Abstract
Without the mass-energy equivalence available on Minkowski spacetime M, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime G to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducial frame, it is possible to combine 3-momentum and kinetic energy into a linear form (particle) or (1,1) tensor (continuum) in a manner that exhibits increased unity of classical mechanics on flat relativistic and non-relativistic spacetimes M and G. As on M, for a material continuum on G, the first law of thermodynamics can be considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate.
Highlights
Traditional points of departure for non-relativistic and relativistic classical mechanics (e.g., [1,2,3])feature distinct pictures of space and time
While Equations (19), (22), and (23) provide an initial understanding of the relationship of inertia-momentum balance to the motion of an individual continuum element, further understanding of the 4-stress Σ is needed. This is obtained from the observation that an element of a material continuum is distinguished from a particle by the fact that, as a piece of matter with non-vanishing extent, it is to be regarded as a tiny thermodynamic system in and of itself
Greater conceptual unity of the relativistic and non-relativistic classical mechanics of material particles and continua is achieved by combining kinetic energy and 3-momentum in a linear form
Summary
Traditional points of departure for non-relativistic and relativistic classical mechanics (e.g., [1,2,3]). 3-momentum and kinetic+internal energy of a material continuum; consists of a vanishing divergence in the absence of external 4-force per baryon; and does so in a conceptually unified way in both the relativistic and non-relativistic cases, that is, on both M and G. This is an equation for the divergence of what I call the “relative energy-momentum flux tensor” S (the adjective “relative” betrays the fact that it is defined in terms of a fixed family of fiducial frames, in such a way that the transformation properties of kinetic energy are not manifest). Such can be the nature of backporting insights from a superseding theory into the one it supersedes
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