Non-relativistic transport from frame-indifferent kinetic theory

TL;DR

The paper develops a covariant, frame-indifferent kinetic theory for non-relativistic gases in Newton–Cartan spacetimes by employing Milne boosts and thermal NC geometry. It constructs a covariant Boltzmann equation, derives conservation laws and hydrostatic equilibrium conditions, and computes zeroth- and first-order hydrodynamics including rotating gases with parity-even and parity-odd transport. The framework resolves a long-standing frame-dependence paradox by showing that constitutive relations must be Milne-invariant and that the fluid rest frame arises from symmetry breaking, enabling a consistent gradient expansion. The results yield explicit transport coefficients for 2D and 3D rotating gases and provide a solid foundation for covariant kinetic theory applicable to rotating quantum fluids and condensed-matter systems. The work also outlines experimental avenues with ultracold atoms and microgravity platforms to probe these geometric transport phenomena.

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.

Figures (1)

• Figure 1: Shifting to the reference frame co-rotating with the fluid, as described in Sec. \ref{['sec:rotate']}. The worldlines of the fluid elements (red dashed lines) at a fixed $\rho$ are plotted for different coordinate systems: $(t,\phi)$ and $(t,\phi')$, and different choices of the reference velocity: $v^\mu$ and $v'^\mu$. The tilt of the $t$ axis is given by the angle between $\partial_t$ and $\partial_\phi$ (left column) or $\partial_{\phi'}$ (right column) calculated using the metric $h_{\mu\nu}$ (top row) or $h'_{\mu\nu}$ (bottom row).