Thermodynamics of Born-Infeld–anti-de Sitter black holes in the grand canonical ensemble

The main objective of this paper is to study thermodynamics and stability of static electrically charged Born-Infeld black holes in AdS space in $D=4$. The Euclidean action for the grand canonical ensemble is computed with the appropriate boundary terms. The thermodynamical quantities such as the Gibbs free energy, entropy and specific heat of the black holes are derived from it. The global stability of black holes are studied in detail by studying the free energy for various potentials. For small values of the potential, we find that there is a Hawking-Page phase transition between a BIAdS black hole and the thermal-AdS space. For large potentials, the black hole phase is dominant and is preferred over the thermal-AdS space. Local stability is studied by computing the specific heat for constant potentials. The nonextreme black holes have two branches: small black holes are unstable and the large black holes are stable. The extreme black holes are shown to be stable both globally as well as locally. In addition to the thermodynamics, we also show that the phase structure relating the mass $M$ and the charge $Q$ of the black holes is similar to the liquid-gas-solid phase diagram.

Figure 1

The function $f(r)$ vs $r$ for $Qb<1/2$. Here $b=1$, $Q=0.1$, $l=\sqrt{3}$ and $G=1$.

Figure 2

The function $f(r)$ vs $r$ for $Qb=1/2$ Here $b=1$, $Q=0.5$, $l=\sqrt{3}$ and $G=1$.

Figure 3

The function $f(r)$ vs $r$ for $Qb>1/2$. Here $b=1$, $Q=2$, $l=\sqrt{3}$ and $G=1$.

Figure 4

The function $M$ vs $Q$ for $b=0.25$. Here $l=\sqrt{3000}$ and $G=1$. The dark line shows the $M=a/2$ bound and the light line shows the $M=M_{\mathrm{ex}}$ bound. “NS” represents naked singularities.

Figure 5

The function $M$ vs $Q$ for RNAdS black holes. Here $l=1$ and $G=1$. The graph is the $M=M_{\mathrm{ex}}$ bound.

Figure 6

The inverse temperature $\beta$ vs horizon radii $r_{+}$ at fixed potential $\Phi =10$. Here $b=2$, $l=1$, $G=1$.

Figure 7

The inverse temperature $\beta$ vs horizon radii $r_{+}$ at fixed potential $\Phi =0.2$. Here $b=2$, $l=1$, $G=1$.

Figure 8

The inverse temperature $\beta$ vs horizon radii $r_{+}$ at fixed potential $\Phi =0$. Here $l=1$ and $G=1$.

Figure 9

The entropy $S$ vs the temperature $T$ at fixed potential $\Phi =10$. Here $l=1$, $b=0.4$ and $G=1$.

Figure 10

The specific heat $C_{\Phi }$ and temperature $T$ vs horizon radii $r_{+}$ at fixed potential $\Phi =10$. Here $l=1$, $b=0.4$ and $G=1$.

Figure 11

The specific heat $C_{\Phi }$ and the temperature $T$ vs horizon radii $r_{+}$ at fixed potential $\Phi =0.3$. Here $l=1$, $b=0.4$ and $G=1$.

Figure 12

The free energy $F$ vs temperature $T$ at fixed potentials $\Phi =0.2$, 0.63, 0.7, 0.91, 1, 1.1, 1.17, 1.3,3. Here $l=1$, $b=0.4$ and $G=1$. The axis intersects at (0,0).

Figure 13

The free energy $F$ vs temperature $T$ for the RNAdS black hole and the BIAdS black hole at their respective critical potentials when the extreme black hole phase appear. The dark curve shows RNAdS black hole and the light curve shows the BIAdS black hole.

Figure 14

The free energy $F$ vs temperature $T$ at potentials $\Phi =0.25$ for BIAdS and RNAdS black holes. Here $l=1$ and $G=1$.