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Non-relativistic gravity and its coupling to matter

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  • Published: 23 June 2020
  • Volume 2020, article number 145, (2020)
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Non-relativistic gravity and its coupling to matter
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  • Dennis Hansen1,
  • Jelle Hartong2 &
  • Niels A. Obers  ORCID: orcid.org/0000-0003-4947-85263,4 

  • 888 Accesses

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A preprint version of the article is available at arXiv.

Abstract

We study the non-relativistic expansion of general relativity coupled to matter. This is done by expanding the metric and matter fields analytically in powers of 1/c2 where c is the speed of light. In order to perform this expansion it is shown to be very convenient to rewrite general relativity in terms of a timelike vielbein and a spatial metric. This expansion can be performed covariantly and off shell. We study the expansion of the Einstein-Hilbert action up to next-to-next-to-leading order. We couple this to different forms of matter: point particles, perfect fluids, scalar fields (including an off-shell derivation of the Schrödinger-Newton equation) and electrodynamics (both its electric and magnetic limits). We find that the role of matter is crucial in order to understand the properties of the Newton-Cartan geometry that emerges from the expansion of the metric. It turns out to be the matter that decides what type of clock form is allowed, i.e. whether we have absolute time or a global foliation of constant time hypersurfaces. We end by studying a variety of solutions of non-relativistic gravity coupled to perfect fluids. This includes the Schwarzschild geometry, the Tolman-Oppenheimer-Volkoff solution for a fluid star, the FLRW cosmological solutions and anti-de Sitter spacetimes.

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References

  1. G. Dautcourt, PostNewtonian extension of the Newton-Cartan theory, Class. Quant. Grav. 14 (1997) A109 [gr-qc/9610036] [INSPIRE].

  2. W. Tichy and E.E. Flanagan, Covariant formulation of the post-1-Newtonian approximation to General Relativity, Phys. Rev. D 84 (2011) 044038 [arXiv:1101.0588] [INSPIRE].

  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].

  4. D. Hansen, J. Hartong and N.A. Obers, Action Principle for Newtonian Gravity, Phys. Rev. Lett. 122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].

  5. E. Cartan, Sur les variétés à connexion affine et la théorie de la rélativité généralisée (premiére partie), Ann. Sci. Éc. Norm. Supér. 40 (1923) 325.

  6. E. Cartan, Sur les variétés à connexion affine et la théorie de la rélativité généralisée (premìere partie) (suite), Ann. Sci. Éc. Norm. Supér. 41 (1924) 1.

  7. A. Trautman, Sur la theorie newtonienne de la gravitation, Compt. Rend. Acad. Sci. Paris 247 (1963) 617.

  8. P. Havas, Four-Dimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity, Rev. Mod. Phys. 36 (1964) 938 [INSPIRE].

  9. 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].

  10. 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].

  11. 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].

  12. J. Hartong and N.A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].

  13. 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].

  14. H.R. Afshar, E.A. Bergshoeff, A. Mehra, P. Parekh and B. Rollier, A Schrödinger approach to Newton-Cartan and Hǒrava-Lifshitz gravities, JHEP 04 (2016) 145 [arXiv:1512.06277] [INSPIRE].

  15. E.A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].

  16. J. Hartong, Y. Lei and N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hǒrava-Lifshitz gravity, Phys. Rev. D 94 (2016) 065027 [arXiv:1604.08054] [INSPIRE].

  17. E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, Newton-Cartan Gravity and Torsion, JHEP 10 (2017) 194 [arXiv:1708.05414] [INSPIRE].

  18. J. Hartong, Y. Lei, N.A. Obers and G. Oling, Zooming in on AdS3 /CFT2 near a BPS bound, JHEP 05 (2018) 016 [arXiv:1712.05794] [INSPIRE].

  19. D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].

  20. K. Jensen, On the coupling of Galilean-invariant field theories to curved spacetime, SciPost Phys. 5 (2018) 011 [arXiv:1408.6855] [INSPIRE].

  21. J. Hartong, E. Kiritsis and N.A. Obers, Lifshitz space-times for Schrödinger holography, Phys. Lett. B 746 (2015) 318 [arXiv:1409.1519] [INSPIRE].

  22. 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].

  23. T. Harmark, J. Hartong and N.A. Obers, Nonrelativistic strings and limits of the AdS/CFT correspondence, Phys. Rev. D 96 (2017) 086019 [arXiv:1705.03535] [INSPIRE].

  24. J. Klusoň, Remark About Non-Relativistic String in Newton-Cartan Background and Null Reduction, JHEP 05 (2018) 041 [arXiv:1803.07336] [INSPIRE].

  25. T. Harmark, J. Hartong, L. Menculini, N.A. Obers and Z. Yan, Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence, JHEP 11 (2018) 190 [arXiv:1810.05560] [INSPIRE].

  26. A.D. Gallegos, U. Gürsoy and N. Zinnato, Torsional Newton Cartan gravity from non-relativistic strings, arXiv:1906.01607 [INSPIRE].

  27. T. Harmark, J. Hartong, L. Menculini, N.A. Obers and G. Oling, Relating non-relativistic string theories, JHEP 11 (2019) 071 [arXiv:1907.01663] [INSPIRE].

  28. R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, ‘Stringy’ Newton-Cartan Gravity, Class. Quant. Grav. 29 (2012) 235020 [arXiv:1206.5176] [INSPIRE].

  29. E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-duality, JHEP 11 (2018) 133 [arXiv:1806.06071] [INSPIRE].

  30. E.A. Bergshoeff, K.T. Grosvenor, C. Simsek and Z. Yan, An Action for Extended String Newton-Cartan Gravity, JHEP 01 (2019) 178 [arXiv:1810.09387] [INSPIRE].

  31. J. Gomis, J. Oh and Z. Yan, Nonrelativistic String Theory in Background Fields, JHEP 10 (2019) 101 [arXiv:1905.07315] [INSPIRE].

  32. E.A. Bergshoeff, J. Gomis, J. Rosseel, C. S¸im¸sek and Z. Yan, String Theory and String Newton-Cartan Geometry, J. Phys. A 53 (2020) 014001 [arXiv:1907.10668] [INSPIRE].

  33. D. Hansen, J. Hartong and N.A. Obers, Gravity between Newton and Einstein, Int. J. Mod. Phys. D 28 (2019) 1944010 [arXiv:1904.05706] [INSPIRE].

  34. D. Hansen, J. Hartong and N.A. Obers, Non-relativistic expansion of the Einstein-Hilbert Lagrangian, in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG15), Rome, Italy, 1–7 July 2018 (2019) [arXiv:1905.13723] [INSPIRE].

  35. M. Cariglia, General theory of Galilean gravity, Phys. Rev. D 98 (2018) 084057 [arXiv:1811.03446] [INSPIRE].

  36. D. Van den Bleeken, Torsional Newton-Cartan gravity and strong gravitational fields, in 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG15), Rome, Italy, 1–7 July 2018 (2019) [arXiv:1903.10682] [INSPIRE].

  37. J.A. de Azcarraga, J.M. Izquierdo, M. Picón and O. Varela, Generating Lie and gauge free differential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity, Nucl. Phys. B 662 (2003) 185 [hep-th/0212347] [INSPIRE].

  38. F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super)algebras through Abelian semigroups, J. Math. Phys. 47 (2006) 123512 [hep-th/0606215] [INSPIRE].

  39. O. Khasanov and S. Kuperstein, (In)finite extensions of algebras from their Inonu-Wigner contractions, J. Phys. A 44 (2011) 475202 [arXiv:1103.3447] [INSPIRE].

  40. E. Bergshoeff, J.M. Izquierdo, T. Ortín and L. Romano, Lie Algebra Expansions and Actions for Non-Relativistic Gravity, JHEP 08 (2019) 048 [arXiv:1904.08304] [INSPIRE].

  41. J. Gomis, A. Kleinschmidt and J. Palmkvist, Galilean free Lie algebras, JHEP 09 (2019) 109 [arXiv:1907.00410] [INSPIRE].

  42. J.A. de Azcárraga, D. Gútiez and J.M. Izquierdo, Extended D = 3 Bargmann supergravity from a Lie algebra expansion, arXiv:1904.12786 [INSPIRE].

  43. 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].

  44. J. Klusoň, Nonrelativistic String Theory σ-model and Its Canonical Formulation, Eur. Phys. J. C 79 (2019) 108 [arXiv:1809.10411] [INSPIRE].

  45. J. Klusoň, (m, n)-String and D1-Brane in Stringy Newton-Cartan Background, JHEP 04 (2019) 163 [arXiv:1901.11292] [INSPIRE].

  46. C.D.A. Blair, A worldsheet supersymmetric Newton-Cartan string, JHEP 10 (2019) 266 [arXiv:1908.00074] [INSPIRE].

  47. J. Klusoň, T-duality of Non-Relativistic String in Torsional Newton-Cartan Background, JHEP 05 (2020) 024 [arXiv:1909.13508] [INSPIRE].

  48. J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].

  49. U.H. Danielsson, A. Guijosa and M. Kruczenski, IIA/B, wound and wrapped, JHEP 10 (2000) 020 [hep-th/0009182] [INSPIRE].

  50. K. Morand and J.-H. Park, Classification of non-Riemannian doubled-yet-gauged spacetime, Eur. Phys. J. C 77 (2017) 685 [Erratum ibid. C 78 (2018) 901] [arXiv:1707.03713] [INSPIRE].

  51. D.S. Berman, C.D.A. Blair and R. Otsuki, Non-Riemannian geometry of M-theory, JHEP 07 (2019) 175 [arXiv:1902.01867] [INSPIRE].

  52. K. Cho and J.-H. Park, Remarks on the non-Riemannian sector in Double Field Theory, Eur. Phys. J. C 80 (2020) 101 [arXiv:1909.10711] [INSPIRE].

  53. K. Cho, K. Morand and J.-H. Park, Stringy Newton Gravity with H -flux, Phys. Rev. D 101 (2020) 064020 [arXiv:1912.13220] [INSPIRE].

  54. K. Jensen, Aspects of hot Galilean field theory, JHEP 04 (2015) 123 [arXiv:1411.7024] [INSPIRE].

  55. M. Geracie, K. Prabhu and M.M. Roberts, Fields and fluids on curved non-relativistic spacetimes, JHEP 08 (2015) 042 [arXiv:1503.02680] [INSPIRE].

  56. J. Hartong, N.A. Obers and M. Sanchioni, Lifshitz Hydrodynamics from Lifshitz Black Branes with Linear Momentum, JHEP 10 (2016) 120 [arXiv:1606.09543] [INSPIRE].

  57. J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Perfect Fluids, SciPost Phys. 5 (2018) 003 [arXiv:1710.04708] [INSPIRE].

  58. J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Hydrodynamic Modes of Homogeneous and Isotropic Fluids, SciPost Phys. 5 (2018) 014 [arXiv:1710.06885] [INSPIRE].

  59. J. Armas, J. Hartong, E. Have, B.F. Nielsen and N.A. Obers, Newton-Cartan Submanifolds and Fluid Membranes, arXiv:1912.01613 [INSPIRE].

  60. 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].

  61. M. Ergen, E. Hamamci and D. Van den Bleeken, Oddity in nonrelativistic, strong gravity, arXiv:2002.02688 [INSPIRE].

  62. J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-Rebolledó, Symmetries of post-Galilean expansions, Phys. Rev. Lett. 124 (2020) 081602 [arXiv:1910.13560] [INSPIRE].

  63. 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].

  64. E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [INSPIRE].

  65. D. Hansen, J. Hartong, N.A. Obers and G. Oling, in preparation.

  66. G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].

  67. N. Ozdemir, M. Ozkan, O. Tunca and U. Zorba, Three-Dimensional Extended Newtonian (Super)Gravity, JHEP 05 (2019) 130 [arXiv:1903.09377] [INSPIRE].

  68. D.M. Peñafiel and P. Salgado-Rebolledó, Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra, Phys. Lett. B 798 (2019) 135005 [arXiv:1906.02161] [INSPIRE].

  69. P. Concha, L. Ravera and E. Rodríguez, Three-dimensional exotic Newtonian gravity with cosmological constant, Phys. Lett. B 804 (2020) 135392 [arXiv:1912.02836] [INSPIRE].

  70. J. Gomis, A. Kleinschmidt, J. Palmkvist and P. Salgado-Rebolledó, Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity, JHEP 02 (2020) 009 [arXiv:1912.07564] [INSPIRE].

  71. H.P. Kuenzle, Canonical dynamics of spinning particles in gravitational and electromagnetic fields, J. Math. Phys. 13 (1972) 739 [INSPIRE].

  72. M. Hassaine and P.A. Horvathy, Field dependent symmetries of a nonrelativistic fluid model, Annals Phys. 282 (2000) 218 [math-ph/9904022] [INSPIRE].

  73. P.A. Horvathy and P.M. Zhang, Non-relativistic conformal symmetries in fluid mechanics, Eur. Phys. J. C 65 (2010) 607 [arXiv:0906.3594] [INSPIRE].

  74. K. Kuchar, Gravitation, geometry, and nonrelativistic quantum theory, Phys. Rev. D 22 (1980) 1285 [INSPIRE].

  75. 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].

  76. C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].

  77. F. Karolyhazy, Gravitation and quantum mechanics of macroscopic objects, Nuovo Cim. A 42 (1966) 390 [INSPIRE].

  78. I. Bialynicki-Birula and J. Mycielski, Nonlinear Wave Mechanics, Annals Phys. 100 (1976) 62 [INSPIRE].

  79. L. Diósi, Gravitation and quantummechanical localization of macroobjects, Phys. Lett. A 105 (1984) 199 [arXiv:1412.0201] [INSPIRE].

  80. L. Diosi, A Universal Master Equation for the Gravitational Violation of Quantum Mechanics, Phys. Lett. A 120 (1987) 377 [INSPIRE].

  81. M. Bahrami, A. Großardt, S. Donadi and A. Bassi, The Schroedinger-Newton equation and its foundations, New J. Phys. 16 (2014) 115007 [arXiv:1407.4370] [INSPIRE].

  82. C. Anastopoulos and B.L. Hu, Problems with the Newton-Schrödinger equations, New J. Phys. 16 (2014) 085007 [arXiv:1403.4921] [INSPIRE].

  83. R. Colella, A.W. Overhauser and S.A. Werner, Observation of gravitationally induced quantum interference, Phys. Rev. Lett. 34 (1975) 1472 [INSPIRE].

  84. V.V. Nesvizhevsky et al., Quantum states of neutrons in the Earth’s gravitational field, Nature 415 (2002) 297 [INSPIRE].

  85. D. Giulini and A. Grossardt, Gravitationally induced inhibitions of dispersion according to the Schródinger-Newton Equation, Class. Quant. Grav. 28 (2011) 195026 [arXiv:1105.1921] [INSPIRE].

  86. D. Carney, P.C.E. Stamp and J.M. Taylor, Tabletop experiments for quantum gravity: a user’s manual, Class. Quant. Grav. 36 (2019) 034001 [arXiv:1807.11494] [INSPIRE].

  87. M. Le Bellac and J.-M. Lévy-Leblond, Galilean Electromagnetism, Nuovo Cim. B 14 (1973) 217.

  88. E.S. Santos, M. de Montigny, F.C. Khanna and A.E. Santana, Galilean covariant Lagrangian models, J. Phys. A 37 (2004) 9771 [INSPIRE].

  89. M. De Montigny and G. Rousseaux, On the electrodynamics of moving bodies at low velocities, Eur. J. Phys. 27 (2006) 755 [physics/0512200] [INSPIRE].

  90. C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].

  91. A. Bagchi, R. Basu and A. Mehra, Galilean Conformal Electrodynamics, JHEP 11 (2014) 061 [arXiv:1408.0810] [INSPIRE].

  92. D. Van den Bleeken and C. Yunus, Newton-Cartan, Galileo-Maxwell and Kaluza-Klein, Class. Quant. Grav. 33 (2016) 137002 [arXiv:1512.03799] [INSPIRE].

  93. G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Symmetries and Couplings of Non-Relativistic Electrodynamics, JHEP 11 (2016) 037 [arXiv:1607.01753] [INSPIRE].

  94. M. Milgrom, A modification of the Newtonian dynamics: implications for galaxy systems, Astrophys. J. 270 (1983) 384 [INSPIRE].

  95. T. Buchert and J. Ehlers, Averaging inhomogeneous Newtonian cosmologies, Astron. Astrophys. 320 (1997) 1 [astro-ph/9510056] [INSPIRE].

  96. G.F.R. Ellis and H. van Elst, Cosmological models: Cargese lectures 1998, NATO Sci. Ser. C 541 (1999) 1 [gr-qc/9812046].

  97. R. Aldrovandi, A.L. Barbosa, L.C.B. Crispino and J.G. Pereira, Non-Relativistic spacetimes with cosmological constant, Class. Quant. Grav. 16 (1999) 495 [gr-qc/9801100] [INSPIRE].

  98. N.E. Chisari and M. Zaldarriaga, Connection between Newtonian simulations and general relativity, Phys. Rev. D 83 (2011) 123505 [Erratum ibid. D 84 (2011) 089901] [arXiv:1101.3555] [INSPIRE].

  99. D. Benisty and E.I. Guendelman, Cosmological Principle in Newtonian Dynamics, arXiv:1902.06511 [INSPIRE].

  100. K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers, Homogeneous Nonrelativistic Geometries as Coset Spaces, Class. Quant. Grav. 35 (2018) 175007 [arXiv:1712.03980] [INSPIRE].

  101. J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].

  102. M. Henneaux and S.-J. Rey, Nonlinear W∞ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].

  103. V.E. Hubeny, S. Minwalla and M. Rangamani, The fluid/gravity correspondence, in Black holes in higher dimensions, pp. 348–383, 2012, arXiv:1107.5780 [INSPIRE].

  104. R.A. Davison, S. Grozdanov, S. Janiszewski and M. Kaminski, Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity, JHEP 11 (2016) 170 [arXiv:1606.06747] [INSPIRE].

  105. E.P. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].

  106. E.P. Verlinde, Emergent Gravity and the Dark Universe, SciPost Phys. 2 (2017) 016 [arXiv:1611.02269] [INSPIRE].

  107. D. Pereñiguez, p-brane Newton-Cartan geometry, J. Math. Phys. 60 (2019) 112501 [arXiv:1908.04801] [INSPIRE].

  108. L. Romano, Non-Relativistic Four Dimensional p-Brane Supersymmetric Theories and Lie Algebra Expansion, arXiv:1906.08220 [INSPIRE].

  109. R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [arXiv:1305.6737] [INSPIRE].

  110. E. Bergshoeff, J. Rosseel and T. Zojer, Newton-Cartan supergravity with torsion and Schrödinger supergravity, JHEP 11 (2015) 180 [arXiv:1509.04527] [INSPIRE].

  111. 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].

  112. 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].

  113. B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].

  114. J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].

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  1. Institut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich, Wolfgang-Pauli-Strasse 27, 8093, Zürich, Switzerland

    Dennis Hansen

  2. School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Peter Guthrie Tait road, Edinburgh, EH9 3FD, UK

    Jelle Hartong

  3. Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91, Stockholm, Sweden

    Niels A. Obers

  4. The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark

    Niels A. Obers

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Hansen, D., Hartong, J. & Obers, N.A. Non-relativistic gravity and its coupling to matter. J. High Energ. Phys. 2020, 145 (2020). https://doi.org/10.1007/JHEP06(2020)145

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  • Received: 04 April 2020

  • Accepted: 26 May 2020

  • Published: 23 June 2020

  • DOI: https://doi.org/10.1007/JHEP06(2020)145

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Keywords

  • Classical Theories of Gravity
  • Space-Time Symmetries
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