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Parallel transport on the sphere

  • Thread starter Pietjuh
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Homework Statement


Consider a closed curve on a sphere. A tangent vector is parallel transported around the curve. Show that the vector is rotated by an angle which is proportional to the solid angle subtended by the area enclosed in the curve.

The Attempt at a Solution


First, I parametrize the curve by a parameter [tex]\lambda[/tex], and write the coordinates as [tex]x^{\mu}(\lambda)[/tex].
It seems like a good idea to look at the parallel transport equation for the tangent vector [tex]V^{\mu}[/tex]:

[tex] \frac{d}{d\lambda}V^{\mu} + \Gamma^{\mu}_{\sigma\rho}\frac{dx^{\sigma}}{d\lambda} = 0[/tex]

My textbook says, that I can solve this equation exactly by using the parallel propagator P, defined by:
[tex]V^{\mu}(\lambda) = P^{\mu}_{\nu}(\lambda,\lambda_0) V^{\mu}(\lambda_0)[/tex]

The form of P is also given by:
[tex]P^{\mu}_{\nu}(\lambda,\lambda_0) = \mathcal{P}\mathrm{exp}\left(-\int^{\lambda}_{\lambda_0} A^{\mu}_{\nu}d\eta\right)[/tex]
where [tex]\mathcal{P}[/tex] is the path ordering symbol, and [tex]A^{\mu}_{\nu} = - \Gamma^{\mu}_{\sigma\nu}\frac{dx^{\sigma}}{d\lambda} [/tex]

So I think that i'm able to rewrite this as:
[tex]
P^{\mu}_{\nu}(\lambda,\lambda_0) = \mathcal{P}\mathrm{exp}\left(\oint \Gamma^{\mu}_{\sigma\nu}dx^{\sigma}\right) = \mathcal{P}\mathrm{exp}\left(\oint A^{\mu}_{\nu}\right)[/tex]

where I now consider A to be the matrix of 'connection 1-forms'. Since I'm looking at a closed loop, the matrix P is basically just a rotation in the 2-dimensional tangent space of the sphere at the point [tex]x^{\nu}(\lambda_0)[/tex], so P is an element of SO(2). So I can also write P as:
[tex]
P^{\mu}_{\nu}(\lambda,\lambda_0) = e^{\alpha(\lambda,\lambda_0) \left(\begin{array}{cc}0&1 \\ -1 & 0 \end{array}\right)}[/tex]

So it seems to me that it is possible to equate the two terms in the exponentials to each other. My idea was then that I would use Stokes's theorem to convert the circular integral of the matrix of one-forms to a surface integral. I could then relate [tex]\alpha[/tex] to this surface integral, so that I hopefully would obtain something proportional to the solid angle.

The problem however, is that I don't really know how to use Stoke's theorem on a matrix of differential 1-forms, and if it is even possible to do. The issue of the path ordering symbol also bothers me a bit.... Could somebody help me out a bit on this please? :confused:
 
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Answers and Replies

  • #2
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Hi I'm also considering a similar problem. Do you mind telling me which textbook you are referring to? Thank you very much in advance.
 

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