polar and time, constant rotation, coordinate
chart coordinates: $x^\tilde{\mu} = \{t, r, \phi\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{t}}, e_{\tilde{r}}, e_{\tilde{\phi}}\}$
embedding coordinates: $u^I = \{t, x, y\}$
embedding basis $e_I = \{e_{t}, e_{x}, e_{y}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_t}^t} = {1}$;
${{{ \tilde{e}}_r}^r} = {1}$;
${{{ \tilde{e}}_{\phi}}^{\phi}} = {1}$
transform from coorinate to basis:
${{{ \tilde{e}}^t}_t} = {1}$;
${{{ \tilde{e}}^r}_r} = {1}$;
${{{ \tilde{e}}^{\phi}}_{\phi}} = {1}$
tensor index associated with coordinate $t$ is index $t$ with operator $e_{t}(\zeta) = $$\frac{\partial \zeta}{\partial t}$
tensor index associated with coordinate $r$ is index $r$ with operator $e_{r}(\zeta) = $$\frac{\partial \zeta}{\partial r}$
tensor index associated with coordinate $\phi$ is index $\phi$ with operator $e_{\phi}(\zeta) = $$\frac{\partial \zeta}{\partial \phi}$
flat metric:
${{{ \eta}_t}_t} = {-1}$;
${{{ \eta}_x}_x} = {1}$;
${{{ \eta}_y}_y} = {1}$
chart in embedded coordinates:
${{ u}^t} = {t}$;
${{ u}^x} = {{{r}} {{cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}$;
${{ u}^y} = {{{r}} {{sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}$
basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_t}^t} = {1}$;
${{{ e}_t}^x} = {-{{{\omega}} \cdot {{r}} {{sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}}$;
${{{ e}_t}^y} = {{{\omega}} \cdot {{r}} {{cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}$;
${{{ e}_r}^x} = {cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}$;
${{{ e}_r}^y} = {sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}$;
${{{ e}_{\phi}}^x} = {-{{{r}} {{sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}}$;
${{{ e}_{\phi}}^y} = {{{r}} {{cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}}}$
${{{ e}^t}_t} = {1}$;
${{{ e}^r}_x} = {cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}$;
${{{ e}^r}_y} = {sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}$;
${{{ e}^{\phi}}_t} = {-{\omega}}$;
${{{ e}^{\phi}}_x} = {{\frac{1}{r}}{({-{sin\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)}})}}$;
${{{ e}^{\phi}}_y} = {{\frac{1}{r}}{({cos\left( {{\phi} + {{{\omega}} \cdot {{t}}}}\right)})}}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right]}}$
basis determinant: ${det(e)} = {r}$
${{{{ c}_a}_b}^c} = {0}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
${{{ g}_t}_t} = {-{({{1} - {{{{\omega}^{2}}} {{{r}^{2}}}}})}}$;
${{{ g}_t}_{\phi}} = {{{\omega}} \cdot {{{r}^{2}}}}$;
${{{ g}_r}_r} = {1}$;
${{{ g}_{\phi}}_t} = {{{\omega}} \cdot {{{r}^{2}}}}$;
${{{ g}_{\phi}}_{\phi}} = {{r}^{2}}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
metric determinant: ${det(g)} = {-{{r}^{2}}}$
${{{{ \Gamma}_a}_b}_c} = {{{\frac{1}{2}}} {{({{{{{{{{ g}_a}_b}_{,c}} + {{{{ g}_a}_c}_{,b}}} - {{{{ g}_b}_c}_{,a}}} + {{{{ c}_a}_b}_c} + {{{{ c}_a}_c}_b}} - {{{{ c}_c}_b}_a}})}}}$
commutation coefficients:
${{{{ c}_a}_b}^c} = {0}$
metric:
${{{ g}_t}_t} = {-{({{1} - {{{{\omega}^{2}}} {{{r}^{2}}}}})}}$;
${{{ g}_t}_{\phi}} = {{{\omega}} \cdot {{{r}^{2}}}}$;
${{{ g}_r}_r} = {1}$;
${{{ g}_{\phi}}_t} = {{{\omega}} \cdot {{{r}^{2}}}}$;
${{{ g}_{\phi}}_{\phi}} = {{r}^{2}}$
metric inverse:
${{{ g}^t}^t} = {-{1}}$;
${{{ g}^t}^{\phi}} = {\omega}$;
${{{ g}^r}^r} = {1}$;
${{{ g}^{\phi}}^t} = {\omega}$;
${{{ g}^{\phi}}^{\phi}} = {\frac{{1} - {{{{\omega}^{2}}} {{{r}^{2}}}}}{{r}^{2}}}$
metric derivative:
${{{{ {\partial g}}_t}_t}_r} = {{{2}} {{r}} {{{\omega}^{2}}}}$;
${{{{ {\partial g}}_t}_{\phi}}_r} = {{{2}} {{\omega}} \cdot {{r}}}$;
${{{{ {\partial g}}_{\phi}}_t}_r} = {{{2}} {{\omega}} \cdot {{r}}}$;
${{{{ {\partial g}}_{\phi}}_{\phi}}_r} = {{{2}} {{r}}}$
1st kind Christoffel:
