Parallel transport approximation

In summary, the conversation discusses the parallel transport equation and the process of taking its derivative with respect to tau. The resulting equation is then substituted into a formula for parallel transporting small distances, and the validity of this method is questioned due to incorrect results in a previous exercise. However, upon locating a mistake, there is no reason to doubt the validity of the method.
  • #1
jostpuur
2,116
19
The parallel transport equation is

[tex]
\frac{d\lambda^{\mu}}{d\tau} = -\Gamma^{\mu}_{\sigma\rho} \frac{dx^{\sigma}}{d\tau} \lambda^{\rho}
[/tex]

If I take the derivative of this with respect to tau, and get

[tex]
\frac{d^2\lambda^{\mu}}{d\tau^2} = -\partial_{\nu}\Gamma^{\mu}_{\sigma\rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\sigma}}{d\tau} \lambda^{\rho} \;- \;\Gamma^{\mu}_{\sigma\rho}\frac{d^2x^{\sigma}}{d\tau^2}\lambda^{\rho} \;-\; \Gamma^{\mu}_{\sigma\rho}\frac{dx^{\sigma}}{d\tau}\frac{d\lambda^{\rho}}{d\tau}
[/tex]
[tex]
= \Big(-\partial_{\nu}\Gamma^{\mu}_{\sigma\rho} \frac{dx^{\nu}}{d\tau}\frac{dx^{\sigma}}{d\tau} \; - \;\Gamma^{\mu}_{\sigma\rho}\frac{d^2x^{\sigma}}{d\tau^2}\;+\; \Gamma^{\mu}_{\sigma\nu}\Gamma^{\nu}_{\alpha\rho} \frac{dx^{\sigma}}{d\tau}\frac{dx^{\alpha}}{d\tau}\Big)\lambda^{\rho}
[/tex]

and then substitute these into

[tex]
\lambda^{\mu}(\tau) = \lambda^{\mu}(0) \;+\; \tau\frac{d\lambda^{\mu}(0)}{d\tau} \;+\; \frac{1}{2}\tau^2\frac{d^2\lambda^{\mu}(0)}{d\tau^2} \;+\; O(\tau^3}),
[/tex]

have I already done something wrong, or is this a valid way to do parallel transport small distances?
 
Last edited:
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  • #2
I was getting wrong answers in one exercise, and started doubting this, but I located the mistake (or a mistake), and right now I have no particular reason to believe that there was anything wrong in this.
 
  • #3


Your approach seems to be correct. You have used the parallel transport equation to find the second derivative of the vector field, which can then be used to approximate the parallel transport over small distances. This is a valid way to do parallel transport as long as the curvature of the space is small and the distance being transported is also small. However, for larger distances or for spaces with significant curvature, this approximation may not hold and a more rigorous approach may be needed.
 

Related to Parallel transport approximation

What is parallel transport approximation?

Parallel transport approximation is a mathematical technique used to approximate the displacement of a vector along a curved path. It is based on the concept of parallel transport, which refers to the idea of moving a vector along a curved path while keeping it parallel to its original direction.

What is the purpose of parallel transport approximation?

The purpose of parallel transport approximation is to simplify the calculation of vector displacement along a curved path. This is especially useful in fields such as physics and engineering, where curved paths are common and accurate calculations of vector displacement are needed.

How does parallel transport approximation work?

Parallel transport approximation works by dividing the curved path into smaller segments and approximating the displacement of the vector along each segment. The sum of these approximations gives an overall approximation of the vector displacement along the entire curved path.

What are the limitations of parallel transport approximation?

One limitation of parallel transport approximation is that it is only an approximation and may not give an accurate result for highly curved paths. Additionally, it may not be suitable for certain types of vector fields, such as those with discontinuities or singularities.

What are the applications of parallel transport approximation?

Parallel transport approximation has many applications in physics, engineering, and computer graphics, where accurate calculations of vector displacement along curved paths are needed. It is also used in fields such as differential geometry and mathematical physics.

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