I have a question about an exercise in Misner, Thorne, and Wheeler's Gravitation. On page 321, exercise 13.5 says
Show that a geodesic of spacetime which is timelike at one event is everywhere timelike. Similarly, show that a geodesic initially spacelike is everywhere spacelike, and a geodesic initially null is everywhere null. [Hint: This is the easiest exercise in the book!]
Ok so the idea is that timelike means that the timelike geodesic have a negative-length 4-velocity, a spacelike would have a positive-length 4-velocity and a null would have a 0-length 4-velocity. So my idea was to use the definition of the geodesic to show that the sign of the length of the 4-velocity doesn't change. The definition of the geodesic translates to a curve that parallel transport conserves the tangent vector to the curve.
My idea was to say that the tangent vector of a spacelike is always in the space parts of spacetime. But it doesn't seem true at all. Finally the Hint doesn't help me much.
I assume there are two kinds of solutions for this problem: one purely abstract with words and one with a more mathematical approach of the concept of geodesics and tangent vectors.