Parallel transport along $\phi = 0$

Question

I am trying to learn how to parallel transport a vector $V^\alpha=(V^\theta_0,V^\phi_0)=\text{constant}$ on the surface of a sphere of radius $R$, starting from $(R,0,0)$ to $(R,\pi/2,0)$ along the line $\phi=0$. The coordinates, therefore are, $x^\alpha=(R,\theta(\lambda),0)$, where $\lambda$ is the parameter along the path. Using the condition of parallel transport, $$ \frac{dV^\alpha}{d\lambda} = -\Gamma_{\beta\nu}^{\alpha}V^\nu\frac{dx^\beta}{d\lambda} $$ After solving the equation of parallel transport I get the following differential equations, $$ \frac{dV^\theta}{d\lambda}=0 $$ $$ \frac{dV^\phi}{d\lambda}=-\frac{\cos \theta}{\sin \theta}V^\phi $$ The $V^\theta$ equation is simple, it just says $V^\theta = \text{constant}$. I am confused about the other equation. The solution is, $$ \ln V^\phi=-\ln \sin \theta + C $$ $$ V^\phi \sin \theta = C $$ Here, $C$ is the constant of integration. When I put the initial conditions $V^\alpha=(V^\theta_0,V^\phi_0)$ at $(R,0,0)$, $C$ just comes out to be zero which implies that $V^\phi$ is zero always, since $\sin \theta(\lambda)$ cannot be zero along the entire path which basically also means that $V^\phi_0=0$.

I do not know how to make sense of this result. I am trying to complete a "triangle" of great circles along the sphere going from $(R,0,0) \rightarrow (R,\pi/2,0) \rightarrow (R,\pi/2,\pi/2)\rightarrow (R,0,\pi/2)$.

Answer 1

The problem is that the coordinates are not well defined at the north pole. There's no way to have a $\phi$ component there; you can only have a $\theta$ component, but that doesn't tell you the direction of the vector. Distinguishing $(R,0,0)$ and $(R, 0, \pi/2)$ also doesn't makes sense, they're the same point.

To avoid this problem, you could for example translate your triangle so it covers $\pi/4 < \theta < 3\pi/4$, or use stereographic coordinates or any other coordinate system that is well behaved at the pole.

Answer 2

REFERENCE : My answer here What is the drawing scheme of the parallel transport of a vector?. $=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=$

See a 3d view of Figure-01 here

Your path of parallel transport is arc of a meridian (a great circle) from the north pole to a point in the equator, see points $\boxed{\boldsymbol{0}},\boxed{\boldsymbol{1}},\boxed{\boldsymbol{2}}$ in Figure-01. That is the vector is transported along a geodesic. This means that it keeps its angle with respect to the geodesic path and its magnitude unchanged. So the components $\mathbf{V}^{\boldsymbol{\theta}}$ and $\mathbf{V}^{\boldsymbol{\phi}}$, that is the projections of $\mathbf{V}$ on the axes $\mathbf{e}_{\boldsymbol{\theta}}$ and $\mathbf{e}_{\boldsymbol{\phi}}$ respectively of the local coordinate system are unchanged. You have found that $\mathbf{V}^{\boldsymbol{\theta}}\boldsymbol{=}\texttt{constant}$. It's all right. But you must find also that $\mathbf{V}^{\boldsymbol{\phi}}\boldsymbol{=}\texttt{constant}$. So, something is going wrong with your differential equation with respect to $\phi$. You must check it.

$=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=\!=$

ADDENDUM :

I think that to avoid the singularities of the north pole you may equivalently well transport the vector along the path $\boxed{\boldsymbol{0}},\boxed{\boldsymbol{1}},\boxed{\boldsymbol{2}}$ on the equator as shown in Figure-02 and after that place the north pole on point $\boxed{\boldsymbol{0}}$ (since no-one knows where is your north pole from the beginning).

The parametric equation of this $90^\circ$-arc with parameter the angle $\phi$ is \begin{equation} \require{cancel} \boldsymbol{\chi}\left(\phi\right)\boldsymbol{=}\left[\:\chi^{\theta}\left(\phi\right)\,,\chi^{\phi}\left(\phi\right)\vphantom{\dfrac{a}{b}}\right]\boldsymbol{=}\left[\:\theta_{0}\,,\frac{\pi}{2}\boldsymbol{-}\phi\right]\boldsymbol{=}\left[\frac{\pi}{2}\,,\frac{\pi}{2}\boldsymbol{-}\phi\right]_{\phi=0}^{\phi=\tfrac{\pi}{2}} \tag{01}\label{01} \end{equation}

Now consider that at point $\boxed{\boldsymbol{0}}$ we have a vector on the tangent plane \begin{equation} \mathbf{V}_0\boldsymbol{=} \begin{bmatrix} V^\theta_0 \vphantom{\dfrac{a}{b}}\\ V^\phi_0 \vphantom{\dfrac{\tfrac{a}{b}}{b}} \end{bmatrix} \tag{02}\label{02} \end{equation} which we want to parallel transport along the arc to point $\boxed{\boldsymbol{2}}$.

