Connection coefficients as derivatives of parallel propagator

In summary, the conversation is about understanding the relationship between the parallel propagator and the connection in terms of a covariant derivative. The person is looking for help understanding how the connection coefficients are the derivatives of the parallel propagator and is asking for suggestions or hints.
  • #1
ianhoolihan
145
0
Hi all,

I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a little trouble.

I've been thinking of the parallel propagator as in http://en.wikipedia.org/wiki/Parall...ng_the_connection_from_the_parallel_transport. I understand how to formulate the covariant derivative
[tex]
\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\Big|_{t=0}.
[/tex]
(Actually not really the second equality -- is this letting [itex]V_{\gamma(0)}=\Gamma(\gamma)_0^0V_{\gamma(0)}[/itex]?)

However, the above link doesn't really show what the connection is. Yet, if you evaluate the last term of the above equation, then using the product rule you've got a term with the derivative of [itex]V[/itex] and a term with the derivative of the parallel propagator. This is what you'd expect for the covariant derivative of a vector, where the connecton coefficients are the derivatives of the parallel propagator.

Nonetheless, I'm unable to make this all work out mathematically, so I was wondering if anyone could give me a hint? That is, on how the connection coefficients are the derivatives of the parallel propagator.

Cheers.
 
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  • #2
Well it looks like lots of people have viewed this thread, so I guess that means some interest. Can anyone shed some light, or offer a suggestion?

Cheers
 

1. What are connection coefficients?

Connection coefficients are mathematical quantities that describe the change in a vector or tensor as it is transported along a curved manifold. They are also known as Christoffel symbols or affine connection coefficients.

2. How are connection coefficients related to parallel propagators?

Connection coefficients are derivatives of the parallel propagator, which is a tool used to transport vectors or tensors along a curved manifold in a parallel manner. The connection coefficients determine the direction and magnitude of the transported vector or tensor at each point on the manifold.

3. What are the applications of connection coefficients?

Connection coefficients are used in various fields of science, including general relativity, differential geometry, and differential equations. They are essential for understanding the behavior of objects in curved spaces and for solving mathematical equations on curved manifolds.

4. How are connection coefficients calculated?

Connection coefficients can be calculated using the metric tensor, which describes the geometry of a curved manifold. The specific method for calculating them depends on the specific manifold and metric being used.

5. What is the relationship between connection coefficients and curvature?

Connection coefficients are closely related to the curvature of a manifold. In fact, the curvature tensor can be expressed in terms of connection coefficients. This relationship is crucial for understanding the geometric properties of a curved space.

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