Non-relativistic transport from frame-indifferent kinetic theory

TL;DR

This paper develops a covariant kinetic theory for non-relativistic gases using Newton-Cartan geometry, resolving a long-standing debate on frame-indifference through a symmetry-based approach.

Contribution

It introduces a novel, frame-indifferent kinetic theory for non-relativistic gases within Newton-Cartan geometry, addressing a fifty-year-old controversy.

Findings

Derived covariant hydrodynamic equations for rotating gases

Extended analysis with gradient expansion for first-order constitutive relations

Provided a modern resolution to the frame-dependence paradox

Abstract

This paper explores the application of Newton-Cartan geometry to the kinetic theory of gases that includes non-relativistic gravitational effects and the principle of general covariance. Starting with an introduction to the basics of Newton-Cartan geometry, we examine the motion of point particles within this framework, leading to a detailed analysis of kinetic theory and the derivation of conservation equations. The equilibrium distribution function is explored, and the example of a rotating gas in a gravitational field is discussed. Further, we develop covariant hydrodynamic equations and extend our analysis through a gradient expansion approach to assess first-order constitutive relations for rotating gases. Finally, we address the frame-dependence paradox, presenting a novel resolution that addresses apparent discrepancies. Our construction resolves a fifty-year-old debate about the frame-indifferent formulation of kinetic theory. The resolution is presented in a modern, symmetry-based approach.