Curved Space and Gravitation. II, Robert W. Brehme, Am. J. Phys. 33, 713 (1965A)
A connection is made between geodesics on a curved, two-dimensional space-time surface and the trajectories of objects falling freely in straight paths in time-independent gravitational fields. The treatment is presented on an elementary level, and the Einstein theory is not closely followed. Both geodesic motion and classical gravitation cause accelerations of objects which are not influenced by the intrinsic properties of the object. The acceleration for geodesic motion is velocity as well as position dependant, whereas for classical gravitation, the acceleration depends only on position. However, by the appropriate change of scale, it is possible to approximate the classical trajectory with geodesic motion to any degree of accuracy. Geodesics on the sphere and cone are given as examples.