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GR is the most successful and physically acceptable theory of gravity, which precisely describes the gravitational interaction in the space-time geometry and the characteristics of matter via the energy-momentum tensor. From a geometrical perspective, the Lorentzian metric tensor
$ g_{\mu \nu} $ is considered to study the smooth manifold that is used to develop the Levi-Civita affine connection$ \Gamma^\lambda _{\mu \nu} $ . To establish a model where the largest family of BH solutions with dynamical torsion and nonmetricity in metric-affine gravity can be found, a propagating traceless nonmetricity tensor must be taken into account in the gravitational action of metric-affine gravity. As a geometrical correction to GR, a quadratic parity-preserving action presenting a dynamical traceless nonmetricity tensor in this situation is given as follows [33−37]:$ \begin{aligned}[b] S=&\int {\rm d}^4 x \sqrt{-g}\bigg\{\mathcal{L}_{\mathrm{m}}+\frac{1}{16 \pi}\Big[-R+2 f_1 \tilde{R}_{(\lambda \rho) \mu \nu} \tilde{R}^{(\lambda \rho) \mu \nu}\\&+2 f_2\left(\tilde{R}_{(\mu \nu)}-\hat{R}_{(\mu \nu)}\right)\left(\tilde{R}^{(\mu \nu)}-\hat{R}^{(\mu \nu)}\right)\Big]\bigg\}, \end{aligned} $
(1) where
$ \tilde{R}^{(\lambda \rho) \mu \nu} $ and$ \tilde{R}_{(\mu\nu)} $ are the affine-connected form of Riemann and Ricci tensors. Here, R denotes the Ricci scalar, g is the determinant of the metric tensor,$\mathcal{L}_{m}$ depicts the matter Lagrangian, and$ f_1 $ ,$ f_2 $ are Lagrangian coefficients. This solution can also be easily generalized to take into account the cosmological constant and Coulomb electromagnetic fields with electric charge ($ q_e $ ) and magnetic charge ($ q_m $ ), which are decoupled from torsion [38, 39]. This assumes the minimal coupling principle.$ \begin{aligned}[b] \tilde{R}^{(\lambda \rho)}_{\mu \nu} =&\frac{1}{2} T_{\mu \nu}^\sigma Q_\sigma^{\lambda \rho}+\tilde{\nabla}_{[\nu} Q_{\mu]}^{\lambda \rho}, \\ \tilde{R}_{(\mu \nu)}-\hat{R}_{(\mu \nu)} =&\tilde{\nabla}_{(\mu} Q_{\nu) \lambda}^\lambda+Q_{\lambda \rho(\mu} Q_{\nu)}^{\lambda \rho}\\&-\tilde{\nabla}_\lambda Q_{(\mu \nu)}^\lambda-Q^{\lambda \rho}{ }_\lambda Q_{(\mu \nu) \rho}+T_{\lambda \rho(\mu} Q_{\nu \rho}^{\lambda \rho}, \end{aligned} $
(2) These variations represent the third Bianchi of GR. By executing changes of the above equations with respect to the co-frame field and the anholonomic interrelation, the following field equations are established:
$ Y 1_\mu^\nu =8 \pi \theta_\mu^\nu $ and$ Y 2^{\lambda \mu \nu} =4 \pi \Delta^{\lambda \mu \nu} $ , where$ Y 1_\mu{ }^\nu $ and$ Y 2^{\lambda \mu \nu} $ are tensor quantities.$ \Delta^{\lambda \mu \nu} $ and$ \theta_\mu{ }^\nu $ are utilized to study the hyper momentum density and canonical energy-momentum tensors of matter, which are expressed as$ \begin{aligned}[b] \Delta^{\lambda \mu \nu} & =\frac{{\rm e}^{a \lambda} e_b \mu}{\sqrt{-g}} \frac{\delta\left(\mathcal{L}_m \sqrt{-g}\right)}{\delta \omega^a{ }_{b \nu}},\\ \theta_\mu^\nu & =\frac{{\rm e}^a{ }_\mu}{\sqrt{-g}} \frac{\delta\left(\mathcal{L}_m \sqrt{-g}\right)}{\delta {\rm e}^a \nu}. \end{aligned} $
(3) Therefore, both matter representations act as sources of the extended gravitational field. In this scenario, metric-affine geometries utilize the Lie algebra of the general linear group GL(4, R) in anholonomic interrelation. This hypermomentum presents its proper decomposition into shear, spin, and dilation currents [36, 37]. Furthermore, the effective gravitational action of the model is provided in terms of these properties. The parameterizations of the spherically symmetric static spacetime are as follows: [39−43]
$ \begin{array}{*{20}{l}} {\rm d}s^2=-\Psi(r){\rm d}t^2+\Psi^{-1}(r){\rm d}r^2 +r^2{\rm d}\theta^2+r^2\sin^2\theta {\rm d}\phi^2. \end{array} $
(4) Compared with the standard case of GR, in the emission process, a matter current coupled to torsion and nonmetricity in a general splitting of the energy levels will potentially affect this spectrum and the efficiency. Interestingly, the performance of a perturbative interpretation on the energy-momentum tensor in vacuum fluctuations of the quantum field coupled to the torsion, as well as nonmetricity tensors of the solution, is used to study the rate of dissipation obtained on its event horizon, which would also cover the further corrections with respect to the system of GR [44, 45]. The metric function (Reissner-Nordström-de Sitter-like) is defined as [37]
$ \Psi(r)=1-\frac{2 m}{r}+\frac{d_1 \kappa_{s}^2-4 e_1 \kappa_{d}^2-2 f_1 \kappa_{\mathrm{sh}}^2+q_{e}^2+q_{m}^2}{r^2}+\frac{\Lambda}{3} r^2, $
(5) which represents the broadest family-charged BH models obtained in metric-affine gravity with real constants
$ e_1 $ and$ d_1 $ . Here,$ \kappa_{\mathrm{sh}} $ ,$\kappa_{s}$ , and$\kappa_{d}$ represent the shear, spin, and dilation charges, respectively. -
A cosmological constant is treated as a thermodynamic variable. After the thermodynamic pressure of the BH is put into the laws of thermodynamics, the cosmological constant is considered as the pressure. From the equation of horizon
$ \Psi(r)=0 $ and pressure$ P=-\dfrac{\Lambda}{8\pi} $ [29, 30], we can deduce the relation between the BH mass m and its event horizon radius,$ r_h $ , which is expressed as follows$ m=\frac{3 d_1 \kappa_{s} ^2-6 f_1 \kappa_{\text{sh}}^2-8 \pi P r_h^4+3 q_e^2+3 q_m^2+3 r_h^2-12 \kappa_d ^2 e_{1} }{6 r_h}. $
(6) The Hawking temperature of the BH related to surface gravity can be obtained as
$\begin{aligned}[b] T=&\frac{\Psi'(r)}{4\pi}=\frac{6-32 \pi P r_h^2}{12 \pi r_h}\\&-\frac{3 d_1 \kappa_{s} ^2-6 f \kappa_{\text{sh}}^2-8 \pi P r_h^4+3 q_e^2+3 q_m^2+3 r_h^2-12 \kappa_d^2 e_{1} }{12 \pi r_h^3}. \end{aligned}$
(7) It has a peak as shown in Figs. 1 and 2 and shifts to right (positive) and increases with increasing
$ P_c $ and$ \kappa_{s} $ . The temperature becomes the absence of the electric charge$ (q= 0) $ . As we increase the values of$ P_c $ and$ \kappa_{s} $ , the local maximum of the Hawking temperature increases, as exhibited in Figs. 1 and 2. Further, the temperature converges when the horizon radius shrinks to zero for the considered BH manifold. The general form of the first law of BH thermodynamics can be written as [29−32, 46, 47]Figure 1. (color online) Plot of temperature T with fixed
$ q_{e}= $ 0.28;$ q_m=0.08 $ ;$ d_1=0.004 $ ;$ f_1=0.313 $ ;$ \kappa _d =0.02 $ ;$ \kappa_s =0.8 $ ; and$ e_{1}=0.4 $ .Figure 2. (color online) Plot of temperature T with fixed
$ q_{e }= $ 0.28;$ q_m=0.08 $ ;$ d_1=0.004 $ ;$ f_1=0.313 $ ;$ \kappa_d =0.02 $ ; and$ e_1=0.4 $ .$ \begin{aligned}[b] {\rm d}M=&T{\rm d}S + V {\rm d}P+ \Phi {\rm d}q_m+ \varphi {\rm d}q_e+\Bbbk_{\rm sh} {\rm d}\kappa_{\rm sh}\\&+\Bbbk_{s}{\rm d}\kappa_{s}+\Bbbk_{d} {\rm d}\kappa_{d}+E1 {\rm d}e_{1}+F_1 {\rm d}f_{1} +D_1 {\rm d} d_{1}, \end{aligned} $
(8) where M, S, V, P, Q, Φ, and φ are the mass, entropy, volume, pressure, magnetic charge, and chemical potential of BH, respectively. They have been treated as thethermodynamic variables corresponding to the conjugating variables
$ \Bbbk_{\rm sh} $ ,$ \Bbbk_{s} $ ,$ \Bbbk_{d} $ ,$ E1 $ , and$ d_1 $ , respectively. The volume and chemical potential of the BH are defined as$ V=\bigg(\frac{\partial M}{\partial P}\bigg)_{S,q_m},\quad \Phi=\bigg(\frac{\partial M}{\partial q_m}\bigg)_{S,P}, $
(9) respectively. The BH entropy with the help of area is defined as [48−50]
$ S=\frac{A}{4}=\pi r_{h}^{2}. $
(10) From Eqs. (6) and (7), the equation of state for the BH can easily be expressed as
$ P=-\frac{d_1 \kappa_{s}^2-2 f_1 \kappa_{\text{sh}}^2+q_e^2+q_m^2+4 \pi r_h^3 T-r_h^2-4 \kappa_d ^2 e_{1} }{8 \pi r_h^4}. $
(11) The red, blue, orange, and black colors indicate the divergence at pressures below the critical pressure. The oscillations of the isotherms at critical temperatures in the
$ P - v_h $ diagram are equivalent to the unstable BHs that are presented by negative heat capacity in this section (Figs. 3 and 4). These divergences are the characteristics of the first-order phase transition that occurs between smaller and larger BHs that are stable and have a positive heat capacity. In response to changes in the value of the parameter$ T_c $ , there is a corresponding shift in the horizontal axis; an increase in this parameter results in a reduction in the critical radius.Figure 3. (color online) Plot of temperature P with fixed
$ q_m= $ 0.0002;$ d_1=0.004 $ ;$ f_1=0.003 $ ;$ \kappa _d =0.01 $ ;$ \kappa _s =0.03 $ ; and$ e _1=0.4 $ .Figure 4. (color online) Plot of temperature P with fixed
$ q_m= $ 0.0002;$ d_1=0.004 $ ;$ f_1=0.003 $ ;$ \kappa _d =0.01 $ ;$ \kappa _s =0.03 $ ; and$ e _1=0.4 $ .The thermodynamic variables V, Φ, φ and the conjugating quantities
$\Bbbk_{\rm sh}$ ,$ \Bbbk_{s} $ ,$ \Bbbk_{d} $ ,$ E1 $ , and$ d_1 $ are obtained from the first law as$ \begin{aligned}[b]& V=\frac{4 \pi r_h^3}{3},\quad \Phi=\frac{q_m}{r},\quad\varphi= \frac{q_e}{r}, \\&\Bbbk_{\rm sh}= \frac{-2 f_1\kappa_{\rm sh} }{r},\quad \Bbbk_{s}=\frac{ d_1\kappa_s }{r},\quad \Bbbk_{d}=\frac{ -4 e_1\kappa_d}{r},\\& E1= \frac{ -4 \kappa_d^2}{r} \;\;\text{and} \;\;D_1= \frac{ \kappa_s^2}{r}. \end{aligned} $
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The most important and basic thermodynamic quantity is the Gibbs free energy of a BH, which can be utilized to explore the small/larger BH phase transition by studying the
$ G-r_h $ and$ G - T $ diagrams. In addition, the Gibbs free energy also helps us to investigate the global stability of a BH. It can be evaluated as [51, 52]$ \begin{array}{*{20}{l}} G=-TS+M. \end{array} $
(13) Using Eqs. (6) and (7) in (13), we obtain
$\begin{aligned}\\[-13pt] G=\frac{\sqrt[3]{\dfrac{\pi }{6}} \left(12 d_1 \kappa_{s} ^2-24 f \kappa_{\text{sh}} ^2+4 \sqrt[3]{\dfrac{6}{\pi }} P v^{4/3}+12 q_e^2+12 q_m^2+\left(\dfrac{6}{\pi }\right)^{2/3} v^{2/3}-48 \kappa_d^2 e_{1} \right)}{8 \sqrt[3]{v}}.\end{aligned} $ (14) We observe the graphical behavior of the phase transitions in the
$ G-r_{h} $ plane as shown in Figs. 5 and 6. It is noted that the Gibbs free energy decreases as the critical radius increases. To calculate the critical thermodynamic properties of a BH, one can use the following condition:Figure 5. (color online) Plot of Gibbs free energy G with fixed
$ q_m=0.003 $ ;$ d_1=0.200 $ ;$ f_1=0.050 $ ;$ \kappa _d =0.010 $ ; and$ e _1=0.050 $ .Figure 6. (color online) Plot of Gibbs free energy G with fixed
$ q_m=0.003 $ ;$ d_1=0.200 $ ;$ f_1=0.050 $ ;$ \kappa _d =0.010 $ ; and$ e _1=0.050 $ .$ \bigg(\frac{\partial P}{\partial \upsilon_{h}}\bigg)_{T}=\bigg(\frac{\partial^{2} P}{\partial \upsilon_{h}^{2}}\bigg)_{T}=0. $
(15) Using Eq. (15), the critical temperature can be expressed as
$ T_{c}=\frac{1}{3 \sqrt{6} \pi \sqrt{d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1} }}. $
(16) From Eq. (15), the critical radius of BH is as follows:
$ \begin{array}{*{20}{l}} \upsilon_{c}= 8 \sqrt{6} \pi \left(d_1 \kappa_{s}^2-2 f_1 \kappa_{\text{sh}}^2+q_e^2+q_m^2-4 \kappa_d^2 e_{1} \right)^{3/2}. \end{array} $
(17) The critical pressure in terms of other parameters takes the following form:
$ P_{c}=\frac{1}{96 \pi \left(-d_1 \kappa_{s}^2+2 f_1 \kappa_{\text{sh}}^2-q_e^2-q_m^2+4 \kappa_d^2 e_1 \right)^2}. $
(18) However, we applied a numerical analysis because calculating the critical numbers analytically is not a simple operation.
To find more data about a phase transition, we studied a thermodynamic quantity such as heat capacity. By applying the standard definition of heat capacity as follows: [46, 53]
$ C_{p}=T \bigg(\frac{\partial S}{\partial T}\bigg)_{P}, $
(19) with a few numerical calculations, one can obtain a dimensionless important relation for the amounts
$ P_c $ ,$ T_c $ , and$ v_c $ . If the expression$ (d_1 \kappa_{s}^2-2 f_1 \kappa_{\text{sh}}^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1})\rightarrow 1.327765310 $ is provided, then our solution satisfies the well-known condition as$ \frac{P_c v_c}{T_c} = 3/8, $
(20) and similar results are studied in the context of the van der Waals equation and in an RN-AdS BH . Therefore, the negative heat capacity that gives the temperamental (unstable) BH is also related to the critical temperature in the
$ P-\upsilon_{h} $ plane. The expressions of volume and entropy of a BH are presented in Eqs. (7) and (10). From Eq. (19), we obtain$ C_{p}=\dfrac{3^{2/3} \sqrt[3]{\dfrac{\pi }{2}} v^{2/3} \left(4 d \kappa_{s} ^2-8 f \kappa_{\text{sh}} ^2+12 \sqrt[3]{\dfrac{6}{\pi }} P v^{4/3}+4 q_e^2+4 q_m^2-\left(\dfrac{6}{\pi }\right)^{2/3} v^{2/3}-16 \kappa_d ^2 e_{1} \right)}{-12 d \kappa_{s} ^2+24 f \kappa_{\text{sh}} ^2+12 \sqrt[3]{\dfrac{6}{\pi }} P v^{4/3}-12 q_e^2-12 q_m^2+\left(\dfrac{6}{\pi }\right)^{2/3} v^{2/3}+48 \kappa_d ^2 e_{1} }. $ (21) It has been discovered that the critical amounts classify the behavior of thermodynamic quantities close to the critical point. In Figs. 7 and 8, for thermodynamically stable BHs, we separate the two cases in which the heat capacity is positive (
$ r _h <r_ c $ ) and the case in which it is negative ($ r_ h> r_ c $ ). The second-order phase transition is implied by the instability areas of BHs, where the heat capacity is discontinuous at the critical temperature$ r _h=r _c $ [54, 55]. It is noted that the heat capacity diverges at$ r_h = 0.50 $ , when$ T_h $ reaches its maximum value as$ T_h=0.24 $ for$ r_h = 1.00 $ ,$ q_m=0.08 $ ,$ d_1=0.004 $ ,$ f_1=0.313 $ ,$ \kappa _d =0.02 $ , and$ e _1=0.4 $ . The critical points in the alternate phase space are obtained by utilizing the standard definition, and we reduced the thermodynamic variables as follows:Figure 7. (color online) Plot of heat capacity with fixed
$ q_m=0.08 $ ;$ d_1=0.004 $ ;$ f_1=0.313 $ ;$ \kappa _d =0.02 $ ; and$ e _1=0.4 $ .Figure 8. (color online) Plot of heat capacity with fixed
$ q_m=0.08 $ ;$ d_1=0.004 $ ;$ f_1=0.313 $ ;$ \kappa _d =0.02 $ ; and$ e _1=0.4 $ .$ T_r=\frac{T}{T_c}, \quad v_r=\frac{v}{v_c}\quad {\rm and} \quad P_r=\frac{P}{P_c}. $
(22) The reduced variables can be written as
$ \begin{aligned}\\[-12pt]T_r=-\frac{3 \sqrt{\dfrac{3}{2}} \sqrt{d_1 \kappa_{s} ^2-2 f \kappa_{\text{sh}} ^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1} } \left(d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+8 \pi P r_h^4+q_e^2+q_m^2-r_h^2-4 \kappa_d ^2 e_{1} \right)}{2 r_h^3}, \end{aligned}$ (23) volume can be obtained as
$ v_r=\frac{r^3}{6 \sqrt{6} \left(d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1} \right)^{3/2}}, $
(24) and pressure is as follows:
$ P_r=-\frac{12 \left(d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1} \right)^2 \left(d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+q_e^2+ q_m^2+4 \pi r_h^3 T-r_h^2-4 \kappa_d^2 e_{1} \right)}{r_h^4}. $ (25) Two adiabatic and two isothermal processes combined to form the Carnot cycle are the hallmark of the most effective heat engine. The single most fundamental and critical feature of the Carnot cycle is that the heat engine efficiency is a function of reservoir temperatures:
$ \eta= 1 -\frac {T_c}{T_h}, $
(26) and as a reservoir can never be at zero temperature, the efficiency cannot be one because
$ T_c $ and$ T_h $ denote cold and hot reservoirs, respectively. Hence, we obtain$ \eta=1+ \frac{2 \sqrt{\dfrac{2}{3}} r_h^3}{3 \sqrt{d_1 \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+q_e^2+q_m^2-4 \kappa_d ^2 e_{1} } \left(d \kappa_{s} ^2-2 f_1 \kappa_{\text{sh}} ^2+8 \pi P r_h^4+q_e^2+q_m^2-r_h^2-4 \kappa_d^2 e_{1} \right)}. $
(27) Now, we study the behavior of the heat engine efficiency η as a function of the the pressure P and entropy S matching to the heat cycle provided in Figs. 9 and 10, for the different values of metric-affine gravity parameters. From these figures, we can observe that the nature of the heat engine efficiency is essentially relying on the metric-affine gravity parameters. In addition, for a given set of input values, the efficiency of the heat engine increases monotonically as the horizon's radius grows. Because of this, larger BHs should expect higher heat-engine efficiency. In other words, they allow for a maximum efficiency curve to be provided for a heat engine by varying only a few fixed parameters (the BH works at the highest efficiency). Here, the local stability is related to the system, but it can be large with small changes in the values of thermodynamic parameters. Thus, the term heat capacity gives information on local stability. In [36], it is stated how the cosmological constant Λ can be studied by treating it as a scale parameter.
