Thermodynamics of non-linearly charged Anti-de Sitter black holes in four-dimensional Critical Gravity

Since its introduction in the late nineties, the idea of the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence [1] has gained momentum in different areas due to its potential to shed light on phenomena that range from superconductivity to quantum computing. This correspondence has also generated the need for the study of theories other than General Relativity due to two main reasons: first, the necessity to have theories that exhibit desired symmetries and properties that match the non-relativistic systems in the context of the correspondence and, secondly, the possibility that these enhanced theories may support a variety of thermodynamically rich AdS black holes, whose holographic role will be to introduce the non-relativistic behavior at finite temperature.

In this context, quadratic curvature gravities have been very successful in providing a plethora of new black hole configurations [2, 3, 4, 5, 6, 7]. A notable example within these theories in 2+1 dimensions is New Massive Gravity [8], a parity-even, renormalisable theory that, at the linearized level, is equivalent to the unitary Fierz-Pauli theory for free massive spin-2 gravitons, where AdS solutions have been previously found [9, 10]. A four-dimensional analogue to this theory is Critical Gravity (CG) [11], which is a ghost-free, renormalizable theory of gravity with quadratic corrections in the curvature¹¹1In general, a theory with quadratic corrections in curvature will propagate massive scalar and spin-2 ghost fields. In CG, the relation between the coupling constants and the cosmological constant leads to a theory in which the scalar field is zero and the spin-2 field becomes massless and with vanishing energy. . A particularity of CG is that its vacuum admits an AdS black hole solution given by

	$\displaystyle ds^{2}$	$\displaystyle=$	$\displaystyle\displaystyle{-\frac{r^{2}}{l^{2}}\left(1-\frac{Ml^{3}}{r^{3}}\right)dt^{2}+\frac{l^{2}}{r^{2}}\frac{dr^{2}}{\left(1-\frac{Ml^{3}}{r^{3}}\right)}}$		(1)
			$\displaystyle+\frac{r^{2}}{l^{2}}\left(dx_{1}^{2}+dx_{2}^{2}\right),$

where $M$ is an integration constant. However, as emphasized by the authors in [11], this solution is massless and has a vanishing entropy. This is also highlighted in [12, 13] in four and six dimensions, where the entropy, as well as global conserved charges of their black holes solutions, vanish identically.

Because of the interest of obtaining black hole solutions that can be framed in the context of the gauge/gravity correspondence, in this work we aim to find new AdS black hole configurations that can be supported by CG and exhibit non-vanishing thermodynamical properties. To achieve this, a nonlinear electrodynamics (NLE) source is employed [14]. The origin of NLE dates to $1912$, the year in which Mie G. explored this formalism for the first time [15]. Some years later, in the thirties, and motivated in part to avoid the well-known singularity of the field of a point particle, Born and Infeld [16, 17, 18] proposed new research , giving rise to the Born-Infeld (BI) theory [19, 20, 21]. Given the complexity to carry out an extension of BI towards solutions of nonlinear equations, this formalism had a stage of stagnation for almost three decades. Nevertheless, around the sixties, J. F. Plebánski presented an outstanding work for NLE in a medium, in which the theory is developed through an antisymmetric conjugate tensor $P^{\mu\nu}$ (known as of Plebánski tensor) and a structure-function $\mathcal{H}=\mathcal{H}(P,Q)$, where $P$ and $Q$ are the invariants that are formed with the antisymmetric conjugate Plebánski tensor. Here, the structure-function $\mathcal{H}(P,Q)$ is associated with the Lagrangian function $\mathcal{L}(F,G)$, that depends on the invariants quadratics constructed from the Maxwell tensor $F_{\mu\nu}$, which can be determined by a Legendre transformation. At the end of the eighties, H. Salazar, A. García, and J. Plebánski found solutions for the equations of NLE coupled to gravity using the formalism [14], in which the BI theory appears as a special case [22]. Some of the benefits provided by the Plebánski formalism is the ability to obtain regular black hole solutions [23, 24, 25, 26, 27], Lifshitz black hole configurations that exist for any value of the dynamic exponent $z>1$ [28], and recently an exact solution of a massive, electrically and magnetically charged, rotating stationary black hole has been found [29, 30, 31]. In General Relativity, NLE has been a valuable tool in order to build exact black hole configurations, some of which exhibit non-standard asymptotic behavior in Einstein’s gravity or in its generalizations, as can be verified in [32, 33, 34, 35, 36, 37, 38]. It is relevant to mention that, in these works, charged black hole solutions that come from nonlinear theories possess interesting thermodynamic properties [39, 40, 41, 42, 43, 44]. All the above shows NLE as an interesting and motivating study field that we aim to explore with the addition of CG. As such, our action of interest will be given by:

$\displaystyle S[g_{\mu\nu},A_{\mu},P^{\mu\nu}]=\int{d}^{4}x\sqrt{-g}(\mathcal{L}_{CG}+\mathcal{L}_{NLE})\,,$

(2)

with

	$\displaystyle\mathcal{L}_{CG}$	$\displaystyle=$	$\displaystyle\frac{1}{2\kappa}\left(R-2\Lambda+\beta_{1}{R}^{2}+\beta_{2}{R}_{\alpha\beta}{R}^{\alpha\beta}\right)\,,$
	$\displaystyle\mathcal{L}_{NLE}$	$\displaystyle=$	$\displaystyle-\frac{1}{2}P^{\mu\nu}F_{\mu\nu}+\mathcal{H}(P)\,,$

where $\Lambda$ is the cosmological constant. As stated above, CG allows for the massive spin-$0$ field to vanish if the coupling constants $\beta_{1}$ and $\beta_{2}$ are restricted to obey the relations

