The parallel transporter in terms of the connection

We can also consider the parallel transport of a vector \({w}\) along an infinitesimal curve \({C}\) with tangent \({v}\). Referring to the previous figure, we see that to order \({\varepsilon}\) the components \({w^{\mu}}\) transform according to

\(\displaystyle \parallel^{\lambda}{}_{\mu}\left(C\right)w^{\mu}=w^{\lambda}-\varepsilon\Gamma^{\lambda}{}_{\mu}\left(v\right)w^{\mu}, \)

where \({v}\) is tangent to the curve \({C}\), and these components are with respect to the frame at the new point after infinitesimal parallel transport. Using this relation, we can build up a frame-dependent expression for the parallel transporter for finite \({C}\) by multiplying terms \({\left(1-\varepsilon\Gamma\left|_{p}\right.\right)}\) where \({\Gamma\left|_{p}\right.}\) is used to denote the matrix \({\Gamma^{\lambda}{}_{\mu}\left(v\left|_{p}\right.\right)}\) evaluated on the tangent \({v\left|_{p}\right.}\) at successive points \({p}\) along \({C}\). The limit of this process is the path-ordered exponential

\begin{aligned}\parallel^{\lambda}{}_{\mu}\left(C\right) & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\left(1-\varepsilon\Gamma\left|_{q-\varepsilon}\right.\right)\left(1-\varepsilon\Gamma\left|_{q-2\varepsilon}\right.\right)\dotsm\left(1-\varepsilon\Gamma\left|_{p+\varepsilon}\right.\right)\left(1-\varepsilon\Gamma\left|_{p}\right.\right)\\ & \equiv P\textrm{exp}\left(-\underset{C}{\int}\Gamma^{\lambda}{}_{\mu}\right),
\end{aligned}

whose definition is based on the expression for the exponential

\(\displaystyle e^{x}=\underset{n\rightarrow\infty}{\textrm{lim}}\left(1+\frac{x}{n}\right)^{n}=\underset{\varepsilon\rightarrow\infty}{\textrm{lim}}\left(1+\varepsilon x\right)^{1/\varepsilon}. \)

Note that the above expression for \({\parallel^{\lambda}{}_{\mu}\left(C\right)}\) exponentiates frame-dependent values in \({gl\left(n,\mathbb{R}\right)}\) to yield a frame-dependent value in \({GL\left(n,\mathbb{R}\right)}\).

An Illustrated Handbook