Thermodynamics of topological nonlinear charged Lifshitz black holes

In this paper, we construct a new class of analytic topological Lifshitz black holes with constant curvature horizon in the presence of a power-law Maxwell field in four or more dimensions. We find that in order to obtain these exact Lifshitz solutions, we need a dilaton and at least three electromagnetic fields. Interestingly enough, we find that the reality of the charge of the electromagnetic field which is needed for having solutions with a curved horizon rules out black holes with a hyperbolic horizon. Next, we study the thermodynamics of these nonlinear charged Lifshitz black holes with spherical and flat horizons by calculating all of the conserved and thermodynamic quantities of the solutions. Furthermore, we obtain a generalized Smarr formula and show that the first law of thermodynamics is satisfied. We also perform a stability analysis in both canonical and grand-canonical ensembles. We find that the solutions are thermally stable in proper ranges of the metric parameters. Finally, we comment on the dynamical stability of the obtained solutions under perturbations in four dimensions.

Figure 1

The function $f(r)$ versus $r$ for $p=0.8$, $z=3$, $l=b=1$, $k=0$, and $r_{\mathrm{ext}}=1$.

Figure 2

The function $f(r)$ versus $r$ for $p=0.8$, $z=3$, $l=b=1$, $k=1$, and $r_{\mathrm{ext}}=1$.

Figure 3

The function $f(r)$ versus $r$ for $p=1/3$ (dashed line), $p=1/4$ (dash-dotted line), $p=0$ (solid line), $z=9$, $m=q=l=b=1$, and $k=0$.

Figure 4

The function $f(r)$ versus $r$ for $p=1/3$ (dashed line), $p=1/4$ (dash-dotted line), $p=0$ (solid line), $z=9$, $m=q=l=b=1$, and $k=1$.

Figure 5

The behavior of $10T$ (solid line), $10({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line), and ${\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $z$ for $k=0$, with $l=b=1$, $q_1=0.1$, $r_{+}=0.6$, $n=4$, and $p=1/3$.

Figure 6

The behavior of $T$ (solid line), ${10}^{-1}({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line) and ${10}^{-1}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $r_{+}$ for $k=0$, with $l=b=1$, $q_1=0.1$, $z=12$, $n=4$, and $p=1/3$.

Figure 7

The behavior of $T$ (solid line) and ${10}^{-3}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $z$ for $k=0$ with $l=b=1$, $q_1=0.2$, $r_{+}=0.55$, $n=5$, and $p=2$.

Figure 8

The behavior of $T$ (solid line) and ${10}^{-5}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $r_{+}$ for $k=0$ with $l=b=1$, $q_1=0.15$, $z=4$, $n=6$, and $p=3$.

Figure 9

The behavior of $10T$ (solid line), $({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line), and ${\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $z$ for $k=1$ with $l=b=1$, $q_1=0.05$, $r_{+}=0.6$, $n=4$, and $p=1/3$.

Figure 10

The behavior of $T$ (solid line), ${10}^{-1}({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line), and ${10}^{-1}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $r_{+}$ for $k=1$, with $l=b=1$, $q_1=0.05$, $z=12$, $n=4$, and $p=1/3$.

Figure 11

The behavior of $T$ (solid line), $({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line), and ${10}^{-3}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $z$ for $k=1$, with $l=b=1$, $q_1=0.1$, $r_{+}=0.6$, $n=4$, and $p=2$.

Figure 12

The behavior of $T$ (solid line), ${10}^{-1}({\partial }^{2}M/\partial S^2{)}_Q$ (dashed line), and ${10}^{-4}{\mathbf{H}}_{S,Q}^M$ (dash-dotted line) versus $r_{+}$ for $k=1$, with $l=b=1$, $q_1=0.1$, $z=1.5$, $n=4$, and $p=2$.