$\displaystyle\beta_{2}=-3\,\beta_{1},\qquad\beta_{1}=-\frac{1}{2\Lambda}.$

(3)

Moreover, the Lagrangian density $\mathcal{L}_{NLE}$ describes the nonlinear behavior of the electromagnetic field $A_{\mu}$ with field strength $F_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The introduction of $P_{\mu\nu}$, which is an antisymmetric secondary field function of the original field $F_{\mu\nu}$, arises from the need to establish a relationship between standard electromagnetic theory with Maxwell’s theory of continuous media.

The source, described by the structure function $\mathcal{H}(P)$ is, of course, real and depends on the invariant formed with the conjugated antisymmetric tensor $P^{\mu\nu}$, which is $P:=\frac{1}{4}P_{\mu\nu}P^{\mu\nu}$, . In general, the structural function also depends on the other invariant $\mathcal{Q}=-\frac{1}{4}P_{\mu\nu}{}^{*}P^{\mu\nu}$ where ^∗ represents the Hodge dual. Here, this invariant is zero because we are interested in static configurations.

The field equations that result from the variation of the action (2) are

	$\displaystyle\nabla_{\mu}P^{\mu\nu}=0,$				(4a)
	$\displaystyle F_{\mu\nu}=\frac{\partial\mathcal{H}}{\partial P}P_{\mu\nu}=\mathcal{H}_{P}P_{\mu\nu}\,,$				(4b)
	$\displaystyle{\color[rgb]{0,0,0}E_{\mu\nu}:=G_{\mu\nu}+\Lambda g_{\mu\nu}+K_{\mu\nu}^{CG}-\kappa T_{\mu\nu}^{NLE}=0,}$				(4c)
where the tensors $K^{CG}_{\mu\nu}$ and $T^{NLE}_{\mu\nu}$ are defined as follows:
	$\displaystyle K_{\mu\nu}^{CG}$	$\displaystyle=$	$\displaystyle 2\beta_{2}\Bigl{(}R_{\mu\rho}R_{\nu}^{\,\rho}-\frac{1}{4}R^{\rho\sigma}R_{\rho\sigma}g_{\mu\nu}\Bigr{)}$
		$\displaystyle+$	$\displaystyle 2\beta_{1}R\Bigl{(}R_{\mu\nu}-\frac{1}{4}Rg_{\mu\nu}\Bigr{)}+\beta_{2}\Bigl{(}\Box R_{\mu\nu}+\frac{1}{2}\Box Rg_{\mu\nu}$
		$\displaystyle-$	$\displaystyle 2\nabla_{\rho}\nabla_{\left(\mu\right.}R_{\left.\nu\right)}^{\,\rho}\Bigr{)}+2\beta_{1}(g_{\mu\nu}\Box R-\nabla_{\mu}\nabla_{\nu}R),$
	$\displaystyle T_{\mu\nu}^{NLE}$	$\displaystyle=$	$\displaystyle{\mathcal{H}_{P}}P_{\mu\alpha}P_{\nu}^{\,\alpha}-g_{\mu\nu}(2P\mathcal{H}_{P}-\mathcal{H})\,,$

with $\beta_{1}$ and $\beta_{2}$ given previously in (3). Note that equation (4a) represents the nonlinear version of Maxwell’s equations, while the constitutive relations are encoded in (4b) and Einstein’s equations are given by (4c).

These equations of motion (4a)-(4c) will lead us to find new AdS black hole configurations in CG coupled with non-linear electrodynamics in section II. Then, in section III, we will show the analysis and characterization of said solutions in terms of the maximum number of horizons. In section IV, the thermodynamical quantities and local stability of the solutions are calculated, and the first law of thermodynamics as well as the Smarr formula are verified. Finally, in section V we present our conclusions and perspectives of this work.

We start the search of new AdS black hole solutions by considering the following asymptotically AdS metric ansatz

$\displaystyle ds^{2}=-\frac{r^{2}}{l^{2}}f(r)dt^{2}+\frac{l^{2}}{r^{2}}\frac{dr^{2}}{f(r)}+\frac{r^{2}}{l^{2}}\left(dx_{1}^{2}+dx_{2}^{2}\right),$

(5)

where $t\in\,(-\infty,+\infty),r>0$, the planar coordinates both are assumed belong to a compact set, this is $0\leq x_{1}\leq\Omega_{x_{1}}$ and $0\leq x_{2}\leq\Omega_{x_{2}}$, and the gravitational potential must satisfy the asymptotic condition

$${\lim_{r\rightarrow+\infty}f(r)=1.}$$

In our case, we propose the structure function $\mathcal{H}$, determining the nonlinear electrodynamics, to be given by

	$\displaystyle\mathcal{H}(P)$	$\displaystyle=$	$\displaystyle\frac{(\alpha_{2}^{2}-3\alpha_{1}\alpha_{3})l^{2}P}{3\kappa}{-\frac{2\alpha_{1}(-2P)^{1/4}}{l\kappa}}$		(6)
		$\displaystyle+$	$\displaystyle{\frac{\alpha_{2}\sqrt{-2P}}{\kappa}},$

where $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are coupling constants.

