Relativistic Free Particles

Abstract

The fundamental assumption in Newtonian mechanics is the absolute nature of space and time. That our empirical knowledge of the physical world points up to the contrary was one of the cornerstones of Einstein’s great critique of space and time. The analysis of Einstein and Lorentz of the empirical and mathematical nature of the physical phenomena established that only space-time, as a four-dimensional manifold, has an invariant physical significance, and that its familiar splitting into space and time is essentially dependent on the observer. Each observer was thus seen to describe only the points of the space-time manifold in his vicinity by four coordinates; the formulas describing the transformation of coordinates of the same space-time point by two different observers involved smooth functions. In other words, space-time became an abstract C∞ manifold and each observer was seen to describe a neighborhood of this manifold in terms of local coordinates. Only statements which retain their meaning in every coordinate system were asserted to have any physical significance, i. e., the global differential geometric statements. Einstein’s theory of gravitation is, for example, a formulation of the theory of gravitational phenomena involving only intrinsic geometric objects like connections and tensor fields.