A generally covariant formulation of Newtonian mechanics and its implications on the interpretation of spacetime theories

This text is divided into three chapters. In the first chapter, the mathematical tools on which the discussion is based are presented, namely, some rudiments of manifold theory. In the second chapter, following the text “Foundations of spacetime theories” by M. Friedman, two different generally covariant formulations of Newtonian mechanics, considered as a four-dimensional spacetime theory, are proposed. In the first version velocity is an absolute notion, that is, independent of the choice coordinate system, while in the second, the one most frequently considered in the context of classical mechanics, Galilean relativity holds. This difference is essentially due to the fact that only in the first version, the one originally advocated by Newton, there is a notion of absolute space. The third chapter comments on Newton's position on absolute spacetime and considers, in contrast, the relationalist perspective in the declinations of Leibniz and Mach. It is then considered the influence the latter had in the formulation and in the philosophical background of the theory of general relativity, explicitly expressed by Einstein himself. Following Friedman, thanks to the clarification made on the concepts of general covariance and absoluteness, it is finally shown that general relativity does not satisfy the relationalist program proposed by Mach. In particular, in light of the possibility of formulating Newtonian mechanics in a generally covariant form, it is shown that the general covariance of a theory and the absoluteness of spacetime are two independent concepts.