Dyonic Taub-NUT-AdS spaces: Phase structures of all horizon geometries

We study phase structures of Lorentzian Dyonic Taub-NUT-AdS spacetimes for different horizon geometries, which could be spherical, flat, or hyperbolic. We check the consistency of our extended thermodynamics approach through satisfying the first law, the Gibbs-Duhem relation, and the generalized Smarr’s relation. Although we study the phase structure for the three cases, we give special attention to the flat and hyperbolic cases since they are known to show no phase transitions and were not studied before. Working in a mixed ensemble, we found that the behaviors of the flat and hyperbolic cases are different from those of a charged black hole. In the latter case, a continuous phase transition occurs at high temperatures and pressures, i.e., above the critical point, but in our cases it occurs at low temperatures and pressures, i.e., below the critical point! Generically, the spherical case is characterized by two critical points with continuous phase transition between them.

Figure 1

The function $g_{yy}$ showing positive values for all $r_h$. The parameter values are $n=1,l=4,m=4,q=0.1$, $\mathrm{p}=0.1$, and $z_{\max }=7$.

Figure 2

The function $g_{\varphi \varphi }$ showing positive values for all $r_h$. The parameter values are $n=1,5l=2,m=4,q=0.1$, $\mathrm{p}=0.1$, and ${\eta }_{\max }=1.5$.

Figure 3

Pressure as a function of $r_h$ as $T$ changes around the critical point. $T$ goes from $1.1T_c$ to $T_c$ to $0.9T_c$. The critical radius occurs at $r_c=0.446$. The thermodynamic parameters take the following values: ${\mathrm{\Phi }}_e=1.1$ $Q_m^h=1.32$, $n=1$, $T_c=0.347$, and $k=-1$.

Figure 4

Gibbs energy vs pressure for $r\in [0.25,6]$, where ${\mathrm{\Phi }}_e=0.50$, $Q_m^h=2.00$, $n=1.98$, $k=-1$, $T_c=0.353$.

Figure 5

P-T phase diagram for ${\mathrm{\Phi }}_e=0.7$, $Q_m^h=1$, $n=1.50$, $k=-1$, $T_c=2.46$, $P_c=0.448$.

Figure 6

Pressure as a function of $r_h$ as $T$ changes around the critical point. $T$ goes from $1.1T_c$ to $T_c$ to $0.9T_c$. The critical radius occurs at $r_c=0.496$. The thermodynamic parameters take the following values: ${\mathrm{\Phi }}_e=1.1$, $Q_m^h=1.32$, $n=1$, $T_c=0.251$, and $k=0$.

Figure 7

Gibbs energy vs pressure for $r\in [0.01,0.3]$, where ${\mathrm{\Phi }}_e=2.00$, $Q_m^h=2.30$, $n=1.1$, $k=0$, $T_c=0.244$.

Figure 8

P-T phase diagram for ${\mathrm{\Phi }}_e=0.10$, $Q_m^h=2.30$, $n=1.1$, $k=0$, $T_c=6.49$, $P_c=1.56$.

Figure 9

P-T phase diagram for ${\mathrm{\Phi }}_e=0$, $Q_m^h=1.30$, $n=1.50$, $k=0$, $T_c=0.902$, $P_c=0.159$.

Figure 10

Pressure as a function of $r_h$ as $T$ changes around the lower critical point for the spherical horizon when ${\mathrm{\Phi }}_e^2<1$. The first-order transition occurs for temperatures $<T_c$. The thermodynamic parameters are ${\mathrm{\Phi }}_e=0.50$, $Q_m^h=2.00$, $n=1.98$, $T_c=0.0363$ and $k=1$.

Figure 11

Pressure as a function of $r_h$ as $T$ changes around the higher critical point for the spherical horizon when ${\mathrm{\Phi }}_e^2<1$. The first-order transition occurs for temperatures $>T_c$. The thermodynamic parameters are ${\mathrm{\Phi }}_e=0.50$, $Q_m^h=2.00$, $n=1.98$, $T_c=0.127$, and $k=1$.

Figure 12

P-T phase diagram for the spherical horizon when ${\mathrm{\Phi }}_e^2<1$. The thermodynamic parameters are ${\mathrm{\Phi }}_e=0.50$, $Q_m^h=2.00$, $n=1.98$, $k=1$.

Figure 13

P-T phase diagram for ${\mathrm{\Phi }}_e=0.5$, $Q_m^h=2.00$, $n=2$, $k=1$.

Figure 14

P-T phase diagram for ${\mathrm{\Phi }}_e=0.5$, $Q_m^h=2.00$, $n=2.1$, $k=1$.