\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

V. Parallel Transport

PT Equation

   A vector ${\mathbf{Y}}(\lambda )$ is said to be parallel transported along ${\mathbf{x}}(\lambda )$, if the coderivative along the curve vanishes

5.5

\[  {\mathbf{u}} \cdot D{\mathbf{Y}} = 0  \]

In view of (5.3) the coordinate form of the PT equation (5.5) is

5.6

\[  \left( {\frac{{d{Y^\nu }}}{{d\lambda }} + \frac{{d{x^\mu }}}{{d\lambda }}\Gamma _{\mu \kappa }^\nu {Y^\kappa }} \right) {{\mathbf{g}}_\nu } = 0  \]

If vectors are decomposed with respect to a tetrad frame, the PT equation takes the form:

5.7

\[  {\mathbf{u}} \cdot \nabla {\mathbf{Y}} + {\mathbf{\omega}}(u) \cdot {\mathbf{Y}} = 0 \]

with ${\mathbf{\omega}}(u): = {u^\mu }{{\mathbf{\omega }}_\mu }$ and ${\mathbf{u}} \cdot \nabla = {u^\mu }{\partial _\mu }$ the directional derivative which acts on the components ${Y^a}$ relative to the tetrad frame.

   Thus, a vector ${\mathbf{Y}}(\lambda )$ is parallel-transported in the ${\mathbf{u}}$-direction, if the variation in the vector components is precisely compensated by the transformation of the transported frame along the given curve. The importance of this procedure is that in this way vectors and tensors that belong to different tangent spaces can be compared with each other. It is worth emphasizing that the affine connection structure of the spacetime manifold is crucial to this purpose.

   The equation of parallel transport is a first-order differential system in the unknowns ${Y^\mu }(\lambda )$ or ${Y^a }(\lambda )$.  When the initial vector ${\mathbf{Y}}({\lambda _0})$ has been specified by the initial data, there exists a unique solution of the equations (5.6), or (5.7), along the curve of transport. The solution depends on the curve ${\mathbf{x}}(\lambda )$. Parallel transport is also dependent on the connection, and different connections lead to different answers. If the connection is metric-compatible, see (1.28), the metric is always parallel transported: ${\mathbf{u}} \cdot D{g_{\mu \nu }} = 0$. It follows that the inner product of two parallel-transported vectors is preserved as is their norm, i.e. parallel transport is an isometry.