Thermodynamics of topological black holes in Brans-Dicke gravity with a power-law Maxwell field

In this paper, we present a new class of higher-dimensional exact topological black hole solutions of the Brans-Dicke theory in the presence of a power-law Maxwell field as the matter source. For this aim, we introduce a conformal transformation which transforms the Einstein-dilaton-power-law Maxwell gravity Lagrangian to the Brans-Dicke-power-law Maxwell theory one. Then, by using this conformal transformation, we obtain the desired solutions. Next, we study the properties of the solutions and conditions under which we have black holes. Interestingly enough, we show that there is a cosmological horizon in the presence of a negative cosmological constant. Finally, we calculate the temperature and charge and then by calculating the Euclidean action, we obtain the mass, the entropy and the electromagnetic potential energy. We find that the entropy does not respect the area law, and also the conserved and thermodynamic quantities are invariant under conformal transformation. Using these thermodynamic and conserved quantities, we show that the first law of black hole thermodynamics is satisfied on the horizon.

Figure 1

The self-interacting potential $10^{-1}V(\mathrm{\Phi })$ versus $\mathrm{\Phi }$ for $n=5$, $p=1.2$, $\alpha =0.6$, $l=b=1$, $\mathrm{\Lambda }=-10$ and $q=10$.

Figure 2

The function $m(r_h)$ versus $r_h$ for $n=4$, $p=1.2$, $\alpha =0.26$, $l=b=1$, $\mathrm{\Lambda }=-6$ and $q=0.6$.

Figure 3

The function $10^{-2}m(r_h)$ versus $r_h$ for $n=5$, $p=1.2$, $\alpha =0.6$, $l=b=1$, $\mathrm{\Lambda }=-10$ and $q=10$.

Figure 4

The function $V(r)$ versus $r$ for $k=0$, $n=4$, $p=1.2$, $\alpha =0.26$, $l=b=1$, $\mathrm{\Lambda }=-6$ and $r_{\mathrm{ext}}=1$.

Figure 5

The function $V(r)$ versus $r$ for $k=-1$, $n=6$, $p=1.1$, $\alpha =1.18$, $l=b=1$, $\mathrm{\Lambda }=15$ and $r_{\mathrm{ext}}=1$.

Figure 6

The function $V(r)$ versus $r$ for $k=1$, $n=5$, $p=1.2$, $\alpha =0.63$, $l=b=1$, $\mathrm{\Lambda }=-10$ and $r_{\mathrm{ext}}=1$.

Figure 7

The function $V(r)$ versus $r$ for $k=-1$, $n=5$, $p=1.2$, $\alpha =0.63$, $l=1$, $b=1.5$, $\mathrm{\Lambda }=-10$ and $r_{\mathrm{ext}}=1$.

Figure 8

The function $V(r)$ versus $r$ for $n=6$, $p=1.1$, $l=b=1$, $\mathrm{\Lambda }=15$, $q=1.5$ and $m=0.5$.

Figure 9

The function $m(r_h)$ versus $r_h$ for $k=1$, $n=6$, $p=1.1$, $\alpha =1.18$, $l=1$, $b=1.4$, $\mathrm{\Lambda }=-15$.

Figure 10

The function $V(r)$ versus $r$ for $k=1$, $n=6$, $p=1.1$, $l=1$, $b=1.4$, $\mathrm{\Lambda }=-15$, $\alpha =1.18$ and $r_{\mathrm{ext}}=1$.