A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

  • Hamid R. Afshar
  • Eric A. Bergshoeff
  • Aditya Mehra
  • Pulastya Parekh
  • Blaise Rollier
Open Access
Regular Article - Theoretical Physics

Abstract

We define a ‘non-relativistic conformal method’, based on a Schrödinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Hořava-Lifshitz gravity by classifying all possible Schrödinger invariant scalar field theories of a complex scalar up to second order in time derivatives.

Keywords

Classical Theories of Gravity Gauge Symmetry 

References

  1. [1]
    M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D 17 (1978) 3179 [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    M. Kaku and P.K. Townsend, Poincaré Supergravity as Broken Superconformal Gravity, Phys. Lett. B 76 (1978) 54 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    S. Ferrara, M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauging the Graded Conformal Group with Unitary Internal Symmetries, Nucl. Phys. B 129 (1977) 125 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).CrossRefMATHGoogle Scholar
  6. [6]
    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie), Annales Sci. Ecole Norm. Sup. 40 (1923) 325.MathSciNetMATHGoogle Scholar
  7. [7]
    E. Cartan, Sur les variétés à connexion affine et la théorie de la relativité généralisée. (première partie) (Suite), Annales Sci. Ecole Norm. Sup. 41 (1924) 1.Google Scholar
  8. [8]
    P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    P. Hořava, Membranes at Quantum Criticality, JHEP 03 (2009) 020 [arXiv:0812.4287] [INSPIRE].ADSMathSciNetGoogle Scholar
  10. [10]
    G. Dautcourt, On the newtonian limit of general relativity, Acta Phys. Polon. B21 (1990) 755.MathSciNetGoogle Scholar
  11. [11]
    S. Mukohyama, Hořava-Lifshitz Cosmology: A Review, Class. Quant. Grav. 27 (2010) 223101 [arXiv:1007.5199] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    T.P. Sotiriou, Hořava-Lifshitz gravity: a status report, J. Phys. Conf. Ser. 283 (2011) 012034 [arXiv:1010.3218] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Trautman, Sur la theorie newtonienne de la gravitation, Compt. Rend. Acad. Sci. 257 (1963) 617.MathSciNetMATHGoogle Scholar
  15. [15]
    H.P. Kuenzle, Galilei and Lorentz structures on space-time — comparison of the corresponding geometry and physics, Annales Poincaré Phys. Theor. 17 (1972) 337 [INSPIRE].MathSciNetGoogle Scholar
  16. [16]
    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
  17. [17]
    C. Leiva and M.S. Plyushchay, Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence, Annals Phys. 307 (2003) 372 [hep-th/0301244] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    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].ADSMathSciNetGoogle Scholar
  20. [20]
    C.P. Herzog, M. Rangamani and S.F. Ross, Heating up Galilean holography, JHEP 11 (2008) 080 [arXiv:0807.1099] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    C. Duval, M. Hassaine and P.A. Horvathy, The Geometry of Schrödinger symmetry in gravity background/non-relativistic CFT, Annals Phys. 324 (2009) 1158 [arXiv:0809.3128] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  25. [25]
    S. Janiszewski and A. Karch, Non-relativistic holography from Hořava gravity, JHEP 02 (2013) 123 [arXiv:1211.0005] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    T. Griffin, P. Hořava and C.M. Melby-Thompson, Lifshitz Gravity for Lifshitz Holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    D.M. Hofman and B. Rollier, Warped Conformal Field Theory as Lower Spin Gravity, Nucl. Phys. B 897 (2015) 1 [arXiv:1411.0672] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    A. Castro, D.M. Hofman and N. Iqbal, Entanglement Entropy in Warped Conformal Field Theories, JHEP 02 (2016) 033 [arXiv:1511.00707] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, arXiv:1408.6855 [INSPIRE].
  30. [30]
    K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    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].ADSMathSciNetGoogle Scholar
  32. [32]
    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
  33. [33]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].
  36. [36]
    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
  37. [37]
    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
  38. [38]
    S. Moroz and C. Hoyos, Effective theory of two-dimensional chiral superfluids: gauge duality and Newton-Cartan formulation, Phys. Rev. B 91 (2015) 064508 [arXiv:1408.5911] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    A.J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43 (1994) 357 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    M. Henkel, M. Pleimling, C. Godreche and J.-M. Luck, Aging and conformal invariance, Phys. Rev. Lett. 87 (2001) 265701 [hep-th/0107122] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Henkel, Phenomenology of local scale invariance: From conformal invariance to dynamical scaling, Nucl. Phys. B 641 (2002) 405 [hep-th/0205256] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    M. Henkel and F. Baumann, Autocorrelation functions in phase-ordering kinetics from local scale-invariance, J. Stat. Mech. 0707 (2007) P07015 [cond-mat/0703226] [INSPIRE].
  43. [43]
    M. Henkel and M. Pleimling, Non-Equilibrium Phase Transitions. Volume 2: Ageing and Dynamical Scaling Far from Equilibrium, Springer, Dordrecht Netherlands (2010).Google Scholar
  44. [44]
    D. Minic and M. Pleimling, Non-relativistic AdS/CFT and Aging/Gravity Duality, Phys. Rev. E 78 (2008) 061108 [arXiv:0807.3665] [INSPIRE].ADSGoogle Scholar
  45. [45]
    J.I. Jottar, R.G. Leigh, D. Minic and L.A. Pando Zayas, Aging and Holography, JHEP 11 (2010) 034 [arXiv:1004.3752] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    D. Minic, D. Vaman and C. Wu, On the 3-point functions of Aging Dynamics and the AdS/CFT Correspondence, Phys. Rev. Lett. 109 (2012) 131601 [arXiv:1207.0243] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    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].ADSMATHGoogle Scholar
  49. [49]
    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  50. [50]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].ADSGoogle Scholar
  53. [53]
    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [INSPIRE].MathSciNetGoogle Scholar
  54. [54]
    A.O. Barut, Conformal Group → Schrödinger Group → Dynamical Group — The Maximal Kinematical Group of the Massive Schrödinger Particle, Helv. Phys. Acta 46 (1973) 496.MathSciNetGoogle Scholar
  55. [55]
    P. Havas and J. Plebánski, Conformal extensions of the Galilei group and their relation to the Schrödinger group., J. Math. Phys 19 (1978) 482.ADSCrossRefMATHGoogle Scholar
  56. [56]
    R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D 42 (1990) 3500 [Erratum ibid. D 48 (1993) 3929] [INSPIRE].
  57. [57]
    M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  58. [58]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].ADSMathSciNetGoogle Scholar
  59. [59]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    R. Banerjee and P. Mukherjee, New approach to nonrelativistic diffeomorphism invariance and its applications, arXiv:1509.05622 [INSPIRE].
  61. [61]
    R. De Pietri, L. Lusanna and M. Pauri, Standard and generalized Newtonian gravities as ‘gauge’ theories of the extended Galilei group. I. The standard theory, Class. Quant. Grav. 12 (1995) 219 [gr-qc/9405046] [INSPIRE].
  62. [62]
    C. Duval and H.P. Kunzle, Minimal Gravitational Coupling in the Newtonian Theory and the Covariant Schrödinger Equation, Gen. Rel. Grav. 16 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  63. [63]
    J. Ehlers, Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie, in Grundlagenprobleme der modernen Physik, J. Nitsch, J. Pfarr and E.W. Stachow eds., Bibliographisches Institut, Mannheim Germany (1981), pp. 65-84.Google Scholar
  64. [64]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  65. [65]
    D. Blas, O. Pujolàs and S. Sibiryakov, Models of non-relativistic quantum gravity: The Good, the bad and the healthy, JHEP 04 (2011) 018 [arXiv:1007.3503] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    D. Blas, O. Pujolàs and S. Sibiryakov, Consistent Extension of Hořava Gravity, Phys. Rev. Lett. 104 (2010) 181302 [arXiv:0909.3525] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    T.P. Sotiriou, M. Visser and S. Weinfurtner, Phenomenologically viable Lorentz-violating quantum gravity, Phys. Rev. Lett. 102 (2009) 251601 [arXiv:0904.4464] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  68. [68]
    E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  69. [69]
    R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  70. [70]
    K. Peeters, A Field-theory motivated approach to symbolic computer algebra, Comput. Phys. Commun. 176 (2007) 550 [cs/0608005] [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Hamid R. Afshar
    • 1
  • Eric A. Bergshoeff
    • 2
  • Aditya Mehra
    • 3
  • Pulastya Parekh
    • 3
  • Blaise Rollier
    • 2
  1. 1.School of PhysicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  3. 3.Indian Institute of Science Education and ResearchPuneIndia

Personalised recommendations