Journal of High Energy Physics

, 2017:194 | Cite as

Newton-Cartan gravity and torsion

  • Eric Bergshoeff
  • Athanasios Chatzistavrakidis
  • Luca Romano
  • Jan Rosseel
Open Access
Regular Article - Theoretical Physics


We compare the gauging of the Bargmann algebra, for the case of arbitrary torsion, with the result that one obtains from a null-reduction of General Relativity. Whereas the two procedures lead to the same result for Newton-Cartan geometry with arbitrary torsion, the null-reduction of the Einstein equations necessarily leads to Newton-Cartan gravity with zero torsion. We show, for three space-time dimensions, how Newton-Cartan gravity with arbitrary torsion can be obtained by starting from a Schrödinger field theory with dynamical exponent z = 2 for a complex compensating scalar and next coupling this field theory to a z = 2 Schrödinger geometry with arbitrary torsion. The latter theory can be obtained from either a gauging of the Schrödinger algebra, for arbitrary torsion, or from a null-reduction of conformal gravity.


Classical Theories of Gravity Space-Time Symmetries 


Open Access

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  1. [1]
    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
  2. [2]
    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
  3. [3]
    D. Van den Bleeken, Torsional Newton-Cartan gravity from the large c expansion of general relativity, Class. Quant. Grav. 34 (2017) 185004 [arXiv:1703.03459] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J.M. Luttinger, Theory of thermal transport coefficients, Phys. Rev. 135 (1964) A1505 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    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
  6. [6]
    M. Geracie, S. Golkar and M.M. Roberts, Hall viscosity, spin density and torsion, arXiv:1410.2574 [INSPIRE].
  7. [7]
    M. Geracie, K. Prabhu and M.M. Roberts, Physical stress, mass and energy for non-relativistic matter, JHEP 06 (2017) 089 [arXiv:1609.06729] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
  11. [11]
    X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view, J. Math. Phys. 57 (2016) 022507 [arXiv:1412.8212] [INSPIRE].
  12. [12]
    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
  13. [13]
    R. Andringa, E. Bergshoeff, S. Panda and M. de Roo, Newtonian gravity and the Bargmann algebra, Class. Quant. Grav. 28 (2011) 105011 [arXiv:1011.1145] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    H.R. Afshar, E.A. Bergshoeff, A. Mehra, P. Parekh and B. Rollier, A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities, JHEP 04 (2016) 145 [arXiv:1512.06277] [INSPIRE].ADSGoogle Scholar
  15. [15]
    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
  16. [16]
    J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan (super)gravity as a non-relativistic limit, Class. Quant. Grav. 32 (2015) 205003 [arXiv:1505.02095] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Eric Bergshoeff
    • 1
  • Athanasios Chatzistavrakidis
    • 1
    • 2
  • Luca Romano
    • 1
  • Jan Rosseel
    • 3
  1. 1.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  2. 2.Division of Theoretical PhysicsRudjer Bošković InstituteZagrebCroatia
  3. 3.Faculty of PhysicsUniversity of ViennaViennaAustria

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