paraboliod, coordinate

chart coordinates:

x^{\tilde{μ}} = {u, v}

chart coordinate basis:

e_{\tilde{μ}} = {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} (ζ) =

\frac{\partial ζ}{\partial u}

tensor index associated with coordinate

v

is index

v

with operator

e_{v} (ζ) =

\frac{\partial ζ}{\partial v}

flat metric:

{η_{x}}_{x} = 1

;

{η_{y}}_{y} = 1

;

{η_{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 ↓ v \to}{[\begin{matrix} 2 & \frac{1}{v} (u) \\ \frac{1}{u} (v) & 2 \end{matrix}]}

{e_{u}}^{I} {e^{u}}_{J} = \overset{I ↓ J \to}{[\begin{matrix} 1 & 0 & - \frac{1}{u} \\ 0 & 1 & - \frac{1}{v} \\ - u & - v & 2 \end{matrix}]}

{c_{a}}_{b}^{c} = 0

{g_{u}}_{v} = {e_{u}}^{I} {e_{v}}^{J} {η_{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} {η_{I}}_{J}

metric determinant:

d e t (g) = 1 + v^{2} + u^{2}

{Γ_{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:

{Γ_{u}}_{u}_{u} = u

;

{Γ_{u}}_{v}_{v} = u

;

{Γ_{v}}_{u}_{u} = v

;

{Γ_{v}}_{v}_{v} = v

connection coefficients / 2nd kind Christoffel:

{Γ^{u}}_{u}_{u} = \frac{u}{1 + v^{2} + u^{2}}

;

{Γ^{u}}_{v}_{v} = \frac{u}{1 + v^{2} + u^{2}}

;

{Γ^{v}}_{u}_{u} = \frac{v}{1 + v^{2} + u^{2}}

;

{Γ^{v}}_{v}_{v} = \frac{v}{1 + v^{2} + u^{2}}

connection coefficients derivative:

{\partial Γ}^{u}_{u}_{u}_{u} = \frac{1 + v^{2} - u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{u}_{u}_{u}_{v} = \frac{- 2 u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{u}_{v}_{v}_{u} = \frac{1 + v^{2} - u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{u}_{v}_{v}_{v} = \frac{- 2 u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{v}_{u}_{u}_{u} = \frac{- 2 u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{v}_{u}_{u}_{v} = \frac{1 - v^{2} + u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{v}_{v}_{v}_{u} = \frac{- 2 u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{\partial Γ}^{v}_{v}_{v}_{v} = \frac{1 - v^{2} + u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

connection coefficients squared:

{(Γ^{2})}^{u}_{u}_{u}_{u} = \frac{u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{u}_{u}_{v}_{u} = \frac{u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{u}_{v}_{u}_{v} = \frac{u^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{u}_{v}_{v}_{v} = \frac{u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{v}_{u}_{u}_{u} = \frac{u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{v}_{u}_{v}_{u} = \frac{v^{2}}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{v}_{v}_{u}_{v} = \frac{u v}{{(1 + v^{2} + u^{2})}^{2}}

;

{(Γ^{2})}^{v}_{v}_{v}_{v} = \frac{v^{2}}{{(1 + v^{2} + u^{2})}^{2}}

Riemann curvature,

♯ ♭ ♭ ♭

{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,

♯ ♯ ♭ ♭

{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,

♯ ♭

{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

♯ ♭

/ 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} + {Γ^{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 ↓}{[\begin{matrix} \ddot{u} \\ \ddot{v} \end{matrix}]} = \overset{a ↓}{[\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}]}

parallel propagators:

[Γ_{u}] = [\begin{matrix} \frac{u}{1 + v^{2} + u^{2}} & 0 \\ \frac{v}{1 + v^{2} + u^{2}} & 0 \end{matrix}]

\int_{u_{L}}^{u_{R}} d u ([\begin{matrix} \frac{u}{1 + v^{2} + u^{2}} & 0 \\ \frac{v}{1 + v^{2} + u^{2}} & 0 \end{matrix}])

[\begin{matrix} l o g (\frac{\sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{\sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}) & 0 \\ l o g ({| \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}}})}) & 0 \end{matrix}]

P_{u}

e^{(- (\int_{u_{L}}^{u_{R}} d u ([\begin{matrix} \frac{u}{1 + v^{2} + u^{2}} & 0 \\ \frac{v}{1 + v^{2} + u^{2}} & 0 \end{matrix}])))}

[\begin{matrix} l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}})})})})})}) & 0 \\ \frac{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})})})})}) + \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}}})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \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}}})})})})} & 1 \end{matrix}]

{P_{u}}^{- 1}

e^{(\int_{u_{L}}^{u_{R}} d u ([\begin{matrix} \frac{u}{1 + v^{2} + u^{2}} & 0 \\ \frac{v}{1 + v^{2} + u^{2}} & 0 \end{matrix}]))}

[\begin{matrix} l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}})})})})})}) & 0 \\ \frac{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \frac{- (\sqrt{- 1 - v^{2}} - u_{R})}{\sqrt{- 1 - v^{2}} + u_{R}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{v}{l o g ({| \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}})})})} {| \frac{- (\sqrt{- 1 - v^{2}} - u_{L})}{\sqrt{- 1 - v^{2}} + u_{L}} |}^{l o g ({| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{v}{l o g ({| \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}})})} {| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- v}{l o g ({| \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}})})})})})}) + \sqrt{| 1 + v^{2} + {u_{R}}^{2} |}}{l o g ({| 1 + v^{2} + {u_{R}}^{2} |}^{(\frac{- \sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}}})})})} {| 1 + v^{2} + {u_{L}}^{2} |}^{(\frac{\sqrt{| 1 + v^{2} + {u_{L}}^{2} |}}{l o g ({| \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}}})})})})} & 1 \end{matrix}]

[Γ_{v}] = [\begin{matrix} 0 & \frac{u}{1 + v^{2} + u^{2}} \\ 0 & \frac{v}{1 + v^{2} + u^{2}} \end{matrix}]

\int_{v_{L}}^{v_{R}} d v ([\begin{matrix} 0 & \frac{u}{1 + v^{2} + u^{2}} \\ 0 & \frac{v}{1 + v^{2} + u^{2}} \end{matrix}])

[\begin{matrix} 0 & l o g ({| \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}}})}) \\ 0 & l o g (\frac{\sqrt{| 1 + {v_{R}}^{2} + u^{2} |}}{\sqrt{| 1 + {v_{L}}^{2} + u^{2} |}}) \end{matrix}]

P_{v}

e^{(- (\int_{v_{L}}^{v_{R}} d v ([\begin{matrix} 0 & \frac{u}{1 + v^{2} + u^{2}} \\ 0 & \frac{v}{1 + v^{2} + u^{2}} \end{matrix}])))}

[\begin{matrix} 1 & l o g ({| \frac{- (\sqrt{- 1 - u^{2}} - v_{L})}{\sqrt{- 1 - u^{2}} + v_{L}} |}^{(\frac{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{(\frac{u \sqrt{| 1 + {v_{R}}^{2} + u^{2} |} \sqrt{- 1 - u^{2}}}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})} {| 1 + {v_{L}}^{2} + u^{2} |}^{(\frac{- u \sqrt{- 1 - u^{2}} \sqrt{| 1 + {v_{R}}^{2} + u^{2} |}}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})}) - u \sqrt{| 1 + {v_{L}}^{2} + u^{2} |}}{l o g (\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}})})})} {| \frac{- (\sqrt{- 1 - u^{2}} - v_{R})}{\sqrt{- 1 - u^{2}} + v_{R}} |}^{(\frac{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{(\frac{u \sqrt{| 1 + {v_{L}}^{2} + u^{2} |} \sqrt{- 1 - u^{2}}}{l o g ({| 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} |})}})})} {| 1 + {v_{L}}^{2} + u^{2} |}^{(\frac{- u \sqrt{- 1 - u^{2}} \sqrt{| 1 + {v_{L}}^{2} + u^{2} |}}{l o g ({| 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} |})}})})}) - u}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})}) \\ 0 & \frac{\sqrt{| 1 + {v_{L}}^{2} + u^{2} |}}{\sqrt{| 1 + {v_{R}}^{2} + u^{2} |}} \end{matrix}]

{P_{v}}^{- 1}

e^{(\int_{v_{L}}^{v_{R}} d v ([\begin{matrix} 0 & \frac{u}{1 + v^{2} + u^{2}} \\ 0 & \frac{v}{1 + v^{2} + u^{2}} \end{matrix}]))}

[\begin{matrix} 1 & l o g ({| \frac{- (\sqrt{- 1 - u^{2}} - v_{L})}{\sqrt{- 1 - u^{2}} + v_{L}} |}^{(\frac{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{(\frac{u \sqrt{| 1 + {v_{L}}^{2} + u^{2} |} \sqrt{- 1 - u^{2}}}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})} {| 1 + {v_{L}}^{2} + u^{2} |}^{(\frac{- u \sqrt{- 1 - u^{2}} \sqrt{| 1 + {v_{L}}^{2} + u^{2} |}}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})}) - u \sqrt{| 1 + {v_{R}}^{2} + u^{2} |}}{l o g (\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}})})})} {| \frac{- (\sqrt{- 1 - u^{2}} - v_{R})}{\sqrt{- 1 - u^{2}} + v_{R}} |}^{(\frac{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{(\frac{u \sqrt{| 1 + {v_{R}}^{2} + u^{2} |} \sqrt{- 1 - u^{2}}}{l o g ({| 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} |})}})})} {| 1 + {v_{L}}^{2} + u^{2} |}^{(\frac{- u \sqrt{- 1 - u^{2}} \sqrt{| 1 + {v_{R}}^{2} + u^{2} |}}{l o g ({| 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} |})}})})}) - u}{l o g ({| 1 + {v_{R}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}} \frac{1}{{| 1 + {v_{L}}^{2} + u^{2} |}^{\sqrt{- 1 - u^{2}}}})})}) \\ 0 & \frac{\sqrt{| 1 + {v_{R}}^{2} + u^{2} |}}{\sqrt{| 1 + {v_{L}}^{2} + u^{2} |}} \end{matrix}]

propagator commutation: