paraboliod, coordinate
chart coordinates: $x^\tilde{\mu} = \{u, v\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{u}}, e_{\tilde{v}}\}$
embedding coordinates: $u^I = \{x, y, z\}$
embedding basis $e_I = \{e_{x}, e_{y}, e_{z}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_u}^u} = {1}$;
${{{ \tilde{e}}_v}^v} = {1}$
transform from coorinate to basis:
${{{ \tilde{e}}^u}_u} = {1}$;
${{{ \tilde{e}}^v}_v} = {1}$
tensor index associated with coordinate $u$ is index $u$ with operator $e_{u}(\zeta) = $$\frac{\partial \zeta}{\partial u}$
tensor index associated with coordinate $v$ is index $v$ with operator $e_{v}(\zeta) = $$\frac{\partial \zeta}{\partial v}$
flat metric:
${{{ \eta}_x}_x} = {1}$;
${{{ \eta}_y}_y} = {1}$;
${{{ \eta}_z}_z} = {1}$
chart in embedded coordinates:
${{ u}^x} = {u}$;
${{ u}^y} = {v}$;
${{ u}^z} = { {-{\frac{1}{2}}} {{({{{u}^{2}} + {{v}^{2}}})}}}$
basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_u}^x} = {1}$;
${{{ e}_u}^z} = {-{u}}$;
${{{ e}_v}^y} = {1}$;
${{{ e}_v}^z} = {-{v}}$
${{{ e}^u}_x} = {1}$;
${{{ e}^u}_z} = {\frac{-1}{u}}$;
${{{ e}^v}_y} = {1}$;
${{{ e}^v}_z} = {\frac{-1}{v}}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 2 & {\frac{1}{v}}{({u})} \\ {\frac{1}{u}}{({v})} & 2\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 & -{\frac{1}{u}} \\ 0 & 1 & -{\frac{1}{v}} \\ -{u} & -{v} & 2\end{matrix}\right]}}$
${{{{ c}_a}_b}^c} = {0}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
${{{ g}_u}_u} = {{1} + {{u}^{2}}}$;
${{{ g}_u}_v} = {{{u}} {{v}}}$;
${{{ g}_v}_u} = {{{u}} {{v}}}$;
${{{ g}_v}_v} = {{1} + {{v}^{2}}}$
${{{ g}_u}_v} = {{{{{ e}_u}^I}} {{{{ e}_v}^J}} {{{{ \eta}_I}_J}}}$
metric determinant: ${det(g)} = {{1} + {{v}^{2}} + {{u}^{2}}}$
${{{{ \Gamma}_a}_b}_c} = {{{\frac{1}{2}}} {{({{{{{{{{ g}_a}_b}_{,c}} + {{{{ g}_a}_c}_{,b}}} - {{{{ g}_b}_c}_{,a}}} + {{{{ c}_a}_b}_c} + {{{{ c}_a}_c}_b}} - {{{{ c}_c}_b}_a}})}}}$
commutation coefficients:
${{{{ c}_a}_b}^c} = {0}$
metric:
${{{ g}_u}_u} = {{1} + {{u}^{2}}}$;
${{{ g}_u}_v} = {{{u}} {{v}}}$;
${{{ g}_v}_u} = {{{u}} {{v}}}$;
${{{ g}_v}_v} = {{1} + {{v}^{2}}}$
metric inverse:
${{{ g}^u}^u} = {\frac{{1} + {{v}^{2}}}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{ g}^u}^v} = {\frac{-{{{u}} {{v}}}}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{ g}^v}^u} = {\frac{-{{{u}} {{v}}}}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{ g}^v}^v} = {\frac{{1} + {{u}^{2}}}{{1} + {{v}^{2}} + {{u}^{2}}}}$
metric derivative:
${{{{ {\partial g}}_u}_u}_u} = {{{2}} {{u}}}$;
${{{{ {\partial g}}_u}_v}_u} = {v}$;
${{{{ {\partial g}}_u}_v}_v} = {u}$;
${{{{ {\partial g}}_v}_u}_u} = {v}$;
${{{{ {\partial g}}_v}_u}_v} = {u}$;
${{{{ {\partial g}}_v}_v}_v} = {{{2}} {{v}}}$
1st kind Christoffel:
${{{{ \Gamma}_u}_u}_u} = {u}$;
${{{{ \Gamma}_u}_v}_v} = {u}$;
${{{{ \Gamma}_v}_u}_u} = {v}$;
${{{{ \Gamma}_v}_v}_v} = {v}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^u}_u}_u} = {\frac{u}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{{ \Gamma}^u}_v}_v} = {\frac{u}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{{ \Gamma}^v}_u}_u} = {\frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}}$;
${{{{ \Gamma}^v}_v}_v} = {\frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}}^u}_u}_u}_u} = {\frac{{1} + {{{v}^{2}} - {{u}^{2}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^u}_u}_u}_v} = {\frac{-{{{2}} {{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^u}_v}_v}_u} = {\frac{{1} + {{{v}^{2}} - {{u}^{2}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^u}_v}_v}_v} = {\frac{-{{{2}} {{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^v}_u}_u}_u} = {\frac{-{{{2}} {{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^v}_u}_u}_v} = {\frac{{{1} - {{v}^{2}}} + {{u}^{2}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^v}_v}_v}_u} = {\frac{-{{{2}} {{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {\partial \Gamma}}^v}_v}_v}_v} = {\frac{{{1} - {{v}^{2}}} + {{u}^{2}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}}^u}_u}_u}_u} = {\frac{{u}^{2}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^u}_u}_v}_u} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^u}_v}_u}_v} = {\frac{{u}^{2}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^u}_v}_v}_v} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^v}_u}_u}_u} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^v}_u}_v}_u} = {\frac{{v}^{2}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^v}_v}_u}_v} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ {(\Gamma^2)}}^v}_v}_v}_v} = {\frac{{v}^{2}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R}^u}_u}_u}_v} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^u}_u}_v}_u} = {\frac{-{{{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^u}_v}_u}_v} = {\frac{{1} + {{v}^{2}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^u}_v}_v}_u} = {\frac{-{({{1} + {{v}^{2}}})}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^v}_u}_u}_v} = {\frac{-{({{1} + {{u}^{2}}})}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^v}_u}_v}_u} = {\frac{{1} + {{u}^{2}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^v}_v}_u}_v} = {\frac{-{{{u}} {{v}}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{{{ R}^v}_v}_v}_u} = {\frac{{{u}} {{v}}}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R}^u}^v}_u}_v} = {\frac{1}{{({{1} + {{u}^{2}} + {{v}^{2}}})}^{2}}}$;
${{{{{ R}^u}^v}_v}_u} = {-{\frac{1}{{({{1} + {{u}^{2}} + {{v}^{2}}})}^{2}}}}$;
${{{{{ R}^v}^u}_u}_v} = {-{\frac{1}{{({{1} + {{u}^{2}} + {{v}^{2}}})}^{2}}}}$;
${{{{{ R}^v}^u}_v}_u} = {\frac{1}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$
Ricci curvature, $\sharp\flat$:
${{{ R}^u}_u} = {\frac{1}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}$;
${{{ R}^v}_v} = {\frac{1}{{({{1} + {{u}^{2}} + {{v}^{2}}})}^{2}}}$
Gaussian curvature:
$\frac{2}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}$
Einstein $\sharp\flat$ / trace-reversed Ricci curvature:
${{{ G}^u}_u} = {-{\frac{1}{{({{1} + {{v}^{2}} + {{u}^{2}}})}^{2}}}}$;
${{{ G}^v}_v} = {-{\frac{1}{{({{1} + {{u}^{2}} + {{v}^{2}}})}^{2}}}}$
divergence: ${{{{ A}^i}_{,i}} + {{{{{{ \Gamma}^i}_i}_j}} {{{ A}^j}}}} = {{{{\frac{\partial {A^{u}}}{\partial u}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{\frac{\partial {A^{v}}}{\partial v}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{v}^{2}}} {{\frac{\partial {A^{u}}}{\partial u}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{v}^{2}}} {{\frac{\partial {A^{v}}}{\partial v}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{u}^{2}}} {{\frac{\partial {A^{u}}}{\partial u}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{u}^{2}}} {{\frac{\partial {A^{v}}}{\partial v}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{A^{v}}}} \cdot {{v}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{{A^{u}}}} \cdot {{u}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{u} \\ \ddot{v}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{-1}} {{u}} {{{\dot{u}}^{2}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{-1}} {{u}} {{{\dot{v}}^{2}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} \\ {{{-1}} {{v}} {{{\dot{u}}^{2}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}} + {{{-1}} {{v}} {{{\dot{v}}^{2}}} {{\frac{1}{{1} + {{v}^{2}} + {{u}^{2}}}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_u]}} = {\left[\begin{matrix} \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} & 0 \\ \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}} & 0\end{matrix}\right]}$
$\int\limits_{{{u_L}}}^{{{u_R}}}d u\left( \left[\begin{matrix} \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} & 0 \\ \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}} & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} log\left( {\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}\right) & 0 \\ log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{({\frac{-{v}}{{{2}} {{\sqrt{{-{1}} - {{v}^{2}}}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{({\frac{v}{{{2}} {{\sqrt{{-{1}} - {{v}^{2}}}}}}})}}}}\right) & 0\end{matrix}\right]$
${ P}_u$ = ${e}^{( -{({\int\limits_{{{u_L}}}^{{{u_R}}}d u\left( \left[\begin{matrix} \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} & 0 \\ \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}} & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right) & 0 \\ \frac{{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right)} + {\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{({\frac{-{v}}{\sqrt{{-{1}} - {{v}^{2}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{({\frac{v}{\sqrt{{-{1}} - {{v}^{2}}}}})}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{({\frac{-{v}}{\sqrt{{-{1}} - {{v}^{2}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{({\frac{v}{\sqrt{{-{1}} - {{v}^{2}}}}})}}}}\right)}})}}}}\right)} & 1\end{matrix}\right]$
${{ P}_u}^{-1}$ = ${e}^{({\int\limits_{{{u_L}}}^{{{u_R}}}d u\left( \left[\begin{matrix} \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} & 0 \\ \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}} & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{{{v}} {{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right) & 0 \\ \frac{{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{v}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{v}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{v}}} {{\frac{1}{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{v}}}}}\right)}})}}}}\right)}}}}\right)}})}}}}\right)} + {\sqrt{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}}}{log\left( {{{{|{{1} + {{v}^{2}} + {{{u_R}}^{2}}}|}^{({\frac{-{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{({\frac{-{v}}{\sqrt{{-{1}} - {{v}^{2}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{({\frac{v}{\sqrt{{-{1}} - {{v}^{2}}}}})}}}}\right)}})}}} {{{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}^{({\frac{\sqrt{|{{1} + {{v}^{2}} + {{{u_L}}^{2}}}|}}{log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_R}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_R}}}}|}^{({\frac{-{v}}{\sqrt{{-{1}} - {{v}^{2}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{v}^{2}}}} - {{u_L}}})}}{{\sqrt{{-{1}} - {{v}^{2}}}} + {{u_L}}}}|}^{({\frac{v}{\sqrt{{-{1}} - {{v}^{2}}}}})}}}}\right)}})}}}}\right)} & 1\end{matrix}\right]$
${{[\Gamma_v]}} = {\left[\begin{matrix} 0 & \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} \\ 0 & \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}\end{matrix}\right]}$
$\int\limits_{{{v_L}}}^{{{v_R}}}d v\left( \left[\begin{matrix} 0 & \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} \\ 0 & \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_L}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_L}}}}|}^{({\frac{-{u}}{{{2}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_R}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_R}}}}|}^{({\frac{u}{{{2}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}})}}}}\right) \\ 0 & log\left( {\frac{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}\right)\end{matrix}\right]$
${ P}_v$ = ${e}^{( -{({\int\limits_{{{v_L}}}^{{{v_R}}}d v\left( \left[\begin{matrix} 0 & \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} \\ 0 & \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 1 & log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_L}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_L}}}}|}^{({\frac{{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({\frac{{{u}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}} {{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({\frac{-{{{u}} {{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}}}\right)} - {{{u}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}})}}}} {{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}})}}}}\right)}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_R}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_R}}}}|}^{({\frac{{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({\frac{{{u}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}})}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}})}}}}}\right)}})}}} {{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({\frac{-{{{u}} {{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}})}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}})}}}}}\right)}})}}}}\right)} - {u}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}}}\right) \\ 0 & \frac{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}\end{matrix}\right]$
${{ P}_v}^{-1}$ = ${e}^{({\int\limits_{{{v_L}}}^{{{v_R}}}d v\left( \left[\begin{matrix} 0 & \frac{u}{{1} + {{v}^{2}} + {{u}^{2}}} \\ 0 & \frac{v}{{1} + {{v}^{2}} + {{u}^{2}}}\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 1 & log\left( {{{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_L}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_L}}}}|}^{({\frac{{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({\frac{{{u}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}} {{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({\frac{-{{{u}} {{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}}}\right)} - {{{u}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}})}}}} {{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}})}}}}\right)}})}}} {{{|{\frac{-{({{\sqrt{{-{1}} - {{u}^{2}}}} - {{v_R}}})}}{{\sqrt{{-{1}} - {{u}^{2}}}} + {{v_R}}}}|}^{({\frac{{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({\frac{{{u}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}} {{\sqrt{{-{1}} - {{u}^{2}}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}})}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}})}}}}}\right)}})}}} {{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({\frac{-{{{u}} {{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}}}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}})}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{({{{\sqrt{{-{1}} - {{u}^{2}}}}} {{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}}})}}}}}\right)}})}}}}\right)} - {u}}{log\left( {{{{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}} {{\frac{1}{{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}^{\sqrt{{-{1}} - {{u}^{2}}}}}}}}\right)}})}}}}\right) \\ 0 & \frac{\sqrt{|{{1} + {{{v_R}}^{2}} + {{u}^{2}}}|}}{\sqrt{|{{1} + {{{v_L}}^{2}} + {{u}^{2}}}|}}\end{matrix}\right]$
propagator commutation: