An exact derivation of both the Schwarzschild metric of general relativity and its equations of motion is made by the use of Newtonian mechanics. Although the form of Newtonian mechanics itself is not modified, the concepts of length and time on which it is based are modified in a manner that allows Newton's laws to be expressed in a non-Euclidean space-time geometry. The lengths used in the laws are defined in terms of local-scale measured distances, rather than the usual coordinate distances. Particle velocities are then defined in terms of these differential scale lengths. The Newtonian law of gravitation is also defined in terms of the gradient of the usual Newtonian potential with respect to these same scale lengths. It is shown that non-Euclidean geometry is imposed by the requirement that a photon in a gravitational field should maintain & constant total energy that is expressed in terms of its frequency, while also having a potential energy that is independent of the geometry of space. These conditions and modifications make it possible to derive equations of motion which are Newtonian, but which can also be reduced to forms that are identical to the Schwarzschild equations of motion for an orbiting particle or a gravitationally deflected photon.