For the purposes of this article, we consider purely electrical configurations, such that $P_{\mu\nu}=2\delta^{t}_{[\mu}\delta_{\nu]}^{r}D(r)$. If we replace the previous ansatz in the nonlinear Maxwell equation (4a) we obtain

$$P_{\mu\nu}=2\delta_{[\mu}^{t}\delta_{\nu]}^{r}\frac{M}{r^{2}}.$$

(7)

Therefore, the electric invariant $P$ is negative definite, since we only consider purely electrical configurations, which reads

$$P=-\frac{M^{2}}{2r^{4}},$$

(8)

where $M$ is a constant of integration related to the electric charge, and $\mathcal{H}(P)$ from (6) is a real function. The electric field is obtained from the constitutive relations (4b), $E\equiv F_{tr}=\mathcal{H}_{p}D$. Using expression (6) for $\mathcal{H}(P)$, the electromagnetic field strength results in

	$\displaystyle F_{\mu\nu}$	$\displaystyle=2\delta^{t}_{[\mu}\delta^{r}_{\nu]}E(r)$
		$\displaystyle=2\delta^{t}_{[\mu}\delta^{r}_{\nu]}\left(\frac{r\alpha_{1}}{l\kappa\sqrt{M}}-\frac{\alpha_{2}}{\kappa}-\frac{l^{2}M\left(3\,\alpha_{1}\,\alpha_{3}-\alpha_{2}^{2}\right)}{3\kappa\,r^{2}}\right).$		(9)

Notice that in order to recover the AdS spacetime asymptotically, the cosmological constant must take the following value,

$$\Lambda=-\frac{3}{l^{2}}.$$

(10)

Let us bring our attention to the fact that the difference between the temporal $E_{t}^{t}$ and radial diagonal $E_{r}^{r}$ components of the mixed version of Einstein’s equations (4c) is proportional to the following fourth-order Cauchy-Euler ordinary differential equation

$\displaystyle r^{4}f^{(4)}+12r^{3}f^{\prime\prime\prime}+36r^{2}f^{\prime\prime}+24rf^{\prime}$

$\displaystyle=$

$\displaystyle 0.$

(11)

Therefore, the solution of the gravitational potential is

$\displaystyle f\left(r\right)=1-C_{1}{\frac{{l}}{r}}+C_{2}{\frac{{l^{2}}}{{r}^{2}}}-C_{3}{\frac{{l^{3}}}{{r}^{3}}},$

(12)

where the fourth integration constant is fixed to comply with the asymptotic behaviour ${\lim_{r\rightarrow+\infty}f(r)=1}$. Additionally, if we replace the expressions in (12) and (8) in the equations of motion (4c) we obtain an equation to determine $\mathcal{H}_{P}$, which can be later integrated to obtain $\mathcal{H}(P)$, given previously in (6). Finally, the remaining equations of motion are satisfied if the previous integration constants $C_{i}$’s from (12) are fixed in terms of the charge-like parameter $M$ through the structural coupling constants as follows

$\displaystyle C_{1}=\alpha_{1}\sqrt{M}\,,\,\,C_{2}=\alpha_{2}M\,,\,\,\mathrm{and}\,\,C_{3}=\alpha_{3}M^{3/2}\,,$

(13)

where the $\alpha_{i}$’s are in the appropriate units in order to the integration constants $C_{i}$’s be dimensionless. The addition of the NLE yields to a rich structure for the metric function $f$ obtained previously in (12)-(13), where the uncharged case is recovered when $\alpha_{1}=\alpha_{2}=0$. This also shows that the linear Maxwell field scenario (that is $\mathcal{H}(P)=P$) is not allowed, which reinforces the necessity to explore other charged theories such as NLE. Moreover, from a physical perspective, this new structure for the metric function $f$ constructed via CG and NLE (2), will allow us to explore solutions with different numbers of horizons, in addition to nonzero thermodynamic quantities, as we will see bellow. These solutions are, to our knowledge, the first example of solutions in four-dimensional CG where their thermodynamic parameters do not vanish.

We have established that the introduction of non-linear electrodynamics when considering CG, results in AdS solutions of the form (5) where

$\displaystyle f\left(r\right)=1-\alpha_{1}\sqrt{M}{\frac{{l}}{r}}+\alpha_{2}M{\frac{{l^{2}}}{{r}^{2}}}-\alpha_{3}M^{3/2}{\frac{{l^{3}}}{{r}^{3}}},$

(14)

provided that $\mathcal{H}(P)$ and $\Lambda$ are given by (6) and (10) respectively. However, this set of expressions only represents a black hole solution if a horizon can be formed, that is, if there exists $r_{h}>0$ such that $f(r_{h})=0$.

To this effect, in the following subsections, we study the conditions in which eq. (14) can vanish, through the analysis of its asymptotic behavior as well as its maxima and minima.

First, let us notice that when $r\rightarrow+\infty$, $f(r)$ will approach 1 asymptotically and, in this regime, $f(r)\simeq 1-\alpha_{1}\sqrt{M}\frac{l}{r}$. As a result, the sign of $\alpha_{1}$ will determine whether the function $f(r)$ approaches the horizontal asymptote from above or from below. Next, let us observe that when $r\rightarrow 0^{+}$, $f(r)\simeq-\alpha_{3}M^{3/2}\frac{l^{3}}{r^{3}}$, that is, the sign of $\alpha_{3}$ will determine if the function $f(r)$ starts increasing from $-\infty$ or decreasing from $+\infty$ in the region $r\in(0,+\infty)$.

Moreover, we can perform an analysis of the extreme values of $f(r)$. From the calculation of $f^{\prime}(r)$ we find that $f(r)$ admits two extreme values which are located at

$\displaystyle r_{ext\,i}$

$\displaystyle=$

$\displaystyle{\frac{\sqrt{M}\,\left({\alpha_{2}}\pm\sqrt{{{\alpha_{2}}}^{2}-3\,{\alpha_{1}}\,{\alpha_{3}}}\right)l}{{\alpha_{1}}}},$

(15)

where the nature of these extreme values (whether they are a maximum or a minimum) will depend on the sign of their evaluation in $f^{\prime\prime}(r)$, that is

$\displaystyle f^{\prime\prime}(r_{ext\,i})=\pm\frac{2\alpha_{1}^{4}\sqrt{\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}}}{(\sqrt{\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}}\pm\alpha_{2})^{4}Ml^{2}},$

(16)

with $i=\{1,2\}$. Here, $r_{ext1}$ (resp. $r_{ext2}$) is associated with positive (resp. negative) sign in equations (15) and (16). Let us notice that from eqns. (15)-(16), the existence of real extreme values is limited to the fulfillment of the condition

$$\alpha_{2}^{2}-3\alpha_{1}\alpha_{3}>0.$$

(17)

At this point, we are ready to analyze each case independently.

Given the structure of these new nonlinearly charged black hole solutions in four dimensions, it is interesting to explore their thermodynamics. As a first step, we will consider the electric charge which reads

$\displaystyle\mathcal{Q}_{e}=\int d\Omega_{2}\left(\frac{r}{l}\right)^{2}n^{\mu}u^{\nu}P_{\mu\nu}=\frac{M\Omega_{2}}{l^{2}}=\frac{\Omega_{2}r_{h}^{2}}{\zeta^{2}l^{4}},$

(19)

where $r_{h}$ is the location of the event (or outer) horizon which can be expressed as $r_{h}=\zeta\sqrt{M}l$, where $\zeta$ is a root of the cubic polynomial

$$\zeta^{3}-\alpha_{1}\zeta^{2}+\alpha_{2}\zeta-\alpha_{3}=0;$$

(20)

$\Omega_{2}$ is the finite volume of the compact planar base manifold given by $\int dx_{1}dx_{2}=\int d\Omega_{2}=\Omega_{2}=\Omega_{x_{1}}\Omega_{x_{2}}$, while $n^{\mu}$ and $u^{\nu}$ are the unit spacelike and timelike normals to a sphere of radius $r$ given by

$\displaystyle{n^{\mu}:=\frac{dt}{\sqrt{-g_{tt}}}=\frac{l}{r\sqrt{f}}dt,\quad u^{\mu}:=\frac{dr}{\sqrt{g_{rr}}}=\frac{r\sqrt{f}}{l}dr.}\quad$

(21)

As a first step to calculate the electric potential we must determine the 4-potential $A_{\mu}$. We achieve this by integrating eq. (II) and taking into account that $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$. As a result the only non-zero component of the 4-potential is given by

$\displaystyle A_{t}(r)={\frac{{r}^{2}{\alpha_{1}}}{2\sqrt{M}l\kappa}}-{\frac{{\alpha_{2}}\,r}{\kappa}}+{\frac{{l}^{2}M\left(3\,{\alpha_{1}}\,{\alpha_{3}}-{{\alpha_{2}}}^{2}\right)}{3\kappa\,r}},\quad$

(22)

where the integration constant in our case is null, and the electric potential ²²2In this work we have used the same definition of the electric potential as [42, 43, 44]. is given by:

	$\displaystyle\Phi_{e}=-A_{t}(r)\Big{|}_{r=r_{h}}$	$\displaystyle=$	$\displaystyle-{\frac{3{\alpha_{1}}\,r_{h}^{2}}{2\sqrt{M}l\kappa}}+{\frac{\left(\alpha_{1}^{2}+\alpha_{2}\right)r_{h}}{\kappa}}$		(23)
		$\displaystyle-$	$\displaystyle{\frac{{\alpha_{1}}\,{\alpha_{2}}\,\sqrt{M}l}{\kappa}}+{\frac{{l}^{2}M\alpha_{2}^{2}}{3\kappa\,r_{h}}},$
		$\displaystyle=$	$\displaystyle\frac{r_{h}}{\kappa}\Big{(}{\alpha_{2}}+\alpha_{1}^{2}-\frac{3}{2}\alpha_{1}\zeta$
		$\displaystyle-$	$\displaystyle{\frac{\alpha_{1}\,\alpha_{2}}{\zeta}}+\frac{1}{3}\,{\frac{\alpha_{2}^{2}}{\zeta^{2}}}\Big{)}.$

On the other hand, to compute the entropy we will consider Wald’s formula [45, 46] which, in our case, yields to

