Solving the same question two ways: Parallel transport vs. the Lie derivative

In summary, the author is looking for an understanding of the Lie derivative of a vector field and how it relates to the transport of a vector on a curve.
  • #1
Markus Kahn
112
14
Homework Statement
Assume we are working with the metric
$$ds^2 = R^2 d\theta^2 + R^2 \sin^2(\theta)d\varphi^2,$$
where ##R=const.##. One can easily check that we then have
$$\Gamma^{\theta}_{\varphi\varphi} = - \sin\theta\cos\theta\quad \text{and}\quad \Gamma^{\varphi}_{\varphi\theta}=\Gamma^{\varphi}_{\theta\varphi}=\cot \theta .$$

a) Consider the parallel transport of a vector ##X=(X_0^\theta,X_0^\varphi)## around a curve ##\gamma=(\theta_0,\varphi)##, where ##\theta_0=const.## and ##\varphi\in [0,2\pi)##, on the 2-sphere.

b) Perform the same transport of the vector using the Lie derivative and compare the results.

(It's been a while since I did this exercise and I'm not 100% sure if I remembered everything correctly, the main point was to perform the same job "transporting a vector", once with parallel transport and once with the Lie derivative...)
Relevant Equations
Parallel transport equation:
$$\dot{v}^i = -\Gamma^{i}_{jk} \dot{\gamma}^{j} v^k.$$
a) I found this part to be quite straight forward. From the Parallel transport equation we obtain the differential equations for the different components of ##X^\mu##:
$$
\begin{align*}
\frac{\partial X^{\theta}}{\partial \varphi} &=X^{\varphi} \sin \theta_{0} \cos \theta_{0}, \\
\frac{\partial X^{\varphi}}{\partial \varphi}&=-X^{\theta} \cot \theta_0.
\end{align*}
$$
Taking the derivative with respect to ##\varphi## of both equations and substituting them into each other we find
$$
\begin{align*}
\frac{\partial^{2} X^{\theta}}{\partial \varphi^{2}}&=-X^{\theta} \cos ^{2} \theta_{0},\\
\frac{\partial^{2} X^{\varphi}}{\partial \varphi^{2}}&=-X^{\varphi} \cos ^{2} \theta_{0}
\end{align*}
$$
so two times the same differential equation. Defining now ##\alpha:= \cos\theta_0##, both equations will be solved by
$$
\begin{aligned}
X^{\theta}(\varphi) &=A \cos \alpha \varphi+B \sin \alpha \varphi ,\\
X^{\varphi}(\varphi) &=C \cos \alpha \varphi+D \sin \alpha \varphi.
\end{aligned}
$$
Initial conditions ##X = (X_0^\theta, X_0^\varphi)## and
$$
\begin{array}{l}
\left.\frac{\partial X^{\theta}}{\partial \varphi}\right|_{\varphi=0}=X_{0}^{\varphi} \sin \theta_{0} \cos \theta_{0} \\
\left.\frac{\partial X^{\varphi}}{\partial \varphi}\right|_{\varphi=0}=-X_{0}^{\theta} \frac{\cos \theta_{0}}{\sin \theta_{0}}
\end{array}
$$
lead us to
$$
\begin{aligned}
X^{\theta}(\varphi) &=X_{0}^{\theta} \cos \left(\varphi \cos \theta_{0}\right)+X_{0}^{\varphi} \sin \theta_{0} \sin \left(\varphi \cos \theta_{0}\right) \\
X^{\varphi}(\varphi) &=X_{0}^{\varphi} \cos \left(\varphi \cos \theta_{0}\right)-\frac{X_{0}^{\theta}}{\sin \theta_{0}} \sin \left(\varphi \cos \theta_{0}\right).
\end{aligned}
$$

b) Here is where the problem starts. I know the definition of the Lie derivative of a tensor field ##T## as
$$
\left(L_{Y} T\right)_{p}=\left.\frac{d}{d t}\right|_{t=0}\left(\left(\tau_{-t}\right)_{*} T_{\tau_{t}(p)}\right),
$$
where ##(\tau_t)_*## is the push-forward. Now the tensor field in this exercise is a vector field so the expression simplifies dramatically,
$$L_Y(X) = [Y,X].$$
The problem is that I don't really know what I'm supposed to do with this and what exactly this has to do with the transport of a vector along a curve..

Any insights are appreciated.
 
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  • #2
Hopefully it helps.If you put some specific objects in the Lie bracket you check whether this objects are commute or not.If [X,Y]=XY-YX =0 the Objects(here vectorfields) commute.This leads to XY=YX otherwise they don't commute.Maybe you can try to interpret the geometry with this new knowledge .You can also check Misner and Wheeler(Gravitation) for a good reference on this field or some nice older post on this board.
 

Related to Solving the same question two ways: Parallel transport vs. the Lie derivative

1. What is the difference between parallel transport and the Lie derivative?

Parallel transport is a geometric concept that describes the movement of a vector or tensor along a curve in a curved space, while the Lie derivative is a mathematical operation that describes the change of a tensor field along a vector field. In simpler terms, parallel transport focuses on the movement of objects in a curved space, while the Lie derivative focuses on the change of objects in a curved space.

2. Which method is more commonly used in scientific research?

Both parallel transport and the Lie derivative are commonly used in scientific research, but the specific method used depends on the problem being studied. Parallel transport is often used in the study of general relativity and differential geometry, while the Lie derivative is commonly used in the study of fluid dynamics and other areas of physics.

3. Can parallel transport and the Lie derivative be used interchangeably?

No, parallel transport and the Lie derivative are not interchangeable. While they both describe the movement of objects in a curved space, they are fundamentally different concepts with different applications. It is important to use the appropriate method for the specific problem being studied.

4. How do parallel transport and the Lie derivative relate to each other?

Parallel transport and the Lie derivative are related through the concept of covariant differentiation. In parallel transport, a vector or tensor is transported along a curve without changing its direction, while in the Lie derivative, a vector or tensor is differentiated along a vector field. Covariant differentiation combines these two concepts and describes the change of a vector or tensor along a curve without changing its direction.

5. Are there any real-world applications of parallel transport and the Lie derivative?

Yes, both parallel transport and the Lie derivative have many real-world applications. Parallel transport is used in the study of general relativity, which has implications for our understanding of the universe. The Lie derivative is used in fluid dynamics, which has applications in weather prediction and aerodynamics. Both methods are also used in the study of differential geometry, which has applications in computer graphics and robotics.

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