Geodesics in geometrothermodynamics (GTD) type II geometry of 4D asymptotically anti-de-Sitter black holes

Abstract

In this work, we study the thermodynamic geodesics of 4D asymptotically anti-de Sitter black holes in the context of the geometrothermodynamics (GTD) type II metric. We construct GTD type II metrics for four different black holes: 4D Kerr–Newman AdS (KN-AdS), Kerr-AdS (K-AdS), Reissner–Nordstrom AdS (RN-AdS), and Dyonic AdS black holes in various ensembles. For K-AdS and RN-AdS black holes, we work in canonical as well as grand canonical ensembles. For KN-AdS and Dyonic AdS black holes, in addition to these two ensembles, we also consider a fixed charged ensemble. For each case, we solve the corresponding geodesic equations numerically and analyze their behavior near the spinodal curves. These spinodal curves, which separate the positive specific heat region from the negative specific heat region, can be treated as the boundary between two black hole phases in the thermodynamic parameter space. The points on the spinodal curve are the points in parameter space at which we reach Davies’ temperature. We find that the geodesics, in all the cases under consideration, are confined to a single phase and exhibit either turning behavior or incompleteness near the spinodal curve (Davies’ temperature). This is universally true for GTD type II geodesics in KN-AdS, K-AdS, RN-AdS, and Dyonic AdS black holes in different ensembles. From this, we conclude that the turning behavior or incompleteness of geodesics in GTD type II geometry can be used as an indicator of phase transitions in 4D asymptotically AdS black holes.

Appendix: Geodesic equations

1.1 Kerr–Newman black hole

Geodesic equations for canonical ensemble:

$$\begin{aligned} &\ddot{s}+ \left[ \frac{ \begin{aligned}&2 \bigl \{ 16 \pi ^7 j^4+4 \pi ^3 \left( 15 s^4+30 \pi s^3+18 \pi ^2 s^2+\left( q^2+2\right) \pi ^4 q^2+3 \pi ^3 \left( 2 s q^2+s\right) \right) j^2 +\bigl \{ 6 s^3+2 \pi s^2 \\&\qquad +\left( 6 q^2-1\right) \pi ^2 s+2 \pi ^3 q^2\bigr \} \left( \pi ^2 q^2+s^2+\pi s\right) ^2\bigr \} \end{aligned} }{ \begin{aligned}&\Bigl \{ 16 \pi ^7 (4 s+3 \pi ) j^4+8 \pi ^3 \bigl \{ 6 s^5+15 \pi s^4+12 \pi ^2 s^3 +3 \left( 2 q^2+1\right) \pi ^3 s^2+2 \left( q^2+2\right) \pi ^4 q^2 s+3 \pi ^5 q^4\bigr \} j^2 \\&\qquad +\left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( 3 \pi ^2 q^2+3 s^2-\pi s\right) \Bigr \} \end{aligned}}\right. \\&\qquad \left. \vphantom{\frac{ \begin{aligned}&2 \bigl \{ 16 \pi ^7 j^4+4 \pi ^3 \left( 15 s^4+30 \pi s^3+18 \pi ^2 s^2+\left( q^2+2\right) \pi ^4 q^2+3 \pi ^3 \left( 2 s q^2+s\right) \right) j^2 +\bigl \{ 6 s^3+2 \pi s^2 \\&\qquad +\left( 6 q^2-1\right) \pi ^2 s+2 \pi ^3 q^2\bigr \} \left( \pi ^2 q^2+s^2+\pi s\right) ^2\bigr \} \end{aligned} }{ \begin{aligned}&\Bigl \{ 16 \pi ^7 (4 s+3 \pi ) j^4+8 \pi ^3 \bigl \{ 6 s^5+15 \pi s^4+12 \pi ^2 s^3 +3 \left( 2 q^2+1\right) \pi ^3 s^2+2 \left( q^2+2\right) \pi ^4 q^2 s+3 \pi ^5 q^4\bigr \} j^2 \\&\qquad +\left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( 3 \pi ^2 q^2+3 s^2-\pi s\right) \Bigr \} \end{aligned}}} -\frac{3}{2 s}+\frac{\left( 6 s^2+6 \pi s+\left( 2 q^2+1\right) \pi ^2\right) s}{-\pi ^4 q^4-4 j^2 \pi ^4+3 s^4+4 \pi s^3+\left( 2 q^2+1\right) \pi ^2 s^2}-\frac{2 \left( 2 \pi ^3 j^2+(2 s+\pi ) \left( \pi ^2 q^2+s^2+\pi s\right) \right) }{4 \pi ^3 (s+\pi ) j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^2}\right] \dot{s}^2 \\ &\qquad + \frac{ \begin{aligned}&8 \pi ^3 \Bigl \{ 64 \pi ^{11} \left( 4 s^2+7 \pi s+3 \pi ^2\right) j^7+48 \pi ^8 (4 s+3 \pi ) \left( \pi ^2 q^2+s^2+\pi s\right) ^2 j^5 \\&\qquad +4 \pi ^3 \bigl \{ 36 s^{10}+171 \pi s^9+\left( 24 q^2+341\right) \pi ^2 s^8+2 \left( 46 q^2+185\right) \pi ^3 s^7+18 \left( 10 q^2+13\right) \pi ^4 s^6 \\&\qquad +\left( -6 q^4+216 q^2+83\right) \pi ^5 s^5+\left( 8 q^6+58 q^4+140 q^2+13\right) \pi ^6 s^4+6 \left( 2 q^4+19 q^2+6\right) \pi ^7 q^2 s^3 \\&\qquad +2 \left( -2 q^4+26 q^2+25\right) \pi ^8 q^4 s^2+3 \left( q^2+12\right) \pi ^9 q^6 s+9 \pi ^{10} q^8\bigr \} j^3 \\&\qquad +\bigl \{ -18 s^7-63 \pi s^6+3 \left( 14 q^2-27\right) \pi ^2 s^5+\left( 53 q^2-45\right) \pi ^3 s^4-3 \pi ^4 \left( 2 q^4-8 q^2+3\right) s^3 \\&\qquad +\left( 7 q^2+5\right) \pi ^5 q^2 s^2+\pi ^6 q^4 \left( 9-2 q^2\right) s+3 \pi ^7 q^6\bigr \} \left( \pi ^2 q^2+s^2+\pi s\right) ^3 j \Bigr \} \dot{j} \dot{s} \end{aligned}}{ \begin{aligned}&\left( \pi ^4 q^4+4 j^2 \pi ^4-3 s^4-4 \pi s^3-\pi ^2 \left( 2 q^2+1\right) s^2\right) \left( 4 \pi ^3 (s+\pi ) j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^2\right) \\&\times \Bigl \{ 16 \pi ^7 (4 s+3 \pi ) j^4+8 \pi ^3 \bigl \{ 6 s^5+15 \pi s^4+12 \pi ^2 s^3+3 \left( 2 q^2+1\right) \pi ^3 s^2+2 \left( q^2+2\right) \pi ^4 q^2 s \\&\qquad +3 \pi ^5 q^4\bigr \} j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( 3 \pi ^2 q^2+3 s^2-\pi s\right) \Bigr \} \end{aligned}}\\&\qquad -\frac{ \begin{aligned}&8 \pi ^4 s \left( \pi ^2 q^2+s^2+\pi s\right) \Bigl \{ 16 \pi ^6 \left( 2 s^2+3 \pi s+\pi ^2\right) \left( \pi ^2 q^2-s^2-\pi s\right) j^4+4 \pi ^2 \bigl \{ 18 s^8+75 \pi s^7 \\&\qquad +\left( 14 q^2+123\right) \pi ^2 s^6+\left( 49 q^2+99\right) \pi ^3 s^5+\left( -2 q^4+64 q^2+39\right) \pi ^4 s^4 \\&\qquad +\left( q^4+37 q^2+6\right) \pi ^5 s^3+\left( 2 q^4+7 q^2+8\right) \pi ^6 q^2 s^2+\left( 3 q^2+4\right) \pi ^7 q^4 s+2 \pi ^8 q^6\bigr \} j^2 \\&\qquad +\left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( \pi ^4 q^4+2 \pi ^3 s q^2-s^4+2 \left( 4 q^2-1\right) \pi s^3+\left( 8 q^2-1\right) \pi ^2 s^2\right) \Bigr \} \dot{j}^2 \end{aligned}}{ \begin{aligned}&\left( \pi ^4 q^4+4 j^2 \pi ^4-3 s^4-4 \pi s^3-\pi ^2 \left( 2 q^2+1\right) s^2\right) \bigl \{ 4 \pi ^3 (s+\pi ) j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^2\bigr \} \\&\times \Bigl \{ 16 \pi ^7 (4 s+3 \pi ) j^4+8 \pi ^3 \bigl \{ 6 s^5+15 \pi s^4+12 \pi ^2 s^3+3 \left( 2 q^2+1\right) \pi ^3 s^2+2 \left( q^2+2\right) \pi ^4 q^2 s \\&\qquad +3 \pi ^5 q^4\bigr \} j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( 3 \pi ^2 q^2+3 s^2-\pi s\right) \Bigr \} \end{aligned}}=0, \end{aligned}$$

(92)

and

$$\begin{aligned} \ddot{j}- \frac{ \begin{aligned}&16 j (s+\pi ) s^2 \left( \pi ^2 q^2+s^2+\pi s\right) ^2 \Bigl \{ 4 \pi ^7 (s+\pi ) j^2+\pi ^3 (2 s+\pi ) \bigl \{ \pi ^4 q^4-2 \pi ^3 s q^2-3 s^4-6 \pi s^3 \\&\qquad -\pi ^2 \left( 2 q^2+3\right) s^2\bigr \}\Bigr \} \dot{j}^2+4 s \left( \pi ^2 q^2+s^2+\pi s\right) \Bigl \{ 16 \pi ^7 \left( 2 s^2+3 \pi s+\pi ^2\right) \left( \pi ^2 q^2-s^2-\pi s\right) j^4 \\&\qquad +4 \pi ^3 \bigl \{ 18 s^8+75 \pi s^7+\left( 14 q^2+123\right) \pi ^2 s^6+\left( 49 q^2+99\right) \pi ^3 s^5+\left( -2 q^4+64 q^2+39\right) \pi ^4 s^4 \\&\qquad +\left( q^4+37 q^2+6\right) \pi ^5 s^3+\left( 2 q^4+7 q^2+8\right) \pi ^6 q^2 s^2+\left( 3 q^2+4\right) \pi ^7 q^4 s+2 \pi ^8 q^6\bigr \} j^2 \\&\qquad +\pi \left( \pi ^2 q^2+s^2+\pi s\right) ^3 \left( \pi ^4 q^4+2 \pi ^3 s q^2-s^4+2 \left( 4 q^2-1\right) \pi s^3+\left( 8 q^2-1\right) \pi ^2 s^2\right) \Bigr \} \dot{s} \dot{j} \\&\qquad -j \Bigl \{ 64 \pi ^{11} \left( 4 s^2+7 \pi s+3 \pi ^2\right) j^6+48 \pi ^8 (4 s+3 \pi ) \left( \pi ^2 q^2+s^2+\pi s\right) ^2 j^4+4 \pi ^3 \bigl \{ 36 s^{10} \\&\qquad +171 \pi s^9+\left( 24 q^2+341\right) \pi ^2 s^8+2 \left( 46 q^2+185\right) \pi ^3 s^7+18 \left( 10 q^2+13\right) \pi ^4 s^6 \\&\qquad +\left( -6 q^4+216 q^2+83\right) \pi ^5 s^5+\left( 8 q^6+58 q^4+140 q^2+13\right) \pi ^6 s^4+6 \left( 2 q^4+19 q^2+6\right) \pi ^7 q^2 s^3 \\&\qquad +2 \left( -2 q^4+26 q^2+25\right) \pi ^8 q^4 s^2 +3 \left( q^2+12\right) \pi ^9 q^6 s+9 \pi ^{10} q^8\bigr \} j^2+\bigl \{ -18 s^7-63 \pi s^6 \\&\qquad +3 \left( 14 q^2-27\right) \pi ^2 s^5+\left( 53 q^2-45\right) \pi ^3 s^4-3 \pi ^4 \left( 2 q^4-8 q^2+3\right) s^3 +\left( 7 q^2+5\right) \pi ^5 q^2 s^2 \\&\qquad +\pi ^6 q^4 \left( 9-2 q^2\right) s+3 \pi ^7 q^6\bigr \} \left( \pi ^2 q^2+s^2+\pi s\right) ^3\Bigr \} \dot{s}^2 \end{aligned}}{ \begin{aligned}&4 s^2 (s+\pi ) \left( \pi ^2 q^2+s^2+\pi s\right) ^2 \left( \pi ^4 q^4+4 j^2 \pi ^4-3 s^4-4 \pi s^3-\pi ^2 \left( 2 q^2+1\right) s^2\right) \\&\times \left( 4 \pi ^3 (s+\pi ) j^2+\left( \pi ^2 q^2+s^2+\pi s\right) ^2\right) \end{aligned}}=0 \end{aligned}$$

(93)

where $\dot{s}=\frac{\partial s}{\partial \tau }$, $\ddot{s}=\frac{\partial ^2 s}{\partial \tau ^2}$, $\dot{j}=\frac{\partial j}{\partial \tau }$, and $\ddot{j}=\frac{\partial ^2 j}{\partial \tau ^2}$ with $\tau$ being the affine parameter. Similar notation is used for the rest of the appendix.

Geodesic equations for grand canonical ensemble (in s vs $\omega$ space):

