Topological black hole in the theory with nonminimal derivative coupling with power-law Maxwell field and its thermodynamics

We obtain topological black hole solutions in scalar-tensor gravity with nonminimal derivative coupling between scalar and tensor components of gravity and a power-law Maxwell field minimally coupled to gravity. The obtained solutions can be treated as a generalization of previously derived charged solutions with standard Maxwell action [X.-H. Feng, H.-S. Liu, H. Lu, and C. N. Pope, Phys. Rev. D 93, 044030 (2016)]. We examine the behavior of obtained metric functions for some asymptotic values of distance and coupling. To obtain information about singularities of the metrics we calculate Kretschmann scalar. We also examine the behavior of gauge potential and show that it is necessary to impose some constraints on parameter of nonlinearity in order to obtain reasonable behavior of the filed. The next part of our work is devoted to the examination of black hole thermodynamics. Namely, we obtain the black hole’s temperature and investigate it in general as well as in some particular cases. To introduce entropy, we use the well-known Wald procedure which can be applied to quite general diffeomorphism-invariant theories. We also extend thermodynamic phase space by introducing thermodynamic pressure related to cosmological constant and as a result we derive generalized first law and Smarr relation. The extended thermodynamic variables also allow us to construct Gibbs free energy and its examination gives important information about thermodynamic stability and phase transitions. We also calculate heat capacity of the black holes which demonstrates variety of behavior for different values of allowed parameters.

Figure 1

Metric functions $U(r)$ for linear ($p=1$, solid curve) and conformal ($p=\frac{n+1}{4}$, dashed curve) types of gauge field. All the other parameters are equal for both cases; namely, $n=4$, $\alpha =0.2$, $\eta =0.4$, $ϵ=1$, $\mathrm{\Lambda }=-2$, $q=0.2$ and $\mu =1$.

Figure 2

Mass parameter $\mu$ as a function of the horizon radius $r_{+}$ for linear ($p=1$, solid curve) and conformal ($p=\frac{n+1}{4}$, dashed curve) types of gauge field. All the other parameters are equal for both cases, namely, $n=4$, $\alpha =0.2$, $\eta =0.4$, $ϵ=1$, $\mathrm{\Lambda }=-2$, $q=0.2$.

Figure 3

Black hole temperature $T$ as a function of the horizon radius for linear $p=1$ (the left graph) and conformal $p=\frac{n+1}{4}$ (the right one). For both graphs the solid curves represent almost critical value of the cosmological constant (${\mathrm{\Lambda }}_c$), the dotted curves correspond to the values of $\mathrm{\Lambda }$ above the critical one, the dashed curves correspond to the values below the critical. For both graphs some parameters are the same, namely, $n=4$, $\alpha =0.2$, $\eta =0.4$, $ϵ=1$, and $q=0.4$. For the left graph correspondence of curves is as follows (from the lowest the highest pressures): $\mathrm{\Lambda }=-3$, ${\mathrm{\Lambda }}_c=-5.7$, $\mathrm{\Lambda }=-8$ and for the right graph we have the following choice of the parameters: $\mathrm{\Lambda }=-5$, ${\mathrm{\Lambda }}_c=-7.4$, $\mathrm{\Lambda }=-10$.

Figure 4

Gibbs free energy $G$ as a function of temperature $T$ for linear $p=1$ (the left graph) and conformal $p=\frac{n+1}{4}$ (the right graph) cases for several different values of pressure. For both graphs the solid curves represent almost critical value of pressure (${\mathrm{\Lambda }}_c$), the dotted curves correspond to the pressure above the critical one, the dashed and dash-dotted curves correspond to the pressures below the critical values. For both graphs, some parameters are the same, namely, $n=4$, $\alpha =0.2$, $\eta =0.4$, $ϵ=1$, and $q=0.4$. For the left graph, correspondence of curves is as follows (from the lowest the highest pressures): $\mathrm{\Lambda }=-0.8$, $\mathrm{\Lambda }=-3$, ${\mathrm{\Lambda }}_c=-5.7$ and $\mathrm{\Lambda }=-8$. For the right graph, we have the following choice of the parameters: $\mathrm{\Lambda }=-0.8$, $\mathrm{\Lambda }=-3$, ${\mathrm{\Lambda }}_c=-7.4$ and $\mathrm{\Lambda }=-10$.

Figure 5

Comparison of heat capacities for linear (solid curve) and conformal (dashed curve) cases. All the fixed parameters are the same, namely, $n=4$, $ϵ=1$, $\alpha =0.2$, $\eta =0.4$, $\mathrm{\Lambda }=-2$, and $q=0.2$.

Figure 6

Heat capacity $C_P$ for conformal case when the discontinuity disappears. Increasing of the pressure $P$ leads to decreasing of the local maximum of heat capacity. The correspondence of the curves is as follows: solid line corresponds to $\mathrm{\Lambda }=-2.1$, dashed curve corresponds to $\mathrm{\Lambda }=-2.15$ and the dotted curve corresponds to $\mathrm{\Lambda }=-2.25$. Other parameters are the same, namely, $n=4$, $ϵ=1$, $\alpha =0.2$, $\eta =0.4$, and $q=0.4$.