Covariance of Lagrangian Density Function

Abstract

During the last two centuries, the concept of absolute spacetime has been extended in two main directions, and accordingly the definition of a continuum has following more or less these evolutions. Galilean physics, and namely the Newton mechanics, is mainly based on the existence of an absolute rigid space and time. For both special and general relativistic physics, Einstein and numerous other authors which were involved in, revised the concept of space and time into spacetime by relativizing the time (Minkowski spacetime) and by transforming of absolute and rigid space into a variable and dynamical four-metric to model the interaction of Einstein spacetime and matter. Further extension of the relativistic continuum physics was obtained when Cartan added the torsion as dynamical variable to obtain the Einstein–Cartan spacetime. More generally, the basic geometry underlying any physics theory may thus be proposed to include metric and affine connection, not necessarily associated to metric (metric affine-manifold). Lagrangian we are interested in are function defined on such a manifold.