${{{{ \Gamma}_t}_t}_r} = {{{r}} {{{\omega}^{2}}}}$;
${{{{ \Gamma}_t}_r}_t} = {{{r}} {{{\omega}^{2}}}}$;
${{{{ \Gamma}_t}_r}_{\phi}} = {{{\omega}} \cdot {{r}}}$;
${{{{ \Gamma}_t}_{\phi}}_r} = {{{\omega}} \cdot {{r}}}$;
${{{{ \Gamma}_r}_t}_t} = {-{{{r}} {{{\omega}^{2}}}}}$;
${{{{ \Gamma}_r}_t}_{\phi}} = {-{{{\omega}} \cdot {{r}}}}$;
${{{{ \Gamma}_r}_{\phi}}_t} = {-{{{\omega}} \cdot {{r}}}}$;
${{{{ \Gamma}_r}_{\phi}}_{\phi}} = {-{r}}$;
${{{{ \Gamma}_{\phi}}_t}_r} = {{{\omega}} \cdot {{r}}}$;
${{{{ \Gamma}_{\phi}}_r}_t} = {{{\omega}} \cdot {{r}}}$;
${{{{ \Gamma}_{\phi}}_r}_{\phi}} = {r}$;
${{{{ \Gamma}_{\phi}}_{\phi}}_r} = {r}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^r}_t}_t} = {-{{{r}} {{{\omega}^{2}}}}}$;
${{{{ \Gamma}^r}_t}_{\phi}} = {-{{{\omega}} \cdot {{r}}}}$;
${{{{ \Gamma}^r}_{\phi}}_t} = {-{{{\omega}} \cdot {{r}}}}$;
${{{{ \Gamma}^r}_{\phi}}_{\phi}} = {-{r}}$;
${{{{ \Gamma}^{\phi}}_t}_r} = {{\frac{1}{r}}{({\omega})}}$;
${{{{ \Gamma}^{\phi}}_r}_t} = {{\frac{1}{r}}{({\omega})}}$;
${{{{ \Gamma}^{\phi}}_r}_{\phi}} = {\frac{1}{r}}$;
${{{{ \Gamma}^{\phi}}_{\phi}}_r} = {\frac{1}{r}}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}}^r}_t}_t}_r} = {-{{\omega}^{2}}}$;
${{{{{ {\partial \Gamma}}^r}_t}_{\phi}}_r} = {-{\omega}}$;
${{{{{ {\partial \Gamma}}^r}_{\phi}}_t}_r} = {-{\omega}}$;
${{{{{ {\partial \Gamma}}^r}_{\phi}}_{\phi}}_r} = {-{1}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_t}_r}_r} = {\frac{-{\omega}}{{r}^{2}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_r}_t}_r} = {\frac{-{\omega}}{{r}^{2}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_r}_{\phi}}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_{\phi}}_r}_r} = {-{\frac{1}{{r}^{2}}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}}^r}_t}_t}_r} = {-{{\omega}^{2}}}$;
${{{{{ {(\Gamma^2)}}^r}_t}_{\phi}}_r} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_t}_t} = {-{{\omega}^{2}}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_t}_{\phi}} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\phi}}_t} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\phi}}_{\phi}} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_t}_r} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_{\phi}}_r} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_t}_t}_t} = {-{{\omega}^{3}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_t}_t}_{\phi}} = {-{{\omega}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_t}_r}_r} = {\frac{\omega}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_t}_{\phi}}_t} = {-{{\omega}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_t}_{\phi}}_{\phi}} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_r}_t} = {\frac{\omega}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_r}_{\phi}} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_t}_t} = {-{{\omega}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_t}_{\phi}} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_r}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\phi}}_t} = {-{\omega}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\phi}}_{\phi}} = {-{1}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R}^a}_b}_c}_d} = {0}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R}^a}^b}_c}_d} = {0}$
Ricci curvature, $\sharp\flat$:
${{{ R}^a}_b} = {0}$
Gaussian curvature:
$0$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}}^a}_b} = {0}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G}^a}_b} = {0}$
Schouten, $\sharp\flat$:
${{{ P}^a}_b} = {0}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C}^a}^b}_c}_d} = {0}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C}_a}_b}_c}_d} = {0}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P}^a}^b}_c}_d} = {0}$
