Thermodynamics of ($2+1$)-dimensional Coulomb-like black holes from nonlinear electrodynamics with a traceless energy momentum tensor

In this work we study the thermodynamics of a ($2+1$)-dimensional static black hole under a nonlinear electric field. In addition to standard approaches, we investigate black hole thermodynamic geometry. We compute Weinhold and Ruppeiner metrics and compare the thermodynamic geometries with the standard description for black hole thermodynamics. We further consider the cosmological constant as an additional extensive thermodynamic variable. For thermodynamic equilibrium in three-dimensional space, we compute heat engine efficiency and show that it may be constructed with this black hole.

Figure 1

The behavior of metric function $f(r)$, with $M=1$, $q=1$, for different values of cosmological constant. Note that when $\mathrm{\Lambda }=-1$, there is a naked singularity.

Figure 2

The behavior of the metric function $f(r)$, with $M=1$, $q=1$, for different values of the cosmological constant $0>\mathrm{\Lambda }>-\frac{M^3}{12q^4}$. Here, we observe a black hole with two horizons. Note the extremal configuration at $\mathrm{\Lambda }=-\frac{1}{12}$.

Figure 3

The behavior of the metric function $f(r)$, with $M=1$, $q=1$, for different values of the cosmological constant when $\mathrm{\Lambda }<-\frac{M^3}{12q^4}$. The region is naked singularities: one real negative solution for the horizon, and two complex solutions. The plot includes extremal case $\mathrm{\Lambda }=-\frac{1}{12}$.

Figure 4

Lapse function behavior as a function of $r$ (top panel), and temperature as function of the event horizon $r_h$ (bottom panel). Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charge $q$. When $q=0$, the temperature profile of BTZ black hole is recovered, and $q=(1/6{)}^{1/4}$ corresponds to the extremal case.

Figure 5

Heat capacity as function of event horizon $r_h$. Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charged $q$. When $q=0$, we recover the heat capacity profile of a static BTZ black hole.

Figure 6

This plot shows $\frac{{\partial }^{2}M(S,q)}{\partial S^2}$ as function of event horizon $r_h$. Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charge $q$. When $q=0$, the $\frac{{\partial }^{2}M(S,q)}{\partial S^2}$ profile of a BTZ black hole is obtained.

Figure 7

Temperature as a function of event horizon $r_h$ (right panel) for a charged BTZ black hole. Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charge $q$. When $q=0$, the temperature profile of BTZ black holes is recovered.

Figure 8

Heat capacity as function of the event horizon $r_h$. Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charge $q$. When $q=0$, we recover the heat capacity profile of static BTZ black holes.

Figure 9

$R_R$ as a function of the event horizon $r_h$. Here, we consider $\mathrm{\Lambda }=-0.5$ and different values of electric charge $q$. When $q=0$, the $R_R$ profile of a BTZ static black hole is obtained.

Figure 10

Isothermal curves for charged monlinear Coulomb-like black holes.