Combining 3-momentum and kinetic energy on Galilei/Newton spacetime

TL;DR

This paper explores how to unify 3-momentum and kinetic energy in a single mathematical framework on Galilei/Newton spacetime, drawing parallels with relativistic spacetime and offering new insights into classical mechanics.

Contribution

It introduces a method to combine 3-momentum and kinetic energy into a linear form or tensor on Galilei/Newton spacetime, enhancing the unity of classical mechanics.

Findings

Unified representation of momentum and energy on $ ext{G}$

First Law of Thermodynamics as a consequence of energy-momentum dynamics

Increased conceptual unity between relativistic and non-relativistic mechanics

Abstract

Without the mass-energy equivalence available on Minkowski spacetime $\mathbb{M}$, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime $\mathbb{G}$ 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 $\mathbb{M}$ and $\mathbb{G}$. As on $\mathbb{M}$, for a material continuum on $\mathbb{G}$, the First Law of Thermodynamics can be considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate.