The Dynamics of Continuous Media and the Notion of an Affine Connection on Space-Time

Abstract

1. Newton’s foundation of classical mechanics rests on the concepts of absolute time and absolute space. Thus, analytically, any event can be labelled in time and space provided a choice is made of an origin, a unit of time, and a frame of spatial coordinates. For example, the frame might originate at the center of mass of the solar system and its axes might point towards fixed stars. Of course, any other frame which is invariantly related to this one would be also admissible. As is well known, the laws of mechanics remain unaffected if the frame of spatial coordinates is made to undergo a rectilinear, uniform translation with respect to Newton’s absolute space, keeping the absolute time undisturbed. One is thus led to the notion of Galilean frames of reference. The principle of inertia may be stated as follows: in absence of interactions with other bodies, the velocity of a point mass remains constant in direction and magnitude in any Galilean frame. The fact that the validity of this principle in one Galilean frame implies its validity in any other Galilean frame follows immediately from the transformation laws governing these frames. Let us label a point in space by arbitrary Cartesian coordinates,1 not necessarily orthogonal. Then the transformation laws are as follows: