Factors for contravariant components with transported vector

[fab13]

I am currently coding a small application that reproduces the transport of a vector along a geodesic on a 2D sphere.

Here's a capture of this application :

You can see as pink vectors the vectors of curvilinear coordinates and in cyan the transported vector.

The transport of vector along the geodesic (in yellow) is well reproduced from algorithmic and geometric point of view (with the using of Matrix rotation and angles computation).

Now I would like to check the validity of this transport from an analytical point of view with the equations of transport on geodesic.

For this, I am using the equations deduced from the definition of parallel transport :

$$\text{d}v^{\theta} = v^{\varphi}\sin\theta\,\cos\theta\,\text{d}\varphi\quad\quad (1)$$

and

$$\text{d}v^{\varphi} = -\cot\theta\,(v^{\theta}\text{d}\varphi + v^{\varphi}\text{d}\theta)\quad\quad (2)$$

In my simulation, I take as a convention that $e_{\theta}$ and $e_{\varphi}$ have a " norm equal to 1 ", so I can write the vector transported like :

$$\vec{V}=V^{\theta}\,\vec{e_{\theta}}+V^{\varphi}\,\vec{e_{\varphi}}$$

and take : $$||\vec{e_{\theta}}|| = 1$$ and $$||\vec{e_{\varphi}}|| = 1$$

Actually, I wonder if i have to apply factors and if yes, how to apply them (on equations above (1) and (2)). I speak about the factors that come from the definition of curvilinear basis vectors, i.e with Einstein notation :

$$\text{d}\vec{OM}=\dfrac{\partial\vec{OM}}{\partial y^{i}}\text{d}y^{i}$$

with the curvilinear basis vector $\vec{e_{i}}= \dfrac{\partial\vec{OM}}{\partial y^{i}}$

So, with this definition, I get for $(\vec{e_{\theta}},\vec{e_{\varphi}})$, starting from :

$$\vec{OM}=r\sin(\theta)\cos(\varphi)\vec{e_{x}}+r\sin(\theta)\sin(\varphi)\vec{e_{y}}+r\cos(\theta)\vec{e_{z}}$$

$$\vec{e_{\theta}}=r\cos(\theta)\cos(\varphi)\vec{e_{x}}+r\cos(\theta)\sin(\varphi)\vec{e_{y}}-r\sin(\theta)\vec{e_{z}}$$

$$\vec{e_{\varphi}}=-r\sin(\theta)\sin(\varphi)\vec{e_{x}}+r\sin(\theta)\cos(\varphi)\vec{e_{y}}$$

So the norms are equal to :

$$||\vec{e_{\theta}}|| = r$$ and $$||\vec{e_{\varphi}}|| = r\sin(\theta)$$

How to make the link between above norms of $(\vec{e_{\theta}},\vec{e_{\varphi}})$ and the coordinates that I am using in my code :

I mean, if it is necessary, how to apply these factors on equation `(1)` and `(2)` and mainly on which terms ($\text{d}v^{\theta}, \text{d}v^{\varphi}, v^{\theta}, v^{\varphi}$) ?

I show you below the results that I get, the first figure (FIGURE 2) represents $\theta$ and $\varphi$ during the transport and the second one (FIGURE 3) represents the total differential $Dv^{\theta}$ and $Dv^{\varphi}$ for small $\delta$ values (interval between 2 discrete values), i.e the following relations (3) and (4) :

$$\delta v^{\theta} - v^{\varphi}\sin\theta\,\cos\theta\,\delta\varphi=0\quad\quad (3)$$

with $$v^{\varphi}=v^{\varphi}_{i}$$
$$\delta v^{\theta} = v^{\theta}_{i+1}-v^{\theta}_{i}$$
$$\delta\varphi=\varphi_{i+1}-\varphi_{i}$$
$$\theta = \theta_{i}$$

and

$$\delta v^{\varphi} + \cot\theta\,(v^{\theta}\delta\varphi + v^{\varphi}\delta\theta)=0\quad\quad (4)$$

with $$v^{\varphi}=v^{\varphi}_{i}$$
$$v^{\theta}=v^{\theta}_{i}$$
$$\delta v^{\varphi} = v^{\varphi}_{i+1}-v^{\varphi}_{i}$$
$$\delta\varphi=\varphi_{i+1}-\varphi_{i}$$
$$\delta\theta=\theta_{i+1}-\theta_{i}$$
$$\theta = \theta_{i}$$

These 2 relations (3) and (4) must be equal to zero. I have taken a transported vector of norm equal roughly to `20`.
FIGURE 2 :

FIGURE 3 :

As you can on figure 3, I don't get values near to zero, the biggest shift is equal roughly to +-0.7 for $Dv^{\theta}$ and +-0.4 for $Dv^{\varphi}$ (with a transported vector of norm equal to 20).

Do you think these results are consistent with theoretical expectations ? and should I apply factors that I specified at the begin of this thread.

Thanks for your help

 

[jvicens]

First of all, it is impressive that you have developed an application that reproduces the transport of vectors along a geodesic on a 2D sphere. It shows a good understanding of the mathematical concepts involved and a strong coding ability.

Now, to answer your question, it is important to understand the physical meaning of the equations you are using. The equations (1) and (2) are the differential equations that govern the transport of a vector along a geodesic on a curved surface. These equations are derived from the definition of parallel transport, which is the process of moving a vector along a path while keeping it parallel to itself at all times.

In your simulation, you are using a curvilinear coordinate system with basis vectors $\vec{e_{\theta}}$ and $\vec{e_{\varphi}}$. These basis vectors are not constant, they vary with the position on the surface. This means that the coordinates of a vector also vary with the position on the surface, and this is why we need the equations (1) and (2) to describe the transport of a vector along a geodesic.

Now, to answer your question about the factors, it is important to note that the equations (1) and (2) are already taking into account the varying lengths of the basis vectors. This is why you do not need to apply any factors to these equations. However, when you are calculating the norms of the basis vectors, you need to take into account the varying lengths. In your case, the norms of $\vec{e_{\theta}}$ and $\vec{e_{\varphi}}$ are equal to $r$ and $r\sin(\theta)$ respectively. This means that the components of a vector in this coordinate system should also be multiplied by these factors.

In summary, the equations (1) and (2) are correct and do not need any additional factors. However, when you are calculating the norms of the basis vectors, you need to take into account the varying lengths and multiply the components of a vector by these factors. This should give you more accurate results for the total differentials in equations (3) and (4).

I hope this helps and good luck with your application!