Applications of Thermal Geometries of Black Hole in Metric-Affine Gravity

Abstract

We study the thermodynamical behavior, stability and micro structures of black hole in metric-affine gravity. We investigate the effect of different values of the space-time parameters on the stability and compatibility of phase transition points of the black hole. We extend our study to discuss the different curvature scalars of Weinhold, Ruppeiner, Geometrothermodynamics (GTD) and Hendi-Panahiyah-Eslam-Momennia (HPEM). Meanwhile, we give the analytical expression of the thermodynamic curvatures of the black hole in metric-affine gravity, which display that the black hole may be governed by the repulsion on low temperature region along with attraction on high temperature region, qualitatively. It is presented that the negative thermodynamic curvature is provides the information of attractive interaction between black hole molecules in metric-affine gravity.

Appendix

The curvature scalars have been calculated in this direction as:

$$\begin{aligned} ds^2\!=\!{\left\{ \begin{array}{ll} m g_{a b}^W d X^b d X^a \Rightarrow \text{ Weinhold } , \\ -T^{-1} m g_{a b}^W d X^b d X^a \Rightarrow \text{ Ruppeiner } , \\ (-m_{SS}dS^2+m_{ \kappa _d \kappa _d}d \kappa _d^2+m_{ \kappa _{sh} \kappa _{sh}}d \kappa _{sh}^2)\frac{S m_S}{(\frac{\partial ^2 m}{\partial \kappa _d^2}\frac{\partial ^2 m}{\partial \kappa _{sh}^2})^3} \Rightarrow \text {HPEM} ,\\ (-S m_{SS}+\kappa _d m_{\kappa _d \kappa _d}+ \kappa _{sh} m_{\kappa _{sh} \kappa _{sh}})(-m_{SS}dS^2+m_{ \kappa _d \kappa _d}d \kappa _d^2+m_{ \kappa _{sh} \kappa _{sh}}d \kappa _{sh}^2) \Rightarrow \text {GTD}. \end{array}\right. } \end{aligned}$$

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Some abbreviations in the Section 4 are defined as:

$$\begin{aligned} A= & {} d_1 \kappa _s ^2+q_{\epsilon }^2+q_m^2-4 \kappa _d ^2 \epsilon _1,\\ B= & {} 6 q_{\epsilon }^2 S^2+6 q_m^2 S^2+72 \kappa _d^2 S^2 \epsilon _1 +64 \kappa _{sh} \kappa _d S \epsilon _1 +21 \kappa _{sh} ^2 \epsilon _1,\\ C= & {} -3 d_1 \kappa _s ^2+6 f_1 \kappa _{sh} ^2+16 f_1 \kappa _d^2 S^2+16 f_1 \kappa _{sh} \kappa _dS -3 q_{\epsilon }^2-3 q_m^2+12 \kappa _d^2 \epsilon _1.\\ \zeta= & {} 9 \pi d_1 \kappa _s ^2+36 \pi d_1 \kappa _s ^2 S-54 \pi f_1 \kappa _{sh} ^2-72 \pi f_1 \kappa _{sh} ^2 S-96 P S^3-152 P S^2\\{} & {} +36 \pi q_{\epsilon }^2 S+9 \pi q_{\epsilon }^2+36 \pi q_m^2 S+9 \pi q_m^2-12 S^2-11 S-36 \pi \kappa _d^2 \epsilon _1,\\ \eta= & {} \pi d_1 \kappa _s^2- 2 \pi f_1 \kappa _{sh}^2+8 P S^2+\pi q_{\epsilon }^2+\pi q_m^2-S-4 \pi \kappa _d ^2 \epsilon _1.\\ \digamma= & {} d_1 \kappa _s ^2-10 f_1 \kappa _{sh} ^2+q_{\epsilon }^2+q_m^2+12 \kappa _d ^2 \epsilon _1 ,\\ \beth= & {} d_1 \kappa _s ^2+4 d_1 \kappa _s ^2 S-6 f_1 \kappa _{sh} ^2-8 f \kappa _{sh} ^2 S+q_{\epsilon }^2 (4 S+1)+4 q_m^2 S+q_m^2-4 \kappa _d ^2 \epsilon _1.\\ \end{aligned}$$