Connection coefficients as derivatives of parallel propagator


by ianhoolihan
Tags: coefficients, connection, derivatives, parallel, propagator
ianhoolihan
ianhoolihan is offline
#1
Mar25-12, 07:33 PM
P: 145
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/Paralle...llel_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.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
ianhoolihan
ianhoolihan is offline
#2
Mar26-12, 03:39 PM
P: 145
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


Register to reply

Related Discussions
Covariant derivative of connection coefficients? Special & General Relativity 14
Table of Connection Coefficients Special & General Relativity 6
Spherical connection coefficients Advanced Physics Homework 2
Vector parallel propagator on a CTC in GR General Physics 0
Connection coefficients entering differential operators Differential Geometry 1