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V. Parallel Transport
Consider a smooth spacetime curve
${\mathbf{x}}(\lambda )$ parametrized by a parameter $\lambda $. The
tangent vectors to this curve are given by
\[ {\mathbf{u}}(\lambda )
: = \frac{{d{\mathbf{x}}}}{{d\lambda }} = \frac{{\partial
{\mathbf{x}}}} {{\partial {x^\mu }}}\frac{{d{x^\mu }}}{{d\lambda }}
= {u^\mu }{\partial _\mu }{\mathbf{x}} = {u^\mu }(\lambda
){{\mathbf{g}}_\mu }({\mathbf{x}}(\lambda )) \]
with respect to the coordinate basis along the curve.
For a vector ${\mathbf{Y}}(\lambda ): =
{\mathbf{Y}}({\mathbf{x}}(\lambda ))$, the directional
coderivative along the curve is defined as
\[
\frac{{D{\mathbf{Y}}}}{{d\lambda }}: = {u^\mu }{D_\mu }{\mathbf{Y}}
= {\mathbf{u}} \cdot D{\mathbf{Y}} \]
This is a natural generalization of ${D_\mu }: = {{\mathbf{g}}_\mu
} \cdot D$, the coderivative along the basis vector
${{\mathbf{g}}_\mu }$. In the literature the directional derivative
${\mathbf{u}} \cdot D$ is often denoted as (bold) ${\nabla
_{\mathbf{u}}}$.
Any vector ${\mathbf{Y}}$ can be expanded in terms of the basis
vectors of the coordinate frame as ${\mathbf{Y}} = {Y^\nu
}{{\mathbf{g}}_\nu }$. In view of (1.17) the coordinate form of
(5.2) is then
\[ {u^\mu }{D_\mu
}{\mathbf{Y}} = {u^\mu }({\partial _\mu }{Y^\nu } + \Gamma _{\mu
\kappa }^\nu {Y^\kappa }) {{\mathbf{g}}_\nu } \]
If the same vector is expanded with respect to the local tetrad
frame $\{ {{\mathbf{e}}_a}\} $, the connection is determined
by the bivector (4.5):
\[ {u^\mu }{D_\mu
}{\mathbf{Y}} = ({u^\mu }{\partial _\mu }{Y^a}) {{\mathbf{e}}_a} +
{u^\mu }{{\bs{\omega }}_\mu } \cdot {\mathbf{Y}} \]