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

TL;DR

This paper examines the definitions of stress, momentum, mass, and energy currents in non-relativistic matter theories, highlighting ambiguities in defining the energy current at fixed torsion.

Contribution

It clarifies the distinction between stress-energy tensor definitions in non-relativistic theories and discusses the inherent ambiguities in defining the energy current.

Findings

The physical non-relativistic stress, momentum, and mass currents are defined by variation at fixed torsion.

The energy current is naturally defined at fixed connection, not at fixed torsion.

Ambiguities in defining the energy current cannot be resolved from background data or conservation laws.

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.