Dynamical Content of Differential Geometry

Abstract

A formulation of the notion of a Cartesian basis for a vector space suitably adapted to the local differential vector space at a space-time point leads to this description of and derivation for the law of motion of a mass point: There exists a class of equivalent 4-dimensional Cartesian frames in which the trajectory is a straight line, the velocity being constant along the trajectory. The lived-in coordinate system, in which the trajectory is in general not a straight line, is in general not a Cartesian coordinate system. The equation for the motion of the point is purely a statement of differential geometry—an integrability condition—and the forces which are responsible for the deviation from constant velocity are immediately derivable from the metric tensor of the non-Cartesian system. The equation of motion and the relations among dynamical variables are more relativistic than Newtonian, but the metric tensor is positive definite; no indefinite metric is needed. A Newtonian equation of motion is an approximation based on small velocity and it covers the general case of motion in a noninertial system in an external field of force. Intrinsic mass is identified as a constant of the motion and, although the relevancy of mass is somewhat limited in this description of the motion of a single point in an immutable force field, it is seen that the effective mass of the point contains a contribution by the potential energy.