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One of the best-known and classical physical processes to explain the change in temperature of a gas flowing from a high-pressure to a reduced pressure section through a porous plug is called the Joule-Thomson expansion. The main focus is on the gas expansion process, which expresses the cooling effect (when the temperature drops) and the heating effect (when the temperature increases), with the enthalpy remaining constant throughout the process. This change depends upon the Joule-Thomson coefficient: [28, 56]
$ \mu_{JT}=\bigg(\frac{\partial T}{\partial P}\bigg)_{H}=\frac{1}{C_{p}}\bigg[T\bigg(\frac{\partial V}{\partial T}\bigg)_{p}-V\bigg]. $
(28) Using Eqs. (7), (9), (21), and (28), the coefficient is calculated as follows:
$ \mu_{JT} = \frac{4 r_h \left(3 d_1 \kappa_{s} ^2-6 f_1 \kappa_{\text{sh}} ^2 + 8 \pi P r_h^4 + 3 q_e^2 + 3 q_m^2 - 2 r_h^2 - 12 \kappa_d ^2 e_{1} \right)}{3 \left(d_1 \kappa_{s} ^2-2 f \kappa_{\text{sh}} ^2+8 \pi P r_h^4+q_e^2+q_m^2-r_h^2-4 \kappa_d ^2 e_{1} \right)}. $
(29) The study of the Joule-Thomson coefficient versus the horizon
$ r_h $ is shown in Figs. 11, 12, 13, and 14. We set$ d_1=0.004 $ ,$ f_1=0.313 $ ,$\kappa _{\rm sh} =0.05$ ,$ \kappa _d =0.02 $ ,$ \kappa _s =0.8 $ , and$ e _1=0.4 $ in that order. There exist both divergence points and zero points for different variations of$ \kappa _d $ ,$ \kappa _s $ , and$\kappa _{\rm sh}$ respectively. It is clear from a comparison of these figures that the zero point of the Hawking temperature and the divergence point of the Joule-Thomson coefficient is the same. This point of divergence gives information on the Hawking temperature and corresponds to the most extreme BHs. From Eq. (29) and utilizing the well-known condition$ \mu_{JT}=0 $ , the temperature inversion occurs asFigure 11. (color online) Joule-Thomson coefficient
$ \mu_{JT} $ plane with fixed$ d_1=0.03 $ ;$ f_1=0.01 $ ;$ \kappa _{d} =0.02 $ ;$ \kappa _s =0.10 $ ; and$ e _1=0.04 $ .Figure 12. (color online) Joule-Thomson coefficient
$ \mu_{JT} $ with fixed$ d_1=0.03 $ ;$ f_1=0.01 $ ;$ \kappa _{d} =0.02 $ ;$ \kappa _s =0.10 $ ; and$ e _1=0.04 $ .Figure 13. (color online) Joule-Thomson coefficient
$ \mu_{JT} $ with fixed$ d_1=0.03 $ ;$ f_1=0.01 $ ;$\kappa _{\rm sh} =0.10$ ;$ \kappa _d =0.02 $ ; and$ e _1=0.04 $ .Figure 14. (color online) Joule-Thomson coefficient
$ \mu_{JT} $ with fixed$ d_1=0.03 $ ;$ f_1=0.01 $ ;$\kappa _{\rm sh} =0.10$ ;$ \kappa _s =0.10 $ ; and$ e _1=0.04 $ .$ T_i=\frac{3 d_1 \kappa_{s} ^2-6 f_1 \kappa_{\text{sh}} ^2-8 \pi P r_h^4+3 q_e^2+3 q_m^2-r_h^2-12 \kappa_d ^2 e_{1}}{12 \pi r_h^3}. $
(30) Because the Joule-Thomson expansion is an isenthalpic process, it is important to analyze the isenthalpic curves of BHs under metric-affine gravity, which are depicted in Figs. 15−18. Thus, we study the isenthalpic curves (
$ T_i -P_i $ plane) by assuming different values of BH mass, which are investigated in Eq. (29) with a larger root of$ r_h $ . We show the isenthalpic and inversion curves of BHs in metric-affine gravity and the result is consistent [57−60]. The heating and cooling zones are characterized by the inversion curve, and the isenthalpic curves possess positive slopes above the inversion curve. In contrast, the pressure always falls in a Joule-Thomson expansion and the slope changes sign when heating occurs below the inversion curve. The heating process appears at higher temperatures, as indicated by the negative slope of the constant mass curves in the Joule-Thomson expansion. When temperatures drop, cooling begins, which is linked to the positive slope of the constant mass curves. From above equation, one can deduce the inversion pressure asFigure 15. (color online) Isenthalpic curves
