I am new to general relativity, and trying to understand parallel transport.
I want to transport a vector along a line of longitude, on a sphere, starting at the equator, and ending (just shy) of the North Pole. I want the vector to point south initially, in the $\theta$ direction.
If I understand Foster & Nightingale correctly, when parallel transporting a vector along a geodesic, the direction of the vector should not change.
But at the starting point (equator), the vector points "down/vertically," while at the ending point (North Pole) it is pointing horizontally. So the direction has changed?!
Is it because I am thinking "globally?" For example, if I look at the local coordinate system at the starting point, the vector is pointing in the $\theta$ direction. At the end point (just shy of the North Pole), although "horizontal" now, if I go to the local coordinate system, the vector is still pointing in the $\theta$ direction. In this sense, its direction has not changed. Is this the correct way to view things?