Thermodynamics of rotating Bardeen black holes: Phase transitions and thermodynamics volume

We investigate the thermodynamical properties of the rotating Bardeen black holes characterized by mass $m$, the spin parameter $a$, and the magnetic charge $g$. We calculate exact expressions of the Hawking temperature, mass, entropy, and heat capacity. The black hole mass is minimum at radius $r_{+}^E$, where both the heat capacity and temperature vanish with a stable remnant. Also, there exists a critical radius $r_{+}^C$ of multiple orders, where the heat capacity diverges, suggesting that the black hole is thermodynamically stable in the range $r_{+}^E<r_{+}<r_{+}^C$. We also analyze the extended phase space thermodynamics of the rotating Bardeen–anti-de Sitter black holes. Treating the cosmological constant $\mathrm{\Lambda }$ and charge $g$ as thermodynamic variables, we derive the generalized first law in the extended phase space to study the critical phenomena of the black holes. The Ehrenfest scheme for the $P-V$ criticality of the rotating Bardeen black holes in anti-de Sitter spaces is examined. The Clausius-Clapeyron-Ehrenfest equations confirming the second-order phase transitions of the van der Waals fluid is no longer valid.

Figure 1

The plot showing the behavior of Hawking temperature $T_{+}$ vs horizon radius $r_{+}$ for rotating Bardeen black holes with a comparison of the temperature of Bardeen black holes ($a=0$) and the Kerr black holes ($g=0$).

Figure 2

The plot showing the behavior of capacity $C_{+}$ vs horizon radius $r_{+}$ for rotating Bardeen black holes with a comparison of the heat capacity of Bardeen black holes ($a=0$) and the Kerr black holes ($g=0$).

Figure 3

Left: Plot showing the behavior of inner horizon $r_{-}$ and outer horizon $r_{+}$ for rotating Bardeen-AdS black holes. Two horizons coincide at $r_E=r_{-}=r_{+}$. Right: Plot showing the behavior of entropy at inner horizon $r_{-}$ and at outer horizon $r_{+}$ for rotating Bardeen-AdS black holes. Two entropies coincide at $r_E=r_{-}=r_{+}$.

Figure 4

Plot showing the behavior of $M$ vs $S$ for rotating Bardeen-AdS black holes.

Figure 5

Plot showing the behavior of $T$ vs $S$ for rotating Bardeen-AdS black holes.

Figure 6

Plot showing the behavior of $G$ vs $S$ for rotating Bardeen-AdS black holes.

Figure 7

Plot showing the behavior of $G$ vs $T$ for a rotating Bardeen-AdS black hole for $P>P_c$ (green dotted line), $P=P_c$ (blue dotted line) and $P<P_c$ (red dotted line), where $P_c$ is the critical pressure at $J=0$.

Figure 8

Plot showing the behavior of $C_P$ vs $S$ for rotating Bardeen-AdS black holes.

Figure 9

Plot showing the behavior of $\alpha$ vs $S$ for rotating Bardeen-AdS black holes.

Figure 10

Plot showing the behavior of ${\kappa }_T$ vs $S$ for rotating Bardeen-AdS black holes.