	$\displaystyle\mathcal{S}_{W}$	$\displaystyle{:=}$	$\displaystyle-2\pi\int_{{H}}d^{2}x\sqrt{|h|}\left(\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\mu\nu\sigma\rho}}\,\varepsilon_{\mu\nu}\,\varepsilon_{\sigma\rho}\right),$		(24)
		$\displaystyle=$	$\displaystyle\frac{2\left(3\,\alpha_{1}\sqrt{M}r_{h}-2\,\alpha_{2}Ml\right)\pi\Omega_{2}}{3\,\kappa\,l},$
		$\displaystyle=$	$\displaystyle\frac{2\Omega_{2}\pi}{\kappa}\left(\frac{r_{h}}{l}\right)^{2}\left(\frac{\alpha_{1}}{\zeta}-\frac{2\alpha_{2}}{3\zeta^{2}}\right),$

where

	$\displaystyle\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\alpha\beta\gamma\delta}}$	$\displaystyle=$	$\displaystyle\frac{1}{4\kappa}\,\Big{(}g^{\alpha\gamma}g^{\beta\delta}-g^{\alpha\delta}g^{\beta\gamma}\Big{)}$
		$\displaystyle+$	$\displaystyle\frac{\beta_{1}}{2\kappa}\,R\left(g^{\alpha\gamma}g^{\beta\delta}-g^{\alpha\delta}g^{\beta\gamma}\right)$
		$\displaystyle+$	$\displaystyle\frac{\beta_{2}}{4\kappa}\,\left(g^{\beta\delta}R^{\alpha\gamma}-g^{\beta\gamma}R^{\alpha\delta}-g^{\alpha\delta}R^{\beta\gamma}+g^{\alpha\gamma}R^{\beta\delta}\right),$

with $\beta_{1}$ and $\beta_{2}$ subject to (3), and the integral is evaluated on a 2-dimensional spacelike surface $H$ (the bifurcation surface) characterized by the fact that the timelike Killing vector $\partial_{t}=\xi^{\mu}\partial_{\mu}$ vanishes, $|h|$ denotes the determinant of the induced metric on ${H}$, $\varepsilon_{\mu\nu}$ represents the binormal antisymmetric tensor constructed via the wedge product of the unit normal vectors $n^{\mu}$ and $u^{\mu}$ from (21), normalized as $\varepsilon_{\mu\nu}\varepsilon^{\mu\nu}=-2$. Additionally, the Hawking temperature takes the form

	$\displaystyle T{:=}\frac{k}{2\pi}\Big{|}_{r=r_{h}}$	$\displaystyle=$	$\displaystyle\frac{3r_{h}}{4\pi l^{2}}-\frac{\alpha_{1}\sqrt{M}}{2\pi l}+\frac{\alpha_{2}M}{4\pi r_{h}},$		(25)
		$\displaystyle=$	$\displaystyle\frac{r_{h}}{4\pi l^{2}}\left(3-\frac{2\alpha_{1}}{\zeta}+\frac{\alpha_{2}}{\zeta^{2}}\right),$

where $k$ is the surface gravity which reads

$$k=\sqrt{-\frac{1}{2}\left(\nabla_{\mu}\xi_{\nu}\right)\left(\nabla^{\mu}\xi^{\nu}\right)}.$$

Finally, to calculate the mass of these charged AdS black hole configurations we will consider the approach described in [47, 48], corresponding to an off-shell prescription of the Abbott-Desser-Tekin (ADT) procedure [49, 50, 51]. The choice of this method to calculate conserved charges is ideal for CG due to the presence of quadratic curvature terms in its gravitational action.

The main elements of the quasilocal method are the surface term

	$\displaystyle\Theta^{\mu}$	$\displaystyle=$	$\displaystyle 2\sqrt{-g}\Biggl{[}\left(\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\mu\alpha\beta\gamma}}\right)\nabla_{\gamma}\delta g_{\alpha\beta}-\delta g_{\alpha\beta}\nabla_{\gamma}\left(\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\mu\alpha\beta\gamma}}\right)$		(26)
		$\displaystyle+$	$\displaystyle\frac{1}{2}\,\left(\frac{\delta\mathcal{L}_{NLE}}{\delta\left(\partial_{\mu}A_{\nu}\right)}\right)\delta A_{\nu}\Biggr{]},$

and the Noether potential

	$\displaystyle K^{\mu\nu}$	$\displaystyle=$	$\displaystyle\sqrt{-g}\Bigg{[}2\left(\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\mu\nu\rho\sigma}}\right)\nabla_{\rho}\xi_{\sigma}-4\xi_{\sigma}\nabla_{\rho}\left(\frac{\delta\mathcal{L}_{\mathrm{grav}}}{\delta R_{\mu\nu\rho\sigma}}\right)$		(27)
		$\displaystyle-$	$\displaystyle\left(\frac{\delta\mathcal{L}_{NLE}}{\delta\left(\partial_{\mu}A_{\nu}\right)}\right)\xi^{\sigma}A_{\sigma}\Bigg{]}.$

With all the above, using a parameter $s\in[0,1]$, we interpolate between the charged solution at $s=1$ and the asymptotic one at $s=0$, resulting in the quasilocal charge:

$\displaystyle{\mathcal{M}(\xi)=\int_{B}d^{2}x_{\mu\nu}\left(\Delta K^{\mu\nu}(\xi)-2\xi^{[\mu}\int^{1}_{0}ds\Theta^{\nu]}\right),}$

where $\Delta K^{\mu\nu}(\xi)\equiv K^{\mu\nu}_{s=1}(\xi)-K^{\mu\nu}_{s=0}(\xi)$ is the difference of the Noether potential between the interpolated solutions. For this particular case the mass reads

