Carroll versus Galilei gravity

  • Eric Bergshoeff
  • Joaquim Gomis
  • Blaise Rollier
  • Jan Rosseel
  • Tonnis ter Veldhuis
Open Access
Regular Article - Theoretical Physics

Abstract

We consider two distinct limits of General Relativity that in contrast to the standard non-relativistic limit can be taken at the level of the Einstein-Hilbert action instead of the equations of motion. One is a non-relativistic limit and leads to a so-called Galilei gravity theory, the other is an ultra-relativistic limit yielding a so-called Carroll gravity theory. We present both gravity theories in a first-order formalism and show that in both cases the equations of motion (i) lead to constraints on the geometry and (ii) are not sufficient to solve for all of the components of the connection fields in terms of the other fields. Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier. We point out a few noteworthy differences between Carroll and Galilei gravity and give some examples of matter couplings.

Keywords

Space-Time Symmetries Classical Theories of Gravity 

Notes

Open Access

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References

  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    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
  5. [5]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    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
  7. [7]
    A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    U.H. Danielsson, A. Guijosa and M. Kruczenski, IIA/B, wound and wrapped, JHEP 10 (2000) 020 [hep-th/0009182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    J. Gomis, J. Gomis and K. Kamimura, Non-relativistic superstrings: A New soluble sector of AdS 5 × S 5, JHEP 12 (2005) 024 [hep-th/0507036] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    E.A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett. 116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    T. Banks and W. Fischler, Holographic Space-time Models of Anti-deSitter Space-times, arXiv:1607.03510 [INSPIRE].
  14. [14]
    C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP 02 (2017) 049 [arXiv:1611.00026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Gomis and P.K. Townsend, The Galilean Superstring, JHEP 02 (2017) 105 [arXiv:1612.02759] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Bañados, R. Troncoso and J. Zanelli, Higher dimensional Chern-Simons supergravity, Phys. Rev. D 54 (1996) 2605 [gr-qc/9601003] [INSPIRE].
  17. [17]
    J.M. Lévy-Leblond, Une nouvelle limite non-relativiste du group de Poincaré, Ann. Inst. H. Poincaré 3 (1965) 1.MathSciNetMATHGoogle Scholar
  18. [18]
    V.D. Sen Gupta, On an Analogue of the Galileo Group, Nuovo Cim. 54 (1966) 512.ADSCrossRefGoogle Scholar
  19. [19]
    M. Henneaux, Geometry of Zero Signature Space-times, Bull. Soc. Math. Belg. 31 (1979) 47 [INSPIRE].MathSciNetMATHGoogle Scholar
  20. [20]
    M. Henneaux, M. Pilati and C. Teitelboim, Explicit Solution for the Zero Signature (Strong Coupling) Limit of the Propagation Amplitude in Quantum Gravity, Phys. Lett. B 110 (1982) 123 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Niedermaier, The gauge structure of strong coupling gravity, Class. Quant. Grav. 32 (2015) 015007 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    M. Niedermaier, The dynamics of strong coupling gravity, Class. Quant. Grav. 32 (2015) 015008 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    M. Niedermaier, A geodesic principle for strong coupling gravity, Class. Quant. Grav. 32 (2015) 215022 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, ’Stringy’ Newton-Cartan Gravity, Class. Quant. Grav. 29 (2012) 235020 [arXiv:1206.5176] [INSPIRE].
  25. [25]
    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
  26. [26]
    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
  27. [27]
    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].
  28. [28]
    E. Bergshoeff, J. Gomis and G. Longhi, Dynamics of Carroll Particles, Class. Quant. Grav. 31 (2014) 205009 [arXiv:1405.2264] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    M. Le Bellac and J.-M. Lévy-Leblond, Galilean Electromagnetism, Nuovo Cim. B 14 (1973) 217.ADSCrossRefGoogle Scholar
  32. [32]
    G. Rousseaux, Forty Years of Galilean Electromagnetism (1973-2013), Eur. Phys. J. Plus 128 (2013) 1.CrossRefGoogle Scholar
  33. [33]
    A. Bagchi, R. Basu and A. Mehra, Galilean Conformal Electrodynamics, JHEP 11 (2014) 061 [arXiv:1408.0810] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Galilean Yang-Mills Theory, JHEP 04 (2016) 051 [arXiv:1512.08375] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    E. Bergshoeff, J. Rosseel and T. Zojer, Non-relativistic fields from arbitrary contracting backgrounds, Class. Quant. Grav. 33 (2016) 175010 [arXiv:1512.06064] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    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].MathSciNetCrossRefGoogle Scholar
  37. [37]
    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
  38. [38]
    B. Cardona, J. Gomis and J.M. Pons, Dynamics of Carroll Strings, JHEP 07 (2016) 050 [arXiv:1605.05483] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional Spin-3 Theories Based on General Kinematical Algebras, JHEP 01 (2017) 114 [arXiv:1612.02277] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    S. Golkar, D.X. Nguyen, M.M. Roberts and D.T. Son, Higher-Spin Theory of the Magnetorotons, Phys. Rev. Lett. 117 (2016) 216403 [arXiv:1602.08499] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    C. Hoyos and D.T. Son, Hall Viscosity and Electromagnetic Response, Phys. Rev. Lett. 108 (2012) 066805 [arXiv:1109.2651] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Eric Bergshoeff
    • 1
  • Joaquim Gomis
    • 2
  • Blaise Rollier
    • 1
  • Jan Rosseel
    • 3
  • Tonnis ter Veldhuis
    • 1
  1. 1.Centre for Theoretical PhysicsUniversity of GroningenGroningenThe Netherlands
  2. 2.Departament de Física Cuàntica i Astrofísica and Institut de Ciències del CosmosUniversitat de BarcelonaBarcelonaSpain
  3. 3.Faculty of PhysicsUniversity of ViennaViennaAustria

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