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  • Letter
  • Open Access

Galilean first-order formulation for the nonrelativistic expansion of general relativity

Dennis Hansen1, Jelle Hartong2, Niels A. Obers3,4, and Gerben Oling3,*

  • 1Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
  • 2School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
  • 3The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
  • 4Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

  • *Corresponding author. gerben.oling@nbi.ku.dk
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Phys. Rev. D 104, L061501 – Published 22 September, 2021

DOI: https://doi.org/10.1103/PhysRevD.104.L061501

Abstract

We reformulate the Palatini action for general relativity in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the GL(n,R) structure group of a general frame bundle. This leads to a novel Palatini-type formulation of general relativity that provides a natural starting point for a first-order nonrelativistic expansion. In doing so, we show how a comparison of Lorentzian and Newton-Cartan metric compatibility explains the appearance of torsion in the nonrelativistic expansion.

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References (38)

  1. D. Van den Bleeken, Classical Quant. Grav. 34, 185004 (2017).
  2. D. Hansen, J. Hartong, and N. A. Obers, Phys. Rev. Lett. 122, 061106 (2019).
  3. D. Hansen, J. Hartong, and N. A. Obers, J. High Energy Phys. 06 (2020) 145.
  4. G. Dautcourt, Classical Quant. Grav. 14, A109 (1997).
  5. W. Tichy and E. E. Flanagan, Phys. Rev. D 84, 044038 (2011).
  6. R. Andringa, E. Bergshoeff, S. Panda, and M. de Roo, Classical Quant. Grav. 28, 105011 (2011).
  7. M. H. Christensen, J. Hartong, N. A. Obers, and B. Rollier, Phys. Rev. D 89, 061901 (2014).
  8. J. Hartong and N. A. Obers, J. High Energy Phys. 07 (2015) 155.
  9. E. Cartan, Ann. Éc. Norm. Super. 40, 325 (1923).
  10. E. Cartan, Ann. Éc. Norm. Super. 41, 1 (1924).
  11. D. T. Son, arXiv:1306.0638.
  12. K. Jensen, SciPost Phys. 5, 011 (2018).
  13. J. Hartong, E. Kiritsis, and N. A. Obers, Phys. Lett. B 746, 318 (2015).
  14. M. Geracie, D. T. Son, C. Wu, and S.-F. Wu, Phys. Rev. D 91, 045030 (2015).
  15. R. Andringa, E. Bergshoeff, J. Gomis, and M. de Roo, Classical Quant. Grav. 29, 235020 (2012).
  16. T. Harmark, J. Hartong, and N. A. Obers, Phys. Rev. D 96, 086019 (2017).
  17. E. Bergshoeff, J. Gomis, and Z. Yan, J. High Energy Phys. 11 (2018) 133.
  18. D. Hansen, J. Hartong, and N. A. Obers, Int. J. Mod. Phys. D 28, 1944010 (2019).
  19. D. Hansen, J. Hartong, and N. A. Obers, arXiv:1905.13723.
  20. T. Jacobson and A. Mohd, Phys. Rev. D 92, 124010 (2015).
  21. F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Phys. Rep. 258, 1 (1995).
  22. J. B. Jiménez, L. Heisenberg, and T. S. Koivisto, Universe 5, 173 (2019).
  23. B. Julia and S. Silva, Classical Quant. Grav. 15, 2173 (1998).
  24. N. Dadhich and J. M. Pons, Gen. Relativ. Gravit. 44, 2337 (2012).

  25. See Ref. [26] for a discussion on intrinsic torsion of nonrelativistic geometries in the language of G structures.

  26. J. Figueroa-O’Farrill, arXiv:2009.01948.
  27. See the Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevD.104.L061501 for an algebraic perspective on the Lorentzian and Galilean decomposition of a general affine connection.
  28. M. Le Bellac and J.-M. Lévy-Leblond, Nuovo Cimento B 14, 217 (1973).
  29. E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel, and T. ter Veldhuis, J. High Energy Phys. 03 (2017) 165.
  30. J. Hartong, J. High Energy Phys. 08 (2015) 069.
  31. L. Donnay and C. Marteau, Classical Quant. Grav. 36, 165002 (2019).
  32. C. Duval, G. W. Gibbons, and P. A. Horvathy, Classical Quant. Grav. 31, 092001 (2014).
  33. L. Ciambelli, R. G. Leigh, C. Marteau, and P. M. Petropoulos, Phys. Rev. D 100, 046010 (2019).
  34. C. Duval, G. W. Gibbons, P. A. Horvathy, and P. M. Zhang, Classical Quant. Grav. 31, 085016 (2014).
  35. D. Hansen, N. A. Obers, G. Oling, and B. Søgaard (to be published).
  36. M. Cariglia, Phys. Rev. D 98, 084057 (2018).
  37. A. Guerrieri and R. F. Sobreiro, J. High Energy Phys. 03 (2021) 104.
  38. J. Beltrán Jiménez, L. Heisenberg, and T. S. Koivisto, J. Cosmol. Astropart. Phys. 08 (2018) 039.

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Phys. Rev. D 104, L061501– Published 22 September, 2021
  • Received 21 December 2020
  • Accepted 25 July 2021
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