$\displaystyle\mathcal{M}$

$\displaystyle=$

$\displaystyle\frac{\alpha_{1}\alpha_{2}M^{\frac{3}{2}}\Omega_{2}}{9l\kappa}=\frac{\alpha_{1}\alpha_{2}r_{h}^{3}\Omega_{2}}{9l^{4}\kappa\zeta^{3}}.$

(28)

Notice that for the Wald entropy $\mathcal{S}_{W}$ (24), as well as the mass $\mathcal{M}$ (28), the presence of the coupling constants $\alpha_{1}$ and $\alpha_{2}$ is providential, reinforcing the importance of the non-linear electrodynamic as a matter source with this gravity theory. In fact, when $\alpha_{1}=\alpha_{2}=0$, the vector potential $A_{t}(r)$ (22) vanishes, recovering the well known four-dimensional Schwarzschild- AdS black hole with a planar base manifold in CG, whose extensive thermodynamical quantities $\mathcal{M}$ and $\mathcal{S}_{W}$ are null, while the Hawking Temperature is $T=3r_{h}/(4\pi l^{2})$.

Just for completeness, from eqns. (28), (24) and (19) we have:

	$\displaystyle\delta\mathcal{M}$	$\displaystyle=$	$\displaystyle\frac{\alpha_{1}\alpha_{2}r_{h}^{2}\Omega_{2}}{3l^{4}\kappa\zeta^{3}}\,\delta r_{h},$
	$\displaystyle\delta\mathcal{S}_{W}$	$\displaystyle=$	$\displaystyle\frac{4\Omega_{2}\pi r_{h}}{\kappa l^{2}}\left(\frac{\alpha_{1}}{\zeta}-\frac{2\alpha_{2}}{3\zeta^{2}}\right)\,\delta r_{h},$
	$\displaystyle\delta\mathcal{Q}_{e}$	$\displaystyle=$	$\displaystyle\frac{2\Omega_{2}r_{h}}{\zeta^{2}l^{4}}\,\delta r_{h},$

and together with the electric potencial $\Phi_{e}$ (23) as well as the Hawking temperature $T$ (25), a first law of the black holes thermodynamics

$\displaystyle\delta\mathcal{M}=T\delta\mathcal{S}_{W}+\Phi_{e}\delta\mathcal{Q}_{e},$

(29)

arises. Together with the above, through the thermodynamical parameters (19), (23)-(25), we can express the mass (28) as a function of the extensive thermodynamical quantities $\mathcal{S}_{W}$ and $\mathcal{Q}_{e}$ in the following form

	$\displaystyle\mathcal{M}(\mathcal{S}_{W},\mathcal{Q}_{e})$	$\displaystyle=$	$\displaystyle{\frac{\sqrt{6}\,\sqrt{\kappa}\,{\mathcal{S}_{W}}^{3/2}\left(3\,{\zeta}^{2}-2\,\alpha_{1}\zeta+\alpha_{2}\right)}{12\sqrt{\Omega_{{2}}}{\pi}^{3/2}\,\sqrt{3\,\alpha_{1}\zeta-2\,\alpha_{2}}\,l\zeta}}$		(30)
		$\displaystyle+$	$\displaystyle\frac{\mathcal{Q}_{e}^{3/2}l^{2}\zeta\Psi}{9\sqrt{\Omega_{2}}\kappa},$

with

$$\Psi=6\,\alpha_{2}+6\alpha_{1}^{2}-9\,\alpha_{1}\zeta-{\frac{6\alpha_{2}\alpha_{1}}{\zeta}}+\frac{2\alpha_{2}^{2}}{{\zeta}^{2}},$$

(31)

where it is straightforward to verify that the intensive parameters

$$T=\left(\frac{\partial\mathcal{M}}{\partial\mathcal{S}_{W}}\right)_{{\mathcal{Q}}_{e}},\qquad\Phi_{e}=\left(\frac{\partial\mathcal{M}}{\partial\mathcal{Q}_{e}}\right)_{{\mathcal{S}}_{W}},$$

are consistent with the expressions (23) and (25) (where the subindices stand for at constant electric charge ${{\mathcal{Q}}_{e}}$, and at constant entropy ${{\mathcal{S}}_{W}}$ respectively). Additionally, under a rescaling with a nonzero parameter $\lambda$, equation (30) becomes

$$\mathcal{M}(\lambda\mathcal{S}_{W},\lambda\mathcal{Q}_{e})=\lambda^{\frac{3}{2}}\mathcal{M}(\mathcal{S}_{W},\mathcal{Q}_{e}),$$

yielding to a four-dimensional Smarr formula [52]

$\displaystyle\mathcal{M}=\frac{2}{3}\left(T\mathcal{S}_{W}+\Phi_{e}\mathcal{Q}_{e}\right),$

(32)

which corresponds to a particular case of the higher-dimensional situation [40]

$\displaystyle\mathcal{M}=\left(\frac{D-2}{D-1}\right)\left(T\mathcal{S}_{W}+\Phi_{e}\mathcal{Q}_{e}\right),$

(33)

where $D$ is the dimension of the space-time, highly explored in [53, 54, 55].

