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.

arXiv comment31 pages, one appendix. Minor clarifications added and typos fixed

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• Supergravity model of the Haldane-Rezayi fractional quantum Hall state
• Non-Relativistic Supergeometry in the Moore-Read Fractional Quantum Hall State
• Extended Nappi-Witten Geometry for the Fractional Quantum Hall Effect
• Trace anomaly for non-relativistic fermions
• Covariant Poisson's equation in torsional Newton-Cartan gravity
• On the Gravitational Wave to Matter Coupling of Superfluid Fermi Gases Near Unitarity