divergence: ${{{{ A}^i}_{,i}} + {{{{{{ \Gamma}^i}_i}_j}} {{{ A}^j}}}} = {{\frac{\partial {A^{t}}}{\partial t}} + {\frac{\partial {A^{r}}}{\partial r}} + {\frac{\partial {A^{\phi}}}{\partial \phi}} + {{{{A^{r}}}} \cdot {{\frac{1}{r}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{t} \\ \ddot{r} \\ \ddot{\phi}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} 0 \\ {{{2}} {{r}} {{\dot{\phi}}} \cdot {{\dot{t}}} \cdot {{\omega}}} + {{{r}} {{{\dot{\phi}}^{2}}}} + {{{r}} {{{\dot{t}}^{2}}} {{{\omega}^{2}}}} \\ {{{-2}} {{\dot{r}}} \cdot {{\dot{\phi}}} \cdot {{\frac{1}{r}}}} + {{{-2}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{\omega}} \cdot {{\frac{1}{r}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_t]}} = {\left[\begin{matrix} 0 & 0 & 0 \\ -{{{r}} {{{\omega}^{2}}}} & 0 & -{{{\omega}} \cdot {{r}}} \\ 0 & {\frac{1}{r}}{({\omega})} & 0\end{matrix}\right]}$
$\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{r}} {{{\omega}^{2}}}} & 0 & -{{{\omega}} \cdot {{r}}} \\ 0 & {\frac{1}{r}}{({\omega})} & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & 0 & 0 \\ {{r}} {{{\omega}^{2}}} {{({{{t_L}} - {{t_R}}})}} & 0 & {{\omega}} \cdot {{r}} {{({{{t_L}} - {{t_R}}})}} \\ 0 & {\frac{1}{r}}{({-{{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}})} & 0\end{matrix}\right]$
${ P}_t$ = ${ⅇ}^{( -{({\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{r}} {{{\omega}^{2}}}} & 0 & -{{{\omega}} \cdot {{r}}} \\ 0 & {\frac{1}{r}}{({\omega})} & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ -{{{\omega}} \cdot {{r}} {{sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}}} & cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right) & -{{{r}} {{sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}}} \\ -{{{\omega}} \cdot {{({{1} - {cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}})}}} & {\frac{1}{r}}{({sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)})} & cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)\end{matrix}\right]$
${{ P}_t}^{-1}$ = ${ⅇ}^{({\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{r}} {{{\omega}^{2}}}} & 0 & -{{{\omega}} \cdot {{r}}} \\ 0 & {\frac{1}{r}}{({\omega})} & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ {{\omega}} \cdot {{r}} {{sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}} & cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right) & {{r}} {{sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}} \\ -{{{\omega}} \cdot {{({{1} - {cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}})}}} & {\frac{1}{r}}{({-{sin\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)}})} & cos\left( {{{\omega}} \cdot {{({{{t_L}} - {{t_R}}})}}}\right)\end{matrix}\right]$
${{[\Gamma_r]}} = {\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\frac{1}{r}}{({\omega})} & 0 & \frac{1}{r}\end{matrix}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\frac{1}{r}}{({\omega})} & 0 & \frac{1}{r}\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ log\left( {{{\frac{1}{{{r_L}}^{\omega}}}} {{{{r_R}}^{\omega}}}}\right) & 0 & log\left( {{\frac{1}{{r_L}}}{({{r_R}})}}\right)\end{matrix}\right]$
${ P}_r$ = ${ⅇ}^{( -{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\frac{1}{r}}{({\omega})} & 0 & \frac{1}{r}\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{-{({{1} + {log\left( {{{{{r_L}}^{({\frac{-{{r_L}}}{log\left( {{{{{r_L}}^{log\left( {{{{{r_L}}^{({\frac{{{\omega}} \cdot {{{r_R}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{-{{{\omega}} \cdot {{{r_R}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}} {{{{r_R}}^{log\left( {{{{{r_L}}^{({\frac{-{{{\omega}} \cdot {{{r_R}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{{{\omega}} \cdot {{{r_R}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{{r_R}}^{({\frac{{r_L}}{log\left( {{{{{r_L}}^{log\left( {{{{{r_L}}^{({\frac{{{\omega}} \cdot {{{r_R}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{-{{{\omega}} \cdot {{{r_R}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}} {{{{r_R}}^{log\left( {{{{{r_L}}^{({\frac{-{{{\omega}} \cdot {{{r_R}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{{{\omega}} \cdot {{{r_R}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right)}})}}{log\left( {\frac{{{r_L}}^{({\frac{1}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}{{{r_R}}^{({\frac{1}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}\right)} & 0 & {\frac{1}{{r_R}}}{({{r_L}})}\end{matrix}\right]$
${{ P}_r}^{-1}$ = ${ⅇ}^{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\frac{1}{r}}{({\omega})} & 0 & \frac{1}{r}\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{-{({{1} + {log\left( {{{{{r_L}}^{({\frac{-{{r_R}}}{log\left( {{{{{r_L}}^{log\left( {{{{{r_L}}^{({\frac{{{\omega}} \cdot {{{r_L}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{-{{{\omega}} \cdot {{{r_L}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}} {{{{r_R}}^{log\left( {{{{{r_L}}^{({\frac{-{{{\omega}} \cdot {{{r_L}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{{{\omega}} \cdot {{{r_L}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{{r_R}}^{({\frac{{r_R}}{log\left( {{{{{r_L}}^{log\left( {{{{{r_L}}^{({\frac{{{\omega}} \cdot {{{r_L}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{-{{{\omega}} \cdot {{{r_L}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}} {{{{r_R}}^{log\left( {{{{{r_L}}^{({\frac{-{{{\omega}} \cdot {{{r_L}}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}} {{{{r_R}}^{({\frac{{{\omega}} \cdot {{{r_L}}}}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right)}})}}{log\left( {\frac{{{r_L}}^{({\frac{1}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}{{{r_R}}^{({\frac{1}{log\left( {{{{{r_L}}^{\omega}}} {{\frac{1}{{{r_R}}^{\omega}}}}}\right)}})}}}\right)} & 0 & {\frac{1}{{r_L}}}{({{r_R}})}\end{matrix}\right]$
${{[\Gamma_\phi]}} = {\left[\begin{matrix} 0 & 0 & 0 \\ -{{{\omega}} \cdot {{r}}} & 0 & -{r} \\ 0 & \frac{1}{r} & 0\end{matrix}\right]}$
$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{\omega}} \cdot {{r}}} & 0 & -{r} \\ 0 & \frac{1}{r} & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & 0 & 0 \\ {{\omega}} \cdot {{r}} {{({{{\phi_L}} - {{\phi_R}}})}} & 0 & {{r}} {{({{{\phi_L}} - {{\phi_R}}})}} \\ 0 & {\frac{1}{r}}{({-{({{{\phi_L}} - {{\phi_R}}})}})} & 0\end{matrix}\right]$
${ P}_{\phi}$ = ${ⅇ}^{( -{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{\omega}} \cdot {{r}}} & 0 & -{r} \\ 0 & \frac{1}{r} & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ -{{{\omega}} \cdot {{r}} {{sin\left( {{{\phi_L}} - {{\phi_R}}}\right)}}} & cos\left( {{{\phi_L}} - {{\phi_R}}}\right) & -{{{r}} {{sin\left( {{{\phi_L}} - {{\phi_R}}}\right)}}} \\ -{{{\omega}} \cdot {{({{1} - {cos\left( {{{\phi_L}} - {{\phi_R}}}\right)}})}}} & {\frac{1}{r}}{({sin\left( {{{\phi_L}} - {{\phi_R}}}\right)})} & cos\left( {{{\phi_L}} - {{\phi_R}}}\right)\end{matrix}\right]$
${{ P}_{\phi}}^{-1}$ = ${ⅇ}^{({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}d \phi\left( \left[\begin{matrix} 0 & 0 & 0 \\ -{{{\omega}} \cdot {{r}}} & 0 & -{r} \\ 0 & \frac{1}{r} & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 1 & 0 & 0 \\ {{\omega}} \cdot {{r}} {{sin\left( {{{\phi_L}} - {{\phi_R}}}\right)}} & cos\left( {{{\phi_L}} - {{\phi_R}}}\right) & {{r}} {{sin\left( {{{\phi_L}} - {{\phi_R}}}\right)}} \\ -{{{\omega}} \cdot {{({{1} - {cos\left( {{{\phi_L}} - {{\phi_R}}}\right)}})}}} & {\frac{1}{r}}{({-{sin\left( {{{\phi_L}} - {{\phi_R}}}\right)}})} & cos\left( {{{\phi_L}} - {{\phi_R}}}\right)\end{matrix}\right]$
propagator commutation: