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In this section, we examine the NLED BH solution [61] that comes from the action of NLED theories that have been coupled with gravity:
$ S= \frac{1}{16 \pi} \int \sqrt{-g}\Big(R + K(\psi)\Big) {\rm d}^{4}x, $
(1) where
$ \begin{array}{*{20}{l}} \psi = F_{\mu\nu} F^{\mu\nu}, \,\,\, F_{\mu\nu}= A_{\nu;\mu}- A_{\mu;\nu}. \end{array} $
(2) Here,
$ A_{\mu} $ gives the Maxwell field, Ricci scalar R, and ψ function$ K(\psi) $ . The field equations of Einstein are given by the change of action relating to the metric$ G_{\mu \nu} = -2K_{, \psi} F_{\mu \lambda} F_{\nu}^{\lambda}+ \frac{1}{2}g_{\mu\nu}K,\quad\quad K_{, \psi}\equiv \frac{{\rm d}K}{{\rm d}\psi}. $
(3) The Maxwell general equations are determined by the field:
$ \begin{array}{*{20}{l}} (K_{, \psi} F^{\mu\nu})_{;\mu}=0. \end{array} $
(4) The spherically symmetric and static background metric [61] is given by
$ {\rm d}s^{2}= -N(r){\rm d}t^{2}+ \frac{1}{N(r)}{\rm d}r^{2}+ f(r)^{2}{\rm d}\Omega_{2}^{2}, $
(5) The only non-zero component of the Maxwell field tensor
$ A_{\mu} $ is given by$ \begin{array}{*{20}{l}} A_{0}= -\phi(r), \end{array} $
(6) $ \begin{array}{*{20}{l}} \psi = -2\varphi'^{2}. \end{array} $
(7) The field equations of Einstein and the generalized equations of Maxwell [61] were also handled:
$ \frac{-N'f'}{f}-\frac{2N f''}{f} + \frac{1}{f^{2}}-\frac{N f'^{2}}{f^{2}}= 2K_{,\psi} \phi'^{2}+ \frac{1 }{2}K, $
(8) $ \frac{-N'f'}{f}+ \frac{1}{f^{2}}-\frac{N f'^{2}}{f^{2}} = 2K_{,\psi} \phi'^{2}+ \frac{1 }{2}K, $
(9) $ \frac{-N'f'}{f}+\frac{Nf''}{f} +\frac{1}{2} N'' = - \frac{1 }{2}K, $
(10) $ \begin{array}{*{20}{l}} (f''K_{, \psi}\phi')' = 0. \end{array} $
(11) The prime symbol denotes differentiation with respect to r. The Maxwell field of Eq. (11) defines the electric charge:
$ \begin{array}{*{20}{l}} Q = f^{2} \phi' K_{, \psi}. \end{array} $
(12) From the difference of Eqs. (8) and (9), we obtain
$ \begin{array}{*{20}{l}} f''(r)=0. \end{array} $
(13) As a result, the solution to f is
$ \begin{array}{*{20}{l}} f(r)=r. \end{array} $
(14) We assume the Maxwell Lagrangian K form to solve the field equations:
$ \begin{eqnarray} K= 2\sqrt{2\alpha}\sqrt{-\psi}-\psi- 2\Lambda, \end{eqnarray} $
(15) where α has the dimension of length. The theory is reduced to Maxwell's equation when
$ \alpha=0 $ (see Eq. (2) for details), and$ 2\Lambda $ is the cosmological constant factor. Ultimately,$ \begin{eqnarray} \phi=r\sqrt{\alpha}+\frac{Q}{r}, \end{eqnarray} $
(16) $ N(r)= 1-\frac{2 M}{r}+\frac{Q^2}{r^2} +\frac{r^2}{l^2}+2 \sqrt{\alpha } Q-\frac{\alpha r^2}{3}. $
(17) Here Q, α, and Λ are the electric charge, coupling constant, and cosmological constant, respectively. If one vanishing
$ \alpha\rightarrow 0 $ , its nothing but just RN-AdS solution, and$ Q=0 $ reveals the Schwarzschild AdS BH solution. -
Now, we discuss the thermodynamical properties of the NLED BH [61] .
Using Eq. (17) with
$ N(r)=0 $ , one may determine the physical characteristics of a BH event horizon. The equations are solved by changing the values of the parameters Q, α, and$ l=20 $ . The behavior of the discovered solution for a fixed value of α is depicted in Figs. 1–3. It is evident that as the charge Q is increased, the BH size grows. From Figs. 1 and 2, we can deduce from the answer that it shows a naked singularity at$ Q=1.1 $ and$ Q=1.2 $ , which means that it gives no event horizon, hence no black holes, when$ Q>Q_{c} $ ; one double zero, which corresponds to an extreme BH, for$ Q=Q_{c} $ ; and two simple zeros, corresponding to two event horizons, which denote a non-extreme BH, for$ Q<Q_{c} $ . In Fig. 3 , as$ \alpha=0 $ and$ Q=0 $ demonstrate event horizons of the RN-BH and the Schwarzschild BH, respectively, two categories of BH solutions appear: extremal and non-extremal.Figure 1. (color online) The graph of N as a function of r for
$ \alpha=0.030 $ ,$ l=20 $ , and$ M=1 $ .Figure 2. (color online) The graph of N as a function of r for
$ \alpha=0.040 $ ,$ l=20 $ , and$ M=1 $ .Taking
$ N(r_{+})=0 $ enables us to determine the BH's mass and obtain$ M=\frac{1}{2} r_+ \left(\frac{r_+^2}{l^2}+\frac{Q^2}{r_+^2}+2 \sqrt{\alpha } Q-\frac{\alpha r_+^2}{3}+1\right). $
(18) We obtain the BH temperature from
$ T=\dfrac{k}{2\pi}=\dfrac{N'(r_{+})}{4\pi} $ , where k is the surface gravity. Thus, the temperature takes the following form:$ T=\frac{1}{{4 \pi r_+}}\Big(1+\frac{3 r_+^2}{l^2}-\frac{Q^2}{r_+^2}+2 \alpha Q-\alpha r_+^2\Big). $
(19) In Figs. 4, 5, 6 and 7, we plot the NLED charged BH Hawking temperature for a small radius BH, as well as a large radius BH, for different values of the charge Q and coupling constant α with
$ l=20 $ . From Figs. 4 and 5, it is clear that the Hawking temperature of the BH increases (decreases) to the maximum (minimum) value when the charge Q decreases (increases) with the increase in the horizon radius. It is worthy to mention that, with a fixed NLED coupling constant$\alpha=0.0020,~0.0030$ and the electric charge start varying from$ Q=0 $ to$ Q=1.3, $ the Hawking temperature of Schwarzschild and RN BHs is explained. Figures 6 and 7 reveal the behavior of the Hawking temperature with fixed values of the charge, i.e.,$ Q=0.5 $ and$ 1 $ , and the NLED coupling parameter varying from$ \alpha=0.0010 $ to$ \alpha=0.0060 $ , describe the increasing behavior of the Hawking emperature as the coupling parameter decreases with the increase in the horizon radius (from small to large BH), and exhibit the universal pattern of stability. Hence, the system remains stable for small as well as large BHs.Figure 4. (color online) The graph of T with respect to
$ r_{+} $ for$ \alpha=0.0020 $ and$ l=20 $ .Figure 5. (color online) The graph of T with respect to
$ r_{+} $ for$ \alpha=0.0030 $ and$ l=20 $ .The BH entropy is derived from
${\rm d}M=T{\rm d}S,$ which is integrated to give$S=\int\dfrac{{\rm d}M}{T}{\rm d}r_{+},$ $ \begin{array}{*{20}{l}} S=\pi r_+^{2}. \end{array} $
(20) The concept of heat capacity can be studied as
$ C=\dfrac{\partial M}{\partial T} $ , which simplifies to$ C=\frac{2 \pi r_+^2 \left(l^2 \left(Q^2-2 \sqrt{\alpha } Q r_+^2+\alpha r_+^4-r_+^2\right)-3 r_+^4\right)}{l^2 \left(-3 Q^2+2 \sqrt{\alpha } Q r_+^2+\alpha r_+^4+r_+^2\right)-3 r_+^4}. $
(21) We now analyze the thermodynamical consistency of the BH solution to Einstein gravity with the NLED field. For this purpose, we turn our attention by reckoning that heat capacity indicates BH solution thermal stability. The thermodynamic stability of the BH is contingent on the behavior of the heat capacity. If
$ C > 0, $ the thermodynamical system is locally stable, whereas it become unstable when the specific heat is$ C < 0 $ . To identify the stability of the specific heat, the graphical representation as shown in Figs. 8–11. We have plotted specific heat C versus$ r_{+} $ for different amounts of electric charge Q and NLED coupling constants$ \alpha. $ If the electric charge is$ Q=0 $ , the heat capacity is that of the Schwarzschild AdS BH. Corresponding to the maximum temperature, critical values discontinue heat capacity curves$ r_{+} = r_{c} $ . At critical values, the specific heat diverges, as shown in Figs. 8 and 10. We find that as the electric charge increases, so does the critical radius. In Figs. 9 and 11, express the stability at initial phase and the second phase change occurs from unstable region to stable region for large BH with variations in the charge Q and fixed values of the NLED parameter α. A similar behavior is observed in Figs. 12 and 13, where the NLED parameter is varied with a fixed value of the charge parameter Q. As a consequence, the large BH in the presence of charge and the coupling parameter tends to be in the stable region compared with the small BH.Figure 8. (color online) The graph of C with respect to
$ r_{+} $ for$ \alpha=0.030 $ and$ l=20 $ .Figure 9. (color online) The graph of C with respect to
$ r_{+} $ for$ \alpha=0.030 $ and$ l=20 $ .Free energy is also known as Gibbs free energy (GFE). Using thermodynamical quantities such as the temperature, mass, and entropy of the BH, the relation
$ G= M-TS, $
takes the form
$ G= \frac{1}{12} \Big(r_+^3(\alpha-8\pi P)+\frac{9Q^2}{r_+}-6\alpha-2\sqrt{\alpha}Qr_++3r_+\Big). $
(22) To recognize the thermodynamic stability of the BH, we must investigate the GFE interactions. The effect of BH parameters plays an important role in the stability of transition. To analyze the behavior of the GFE, we draw a plot of the GFE G as a function of the BH temperature. We vary the electric charge Q and fix
$ \alpha= 1 $ and$ \alpha= 0.5 $ . Figures 14 and 15 show the curves associated with different values of pressure. The positive region of the GFE shows stability, while the negative region of the GFE indicates thermodynamic instability. Hence, the BHs with negative GFE release energy to the surroundings to obtain the low energy state, and BHs with a positive GFE are globally stable. -
The extensive state space pressure takes the form
$ P= \frac{-\Lambda}{8\pi} = \frac{3}{8\pi l^{2}}, $
and using Eq. (19), we obtain
$ P(r_+, T)= \Big(\frac{\alpha }{8 \pi }+\frac{Q^2}{8 \pi r_+^4}-\frac{\alpha Q}{4 \pi r_+^2}-\frac{1}{8 \pi r_+^2}+\frac{T}{2 r_+}\Big),\,\,\,\,\,\, v=2r_+. $
(23) Additionally,
$ P(v, T)= \Big(\frac{\alpha}{8\pi}+\frac{2Q^2}{\pi v^4}-\frac{\alpha Q}{\pi v^2}+\frac{T}{v}-\frac{1}{2\pi v^2}\Big). $
(24) To find the critical values, we use
$ \frac{\partial P}{\partial r_+}|_{T} =0 \,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\, \frac{\partial^{2} P}{\partial r_+^{2}}|_{T} =0. $
(25) To comprehend the phase transition of the BH solution, we are looking for specific properties. Using the equations for pressure, volume, and temperature, we plot
$ P\text{-}V $ diagrams in Figs. 16–21 to evaluate the critical behavior and the critical characteristics of the NLED AdS BH. From Figs. 16 and 19, it is obvious that a thermodynamic system behaves as an ideal gas and is stable when$ T > T_{c} $ . When$ T = T_{c} $ , the critical isotherm has an inflection point, and when$ T <T_{c} $ , the unstable zone exists in the system and we observe that the small/large BH phase transition occurs. To understand this more clearly, a phase transition is displayed in the right plots in Figs. 17 and 20. It may noted that$ P\text{-}V $ criticality has stable and unstable regions. The stability can be seen in the regions$ V\in[0,a] $ and$ [b, \infty] $ corresponding to the small and large BHs, respectively. Meanwhile, there is instability in the region$ V \in[a, b] $ , and the phase transition can coexist for the unstable region of small and large BHs. To show the impact of the NLED parameter α in$P\text{-}V$ criticality is plotted in Figs. 18 and 21 to show phase transition. The$P\text{-}V$ diagrams in Figs. 16 and 19 show that the stationary points of inflection are at the same locations where the critical points of the equation of the state occur. Now, to determine the critical values, we differentiate Eq. (23) and obtainFigure 17. (color online) The graph of P as a function of V for
$ Q=0.50 $ ,$ \alpha=0.20 $ , and$ T=0.06828 $ .Figure 20. (color online) The graph of P as a function of V for
$ Q=1 $ ,$ \alpha=0.30 $ , and$ T=0.07704 $ .$ r_{c}=\frac{\sqrt{6} Q}{\sqrt{2 \alpha Q+1}}, $
(26) $ V_{c}=\frac{2 \left(\sqrt{6} Q\right)}{\sqrt{2 \alpha Q+1}}, $
(27) $ T_{c}= \frac{\frac{4 \sqrt{6} \alpha ^2 Q^2}{\sqrt{2 \alpha Q+1}}+\frac{4 \sqrt{6} \alpha Q}{\sqrt{2 \alpha Q+1}}+\frac{\sqrt{6}}{\sqrt{2 \alpha Q+1}}}{18 \pi Q}, $
(28) $ P_{c}=\frac{9 Q^3 \sqrt{2 \alpha Q+1}+9 \alpha Q r_+^4 \sqrt{2 \alpha Q+1}+2 \sqrt{6} r_+^3 (2 \alpha Q+1)^2-9 Q r_+^2 (2 \alpha Q+1)^{3/2}}{72 \pi Q r_+^4 \sqrt{2 \alpha Q+1}}. $ (29) Further, using Eqs. (27)–(29), we obtain
$V_{c}= 2.14834$ ,$ T_{c}= {0.4031}/{\pi } $ , and$ P_{c}= 0.034350. $ The next step is to investigate the critical exponents, which demonstrate the feature of phase transitions generally. In the vicinity of the critical point, there are four critical exponents that describe the specific heat, parameter order, isothermal compressibility, and critical isotherm, i.e., α, β, γ, and δ, which are frequently used to define the phase transition, e.g., the van der Waals phase transition. We define$ \tau= \frac{T}{T_{c}}-1 = t-1, $
where
$ t=\frac{T}{T_{c}}, $
(30) $ \omega = \frac{v-v_{c}}{v} = \frac{v}{v_{c}}-1 = \phi -1, $
(31) Here,
$ \phi = \frac{v}{v_{c}}. $
$ p = \frac{P}{P_{c}}, \,\,\,\,\,\,\,\Rightarrow P=pP_{c}. $
The critical exponents are identified as
$ t=1+ \tau \,\,\, \phi= 1+\omega. $ $ \begin{array}{*{20}{l}} p\approx 0.9514+ {\tau}-{\tau \omega}- {1.972\omega^{3}}+ O(\tau \omega^{2}, \omega^{4} ). \end{array} $
(32) The black hole sustains a step change from small to large, and the temperature, pressure, and thermodynamic volume remain constant. The equation of state Eq. (32) holds:
$ \begin{aligned}[b] p =& 0.9514+ {\tau}-{\tau \omega_{s}}- {1.972\omega_{s}^{3}}\\=& 0.9514+ {\tau}-{\tau \omega_{l}^{3}}- {1.972\omega_{l }^{3}}. \end{aligned} $
(33) The Maxwell's area law (
$\oint v {\rm d}p=0$ ) during phase transition holds:$ \begin{array}{*{20}{l}} \int_{\omega_{s}}^{\omega_{l}} \omega {\rm d}p= \int_{\omega_{s}}^{\omega_{l}}(\omega(p_{c}+5.916 \omega^{2}))\omega {\rm d}p, \end{array} $
(34) After a certain calculation, one can obtain
$ \begin{array}{*{20}{l}} \omega_{l}=-\omega_{s}, \end{array} $
(35) and we obtain
$ \omega_{l}=\sqrt{1.333 \tau}. $
$ 1. $ The specific heat at a constant value$ S = \pi r^{2} $ is governed by$ "\alpha_{1}" $ $ \begin{array}{*{20}{l}} C_{v}= T \frac{\partial S} {\partial T}|_{v} \varpropto |\tau|,^{-\alpha_{1}} \end{array} $
(36) $ \Rightarrow \frac{\partial S}{\partial T}=0. $
The entropy S is autonomously from T; thus, we obtain
$\alpha_{1} = 0.$ 2. Exponent β describes the order parameter
$ \eta = V_{s} - V_{l} $. $ \begin{array}{*{20}{l}} \eta = V^{s}- V_{l} \varpropto |\tau|^{\beta}, \end{array} $
(37) which simplifies to
$ \eta = V_{s} - V_{l} = V_{c}(1+\omega_{s})- V_{c}(1+\omega_{l}), $
$ \eta = |\tau|^{\frac{1}{2}}, $
and thus,
$ \Rightarrow \beta = \frac{1}{2}. $
(38) $ 3. $ The exponent γ determines the isothermal compressibility$ k\tau $ . From Eq. (32), we have$ k\tau = \frac{-1}{V} \frac{\partial V}{\partial P}|_{T} \varpropto |\tau|^{-\gamma}, $
(39) $ k\tau = \frac{-1}{V} \frac{\partial V}{\partial P}|_{T} \varpropto |\tau|^{-1},$
(40) and thus,
$ \begin{array}{*{20}{l}} \Rightarrow \gamma = 1. \end{array} $
(41) $ 4. $ As$ T=T_{c} $ , a critical isotherm is given by$ \begin{array}{*{20}{l}} |P-P_{c}|_{T_{c}} \varpropto |V-V_{c}|^{\delta}, \end{array} $
(42) $ \begin{array}{*{20}{l}} \alpha_{1} + \beta (\delta +1) = 2, \end{array} $
(43) which implies that
$ \delta = 3 $ . Hence, the critical elements connected to the exact BH with the NLED field include critical behavior, phase transition, and critical exponents, which is a universal pattern of phase transition.
Extended phase space thermodynamics of black hole with non-linear electrodynamic field
- Received Date: 2022-12-07
- Available Online: 2023-06-15
Abstract: This paper deals with the thermodynamical properties of the black hole formulated in Einstein's theory of relativity associated with a nonlinear electromagnetic field. The transition of the black hole is analyzed using the mass, electric charge, coupling constant, and cosmological constant. We examine the thermodynamical aspects of exact black hole solutions to compute the black hole mass, temperature, entropy, Gibbs free energy, specific heat, and critical exponents in the phase space. Further, we study the stability of the black hole solution using the specific heat and Gibbs free energy. We examine the first and second phase changes and show a P-V criticality, which is similar to the van der Waals phase change. We also examine the equation of the state and the critical exponents.