For the parallel transport we use the equation provided by the OP \begin{equation} \frac{\mathrm dV^\alpha}{\mathrm d\phi} \boldsymbol{=} \boldsymbol{-}\Gamma_{\beta\nu}^{\alpha}V^\nu\frac{\mathrm d\chi^\beta}{\mathrm d\phi} \qquad \left(\alpha,\beta,\nu \boldsymbol{=}\theta,\phi\right) \tag{03}\label{03} \end{equation} where $\Gamma_{\beta\nu}^{\alpha}$ the Christoffer symbols. Above equation could be expressed as \begin{equation} \dfrac{\mathrm d\mathbf{V}}{\mathrm d\phi} \boldsymbol{=} \begin{bmatrix} \dfrac{\mathrm dV^\theta}{\mathrm d\phi}\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \dfrac{\mathrm dV^\phi}{\mathrm d\phi}\vphantom{\dfrac{a}{b}} \end{bmatrix} \boldsymbol{=}\boldsymbol{-} \begin{bmatrix} \Bigl< \boldsymbol{\Gamma^{\theta}}\mathbf{V},\dfrac{\mathrm d\boldsymbol{\chi}}{\mathrm d\phi}\Bigr> \vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \Bigl< \boldsymbol{\Gamma^{\phi}}\mathbf{V},\dfrac{\mathrm d\boldsymbol{\chi}}{\mathrm d\phi}\Bigr> \vphantom{\dfrac{a}{\dfrac{a}{b}}} \end{bmatrix} \tag{04}\label{04} \end{equation} where from \eqref{01} \begin{equation} \dfrac{\mathrm d\boldsymbol{\chi}}{\mathrm d\phi}\boldsymbol{=}\dfrac{\mathrm d}{\mathrm d\phi} \begin{bmatrix} \chi^{\theta}\left(\phi\right)\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \chi^{\phi}\left(\phi\right)\vphantom{\dfrac{a}{\tfrac{a}{b}}} \end{bmatrix} \boldsymbol{=}\dfrac{\mathrm d}{\mathrm d\phi} \begin{bmatrix} \dfrac{\pi}{2}\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \dfrac{\pi}{2}\boldsymbol{-}\phi\vphantom{\dfrac{a}{\tfrac{a}{b}}} \end{bmatrix} \boldsymbol{=}\boldsymbol{-} \begin{bmatrix} \:\:0\:\:\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \:\:1\:\:\vphantom{\dfrac{a}{\tfrac{a}{b}}} \end{bmatrix} \tag{05}\label{05} \end{equation} and $\boldsymbol{\Gamma^{\theta}},\boldsymbol{\Gamma^{\phi}}$ the following matrices of Christoffer symbols \begin{align} \boldsymbol{\Gamma^{\theta}}\boldsymbol{=}\Gamma^{\theta}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} \: \Gamma^{\theta}_{\theta\theta} & \hphantom{=}\Gamma^{\theta}_{\theta\phi} \:\vphantom{\dfrac{a}{b}}\\ \: \Gamma^{\theta}_{\phi\theta} & \hphantom{=}\Gamma^{\theta}_{\phi\phi} \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{06a}\label{06a}\\ \boldsymbol{\Gamma^{\phi}}\boldsymbol{=}\Gamma^{\phi}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} \: \Gamma^{\phi}_{\theta\theta} & \hphantom{=}\Gamma^{\phi}_{\theta\phi} \:\vphantom{\dfrac{a}{b}}\\ \: \Gamma^{\phi}_{\phi\theta} & \hphantom{=}\Gamma^{\phi}_{\phi\phi} \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{06b}\label{06b} \end{align} The Christoffer symbols are expressed through the components of the metric tensor $\mathbf{g}$ \begin{equation} \Gamma_{\beta\nu}^{\alpha}\boldsymbol{=} \frac{1}{2}\sum\limits_{k\boldsymbol{=}\theta,\phi} g^{\alpha k} \left(\frac{\partial g_{k\nu}}{\partial \chi^{\beta }}\boldsymbol{+}\frac{\partial g_{\beta k}}{\partial \chi^{\nu}}\boldsymbol{-}\frac{\partial g_{\beta \nu}}{\partial \chi^{k}}\right) \tag{07}\label{07} \end{equation} From the infinitesimal displacement on the sphere \begin{equation} \left(\mathrm ds\right)^2\boldsymbol{=}R^2\left(\mathrm d\theta\right)^2\boldsymbol{+}R^2\sin^2\theta\left(\mathrm d\phi\right)^2 \tag{08}\label{08} \end{equation} the metric tensor is \begin{align} \mathbf{g}\boldsymbol{=}g_{ij}& \boldsymbol{=} \begin{bmatrix} \: g_{\theta\theta} & \hphantom{=}g_{\theta\phi} \:\vphantom{\dfrac{a}{b}}\\ \: g_{\phi\theta} & \hphantom{=}g_{\phi\phi} \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \boldsymbol{=} \begin{bmatrix} \hphantom{r^2}R^2\hphantom{r^2\theta} & \hphantom{r^2}0\hphantom{r^2\theta} \vphantom{\dfrac{a}{b}}\\ \hphantom{r^2}0\hphantom{r^2\theta} & R^2\sin^2\theta \vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{09a}\label{09a}\\ \mathbf{g}^{\boldsymbol{-}1}\boldsymbol{=}g^{ij}& \boldsymbol{=} \begin{bmatrix} \: g^{\theta\theta} & \hphantom{=}g^{\theta\phi} \:\vphantom{\dfrac{a}{b}}\\ \: g^{\phi\theta} & \hphantom{=}g^{\phi\phi} \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \boldsymbol{=} \begin{bmatrix} \hphantom{r^2}R^{\boldsymbol{-}2}\hphantom{r^2\theta} & \hphantom{r^2}0\hphantom{r^2\theta} \vphantom{\dfrac{a}{b}}\\ \hphantom{r^2}0\hphantom{r^2\theta} & R^{\boldsymbol{-}2}\sin^{\boldsymbol{-}2}\theta \vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{09b}\label{09b} \end{align} So for the elements of $\boldsymbol{\Gamma^{\theta}}$ ^{\begin{align} \Gamma^{\theta}_{\theta\theta} & \boldsymbol{=} \frac12\left[g^{\theta\theta} \left(\cancelto{0}{\frac{\partial g_{\theta\theta}}{\partial \theta}}\boldsymbol{+}\cancelto{0}{\frac{\partial g_{\theta\theta}}{\partial \theta}}\boldsymbol{-}\cancelto{0}{\frac{\partial g_{\theta\theta}}{\partial \theta}}\right)\boldsymbol{+} \cancelto{0}{g^{\theta\phi}} \left(\frac{\partial g_{\phi\theta}}{\partial \theta}\boldsymbol{+}\frac{\partial g_{\theta\phi}}{\partial \theta}\boldsymbol{-}\frac{\partial g_{\theta\theta}}{\partial \phi}\right)\right]\boldsymbol{=}0 \tag{10a}\label{10a}\\ \Gamma^{\theta}_{\theta\phi} & \boldsymbol{=} \frac12\left[g^{\theta\theta} \left(\cancel{\frac{\partial g_{\theta\phi}}{\partial \theta}}\boldsymbol{+}\cancelto{0}{\frac{\partial g_{\theta\theta}}{\partial \phi}}\boldsymbol{-}\cancel{\frac{\partial g_{\theta\phi}}{\partial \theta}}\right)\boldsymbol{+} \cancelto{0}{g^{\theta\phi}} \left(\frac{\partial g_{\phi\phi}}{\partial \theta}\boldsymbol{+}\frac{\partial g_{\theta\phi}}{\partial \phi}\boldsymbol{-}\frac{\partial g_{\theta\phi}}{\partial \phi}\right)\right]\boldsymbol{=}0 \tag{10b}\label{10b}\\ \Gamma^{\theta}_{\phi\theta} & \boldsymbol{=} \Gamma^{\theta}_{\theta\phi}\boldsymbol{=}0 \tag{10c}\label{10c} \\ \Gamma^{\theta}_{\phi\phi} & \boldsymbol{=} \frac12\left[\cancelto{^{R^{-2}}}{g^{\theta\theta}}\left(\frac{\partial \cancelto{0}{g_{\theta\phi}}}{\partial \phi}\boldsymbol{+}\frac{\partial \cancelto{0}{g_{\phi\theta}}}{\partial \phi}\boldsymbol{-}\cancelto{^{2R^2\sin\theta\cos\theta}}{\frac{\partial g_{\phi\phi}}{\partial \theta}}\right)\boldsymbol{+}\cancelto{0}{g^{\theta\phi}} \left(\frac{\partial g_{\phi\phi}}{\partial \phi}\boldsymbol{+}\frac{\partial g_{\phi\phi}}{\partial \phi}\boldsymbol{-}\frac{\partial g_{\phi\phi}}{\partial \phi}\right)\right] \nonumber\\ &\boldsymbol{=}\boldsymbol{-}\sin\theta\cos\theta \tag{10d}\label{10d} \end{align}} Finding also by this way the elements of $\boldsymbol{\Gamma^{\phi}}$ we have the following two $2\times 2$ symmetric matrices \begin{align} \boldsymbol{\Gamma^{\theta}}\boldsymbol{=}\Gamma^{\theta}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} 0\hphantom{r^2\theta} & \hphantom{r^2}0\hphantom{r^2\theta} \vphantom{\dfrac{a}{b}}\\ 0\hphantom{r^2\theta} & \boldsymbol{-}\dfrac{\sin2\theta}{2} \vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{11a}\label{11a}\\ \boldsymbol{\Gamma^{\phi}}\boldsymbol{=}\Gamma^{\phi}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} 0 & \hphantom{r^2}\cot\theta \vphantom{\dfrac{a}{b}}\\ \cot\theta & \hphantom{r^2} 0 \vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{11b}\label{11b} \end{align} But in our case $\theta\boldsymbol{=}\theta_{0}\boldsymbol{=}\pi/2$ so \begin{align} \boldsymbol{\Gamma^{\theta}}\boldsymbol{=}\Gamma^{\theta}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} 0\hphantom{r^2\theta} & \hphantom{r^2}0\hphantom{r^2\theta} \vphantom{\dfrac{a}{b}}\\ 0\hphantom{r^2\theta} & \boldsymbol{-}\dfrac{\sin2\theta}{2} \vphantom{\dfrac{a}{b}} \end{bmatrix} \stackrel{\theta=\pi/2}{\boldsymbol{=\!=\!=}} \begin{bmatrix} \: 0 & \hphantom{=}0 \:\vphantom{\dfrac{a}{b}}\\ \: 0 & \hphantom{=}0 \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{12a}\label{12a}\\ \boldsymbol{\Gamma^{\phi}}\boldsymbol{=}\Gamma^{\phi}_{\beta\nu}& \boldsymbol{=} \begin{bmatrix} 0 & \hphantom{r^2}\cot\theta \vphantom{\dfrac{a}{b}}\\ \cot\theta & \hphantom{r^2} 0 \vphantom{\dfrac{a}{b}} \end{bmatrix} \stackrel{\theta=\pi/2}{\boldsymbol{=\!=\!=}} \begin{bmatrix} \: 0 & \hphantom{=}0 \:\vphantom{\dfrac{a}{b}}\\ \: 0 & \hphantom{=}0 \:\vphantom{\dfrac{a}{b}} \end{bmatrix} \tag{12b}\label{12b} \end{align} and \eqref{04} yields \begin{equation} \dfrac{\mathrm d\mathbf{V}}{\mathrm d\phi} \boldsymbol{=} \begin{bmatrix} \dfrac{\mathrm dV^\theta}{\mathrm d\phi}\vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \dfrac{\mathrm dV^\phi}{\mathrm d\phi}\vphantom{\dfrac{a}{b}} \end{bmatrix} \boldsymbol{=}\boldsymbol{-} \begin{bmatrix} \Bigl< \boldsymbol{\Gamma^{\theta}}\mathbf{V},\dfrac{\mathrm d\boldsymbol{\chi}}{\mathrm d\phi}\Bigr> \vphantom{\dfrac{a}{\dfrac{a}{b}}}\\ \Bigl< \boldsymbol{\Gamma^{\phi}}\mathbf{V},\dfrac{\mathrm d\boldsymbol{\chi}}{\mathrm d\phi}\Bigr> \vphantom{\dfrac{a}{\dfrac{a}{b}}} \end{bmatrix} \boldsymbol{=} \begin{bmatrix} \:\:0\:\:\vphantom{\dfrac{\tfrac{a}{b}}{\dfrac{a}{b}}}\\ \:\:0\:\:\vphantom{\dfrac{\dfrac{a}{b}}{\tfrac{a}{b}}} \end{bmatrix} \tag{13}\label{13} \end{equation} that is \begin{equation} V^\theta\boldsymbol{=}\texttt{constant}\boldsymbol{=}V^\theta_{0}\,, \qquad V^\phi\boldsymbol{=}\texttt{constant}\boldsymbol{=}V^\phi_{0} \tag{14}\label{14} \end{equation}