$$\begin{aligned} \ddot{s}-\frac{ \begin{aligned}&4 \Bigl \{ -9 \left( \omega ^2-1\right) ^4 s^6+\left( \omega ^2-1\right) ^3 \bigl \{ 4 \left( 3 \omega ^2+1\right) \phi ^2-10 \omega ^2+47\bigr \} \pi s^5+\bigl \{ \left( 2 \omega ^2+1\right) \phi ^4 \\&\qquad +2 \left( 2 \omega ^4-23 \omega ^2-9\right) \phi ^2-2 \omega ^4+47 \omega ^2-100\bigr \} \left( \omega ^2-1\right) ^2 \pi ^2 s^4-5 \pi ^3 \left( \omega ^2-1\right) \bigl \{ \left( 2 \omega ^2+1\right) \phi ^4 \\&\qquad +2 \left( 2 \omega ^4-5 \omega ^2-3\right) \phi ^2-2 \omega ^4+17 \omega ^2-22\bigr \} s^3-\pi ^4 \bigl \{ -\phi ^4\left( 2 \omega ^4+13 \omega ^2+9\right) +\left( -16 \omega ^4+10 \omega ^2+22\right) \phi ^2 \\&\qquad +16 \omega ^4-73 \omega ^2+65\bigr \} s^2+\bigl \{ \left( 8 \omega ^2+7\right) \phi ^4-6 \phi ^2+10 \omega ^2-19\bigr \} \pi ^5 s+2 \left( \phi ^4-1\right) \pi ^6 \Bigr \} (s+\pi )^3 \dot{\omega }^2 s^3 \\&\qquad +4 \Bigl \{ 9 \left( \omega ^2-1\right) ^5 s^7+9 \left( 2 \omega ^2-7\right) \left( \omega ^2-1\right) ^4 \pi s^6+\left( \omega ^2-1\right) ^3 \left( \phi ^4-4 \phi ^2+8 \omega ^4-106 \omega ^2+189\right) \pi ^2 s^5 \\&\qquad -\pi ^3 \left( \omega ^2-1\right) ^2 \left( 9 \phi ^4+4 \left( 4 \omega ^2-3\right) \phi ^2+40 \omega ^4-264 \omega ^2+315\right) s^4-\pi ^4 \left( \omega ^2-1\right) \bigl \{ 2 \left( 7 \omega ^2-13\right) \phi ^4 \\&\qquad +4 \left( 5 \omega ^4-9 \omega ^2+2\right) \phi ^2-85 \omega ^4+356 \omega ^2-315\bigr \} s^3-\pi ^5 \bigl \{ -2 \omega ^6+95 \omega ^4-274 \omega ^2+\left( 8 \omega ^4-50 \omega ^2+34\right) \phi ^4 \\&\qquad +4 \left( 2 \omega ^6-7 \omega ^4+3 \omega ^2+2\right) \phi ^2+189\bigr \} s^2+\bigl \{ \left( 37 \omega ^2-21\right) \phi ^4+4 \left( \omega ^4+2 \omega ^2-3\right) \phi ^2-4 \omega ^4+51 \omega ^2-63\bigr \} \pi ^6 s \\&\qquad +\bigl \{ \left( 12 \omega ^2-5\right) \phi ^4+4 \left( \omega ^2-1\right) \phi ^2+2 \omega ^2-9\bigr \} \pi ^7\Bigr \} (s+\pi ) \omega \dot{s} \dot{\omega } s^3+\bigl \{ 9 \left( \omega ^2-1\right) ^6 s^{10}+9 \left( 3 \omega ^2-8\right) \\&\qquad\times \left( \omega ^2-1\right) ^5 \pi s^9+\left( \omega ^2-1\right) ^4 \bigl \{ \phi ^4-2 \left( \omega ^2+1\right) \phi ^2+28 \omega ^4-190 \omega ^2+253\bigr \} \pi ^2 s^8+\bigl \{ 8 \omega ^6-165 \omega ^4+578 \omega ^2 \\&\qquad +2 \left( \omega ^4-11 \omega ^2+8\right) \phi ^2-\phi ^4 \left( \omega ^2+8\right) -512\bigr \} \left( \omega ^2-1\right) ^3 \pi ^3 s^7-2 \pi ^4 \left( \omega ^2-1\right) ^2 \bigl \{ 20 \omega ^6-210 \omega ^4+497 \omega ^2 \\&\qquad +2 \left( 4 \omega ^2-7\right) \phi ^4+\left( 35 \omega ^4-67 \omega ^2+28\right) \phi ^2 -329\bigr \} s^6-\pi ^5 \left( \omega ^2-1\right) \bigl \{ -93 \omega ^6+605 \omega ^4-1064 \omega ^2 \\&\qquad +\left( 23 \omega ^4-87 \omega ^2+56\right) \phi ^4+2 \left( 26 \omega ^6-121 \omega ^4+151 \omega ^2-56\right) \phi ^2+560\bigr \} s^5-2 \pi ^6 \left( \omega ^2-1\right) \bigl \{ -2 \omega ^6+63 \omega ^4 \\&\qquad -210 \omega ^2+\left( 4 \omega ^4-51 \omega ^2+35\right) \phi ^4 +\left( 8 \omega ^6-61 \omega ^4+123 \omega ^2-70\right) \phi ^2+161\bigr \} s^4 +\bigl \{ \left( 61 \omega ^2-56\right) \phi ^4 \\&\qquad +2 \left( 13 \omega ^4-49 \omega ^2+56\right) \phi ^2-11 \omega ^4+94 \omega ^2-128\bigr \}\left( \omega ^2-1\right) \pi ^7 s^3+\bigl \{ 4 \left( 4 \omega ^4-10 \omega ^2+7\right) \phi ^4 \\&\qquad -2 \left( 9 \omega ^4-33 \omega ^2+28\right) \phi ^2+11 \omega ^4-44 \omega ^2+37\bigr \} \pi ^8 s^2 -\pi ^9 \left( \phi ^2-1\right) ^2 \left( 5 \omega ^2-8\right) s+\left( \phi ^2-1\right) ^2 \pi ^{10}\bigr \} \dot{s}^2 \end{aligned}}{ \begin{aligned}&2 s (s+\pi ) \left( -s \omega ^2+s+\pi \right) \bigl \{ -3 \left( \omega ^2-1\right) ^3 s^5+\left( \omega ^2-1\right) ^2 \left( \phi ^2-4 \omega ^2+11\right) \pi s^4-2 \pi ^2 \left( 2 \phi ^2-5 \omega ^2+7\right) \\&\left( \omega ^2-1\right) s^3 -6 \pi ^3 \left( \phi ^2+1\right) \left( \omega ^2-1\right) s^2-\pi ^4 \left( 4 \phi ^2-1\right) \left( \omega ^2-1\right) s+\left( \phi ^2-1\right) \pi ^5\bigr \} \bigl \{ -3 \left( \omega ^2-1\right) ^2 s^3 \\&\qquad -\pi \left( \omega ^2-1\right) \left( \phi ^2+2 \omega ^2-7\right) s^2 +\left( 2 \phi ^2+3 \omega ^2-5\right) \pi ^2 s+\left( \phi ^2-1\right) \pi ^3\bigr \} \end{aligned}}=0, \end{aligned}$$

(94)

and

$$\begin{aligned} \ddot{\omega }-&\frac{ \begin{aligned}&\omega \Bigl \{ -9 \left( \omega ^2-1\right) ^5 s^7-9 \pi \left( \omega ^2-1\right) ^4 \left( 2 \omega ^2-7\right) s^6-\pi ^2 \left( \omega ^2-1\right) ^3 \left( \phi ^4-4 \phi ^2+8 \omega ^4-106 \omega ^2+189\right) s^5 \\&\qquad +\left( \omega ^2-1\right) ^2 \bigl \{ 9 \phi ^4+4 \left( 4 \omega ^2-3\right) \phi ^2+40 \omega ^4-264 \omega ^2+315\big \} \pi ^3 s^4+\bigl \{ 2 \left( 7 \omega ^2-13\right) \phi ^4 \\&\qquad +4 \left( 5 \omega ^4-9 \omega ^2+2\right) \phi ^2-85 \omega ^4+356 \omega ^2-315\bigr \} \left( \omega ^2-1\right) \pi ^4 s^3+\bigl \{ -2 \omega ^6+95 \omega ^4-274 \omega ^2 \\&\qquad +\left( 8 \omega ^4-50 \omega ^2+34\right) \phi ^4+4 \left( 2 \omega ^6-7 \omega ^4+3 \omega ^2+2\right) \phi ^2+189\bigr \} \pi ^5 s^2+\bigl \{ \left( 21-37 \omega ^2\right) \phi ^4 \\&\qquad -4 \left( \omega ^4+2 \omega ^2-3\right) \phi ^2+4 \omega ^4-51 \omega ^2+63\bigr \} \pi ^6 s+\left( \left( 5-12 \omega ^2\right) \phi ^4-4 \left( \omega ^2-1\right) \phi ^2-2 \omega ^2+9\right) \pi ^7\Bigr \} \dot{s}^2 \end{aligned} }{ \begin{aligned}&\bigl \{ 4 s \left( -s \omega ^2+s+\pi \right) \bigr \} (s+\pi )^2 \bigl \{ 3 \left( \omega ^2-1\right) ^2 s^3+\left( \phi ^2+2 \omega ^2-7\right) \left( \omega ^2-1\right) \pi s^2 \\&\qquad +\left( -2 \phi ^2-3 \omega ^2+5\right) \pi ^2 s-\pi ^3 \left( \phi ^2-1\right) \bigr \} \Bigl \{ \left( \omega ^2-1\right) s^2-\pi \big \{ \left( 2 \omega ^2+1\right) \phi ^2-\omega ^2+2\big \} s-\pi ^2 \left( \phi ^2+1\right) \Bigr \} \end{aligned} } \\&\qquad -\frac{ \begin{aligned}&4 \Bigl \{ 9 \left( \omega ^2-1\right) ^4 s^6-\pi \left( \omega ^2-1\right) ^3 \bigl \{ 4 \left( 3 \omega ^2+1\right) \phi ^2-10 \omega ^2+47\bigr \} s^5-\pi ^2 \left( \omega ^2-1\right) ^2 \bigl \{ \left( 2 \omega ^2+1\right) \phi ^4 \\&\qquad +2 \left( 2 \omega ^4-23 \omega ^2-9\right) \phi ^2-2 \omega ^4+47 \omega ^2-100\bigr \} s^4+5 \bigl \{ \left( 2 \omega ^2+1\right) \phi ^4+2 \left( 2 \omega ^4-5 \omega ^2-3\right) \phi ^2-2 \omega ^4 \\&\qquad +17 \omega ^2-22\bigr \} \left( \omega ^2-1\right) \pi ^3 s^3+\bigl \{ -\phi ^4\left( 2 \omega ^4+13 \omega ^2+9\right) +\left( -16 \omega ^4+10 \omega ^2+22\right) \phi ^2+16 \omega ^4-73 \omega ^2 \\&\qquad +65\bigr \} \pi ^4 s^2 +\bigl \{ -\phi ^4\left( 8 \omega ^2+7\right) +6 \phi ^2-10 \omega ^2+19\bigr \} \pi ^5 s-2 \pi ^6 \left( \phi ^4-1\right) \Bigr \} \dot{s} \dot{\omega } \end{aligned} }{ \begin{aligned}&\bigl \{ 4 s \left( -s \omega ^2+s+\pi \right) \bigr \} \bigl \{ 3 \left( \omega ^2-1\right) ^2 s^3+\left( \phi ^2+2 \omega ^2-7\right) \left( \omega ^2-1\right) \pi s^2 \\&\qquad +\left( -2 \phi ^2-3 \omega ^2+5\right) \pi ^2 s -\pi ^3 \left( \phi ^2-1\right) \bigr \} \left( \left( \omega ^2-1\right) s^2-\pi \left( \left( 2 \omega ^2+1\right) \phi ^2-\omega ^2+2\right) s-\pi ^2 \left( \phi ^2+1\right) \right) \end{aligned} } \\&\qquad +\frac{ \begin{aligned}&4 s^2 \omega \Bigl \{ 3 \left( \omega ^2-1\right) ^3 s^5-\pi \left( \omega ^2-1\right) ^2 \bigl \{ 2 \left( 3 \omega ^2+5\right) \phi ^2-5 \omega ^2+12\bigr \} s^4-\pi ^2 \left( \omega ^2-1\right) \bigl \{ \left( 4 \omega ^2+5\right) \phi ^4 \\&\qquad +\left( 4 \omega ^4+6 \omega ^2-30\right) \phi ^2-2 \omega ^4+15 \omega ^2-18\bigr \} s^3+\bigl \{ \left( 5 \omega ^2+16\right) \phi ^4+\left( -4 \omega ^4+30 \omega ^2-30\right) \phi ^2-4 \omega ^4 \\&\qquad +15 \omega ^2-12\bigr \} \pi ^3 s^2+\bigl \{ \left( 6 \omega ^2+17\right) \phi ^4+2 \left( 4 \omega ^2-5\right) \phi ^2+2 \omega ^2-3\bigr \} \pi ^4 s+6 \pi ^5 \phi ^4\Bigr \} \dot{\omega }^2 \end{aligned} }{ \begin{aligned}&\bigl \{ 4 s \left( -s \omega ^2+s+\pi \right) \bigr \} \bigl \{ -3 \left( \omega ^2-1\right) ^2 s^3-\pi \left( \omega ^2-1\right) \left( \phi ^2+2 \omega ^2-7\right) s^2+\left( 2 \phi ^2+3 \omega ^2-5\right) \pi ^2 s \\&\qquad +\left( \phi ^2-1\right) \pi ^3\bigr \} \bigl \{ -s^2 \left( \omega ^2-1\right) +\left( \left( 2 \omega ^2+1\right) \phi ^2-\omega ^2+2\right) \pi s+\left( \phi ^2+1\right) \pi ^2\bigr \} \end{aligned} } =0. \end{aligned}$$

(95)

Geodesic equations for grand canonical ensemble (in s vs $\phi$ space):

$$\begin{aligned} \ddot{s}-\frac{ \begin{aligned}&8 \Bigl \{ 6 \left( \omega ^2-1\right) ^3 s^4+\left( \phi ^2+2 \omega ^2-19\right) \left( \omega ^2-1\right) ^2 \pi s^3-3 \pi ^2 \left( \omega ^2-1\right) \left( \phi ^2+2 \omega ^2-7\right) s^2+\bigl \{ \left( \omega ^2+3\right) \phi ^2 \\&\qquad +5 \omega ^2-9\bigr \} \pi ^3 s+\left( \phi ^2-1\right) \pi ^4\Bigr \} \pi \left( -s \omega ^2+s+\pi \right) ^2 s^2 \dot{\phi }^2 (s+\pi )^3-8 \pi ^2 s \phi \left( -s \omega ^2+s+\pi \right) \\&\qquad\times \bigl \{ - s^6 \left( \phi ^2-\omega ^2-1\right) \left( \omega ^2-1\right) ^3 +2 \left( 3 \phi ^2+2 \omega ^2-3\right) \left( \omega ^2-1\right) ^2 \pi s^5+\left( \omega ^2-1\right) \bigl \{ \left( 7 \omega ^2-15\right) \phi ^2 \\&\qquad +5 \left( 2 \omega ^4-5 \omega ^2+3\right) \bigr \} \pi ^2 s^4+4 \left( \phi ^2+\omega ^2-1\right) \left( \omega ^4-6 \omega ^2+5\right) \pi ^3 s^3-\pi ^4 \left( \omega ^2-1\right) \left( 15 \phi ^2+7 \omega ^2-15\right) s^2 \\&\qquad -2 \pi ^5 \left( \phi ^2-1\right) \left( 2 \omega ^2-3\right) s+\left( \phi ^2-1\right) \pi ^6\bigr \} \dot{s} \dot{\phi } (s+\pi )+\bigl \{ 9 \left( \omega ^2-1\right) ^6 s^{10}+9 \left( 3 \omega ^2-8\right) \left( \omega ^2-1\right) ^5 \pi s^9 \\&\qquad +\left( \omega ^2-1\right) ^4 \bigl \{ \phi ^4-2 \left( \omega ^2+1\right) \phi ^2+28 \omega ^4-190 \omega ^2+253\bigr \} \pi ^2 s^8+\bigl \{ 8 \omega ^6-165 \omega ^4+578 \omega ^2 \\&\qquad +2 \left( \omega ^4-11 \omega ^2+8\right) \phi ^2-\phi ^4 \left( \omega ^2+8\right) -512\bigr \} \left( \omega ^2-1\right) ^3 \pi ^3 s^7-2 \pi ^4 \left( \omega ^2-1\right) ^2 \bigl \{ 20 \omega ^6-210 \omega ^4 \\&\qquad +497 \omega ^2+2 \left( 4 \omega ^2-7\right) \phi ^4+\left( 35 \omega ^4-67 \omega ^2+28\right) \phi ^2-329\bigr \} s^6-\pi ^5 \left( \omega ^2-1\right) \bigl \{ -93 \omega ^6+605 \omega ^4 \\&\qquad -1064 \omega ^2+\left( 23 \omega ^4-87 \omega ^2+56\right) \phi ^4+2 \left( 26 \omega ^6-121 \omega ^4+151 \omega ^2-56\right) \phi ^2+560\bigr \} s^5-2 \pi ^6 \left( \omega ^2-1\right) \\&\qquad\times \bigl \{ -2 \omega ^6+63 \omega ^4-210 \omega ^2+\left( 4 \omega ^4-51 \omega ^2+35\right) \phi ^4+\left( 8 \omega ^6-61 \omega ^4+123 \omega ^2-70\right) \phi ^2+161\bigr \} s^4 \\&\qquad +\bigl \{ \left( 61 \omega ^2-56\right) \phi ^4+2 \left( 13 \omega ^4-49 \omega ^2+56\right) \phi ^2-11 \omega ^4+94 \omega ^2-128\bigr \} \left( \omega ^2-1\right) \pi ^7 s^3 \\&\qquad +\bigl \{ 4 \left( 4 \omega ^4-10 \omega ^2+7\right) \phi ^4-2 \left( 9 \omega ^4-33 \omega ^2+28\right) \phi ^2+11 \omega ^4-44 \omega ^2+37\bigr \} \pi ^8 s^2-\pi ^9 \left( \phi ^2-1\right) ^2 \left( 5 \omega ^2-8\right) s \\&\qquad +\left( \phi ^2-1\right) ^2 \pi ^{10}\bigr \} \dot{s}^2 \end{aligned} }{ \begin{aligned}&2 s (s+\pi ) \left( -s \omega ^2+s+\pi \right) \bigl \{ -3 \left( \omega ^2-1\right) ^3 s^5+\left( \omega ^2-1\right) ^2 \left( \phi ^2-4 \omega ^2+11\right) \pi s^4-2 \pi ^2 \left( 2 \phi ^2-5 \omega ^2+7\right) \\&\times \left( \omega ^2-1\right) s^3 -6 \pi ^3 \left( \phi ^2+1\right) \left( \omega ^2-1\right) s^2-\pi ^4 \left( 4 \phi ^2-1\right) \left( \omega ^2-1\right) s+\left( \phi ^2-1\right) \pi ^5\bigr \} \bigl \{ -3 \left( \omega ^2-1\right) ^2 s^3 \\&\qquad -\pi \left( \omega ^2-1\right) \left( \phi ^2+2 \omega ^2-7\right) s^2 +\left( 2 \phi ^2+3 \omega ^2-5\right) \pi ^2 s+\left( \phi ^2-1\right) \pi ^3\bigr \} \end{aligned} }=0, \end{aligned}$$

(96)

and

$$\begin{aligned} \begin{aligned} \ddot{\phi }&-\frac{ \begin{aligned}&\dot{s} \dot{\phi } \Bigl \{ 6 s^4 \left( \omega ^2-1\right) ^3+\pi s^3 \left( \omega ^2-1\right) ^2 \left( 2 \omega ^2+\phi ^2-19\right) -3 \pi ^2 s^2 \left( \omega ^2-1\right) \left( 2 \omega ^2+\phi ^2-7\right) +\pi ^3 s \bigl \{ 5 \omega ^2 \\&\qquad +\left( \omega ^2+3\right) \phi ^2-9\bigr \} +\pi ^4 \left( \phi ^2-1\right) \Bigr \} \end{aligned} }{s \left( -s \omega ^2+s+\pi \right) \bigl \{ 3 s^3 \left( \omega ^2-1\right) ^2+\pi s^2 \left( \omega ^2-1\right) \left( 2 \omega ^2+\phi ^2-7\right) +\pi ^2 s \left( -3 \omega ^2-2 \phi ^2+5\right) -\pi ^3 \left( \phi ^2-1\right) \bigr \} } \\ \\&\qquad +\frac{ \begin{aligned}&\pi \dot{s}^2 \phi \bigl \{ -s^6 \left( \omega ^2-1\right) ^3 \left( -\omega ^2+\phi ^2-1\right) +2 \pi s^5 \left( \omega ^2-1\right) ^2 \left( 2 \omega ^2+3 \phi ^2-3\right) +\pi ^2 s^4 \left( \omega ^2-1\right) \\&\times \bigl \{ 5 \left( 2 \omega ^4-5 \omega ^2+3\right) +\left( 7 \omega ^2-15\right) \phi ^2\bigr \} +4 \pi ^3 s^3 \left( \omega ^4-6 \omega ^2+5\right) \left( \omega ^2+\phi ^2-1\right) -\pi ^4 s^2 \left( \omega ^2-1\right) \\&\times \left( 7 \omega ^2+15 \phi ^2-15\right) -2 \pi ^5 s \left( 2 \omega ^2-3\right) \left( \phi ^2-1\right) +\pi ^6 \left( \phi ^2-1\right) \bigr \} \end{aligned} }{ \begin{aligned}&4 s^2 (s+\pi )^2 \left( -s \omega ^2+s+\pi \right) ^2 \bigl \{ 3 s^3 \left( \omega ^2-1\right) ^2+\pi s^2 \left( \omega ^2-1\right) \left( 2 \omega ^2+\phi ^2-7\right) \\&\qquad +\pi ^2 s \left( -3 \omega ^2-2 \phi ^2+5\right) -\pi ^3 \left( \phi ^2-1\right) \bigr \} \end{aligned} } \\ \\&\qquad +\frac{\pi \phi \dot{\phi }^2 \left( s^2 \omega ^2-(s+\pi )^2\right) }{(3 s+2 \pi ) s^2 \omega ^4+s \omega ^2 \left( \pi s \phi ^2-3 (s+\pi ) (2 s+\pi )\right) -(s+\pi )^2 \left( \pi \left( \phi ^2-1\right) -3 s\right) } =0. \end{aligned} \end{aligned}$$

(97)

Geodesic equations for fixed charged ensemble (in s vs $\omega$ space):

$$\begin{aligned} \ddot{s}- \frac{ \begin{aligned}&-4 (s+\pi )^2 \Bigl \{ 9 \left( \omega ^2-1\right) ^2 s^7+\left( 28 \omega ^4-75 \omega ^2+47\right) \pi s^6+\bigl \{31 \omega ^4-123 \omega ^2+2 \left( \omega ^2-1\right) ^2 q^2+100\bigr \} \pi ^2 s^5 \\&+\bigl \{ 14 \omega ^4-99 \omega ^2+2 \left( 2 \omega ^4-7 \omega ^2+5\right) q^2+110\bigr \} \pi ^3 s^4+\bigl \{ \left( \omega ^2-1\right) ^2 q^4+2 \left( \omega ^4-10 \omega ^2+9\right) q^2+2 \omega ^4 \\&-39 \omega ^2+65\bigr \} \pi ^4 s^3+\left( -3 \left( \omega ^2-1\right) q^4-14 \left( \omega ^2-1\right) q^2-6 \omega ^2+19\right) \pi ^5 s^2+\bigl \{ -3 \left( \omega ^2-1\right) q^4 \\&-4 \left( \omega ^2-1\right) q^2+2\bigr \} \pi ^6 s+\pi ^7 q^4 \left( 1-2 \omega ^2\right) \Bigr \} \dot{\omega }^2 s^4-4 \omega \Bigl \{ -9 \left( \omega ^2-1\right) ^3 s^{10}-9 \pi \left( \omega ^2-1\right) ^2 \left( 4 \omega ^2-7\right) s^9 \\& -\pi ^2 \left( \omega ^2-1\right) \bigl \{ 53 \omega ^4-214 \omega ^2+6 \left( \omega ^2-1\right) ^2 q^2+189\bigr \} s^8+\bigl \{ -34 \omega ^6+261 \omega ^4-534 \omega ^2 \\& -2 q^2 \left( \omega ^2-1\right) ^2 \left( 4 \omega ^2-13\right) +315\bigr \} \pi ^3 s^7+\bigl \{ -8 \omega ^6+126 \omega ^4-401 \omega ^2+3 \left( \omega ^2-1\right) ^3 q^4 \\& -2 q^2 \left( \omega ^6-15 \omega ^4+39 \omega ^2-25\right) +315\bigr \} \pi ^4 s^6+\bigl \{ -9 \left( \omega ^2-1\right) ^2 q^4+\left( 6 \omega ^4-60 \omega ^2+62\right) q^2+24 \omega ^4 \\&-166 \omega ^2+189\bigr \} \pi ^5 s^5+\bigl \{ 4 \left( \omega ^2-1\right) q^4+\left( 58-42 \omega ^2\right) q^2-33 \omega ^2+63\bigr \} \pi ^6 s^4+\bigl \{ -14 \left( \omega ^2-1\right) q^4 \\& +\left( 38-24 \omega ^2\right) q^2-2 \omega ^2+9\bigr \} \pi ^7 s^3+\bigl \{ \left( 21-13 \omega ^2\right) q^2-6 \omega ^2+14\bigr \} \pi ^8 q^2 s^2+\bigl \{ \left( 11-4 \omega ^2\right) q^2+2\bigr \} \pi ^9 q^2 s \\& +2 \pi ^{10} q^4\Bigr \} \dot{s} \dot{\omega } s^2+A \end{aligned} }{\begin{aligned}&2 s \left( -s \omega ^2+s+\pi \right) \Bigl \{ 3 \left( \omega ^2-1\right) s^4+\left( 5 \omega ^2-7\right) \pi s^3+\bigl \{ -q^2\left( \omega ^2-1\right) +2 \omega ^2-5\bigr \} \pi ^2 s^2+\left( 2 q^2-1\right) \pi ^3 s \\&\qquad +\pi ^4 q^2\Bigr \} \Bigl \{ 3 \left( \omega ^2-1\right) ^2 s^6+\left( 7 \omega ^4-18 \omega ^2+11\right) \pi s^5+\bigl \{ 3 \left( \omega ^2-1\right) q^2+4 \omega ^2-14\bigr \} \left( \omega ^2-1\right) \pi ^2 s^4 \\&\qquad -6 \pi ^3 \left( 2 q^2+1\right) \left( \omega ^2-1\right) s^3-\pi ^4 \bigl \{ 2 \left( 5 \omega ^2-9\right) q^2+1\bigr \} s^2+3 \pi ^6 q^2-\pi ^5 \left( 4 \left( \omega ^2-3\right) s q^2+s\right) \Bigr \} \end{aligned} } =0, \end{aligned}$$

(98)

and

$$\begin{aligned} \ddot{\omega }-\frac{-4 \Bigl \{ 3 \left( \omega ^2-1\right) s^4+\left( 5 \omega ^2-6\right) \pi s^3+\bigl \{ -q^2\left( \omega ^2-1\right) +2 \omega ^2-3\bigr \} \pi ^2 s^2+3 \pi ^3 q^2 s+2 \pi ^4 q^2\Bigr \} s \omega \dot{\omega }^2+B-C}{\begin{aligned}&4 \left( -s \omega ^2+s+\pi \right) \Bigl \{ 3 \left( \omega ^2-1\right) s^4+\left( 5 \omega ^2-7\right) \pi s^3+\bigl \{ -q^2\left( \omega ^2-1\right) \\&\qquad +2 \omega ^2-5\bigr \} \pi ^2 s^2+\left( 2 q^2-1\right) \pi ^3 s+\pi ^4 q^2\Bigr \} \end{aligned}}=0, \end{aligned}$$

(99)

where

$$\begin{aligned} A&= {} \frac{ \begin{aligned}&\Bigl \{ 9 \left( \omega ^2-1\right) ^4 s^{12}+9 \left( 5 \omega ^2-8\right) \left( \omega ^2-1\right) ^3 \pi s^{11}-\pi ^2 \left( \omega ^2-1\right) ^2 \bigl \{ -91 \omega ^4+316 \omega ^2+6 \left( \omega ^2-1\right) ^2 q^2 \\&\quad -253\bigr \} s^{10}+\bigl \{ 91 \omega ^6-545 \omega ^4+958 \omega ^2+2 \left( \omega ^2-1\right) ^2 \left( \omega ^2+28\right) q^2-512\bigr \} \left( \omega ^2-1\right) \pi ^3 s^9 \\&\quad +\bigl \{ 44 \omega ^6-406 \omega ^4+980 \omega ^2+9 \left( \omega ^2-1\right) ^3 q^4+2 \left( 5 \omega ^6+46 \omega ^4-167 \omega ^2+116\right) q^2-658\bigr \} \left( \omega ^2-1\right) \pi ^4 s^8 \\&\quad -\pi ^5 \left( \omega ^2-1\right) \bigl \{ -8 \omega ^6+157 \omega ^4-616 \omega ^2+3 \left( \omega ^2-1\right) ^2 \left( \omega ^2+24\right) q^4-2 q^2 \left( \omega ^6+82 \omega ^4-311 \omega ^2+280\right) \\&\quad +560\bigr \} s^7-2 \pi ^6 \bigl \{ 12 \omega ^6-126 \omega ^4+273 \omega ^2+14 \left( 2 \omega ^2-9\right) \left( \omega ^2-1\right) ^2 q^4+\left( -87 \omega ^6+456 \omega ^4-787 \omega ^2+434\right) q^2 \\&\quad -161\bigr \} s^6+\bigl \{ \left( -37 \omega ^6+386 \omega ^4-853 \omega ^2+504\right) q^4+2 \left( 42 \omega ^6-303 \omega ^4+665 \omega ^2-448\right) q^2+45 \omega ^4-162 \omega ^2 \\&\quad +128\bigr \} \pi ^7 s^5+\bigl \{ \left( -8 \omega ^6+254 \omega ^4-812 \omega ^2+630\right) q^4+2 \left( 8 \omega ^6-105 \omega ^4+345 \omega ^2-308\right) q^2+4 \omega ^4-30 \omega ^2 \\&\quad +37\bigr \} \pi ^8 s^4+\bigl \{ \left( 95 \omega ^4-479 \omega ^2+504\right) q^4+\left( -30 \omega ^4+202 \omega ^2-272\right) q^2-3 \omega ^2+8\bigr \} \pi ^9 s^3 \\&\quad +\bigl \{ 4 \left( 4 \omega ^4-40 \omega ^2+63\right) q^4+\left( 26 \omega ^2-70\right) q^2+1\bigr \} \pi ^{10} s^2+\left( q^2 \left( 72-23 \omega ^2\right) -8\right) \pi ^{11} q^2 s+9 \pi ^{12} q^4\Bigr \} \dot{s}^2 \end{aligned} }{s+\pi }, \end{aligned}$$

(100)

$$\begin{aligned} B&= {} \frac{ \begin{aligned}&\omega \Bigl \{ -9 \left( \omega ^2-1\right) ^3 s^{10}-9 \pi \left( \omega ^2-1\right) ^2 \left( 4 \omega ^2-7\right) s^9-\pi ^2 \left( \omega ^2-1\right) \bigl \{ 53 \omega ^4-214 \omega ^2+6 \left( \omega ^2-1\right) ^2 q^2+189\bigr \} s^8 \\&\quad +\bigl \{ -34 \omega ^6+261 \omega ^4-534 \omega ^2-2 q^2 \left( \omega ^2-1\right) ^2 \left( 4 \omega ^2-13\right) +315\bigr \} \pi ^3 s^7+\bigl \{ -8 \omega ^6+126 \omega ^4-401 \omega ^2 \\&\quad +3 \left( \omega ^2-1\right) ^3 q^4-2 q^2 \left( \omega ^6-15 \omega ^4+39 \omega ^2-25\right) +315\bigr \} \pi ^4 s^6+\bigl \{ -9 \left( \omega ^2-1\right) ^2 q^4+\left( 6 \omega ^4-60 \omega ^2+62\right) q^2 \\&\quad +24 \omega ^4-166 \omega ^2+189\bigr \} \pi ^5 s^5+\bigl \{ 4 \left( \omega ^2-1\right) q^4+\left( 58-42 \omega ^2\right) q^2-33 \omega ^2+63\bigr \} \pi ^6 s^4+\bigl \{ -14 \left( \omega ^2-1\right) q^4 \\&\quad +\left( 38-24 \omega ^2\right) q^2-2 \omega ^2+9\bigr \} \pi ^7 s^3+\bigl \{ \left( 21-13 \omega ^2\right) q^2-6 \omega ^2+14\bigr \} \pi ^8 q^2 s^2+\bigl \{\left( 11-4 \omega ^2\right) q^2+2\bigr \} \pi ^9 q^2 s \\&\quad +2 \pi ^{10} q^4\Bigr \} \dot{s}^2 \end{aligned} }{s^2 (s+\pi )^3 \left( \pi ^2 q^2+s^2+\pi s\right) }, \end{aligned}$$

(101)

and

$$\begin{aligned} C=\frac{ \begin{aligned}&4 \Bigl \{ -9 \left( \omega ^2-1\right) ^2 s^7+\left( -28 \omega ^4+75 \omega ^2-47\right) \pi s^6-\pi ^2 \bigl \{ 31 \omega ^4-123 \omega ^2+2 \left( \omega ^2-1\right) ^2 q^2+100\bigr \} s^5 \\&\qquad -\pi ^3 \bigl \{ 14 \omega ^4-99 \omega ^2+2 \left( 2 \omega ^4-7 \omega ^2+5\right) q^2+110\bigr \} s^4-\pi ^4 \bigl \{ \left( \omega ^2-1\right) ^2 q^4+2 \left( \omega ^4-10 \omega ^2+9\right) q^2 \\&\qquad +2 \omega ^4-39 \omega ^2+65\bigr \} s^3+\bigl \{ 3 \left( \omega ^2-1\right) q^4+14 \left( \omega ^2-1\right) q^2+6 \omega ^2-19\bigr \} \pi ^5 s^2+\bigl \{ 3 \left( \omega ^2-1\right) q^4 \\&\qquad +4 \left( \omega ^2-1\right) q^2-2\bigr \} \pi ^6 s+\left( 2 \omega ^2-1\right) \pi ^7 q^4\Bigr \} \dot{s} \dot{\omega } \end{aligned} }{(s+\pi ) \left( \pi ^2 q^2+s^2+\pi s\right) } \end{aligned}$$

(102)

Geodesic equations for fixed charged ensemble (in s vs q space):