$ (T-P) $ plane with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$\kappa _{\rm sh} =0.05$ ;$ \kappa _d =0.02 $ ; and$ e _1=0.4 $ .Figure 16. (color online) Isenthalpic curves
$ (T-P) $ plane with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$ \kappa _d =0.02 $ ;$ \kappa _s =0.8 $ , and$ e _1=0.4 $ .Figure 17. (color online) Isenthalpic curves
$ (T-P) $ plane with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$\kappa _{\rm sh} =0.05$ ;$ \kappa _d =0.02 $ ;$ \kappa _s =0.8 $ , and$ e _1=0.4 $ .Figure 18. (color online) Isenthalpic curves
$ {T-P} $ plane with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$\kappa _{\rm sh} =0.05$ ;$ \kappa _d =0.02 $ ;$ \kappa _s =0.8 $ ; and$ e _1=0.4 $ .$ P_i =\frac{3 d_1 \kappa_{s} ^2-6 f_1 \kappa_{\text{sh}} ^2+3 q_e^2+3q_m^2-12 \pi r_h^3 Ti-r_h^2-12 \kappa_d^2 e_{1} }{8 \pi r_h^4}. $
(31) The inversion curves for different values,
$ d_1=0.004 $ ,$ f_1=0.313 $ ,$\kappa _{\rm sh} =0.05$ , and$ \kappa _d =0.02 $ , are shown in Figs. 19, 20, 21, and 22. The inversion temperature increases with variations of the important parameters m,$ q_e $ ,$\kappa _{\rm sh}$ , and$ \kappa _s $ , respectively. We can go back to the case of the BH in metric-affine gravity. Compared with the van der Waals fluids, we can observe from Figs. 19−22 that the inversion curve is not closed. From the above results, in the$ Ti-Pi $ plane at low pressure, the inversion temperature$ T_i $ decreases with the increase in charge$ q_e $ and mass m, and it shows the opposite behavior for higher pressure. It is also clear that, unlike the case with van der Waals fluids, the inversion temperature continues to rise monotonically with increasing inversion pressure, and hence, the inversion curves are not closed [59, 60].Figure 19. (color online) Inversion curves
$ (Ti-Pi) $ with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$\kappa _{\rm sh} =0.05$ ;$ \kappa _d =0.02 $ ;$ \kappa _s =0.8 $ ; and$ e _1=0.4 $ .Figure 20. (color online) Inversion curves
$ (Ti-Pi) $ with fixed$ d_1=0.004 $ ;$ f_1=0.313 $ ;$\kappa _{\rm sh} =0.05$ ;$ \kappa _d =0.02 $ ;$ \kappa _s =0.8 $ ; and$ e _1=0.4 $ .
Thermal analysis and Joule-Thomson expansion of black hole exhibiting metric-affine gravity
- Received Date: 2023-07-17
- Available Online: 2024-01-15
Abstract: This study examines a recently hypothesized black hole, which is a perfect solution of metric-affine gravity with a positive cosmological constant, and its thermodynamic features as well as the Joule-Thomson expansion. We develop some thermodynamical quantities, such as volume, Gibbs free energy, and heat capacity, using the entropy and Hawking temperature. We also examine the first law of thermodynamics and thermal fluctuations, which might eliminate certain black hole instabilities. In this regard, a phase transition from unstable to stable is conceivable when the first law order corrections are present. In addition, we study the efficiency of this system as a heat engine and the effect of metric-affine gravity for the physical parameters