# Why is the direction of a vector in the gravitational field of a non-rotating mass changed after a parallel transport around the mass?

If you paralllel transport a vector (for example angular momentum vector, though this is strictly speaking a pseudovector) around a non-rotating spherical mass (like the earth) its direction is changed when it returns to its begin position. The space part of the curvature around the mass can be compared to a very flat cone. I can see how the direction of a vector changes while you parallel transport the vector over a sphere like you can see in the article in the link. But how can you visualise the parallel transport of a vector around the curved space of the non-rotating mass (represented as a flat cone)? I tried it with a round piece of paper. After removing a "piece of cake" I glued the to emerging edges together, so a cone formed, but I couldn't see a change in the direction of a vector when I tried to parallel transport it around the cone. The change should depend on the magnitude of the "piece of cake" (the angle in the centre of the piece of round paper after cutting out the "piece of cake"), but I couldn't see a change in direction.

• What makes you think that for a circular path on the surface of a sphere (i.e. the base of a cone) there is a change in direction? – JMLCarter Feb 4 '17 at 19:28
• @JMLCarter-I didn't mean a circular path on a sphere, wich obviously doesn't change the direction of a vector if parallel transported on a circular path on a sphere. I was referring to the vector that is parallel transported on a sphere in the article behind the link. – descheleschilder Feb 5 '17 at 18:49
• @JMLCarter-see en.wikipedia.org/wiki/Gravity_Probe_B for the flat cone. – descheleschilder Feb 5 '17 at 19:21