For high IQ cells only. How parallel displacement works in General Relativity. Dirac

[Aristotélēs]

Suppose we have a vector A^μ located at a point P. If the space is curved, we cannot give a meaning to a parallel vector at a different point Q, as one can easily see if one thinks of the example of a curved two-dimensional space in a three-dimensional Euclidean space. However, if we take a point P' close to P, there is a parallel vector at P', with an uncertainty of the second order, counting the distance from P to P' as the first order. Thus we can give a meaning to displacing the vector A^μ from P to Q keeping it parallel to itself and keeping the length constant.

We can transfer the vector continuously along a path by this process of parallel displacement. Taking a path from P to Q, we end up with a vector at Q which is parallel to the original vector at P with respect to this path. But a different path would give a different result. There is no absolute meaning to a parallel vector at Q. If we transport the vector at P by parallel displacement around a closed loop, we shall end up with a vector at P which is usually in a different direction