Physical stress, mass, and energy for non-relativistic matter

Abstract

For theories of relativistic matter fields there exist two possible definitions of the stress-energy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stress-energy tensors do not necessarily coincide and it is the latter that corresponds to the Cauchy stress measured in the lab. In this note we discuss the corresponding issue for non-relativistic matter theories. We point out that while the physical non-relativistic stress, momentum, and mass currents are defined by a variation of the action at fixed torsion, the energy current does not admit such a description and is naturally defined at fixed connection. Any attempt to define an energy current at fixed torsion results in an ambiguity which cannot be resolved from the background spacetime data or conservation laws. We also provide computations of these quantities for some simple non-relativistic actions.

Effective field theory for non-relativistic hydrodynamics

“…To this end, we have a choice between two frameworks: Newton-Cartan (NC) backgrounds [30] or null backgrounds [46]. While the former framework is much widely used to couple to Galilean field theories [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], the latter has an advantage that it recasts (d+1)-dimensional Galilean-invariant physics in terms of (d+2)-dimensional relativistic physics that we understand better. Unlike the NC framework, the null background framework also explicitly manifests the Galilean (Milne) boost symmetry.…”

“…The hydrodynamic observables are the mass (particle number) density ρ t , mass (particle number) flux/momentum density ρ i , energy density t , energy flux i , and stress tensor τ ij . These need to be coupled to the so-called Newton-Cartan (NC) structure: mass gauge field (A t , A i ), clock form (n t , n i ), and spatial metric h ij respectively, that describes a curved non-relativistic spacetime [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]; see [30] for a review. 1 However, Galilean boost symmetry (also referred to as Milne boosts in the NC literature) acts on these quantities quite non-trivially and is quite tedious to implement in an effective theory written in the NC language.…”

Trace anomaly for non-relativistic fermions

“…We will sometimes refer to null-reduction method as Discrete Light-Cone Quantization (DLCQ). Useful references about NC geometry include [31][32][33][34][35][36][37][38][39][40]. Galilei invariance for fermions was first studied in [41].…”

Newton-Cartan gravity and torsion

“…However, in condensed matter applications, one often considers gravity not as a dynamical theory but as background fields for determining the response of the system to a geometrical force and for defining a non-relativistic energy and momentum flux. 6 It was pointed out a long time ago in the seminal paper by Luttinger [3] that to describe thermal transport in a resistive medium one needs to consider an auxiliary gravitational field ψ(x) that couples to the energy and is defined by [4]…”