Given these thermodynamical quantities, it is interesting to study this system under small perturbations around the equilibrium. In our case, we will consider the grand canonical ensemble, where the intensive thermodynamical quantities are fixed. With this, we can express the entropy, mass, and charge in functions of $T$ and $\Phi_{e}$ in the following form

	$\displaystyle\mathcal{S}_{W}$	$\displaystyle=$	$\displaystyle{\frac{32}{3}}\,{\frac{{l}^{2}{T}^{2}\Omega_{{2}}{\pi}^{3}{\zeta}^{2}\left(3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\right)}{\left(3\,{\zeta}^{2}-2\,\alpha_{{1}}\zeta+\alpha_{{2}}\right)^{2}\kappa}},$		(34)
	$\displaystyle\mathcal{Q}_{e}$	$\displaystyle=$	$\displaystyle\frac{36\,\Phi_{e}^{2}\kappa^{2}\Omega_{2}}{\Psi^{2}\zeta^{2}{l}^{4}},$		(35)
	$\displaystyle\mathcal{M}$	$\displaystyle=$	$\displaystyle{\frac{64}{9}}\,{\frac{{l}^{2}{T}^{3}\Omega_{{2}}{\pi}^{3}{\zeta}^{2}\left(3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\right)}{\left(3\,{\zeta}^{2}-2\,\alpha_{{1}}\zeta+\alpha_{{2}}\right)^{2}\kappa}}$		(36)
		$\displaystyle+$	$\displaystyle\frac{24\,\Phi_{e}^{3}\kappa^{2}\Omega_{2}}{\Psi^{2}\zeta^{2}{l}^{4}}.$

With this information, we are in a position to determine the local thermodynamical (in)stability of this charged black hole solution under thermal fluctuations through the behavior of the specific heat $C_{\Phi_{e}}$, given by

	$\displaystyle C_{\Phi_{e}}$	$\displaystyle=$	$\displaystyle\left(\frac{\partial\mathcal{M}}{\partial T}\right)_{\Phi_{e}}=T\left(\frac{\partial\mathcal{S}_{W}}{\partial T}\right)_{\Phi_{e}}$		(37)
		$\displaystyle=$	$\displaystyle{\frac{64}{3}}\,{\frac{{l}^{2}{T}^{2}\Omega_{{2}}{\pi}^{3}{\zeta}^{2}\left(3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\right)}{\left(3\,{\zeta}^{2}-2\,\alpha_{{1}}\zeta+\alpha_{{2}}\right)^{2}\kappa}}.$

Here we observe that for $T\geq 0$ or in the same way,

$$\Psi_{1}:=3\zeta^{2}-2\alpha_{1}\zeta+\alpha_{2}\geq 0,$$

(38)

the specific heat becomes non-negative when

$$\Psi_{2}:=3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\geq 0,$$

(39)

which can be interpreted as a locally stable configuration. Nevertheless, it is worth pointing out that, to have a real and well-defined mass according to the expression (30) as well as specific heat (37), only the strict inequalities from (38) and (39) are considered . Therefore, here we can conclude that the introduction of the nonlinear electrodynamics to the CG action induces rich thermodynamical properties. The comment on stability above is consistent with the analysis of the Gibbs Free Energy $G(T,\Phi_{e})=\mathcal{M}-T\mathcal{S}_{W}-\Phi_{e}\mathcal{Q}_{e}$, given by

	$\displaystyle G(T,\Phi_{e})$	$\displaystyle=$	$\displaystyle-{\frac{32}{9}}\,{\frac{{l}^{2}{T}^{3}\Omega_{{2}}{\pi}^{3}{\zeta}^{2}\left(3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\right)}{\left(3\,{\zeta}^{2}-2\,\alpha_{{1}}\zeta+\alpha_{{2}}\right)^{2}\kappa}}$
		$\displaystyle-$	$\displaystyle\frac{12\,\Phi_{e}^{3}\kappa^{2}\Omega_{2}}{\Psi^{2}\zeta^{2}{l}^{4}},$

where its Hessian matrix $H_{ab}:=\partial_{a}\partial_{b}G(T,\Phi_{e})$, with $a,b\in\{T,\Phi_{e}\}$, satisfies the conditions

	$\displaystyle H_{TT}$	$\displaystyle=$	$\displaystyle-{\frac{64}{3}}\,{\frac{{l}^{2}{T}\Omega_{2}{\pi}^{3}{\zeta}^{2}\left(3\,\alpha_{{1}}\zeta-2\,\alpha_{{2}}\right)}{\left(3\,{\zeta}^{2}-2\,\alpha_{{1}}\zeta+\alpha_{{2}}\right)^{2}\kappa}}\leq 0,$
	$\displaystyle H_{\Phi_{e}\Phi_{e}}$	$\displaystyle=$	$\displaystyle-\frac{72\,\Phi_{e}\kappa^{2}\Omega_{2}}{\Psi^{2}\zeta^{2}{l}^{4}}\leq 0,$
	$\displaystyle|H_{ab}|$	$\displaystyle=$	$\displaystyle H_{TT}H_{\Phi_{e}\Phi_{e}}-\left(H_{T\Phi_{e}}\right)^{2}$
		$\displaystyle=$	$\displaystyle\frac{1536\,\Omega_{2}^{2}\,T\,\pi^{3}\left(3\,\alpha_{1}\zeta-2\,\alpha_{2}\right)\kappa\,\Phi_{e}}{{l}^{2}\left(3\,{\zeta}^{2}-2\,\alpha_{1}\zeta+\alpha_{2}\right)^{2}\Psi^{2}}\geq 0,$

if $T,\Phi_{e}\geq 0$, this is if (38)-(39) and

$\displaystyle\Psi=\frac{\alpha_{1}\alpha_{2}}{\zeta}-\frac{\Psi_{1}\Psi_{2}}{\zeta^{2}}\geq 0,$

