Moving in curved space-time

This represents two of the four dimensions. Just as the Earth appears flat to us when we are looking at only a tiny portion of it, this curved space-time will also appear flat if we zoom in and look at only a tiny section. If we approximate this region with the mathematics of flat space-time, we can define an axis for one of the three spatial dimensions and an axis for time. Suppose we move forward along the time axis seen by the observer moving in the direction of the red arrow. Our approximation for flat space-time is now no longer valid. We correct for this by rotating our red arrow along the line 90 degrees to the surface. We can now create a new flat space-time approximation for this new region. We can use this new approximation to again move forward in the direction of the red arrow. If we repeat this over and over again, using infinitesimally small time steps, we get the path that the object will follow through curved space-time. The path appears curved to us, thereby creating the illusion of a gravitational force. Suppose we move a vector by an infinitesimally small amount without changing the direction it is pointing. Every time we do this, we need to take the curvature into account by rotating the vector along the line 90 degrees to the surface. If we repeat this over and over again, we can move a vector over long distances, which we refer to as parallel transport.