The first equality is the definition of the derivative of a function (in this case Vα ) along the curve; the second equality comes from the fact that α μν = 0 at P in these coordinates. But the third equality is a frame-invariant expression and holds in any basis, so it can be

taken as a frame-invariant definition of the parallel-transport of V along U:

The last step uses the notation for the derivative along U introduced in Eq. (3.67).

Geodesics

The most important curves in flat space are straight lines. One of Euclid’s axioms is that two straight lines that are initially parallel remain parallel when extended. What does he mean by ‘extended’? He doesnít mean ‘continued in such a way that the distance between them remains constant’, because even then they could both bend. What he means is that each line keeps going in the direction it has been going in. More precisely, the tangent to the curve at one point is parallel to the tangent at the previous point. In fact, a straight line in Euclidean space is the only curve that parallel-transports its own tangent vector! In a curved space, we can also draw lines that are ‘as nearly straight as possible’ by demanding parallel-transport of the tangent vector. These are called geodesics:

(Note that in a locally inertial system these lines are straight.) In component notation:

Uβ Uα ;β = Uβ Uα ,β + α μβ UμUβ = 0. (6.50)

Now, if we let λ be the parameter of the curve, then Uα = dxα /dλ and Uβ ∂/∂xβ = d/dλ: