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

  • Michael Geracie
  • Kartik Prabhu
  • Matthew M. Roberts
Open Access
Regular Article - Theoretical Physics


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.


Space-Time Symmetries Differential and Algebraic Geometry Effective Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    F. Belinfante, On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields, Physica 7 (1940) 449.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    F.W. Hehl, On the energy tensor of spinning massive matter in classical field theory and general relativity, Rept. Math. Phys. 9 (1976) 55 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    F.W. Hehl, On energy-momentum and spin/helicity of quark and gluon fields, in Proceedings of the 15th Workshop on High Energy Spin Physics (DSPIN-13), Dubna Russia, 8-12 Oct 2013 [arXiv:1402.0261] [INSPIRE].
  4. [4]
    H. Shapourian, T.L. Hughes and S. Ryu, Viscoelastic response of topological tight-binding models in two and three dimensions, Phys. Rev. B 92 (2015) 165131 [arXiv:1505.03868].ADSCrossRefGoogle Scholar
  5. [5]
    G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Torsional Newton-Cartan geometry from the Noether procedure, Phys. Rev. D 94 (2016) 105023 [arXiv:1607.01926] [INSPIRE].ADSGoogle Scholar
  6. [6]
    M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Geracie, K. Prabhu and M.M. Roberts, Curved non-relativistic spacetimes, Newtonian gravitation and massive matter, J. Math. Phys. 56 (2015) 103505 [arXiv:1503.02682] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Luttinger, Theory of thermal transport coefficients, Phys. Rev. 135 (1964) A1505 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Gromov and A.G. Abanov, Thermal Hall effect and geometry with torsion, Phys. Rev. Lett. 114 (2015) 016802 [arXiv:1407.2908] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Ann. Sci. Ecole Norm. Sup. 40 (1923) 325.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie) (suite), Ann. Sci. Ecole Norm. Sup. 41 (1924) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.B. Malament, Topics in the foundations of general relativity and Newtonian gravitation theory, University of Chicago Press, Chicago U.S.A. (2012).CrossRefzbMATHGoogle Scholar
  13. [13]
    C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman, San Francisco U.S.A. (1973).Google Scholar
  14. [14]
    D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [INSPIRE].
  15. [15]
    K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    B. Carter and I.M. Khalatnikov, Canonically covariant formulation of Landau’s Newtonian superfluid dynamics, Rev. Math. Phys. 6 (1994) 277 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Mitra, Weyl rescaled Newton-Cartan geometry and non-relativistic conformal hydrodynamics, arXiv:1508.03207 [INSPIRE].
  18. [18]
    C. Hoyos and D.T. Son, Hall viscosity and electromagnetic response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
  20. [20]
    S. Golkar, D.X. Nguyen and D.T. Son, Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect, JHEP 01 (2016) 021 [arXiv:1309.2638] [INSPIRE].CrossRefGoogle Scholar
  21. [21]
    M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev. D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    M. Geracie, K. Prabhu and M.M. Roberts, Covariant effective action for a Galilean invariant quantum Hall system, JHEP 09 (2016) 092 [arXiv:1603.08934] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan geometry and Lifshitz holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].ADSzbMATHGoogle Scholar
  24. [24]
    M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    J. Hartong, E. Kiritsis and N.A. Obers, Schrödinger invariance from Lifshitz isometries in holography and field theory, Phys. Rev. D 92 (2015) 066003 [arXiv:1409.1522] [INSPIRE].ADSGoogle Scholar
  27. [27]
    J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    E.A. Bergshoeff, J. Hartong and J. Rosseel, Torsional Newton-Cartan geometry and the Schrödinger algebra, Class. Quant. Grav. 32 (2015) 135017 [arXiv:1409.5555] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    J. Hartong, E. Kiritsis and N.A. Obers, Field theory on Newton-Cartan backgrounds and symmetries of the Lifshitz vacuum, JHEP 08 (2015) 006 [arXiv:1502.00228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    R. Banerjee, A. Mitra and P. Mukherjee, A new formulation of non-relativistic diffeomorphism invariance, Phys. Lett. B 737 (2014) 369 [arXiv:1404.4491] [INSPIRE].ADSzbMATHGoogle Scholar
  31. [31]
    R. Banerjee, A. Mitra and P. Mukherjee, Localization of the Galilean symmetry and dynamical realization of Newton-Cartan geometry, Class. Quant. Grav. 32 (2015) 045010 [arXiv:1407.3617] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    R. Banerjee, A. Mitra and P. Mukherjee, General algorithm for nonrelativistic diffeomorphism invariance, Phys. Rev. D 91 (2015) 084021 [arXiv:1501.05468] [INSPIRE].ADSMathSciNetGoogle Scholar
  33. [33]
    R. Banerjee and P. Mukherjee, New approach to nonrelativistic diffeomorphism invariance and its applications, Phys. Rev. D 93 (2016) 085020 [arXiv:1509.05622] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    T. Brauner, S. Endlich, A. Monin and R. Penco, General coordinate invariance in quantum many-body systems, Phys. Rev. D 90 (2014) 105016 [arXiv:1407.7730] [INSPIRE].ADSGoogle Scholar
  35. [35]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].ADSGoogle Scholar
  36. [36]
    S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
  38. [38]
    F. Irgens, Continuum mechanics, Springer Science & Business Media (2008).Google Scholar
  39. [39]
    M. Greiter, F. Wilczek and E. Witten, Hydrodynamic relations in superconductivity, Mod. Phys. Lett. B 3 (1989) 903 [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    D.T. Son, Effective Lagrangian and topological interactions in supersolids, Phys. Rev. Lett. 94 (2005) 175301 [cond-mat/0501658] [INSPIRE].
  41. [41]
    M. Geracie, Galilean geometry in condensed matter systems, Ph.D. Thesis, University of Chicago (2016).Google Scholar
  42. [42]
    M. de Montigny, J. Niederle and A.G. Nikitin, Galilei invariant theories. I. Constructions of indecomposable finite-dimensional representations of the homogeneous Galilei group: directly and via contractions, J. Phys. A 39 (2006) 9365 [math.PH/0604002].
  43. [43]
    J. Niederle and A.G. Nikitin, Galilei invariant theories. II. Wave equations for massive fields, arXiv:0707.3286 [INSPIRE].
  44. [44]
    J.-M. Levy-Leblond, Nonrelativistic particles and wave equations, Commun. Math. Phys. 6 (1967) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    J.F. Fuini, A. Karch and C.F. Uhlemann, Spinor fields in general Newton-Cartan backgrounds, Phys. Rev. D 92 (2015) 125036 [arXiv:1510.03852] [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    X.G. Wen and A. Zee, Shift and spin vector: new topological quantum numbers for the Hall fluids, Phys. Rev. Lett. 69 (1992) 953 [Erratum ibid. 69 (1992) 3000] [INSPIRE].
  47. [47]
    B. Bradlyn and N. Read, Low-energy effective theory in the bulk for transport in a topological phase, Phys. Rev. B 91 (2015) 125303 [arXiv:1407.2911] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    A. Gromov, K. Jensen and A.G. Abanov, Boundary effective action for quantum Hall states, Phys. Rev. Lett. 116 (2016) 126802 [arXiv:1506.07171] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    M. Geracie and D.T. Son, Hydrodynamics on the lowest Landau level, JHEP 06 (2015) 044 [arXiv:1408.6843] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Baggio, J. de Boer and K. Holsheimer, Anomalous breaking of anisotropic scaling symmetry in the quantum Lifshitz model, JHEP 07 (2012) 099 [arXiv:1112.6416] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    K. Jensen, Anomalies for Galilean fields, arXiv:1412.7750 [INSPIRE].
  52. [52]
    I. Arav, S. Chapman and Y. Oz, Lifshitz scale anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  53. [53]
    I. Arav, S. Chapman and Y. Oz, Non-relativistic scale anomalies, JHEP 06 (2016) 158 [arXiv:1601.06795] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    S. Pal and B. Grinstein, Weyl consistency conditions in non-relativistic quantum field theory, JHEP 12 (2016) 012 [arXiv:1605.02748] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    D.T. Son, Is the composite fermion a Dirac particle?, Phys. Rev. X 5 (2015) 031027 [arXiv:1502.03446] [INSPIRE].CrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Michael Geracie
    • 1
  • Kartik Prabhu
    • 2
  • Matthew M. Roberts
    • 3
  1. 1.Center for Quantum Mathematics and Physics (QMAP), Department of PhysicsUniversity of CaliforniaDavisU.S.A.
  2. 2.Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE)Cornell UniversityIthacaU.S.A.
  3. 3.Kadanoff Center for Theoretical PhysicsUniversity of ChicagoChicagoU.S.A.

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