A metric-space formulation of Newtonian fields

Abstract

In the usual curved-space description of gravity, a class of fields is defined which correspond to the fields of Newtonian gravitational theory. Using these fields, the field equations of Newtonian theory are formulated in a four-dimensional metric space. The equations are then modified so that they transform properly under the Lorentz transformation and so that their weak-field approximation is closely analogous to the equations of classical electrodynamics. The resulting equations lead to Newtonian theory in the non-relativistic limit, and they lead rigorously to the Schwarzschild field and to the known relativistic corrections associated with it. Finally, these field equations are compared with Einstein's field equations.