(41)

hold, where $\Psi$ was defined previously in (31) and in order to have a well-defined Gibbs free energy (IV), we consider only the strict inequality from (41). As an example, for $\zeta=1$, which implies that $r_{h}=\sqrt{M}l$, we have that the strict inequalities (38)-(39) and (41) are satisfied when the constants $\alpha_{1}$ and $\alpha_{2}$ belong to the region $\cal{R}$ represented in the Figure 4.

Additionally, it is interesting to note that for this new charged black hole, we can analyze its response under electrical fluctuations, represented by the electric permittivity $\epsilon_{T}$ at a constant temperature, which reads as follow:

$$\epsilon_{T}=\left(\frac{\partial\mathcal{Q}_{e}}{\partial\Phi_{e}}\right)_{T}=\frac{72\,\Phi_{e}\kappa^{2}\Omega_{2}}{\Psi^{2}\zeta^{2}{l}^{4}},$$

which is a non-negative quantity if the strict inequality (41) holds, ensuring local stability [39, 56].

In this work we propose a nonlinear electrodynamics in the $(\mathcal{H},P)$-formalism, which allows us to obtain charged configurations of AdS black holes in four dimensions with a planar base manifold in CG. These configurations have only one integration constant, given by the charge-like parameter $M$, and being parameterized by the structural coupling constants (13). As was explained at the beginning, to our knowledge, these planar configurations are the first example of solutions in four-dimensional CG where their thermodynamic quantities do not vanish.

With respect to the metric function, the structural coupling constants play a very important role in the characterization of these charged solutions and when analyzing condition (17), we conclude that there are five different cases: one represents a black hole with three horizons, two cases represent black holes with two horizons and the other two are single horizon configurations.

Also, in order to find these new charged black holes, the introduction of nonlinear electrodynamics to CG allows us to obtain nonzero thermodynamic properties, thanks to the contributions given by $\alpha_{1}$ and $\alpha_{2}$. Together with the above, these configurations satisfy the four dimensional Smarr relation (32) as well as the First Law (29). Additionally, the Critical-Gravity-non-linear electrodynamics model enjoys local stability under thermal fluctuations, thanks to the non-negativity of the specific heat $C_{\Phi_{e}}$ as well as the Gibbs Free Energy $G$ analysis if (38)-(39) and (41) are satisfied. Supplementing the above, the non-negativity of the electric permittivity $\epsilon_{T}$ shows that our solution is also a locally stable thermodynamic system under electrical fluctuations. It is interesting to note the behavior of $\epsilon_{T}$ for these charged configurations, as it is a non-negativity quantity if $\Phi_{e}>0$, unlike other solutions found in the literature (see for example [54]).

Some natural extensions of this work may include, the exploration of other gravity theories with quadratic contributions. In this sense, a theory that also showcases critical conditions, is given in [57] where the square of the Weyl tensor and the square of the Ricci scalar play the main roles in the action, in the absence of the Einstein Gravity. Another interesting scenario would be to study the higher dimensional case [58], where now the CG Lagrangian takes the form

	$\displaystyle\mathcal{L}$	$\displaystyle=$	$\displaystyle R-2\Lambda+\beta_{1}{R}^{2}+\beta_{2}{R}_{\alpha\beta}{R}^{\alpha\beta}$
		$\displaystyle+$	$\displaystyle\beta_{3}{R}_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu},$

and that the coupling constants are tied as [59]

$\displaystyle\beta_{1}=-\frac{\beta_{2}}{2(D-1)}=\frac{2\beta_{3}}{(D-1)(D-2)}=\frac{1}{4\Lambda(D-3)},$

where the four dimensional case (2)-(3) can be recovered impossing $D=4$ together with the transformation

$$(\beta_{1},\beta_{2},\beta_{3})\mapsto(\beta_{1}-\beta_{3},\beta_{2}+4\beta_{3},0).$$

Given the power of the non-linear electrodynamics as a matter source to find new solutions with a planar base manifold, it would be interesting to study charged black holes where their event horizons enjoy spherical or hyperbolical topologies. It would also be interesting to study charged black hole configurations with non-standard asymptotically behaviors, such as Lifshitz black holes, which were first explored for the uncharged case with CG in [60]. For spherically symmetric metrics, it is possible, as was shown in [61], to obtain a generalization of the Smarr relation as well as the first law of black hole mechanics, being understood from a dual holographic point of view and related to the black hole chemistry [62, 63, 64], where now the cosmological constant $\Lambda$ takes the role as a dynamical variable, unlike the expression found in (32) where we explored charged planar black holes and $\Lambda$ does not appear in an active way.

Finally, from a physical motivation, these nonzero extensive thermodynamical quantities will allow us to explore, from a holographic point of view, the connection between black holes and quantum complexity [65, 66], as well as the effects on shear viscosity [67, 68, 69], where the mass $\mathcal{M}$ and the entropy $\mathcal{S}_{W}$ take a providential role.