed values of $J$, $Q$, and $M$ are listed in each panel.

Figure 7

The $T$-intercepts of the isenthalpic curves as a function of the black hole masses. For the RN–AdS black holes (a), we choose $Q=1$, and the maximum point is located at ($2\sqrt{3}/3$, $1/(6\sqrt{3}\pi )$). For the KN–AdS black holes (b), we choose $J=1$ and $Q=1$, and the maximum point is located at (1.500, 0.02307).

Figure 8

The throttling process of the KN–AdS black holes with different enthalpies (i.e., masses), with $J=1$ and $Q=1$. The inversion curve is a monotonically increasing function and separates the $T–p$ plane into the cooling region ($\mu >0$) and the heating region ($\mu <0$). There is only a minimum inversion temperature $T_{\mathrm{i}}^{\min }=0.01707$, but no maximum one. Five characteristic isenthalpic curves are shown in order: (1) $M=1.29<M_{*}$, the slope of the isenthalpic curve is negative, and this curve always lies in the heating region; (2) $M=M_{*}=1.309$, the isenthalpic curve has the minimum inversion temperature $T_{\mathrm{i}}^{\min }$; (3) $M_{*}<M=1.35<\tilde{M}$, the isenthalpic curve goes up and then down and crosses the inversion curve at its maximum point (0.002028, 0.02265); (4) $M=\tilde{M}=1.500$, the isenthalpic curve has the largest $T$-intercept, $T_0^{\max }=0.02307$; (5) $M=5>\tilde{M}$, the $T$-intercept decreases, and the isenthalpic curve intersects with other four isenthalpic curves. All isenthalpic curves with $M>M_{*}$ will descend and intersect with the $p$-axis at larger pressures, but cannot be shown in this figure.