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

Tetrads in General Relativity

V. Parallel Transport

Geodesics

   A curve ${\mathbf{x}}(\lambda )$ is said to be auto-parallel or a geodesic curve if its tangent vector satisfies the condition

5.8

\[  {u^\mu }{D_\mu }{\mathbf{u}} = {\mathbf{u}} \cdot D{\mathbf{u}} = 0  \]

That is, if it remains parallel to itself along ${\mathbf{x}}(\lambda )$. In GRT, a geodesic curve generalizes the notion of a ‘straight line’ to curved spacetime, namely, by being as straight as possible.

   If the parameter $\lambda $ is changed $\lambda \to \tau $, the tangent vector changes:

5.9

\[  {\mathbf{u}}(\tau ) = \frac{{d{\mathbf{x}}}}{{d\lambda }}\frac{{d\lambda }}{{d\tau }} = {\mathbf{u}}(\lambda )\frac{{d\lambda }}{{d\tau }}{\text{ }}  \]

The auto-parallel character of the curve is not modified by this parameter change if and only if the parameter is affine, that is, $\lambda = a\tau + b$, where $a,b$ are arbitrary constants. This linear scaling of the path amounts to the freedom to change the unit of length/time and to choose any initial point.

   Geodesics may be characterized as a path of extremal length between two given spacetime points $[{{\mathbf{x}}_1},{{\mathbf{x}}_2}]$. From the line element $d{s^2} = d{\mathbf{x}} \cdot d{\mathbf{x}}$, one defines the length of a curve ${\mathbf{x}}(\lambda )$ between the given points by the proper lenght integral

5.10

\[s[{\mathbf{x}}(\lambda )] : = \int\limits_1^2 {ds} = \int\limits_1^2 {\sqrt {\left| {d{\mathbf{x}} \cdot d{\mathbf{x}}} \right|} } = \int\limits_1^2 {\sqrt {\left| {{\mathbf{u}} \cdot {\mathbf{u}}} \right|} } d\lambda \]

where the integral is over the path. If the manifold $\mathcal{M}_4$ is connected, any two spacetime points can be connected by a smooth curve. The geodesic is then that curve that extremizes the proper length $s$. From definition (5.10) it is seen that this result depends neither on the choice of coordinates nor on the parametrization of the curve.