Coordinate-independence of equation for the parallel transport

In summary, the conversation discusses the process of showing that the defining equation for the parallel transport of a contravariant vector is coordinate-independent. The conversation includes a derivation and a correction to the previous attempt at solving the problem. The final result shows that the equation holds true in any coordinate frame.
  • #1
rbwang1225
118
0

Homework Statement


Please show that the defining equation for the parallel transport of a contravariant vector along a curve [itex]\dot{\lambda}^a+\Gamma^a_{bc}\lambda^b\dot{x}^c=0[/itex] is coordinate-independent, given that the transformation formula for the christoffel symbol being ##\Gamma^{a'}_{b'c'}=(\Gamma^{d}_{ef}X^{a'}_d-X^{a'}_{ef})X^{e}_{b'}X^{f}_{c'}##.

The Attempt at a Solution


I have stuck by the following derivation ##X^{a'}_{ab}\dot x^b\lambda^a+X^{a'}_a\dot\lambda^a+(\Gamma^{d}_{ef}X^{a'}_d-X^{a'}_{ef})X^{e}_{b'}X^{f}_{c'}(X^{c'}_{ef}\dot x^fx^e+X^{c'}_e\dot x^e)##, where I can't simplify it to the unprimed equation [itex]\dot{\lambda}^a+\Gamma^a_{bc}\lambda^b\dot{x}^c=0[/itex].
Any advice will be appreciated!
 
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  • #2
Hello!
After think twice, I met some mistakes in the derivation.
First, dummy indexes were repeated in the same term.
Second, tangent vectors are transformed by ##\dot{x} ^{a'}=X^{a'}_a\dot x^a##

Now, I describe my derivation as follows.

The parallel transport equation for contravariant vector ##\lambda ^{a'}## along a curve ##\gamma## parametrized by t
##\dot\lambda^{a'}+\Gamma^{a'}_{b'c'}\lambda^{b'} \dot{x}^{c'}=0##
was transformed as
##\frac{dX^{a'}_a\lambda^a}{dt}+(\Gamma^{d}_{ef}X^{a'}_dX^{e}_{b'}X^f_{c'}-X^e_{b'}X^f_{c'}X^{a'}_{ef})(X^{b'}_b\lambda^bX^{c'}_c\dot x^c)##
##=X^{a'}_{ab}\lambda^a\dot x^b+X^{a'}_a\dot\lambda^a+\Gamma^d_{ef}X^{a'}_d \lambda ^e \dot x^f-\lambda^e \dot x^fX^{a'}_{ef}##
##=X^{a'}_a(\dot\lambda^a+\Gamma^a_{bc}\lambda^b \dot x^b)=0##
Since ##X^{a'}_a ## are arbitrary coefficients of transformation, ##\dot\lambda^a+\Gamma^a_{bc}\lambda^b \dot x^c)=0## in ##x^a## coordinate frame, as desired.

If I have fault, please kindly inform me.
 

1. What is the concept of coordinate-independence in the context of parallel transport?

Coordinate-independence refers to the property of an equation or concept remaining the same regardless of the choice of coordinate system used to describe it. In the case of parallel transport, this means that the equation for calculating the change in a vector's direction or magnitude remains the same regardless of the coordinates used to describe the path along which the vector is transported.

2. Why is it important for the equation for parallel transport to be coordinate-independent?

If the equation for parallel transport were not coordinate-independent, it would mean that the result of the calculation would depend on the choice of coordinates used. This would make it difficult to compare results or make predictions, as different coordinate systems could give different answers. By being coordinate-independent, the equation for parallel transport allows for consistent and accurate calculations regardless of the coordinate system used.

3. How is the coordinate-independence of the equation for parallel transport related to the concept of a covariant derivative?

The coordinate-independence of the equation for parallel transport is closely related to the concept of a covariant derivative. A covariant derivative is a mathematical tool used to calculate the change in a vector as it is transported along a path. By being coordinate-independent, the equation for parallel transport can be expressed in terms of a covariant derivative, allowing for a more elegant and general formulation of the concept.

4. Are there any practical applications of the coordinate-independence of the equation for parallel transport?

Yes, the coordinate-independence of the equation for parallel transport has practical applications in various fields such as physics, engineering, and computer graphics. It is used in the calculation of geodesics (the shortest paths between points) in curved spaces, as well as in the study of electromagnetic fields and fluid flow. In computer graphics, the concept of parallel transport is used to smoothly interpolate between orientations of objects in animation.

5. Is the coordinate-independence of the equation for parallel transport a fundamental property of nature?

Yes, the coordinate-independence of the equation for parallel transport is a fundamental property of nature. It is a result of the intrinsic nature of space and time, as described by Einstein's theory of general relativity. This theory states that the laws of physics should be the same regardless of the choice of coordinate system used to describe them, and the coordinate-independence of the equation for parallel transport is consistent with this fundamental principle.

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