From General Relativity to Classical Dynamics

It was a project for a book, but, as my interests shifted towards a different project, I decided to stop this, and start writing my Elements for Physics. The occasional reader is invited to have a look at the Preface and Table of Contents. It may well be that she/he finds useful some parts of this text. Preface The theory of continuous media is not of universal validity but it is applicable in numerous domains. Not only we can analyze the dynamics of fluids or solids, but we know that even a good model of the space-time itself can be based on the theory. The notions necessary to describe general coordinate systems in Euclidean spaces —or curved surfaces embedded in them— are the same as those needed to describe general (non Euclidean) spaces, and are the object of the differential geometry. As our physical space is not Euclidean, the study of the differential geometry is essential. General spaces may have curvature and torsion, and the tensors needed to describe them satisfy some important identities: the Bianchi identities. As demonstrated by Einstein, the geometrical analysis of space-time introduces some tensors that, when identified with material properties, give —because of mathematical properties valid in any space— not only the fundamental equations of conservation of mass, linear and angular momentum, but also describe the gravitational interactions. The equations so obtained describe the dynamics of continuous media, but leave some degrees of freedom: those that are locked by the constitutive equations that allow the conservation equations to describe different types of physical media with different rheologies. Einstein's gravitation theory identifies mass with space-time curvature. But, originally, the theory assumed that space-time had no torsion. This is because when the theory was developed (1915) [note: give here a historically accurate data] it was not yet realized that, in addition to mass, particles have another intrinsic property: spin (Stern and Gerlach's paper dates from 1922). There has been so many failed attemps to " geometrize " other particle properties, like the electric charge, that I see as a social mystery the fact that it is not widely recognized how the simple dropping of the assumption no torsion leads to a consistent theory where, while mass produces curvature of space-time, spin produces space-time torsion. This theory is not only more general, but more equilibrated that the traditional one (treating in an equivalent way curvature and torsion). An …