$$\begin{aligned} \ddot{s}-\frac{ \begin{aligned}&8 \pi ^2 (s+\pi ) \left( -s \omega ^2+s+\pi \right) ^2 \Bigl \{ -3 \left( \omega ^2-1\right) s^5-9 \pi \left( \omega ^2-1\right) s^4+\bigl \{ - q^2 \left( \omega ^2-1\right) -8 \omega ^2+9\bigr \} \pi ^2 s^3 \\&\qquad +\bigl \{ \left( \omega ^2+3\right) q^2-2 \omega ^2+3\bigr \} \pi ^3 s^2+3 \pi ^4 q^2 s+\pi ^5 q^2\Bigr \} s \dot{q}^2-8 \pi ^2 q \Bigl \{ 3 \left( \omega ^2-1\right) ^3 s^8+4 \left( \omega ^2-5\right) \left( \omega ^2-1\right) ^2 \pi s^7 \\&\qquad -\pi ^2 \left( \omega ^2-1\right) \bigl \{ -\omega ^4+38 \omega ^2+3 \left( \omega ^2-1\right) ^2 q^2-57\bigr \} s^6+2 \bigl \{ -17 \omega ^4+58 \omega ^2+9 \left( \omega ^2-1\right) ^2 q^2-45\bigr \} \pi ^3 s^5\\&\qquad +\bigl \{-18 \omega ^4+83 \omega ^2+\left( 13 \omega ^4-58 \omega ^2+45\right) q^2-85\bigr \}) \pi ^4 s^4+4 \bigl \{ -\omega ^4+8 \omega ^2+\left( \omega ^4-12 \omega ^2+15\right) q^2 \\&\qquad -12\bigr \} \pi ^5 s^3+\bigl \{ \left( 45-21 \omega ^2\right) q^2+5 \left( \omega ^2-3\right) \bigr \} \pi ^6 s^2+3 \pi ^8 q^2-2 \pi ^7 \left( \left( 2 \omega ^2-9\right) s q^2+s\right) \Bigr \} \dot{s} \dot{q}+D \end{aligned} }{\begin{aligned}&2 \Bigl \{ 3 \left( \omega ^2-1\right) s^4+\left( 5 \omega ^2-7\right) \pi s^3+\bigl \{ -q^2\left( \omega ^2-1\right) +2 \omega ^2-5\bigr \} \pi ^2 s^2+\left( 2 q^2-1\right) \pi ^3 s+\pi ^4 q^2\Bigr \} \\&\times \Bigl \{ 3 \left( \omega ^2-1\right) ^2 s^6+\left( 7 \omega ^4-18 \omega ^2+11\right) \pi s^5+\bigl \{ 3 \left( \omega ^2-1\right) q^2+4 \omega ^2-14\bigr \}) \left( \omega ^2-1\right) \pi ^2 s^4 \\&\qquad -6 \pi ^3 \left( 2 q^2+1\right) \left( \omega ^2-1\right) s^3-\pi ^4 \bigl \{ 2 \left( 5 \omega ^2-9\right) q^2+1\bigr \} s^2+3 \pi ^6 q^2-\pi ^5 \bigl \{ 4 \left( \omega ^2-3\right) s q^2+s\bigr \} \Bigr \} \end{aligned} }=0, \end{aligned}$$

(103)

and

$$\begin{aligned} \ddot{q}+\frac{\pi ^2 q \dot{q}^2 \Bigl \{ \frac{s^2 \omega ^2}{(s+\pi )^2}-1\Bigr \}}{\frac{s^2 \omega ^2 \bigl \{ \pi ^2 \left( q^2-2\right) -3 s^2-5 \pi s\bigr \}}{(s+\pi )^2}-\pi ^2 q^2+3 s^2+\pi s}-E+F =0 \end{aligned}$$

(104)

where

$$\begin{aligned} D&= {} \frac{ \begin{aligned}&\Bigl \{ 9 \left( \omega ^2-1\right) ^4 s^{12}+9 \left( 5 \omega ^2-8\right) \left( \omega ^2-1\right) ^3 \pi s^{11}-\pi ^2 \left( \omega ^2-1\right) ^2 \bigl \{ -91 \omega ^4+316 \omega ^2+6 \left( \omega ^2-1\right) ^2 q^2 \\&\qquad -253\bigr \} s^{10}+\bigl \{ 91 \omega ^6-545 \omega ^4+958 \omega ^2+2 \left( \omega ^2-1\right) ^2 \left( \omega ^2+28\right) q^2-512\bigr \} \left( \omega ^2-1\right) \pi ^3 s^9 \\&\qquad +\bigl \{ 44 \omega ^6-406 \omega ^4+980 \omega ^2+9 \left( \omega ^2-1\right) ^3 q^4+2 \left( 5 \omega ^6+46 \omega ^4-167 \omega ^2+116\right) q^2-658\bigr \} \left( \omega ^2-1\right) \pi ^4 s^8 \\&\qquad -\pi ^5 \left( \omega ^2-1\right) \bigl \{ -8 \omega ^6+157 \omega ^4-616 \omega ^2+3 \left( \omega ^2-1\right) ^2 \left( \omega ^2+24\right) q^4-2 q^2 \left( \omega ^6+82 \omega ^4-311 \omega ^2+280\right) \\&\qquad +560\bigr \} s^7-2 \pi ^6 \bigl \{ 12 \omega ^6-126 \omega ^4+273 \omega ^2+14 \left( 2 \omega ^2-9\right) \left( \omega ^2-1\right) ^2 q^4+\big ( -87 \omega ^6+456 \omega ^4-787 \omega ^2 \\&\qquad +434\bigr ) q^2-161\bigr \} s^6+\bigl \{ \left( -37 \omega ^6+386 \omega ^4-853 \omega ^2+504\right) q^4+2 \left( 42 \omega ^6-303 \omega ^4+665 \omega ^2-448\right) q^2 \\&\qquad +45 \omega ^4 -162 \omega ^2+128\bigr \} \pi ^7 s^5+\bigl \{ \left( -8 \omega ^6+254 \omega ^4-812 \omega ^2+630\right) q^4+2 \left( 8 \omega ^6-105 \omega ^4+345 \omega ^2-308\right) q^2 \\&\qquad +4 \omega ^4-30 \omega ^2+37\bigr \} \pi ^8 s^4+\bigl \{ \left( 95 \omega ^4-479 \omega ^2+504\right) q^4+\left( -30 \omega ^4+202 \omega ^2-272\right) q^2-3 \omega ^2+8\bigr \} \pi ^9 s^3 \\&\qquad +\bigl \{ 4 \left( 4 \omega ^4-40 \omega ^2+63\right) q^4+\left( 26 \omega ^2-70\right) q^2+1\bigr \} \pi ^{10} s^2+\left( q^2 \left( 72-23 \omega ^2\right) -8\right) \pi ^{11} q^2 s+9 \pi ^{12} q^4\Bigr \} \dot{s}^2 \end{aligned} }{s (s+\pi ) \left( -s \omega ^2+s+\pi \right) }, \end{aligned}$$

(105)

$$\begin{aligned} E&= {} \frac{ \begin{aligned}&\dot{q} \dot{s} \bigl \{ \pi ^2 s^3 \bigl \{ -q^2 \left( \omega ^2-1\right) -8 \omega ^2+9\bigr \} +\pi ^3 s^2 \bigl \{ q^2 \left( \omega ^2+3\right) -2 \omega ^2+3\bigr \} +3 \pi ^4 q^2 s \\&\qquad +\pi ^5 q^2-3 s^5 \left( \omega ^2-1\right) -9 \pi s^4 \left( \omega ^2-1\right) \bigr \} \end{aligned} }{s (s+\pi ) \left( \pi ^2 s^2 \bigl \{ -q^2 \left( \omega ^2-1\right) +2 \omega ^2-5\bigr \} +\pi ^3 \left( 2 q^2-1\right) s+\pi ^4 q^2+3 s^4 \left( \omega ^2-1\right) +\pi s^3 \left( 5 \omega ^2-7\right) \right) }, \end{aligned}$$

(106)

and

$$\begin{aligned} F=\frac{ \begin{aligned}&q \dot{s}^2 \Bigl \{ -\pi ^2 s^6 \left( \omega ^2-1\right) \bigl \{ 3 q^2 \left( \omega ^2-1\right) ^2-\omega ^4+38 \omega ^2-57\bigr \} +2 \pi ^3 s^5 \bigl \{ 9 q^2 \left( \omega ^2-1\right) ^2-17 \omega ^4+58 \omega ^2-45\bigr \} \\&\qquad +\pi ^4 s^4 \bigl \{ q^2 \left( 13 \omega ^4-58 \omega ^2+45\right) -18 \omega ^4+83 \omega ^2-85\bigr \} +4 \pi ^5 s^3 \bigl \{ q^2 \left( \omega ^4-12 \omega ^2+15\right) -\omega ^4+8 \omega ^2-12\bigr \} \\&\qquad +\pi ^6 s^2 \bigl \{ q^2 \left( 45-21 \omega ^2\right) +5 \left( \omega ^2-3\right) \bigr \} -2 \pi ^7 \bigl \{ q^2 s \left( 2 \omega ^2-9\right) +s\bigr \} +3 \pi ^8 q^2+3 s^8 \left( \omega ^2-1\right) ^3 \\&\qquad +4 \pi s^7 \left( \omega ^2-5\right) \left( \omega ^2-1\right) ^2\Bigr \} \end{aligned}}{ \begin{aligned}&4 s^2 (s+\pi )^2 \left( -s \omega ^2+s+\pi \right) ^2 \Bigl \{ \pi ^2 s^2 \bigl \{ -q^2 \left( \omega ^2-1\right) +2 \omega ^2-5\bigr \} +\pi ^3 \left( 2 q^2-1\right) s+\pi ^4 q^2+3 s^4 \left( \omega ^2-1\right) \\&\qquad +\pi s^3 \left( 5 \omega ^2-7\right) \Bigr \} \end{aligned}} \end{aligned}$$

(107)

1.2 2. Kerr-AdS black hole

Geodesic equations for canonical ensemble:

$$\begin{aligned} \ddot{s}+\frac{ \begin{aligned}&16 \pi ^4 \dot{j}^2 (s+\pi )^3 s^4 \bigl \{ -12 \pi ^2 J^2 s^2 \left( 6 s^3+13 \pi s^2+9 \pi ^2 s+2 \pi ^3\right) +16 \pi ^6 J^4 (2 s+\pi )+(s+\pi )^3 s^4\bigr \} \\&\qquad -16 \pi ^3 \dot{j} (s+\pi ) \dot{s} s \bigl \{ -48 \pi ^8 J^5 s^2 \left( 4 s^2+7 \pi s+3 \pi ^2\right) -4 \pi ^3 J^3 s^4 (s+\pi )^2 \big ( 36 s^3+63 \pi s^2 \\&\qquad +44 \pi ^2 s+13 \pi ^3\big ) -64 \pi ^{11} J^7 (4 s+3 \pi )+9 J (s+\pi )^5 (2 s+\pi ) s^6\bigr \} +\dot{s}^2 \bigl \{ -256 \pi ^{14} J^8 \big ( 16 s^2 \\&\qquad +23 \pi s+9 \pi ^2\big ) -192 \pi ^{11} J^6 s^2 \left( 15 s^3+36 \pi s^2+29 \pi ^2 s+8 \pi ^3\right) -16 \pi ^6 J^4 s^4 (s+\pi )^2 \big ( 144 s^4 \\&\qquad +333 \pi s^3 +292 \pi ^2 s^2+119 \pi ^3 s+22 \pi ^4\big ) +12 \pi ^3 J^2 (s+\pi )^5 \left( 3 s^2+21 \pi s+8 \pi ^2\right) s^6 \\&\qquad -s^8 (s+\pi )^6 \left( 9 s^2+\pi ^2\right) \bigr \} \end{aligned} }{ \begin{aligned}&2 s (s+\pi ) \bigl \{ 4 \pi ^3 J^2+(s+\pi ) s^2\bigr \} \bigl \{ 24 \pi ^3 J^2 (s+\pi )^2 (2 s+\pi ) s^2+16 \pi ^7 J^4 (4 s+3 \pi ) \\&\qquad -(\pi -3 s) s^4 (s+\pi )^3\bigr \} \bigl \{ 4 \pi ^4 J^2-s^2 \left( 3 s^2+4 \pi s+\pi ^2\right) \bigr \} \end{aligned} }=0, \end{aligned}$$

(108)

and

$$\begin{aligned} \ddot{j}-\frac{ \begin{aligned}&64 \pi ^7 \dot{j} J^4 (s+\pi )^2 (2 s+\pi ) s^3 \dot{s}-4 \pi ^3 J^3 s^4 (s+\pi )^2 \bigl \{ 16 \pi ^4 \dot{j}^2 (s+\pi )-\left( 36 s^3+63 \pi s^2+44 \pi ^2 s+13 \pi ^3\right) \dot{s}^2\bigr \} \\&\qquad -48 \pi ^3 \dot{j} J^2 s^5 (s+\pi )^3 \left( 6 s^2+7 \pi s+2 \pi ^2\right) \dot{s}-3 J s^6 (s+\pi )^4 (2 s+\pi ) \bigl \{ 3 (s+\pi ) \dot{s}^2-16 \pi ^3 \dot{j}^2\bigr \} \\&\qquad +4 \pi \dot{j} (s+\pi )^5 s^7 \dot{s}+48 \pi ^8 J^5 \left( 4 s^2+7 \pi s+3 \pi ^2\right) s^2 \dot{s}^2+64 \pi ^{11} J^7 (4 s+3 \pi ) \dot{s}^2 \end{aligned} }{4 s^4 (s+\pi )^3 \left( 4 \pi ^3 J^2+(s+\pi ) s^2\right) \left( s^2 \left( 3 s^2+4 \pi s+\pi ^2\right) -4 \pi ^4 J^2\right) }=0. \end{aligned}$$

(109)

Geodesic equations for grand canonical ensemble:

$$\begin{aligned} \ddot{s}+\frac{ \begin{aligned}&4 (s+\pi )^3 s^3 \dot{\omega }^2 \bigl \{ 9 s^4 \left( \omega ^2-1\right) ^2+\pi s^3 \left( 10 \omega ^4-39 \omega ^2+29\right) +\pi ^2 s^2 \left( 2 \omega ^4-27 \omega ^2+33\right) +3 \pi ^3 s \left( 5-2 \omega ^2\right) \\&\qquad +2 \pi ^4\bigr \} -4 \dot{s} s^3 \omega \dot{\omega } \bigl \{ 9 s^6 \left( \omega ^2-1\right) ^3+27 \pi s^5 \left( \omega ^2-2\right) \left( \omega ^2-1\right) ^2 +\pi ^2 s^4 \left( 26 \omega ^6-159 \omega ^4+268 \omega ^2-135\right) \\&\qquad +2 \pi ^3 s^3 \left( 4 \omega ^6-51 \omega ^4+133 \omega ^2-90\right) -3 \pi ^4 s^2 \left( 8 \omega ^4-45 \omega ^2+45\right) +\pi ^5 s \left( 31 \omega ^2-54\right) +\pi ^6 \left( 2 \omega ^2-9\right) \bigr \} \\&\qquad -\dot{s}^2 \bigl \{ 9 s^8 \left( \omega ^2-1\right) ^4+27 \pi s^7 \left( \omega ^2-2\right) \left( \omega ^2-1\right) ^3+4 \pi ^2 s^6 \left( \omega ^2-1\right) ^2 \left( 7 \omega ^4-34 \omega ^2+34\right) +\pi ^3 s^5 \big ( 8 \omega ^8 \\&\qquad -117 \omega ^6+388 \omega ^4-465 \omega ^2+186\big )-6 \pi ^4 s^4 \left( 4 \omega ^6-29 \omega ^4+50 \omega ^2-25\right) +37 \pi ^5 s^3 \left( \omega ^4-3 \omega ^2+2\right) \\&\qquad +4 \pi ^6 s^2 \left( \omega ^4-6 \omega ^2+6\right) -3 \pi ^7 s \left( \omega ^2-2\right) +\pi ^8\bigr \} \end{aligned} }{ \begin{aligned}&2 s (s+\pi ) \left( -s \omega ^2+s+\pi \right) \bigl \{ 3 s^2 \left( \omega ^2-1\right) +2 \pi s \left( \omega ^2-2\right) -\pi ^2\bigr \} \bigl \{ 3 s^4 \left( \omega ^2-1\right) ^2+4 \pi s^3 \left( \omega ^4-3 \omega ^2+2\right) \\&\qquad -6 \pi ^2 s^2 \left( \omega ^2-1\right) -\pi ^4\bigr \} \end{aligned} }=0, \end{aligned}$$

(110)

and

$$\begin{aligned} \ddot{\omega }+\frac{ \begin{aligned}&-4 (s+\pi )^3 s^3 \omega \dot{\omega }^2 \bigl \{ 3 s \left( \omega ^2-1\right) +\pi \left( 2 \omega ^2-3\right) \bigr \} +\dot{s}^2 \omega \bigl \{ -9 s^5 \left( \omega ^2-1\right) ^3-9 \pi s^4 \left( \omega ^2-1\right) ^2 \left( 2 \omega ^2-5\right) \\&+2 \pi ^2 s^3 \left( -4 \omega ^6+39 \omega ^4-80 \omega ^2+45\right) +2 \pi ^3 s^2 \left( 12 \omega ^4-53 \omega ^2+45\right) +\pi ^4 s \left( 45-29 \omega ^2\right) +\pi ^5 \left( 9-2 \omega ^2\right) \bigr \} \\& +4 (s+\pi )^2 \dot{s} \dot{\omega } \bigl \{ 9 s^4 \left( \omega ^2-1\right) ^2+\pi s^3 \left( 10 \omega ^4-39 \omega ^2+29\right) +\pi ^2 s^2 \left( 2 \omega ^4-27 \omega ^2+33\right) +3 \pi ^3 s \left( 5-2 \omega ^2\right) \\& +2 \pi ^4\bigr \} \end{aligned} }{4 s (s+\pi )^3 \left( -s \omega ^2+s+\pi \right) \left( -3 s^2 \left( \omega ^2-1\right) -2 \pi s \left( \omega ^2-2\right) +\pi ^2\right) }=0. \end{aligned}$$

(111)