Tetrads in General Relativity

V. Parallel Transport

A curve $x (λ)$ is said to be auto-parallel or a geodesic curve if its tangent vector satisfies the condition

5.8

u^{μ} D_{μ} u = u \cdot D u = 0

That is, if it remains parallel to itself along $x (λ)$ . 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 $λ$ is changed $λ \to τ$ , the tangent vector changes:

5.9

u (τ) = \frac{d x}{d λ} \frac{d λ}{d τ} = u (λ) \frac{d λ}{d τ}

The auto-parallel character of the curve is not modified by this parameter change if and only if the parameter is affine, that is, $λ = a τ + b$ , 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 $[x_{1}, x_{2}]$ . From the line element $d s^{2} = d x \cdot d x$ , one defines the length of a curve $x (λ)$ between the given points by the proper lenght integral

5.10

s [x (λ)] := \int_{1}^{2} d s = \int_{1}^{2} \sqrt{| d x \cdot d x |} = \int_{1}^{2} \sqrt{| u \cdot u |} d λ

where the integral is over the path. If the manifold $M_{4}$ 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.