Three Dimensional Einstein’s Gravity

  • Geoffrey Compère
Part of the Lecture Notes in Physics book series (LNP, volume 952)


General Relativity is a very complex theory whose quantization remains elusive. A reduced version of the theory exists: 3-dimensional Einstein’s gravity to which this lecture is dedicated. As a toy-model, it is a very useful framework thanks to which we can experiment some techniques and derive features, some of which extend to the physical 4d case.

In this lecture, we will review the typical properties of 3d gravity, which are mostly due to the vanishing of the Weyl curvature. Next we will turn to the AdS3 phase space: we will describe global features of AdS3 itself, give several elements on the Brown-Henneaux boundary conditions and the resulting asymptotic symmetry group, and finally discuss BTZ black holes. We will show that the asymptotically flat phase space can be obtained from the flat limit of the asymptotically AdS3 phase space. Finally, we will shortly present the Chern-Simons formulation of 3d gravity, which reduces the theory to the one of two non-abelian gauge vector fields.


  1. 1.
    J.D. Brown, M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys. 104, 207–226 (1986). ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113–1133 (1999). arXiv:hep-th/9711200 [hep-th];; [Adv. Theor. Math. Phys. 2, 231 (1998)]
  3. 3.
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Large N field theories, string theory and gravity. Phys. Rep. 323, 183–386 (2000). arXiv:hep-th/9905111 [hep-th]; ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Witten, (2+1)-Dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988).
  5. 5.
    A. Achucarro, P.K. Townsend, A Chern-Simons action for three-dimensional anti-de sitter supergravity theories. Phys. Lett. B 180, 89 (1986). ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Banados, C. Teitelboim, J. Zanelli, The black hole in three-dimensional space-time. Phys. Rev. Lett. 69, 1849–1851 (1992). arXiv:hep-th/9204099 [hep-th]; ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Geometry of the (2+1) black hole. Phys. Rev. D 48(6), 1506–1525 (1993). arXiv:gr-qc/9302012 [gr-qc];; ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Strominger, Black hole entropy from near horizon microstates. J. High Energy Phys. 02, 009 (1998). arXiv:hep-th/9712251 [hep-th];
  9. 9.
    E. Witten, Three-dimensional gravity revisited (2007, Unpublished).
  10. 10.
    A. Maloney, E. Witten, Quantum gravity partition functions in three dimensions. J. High Energy Phys. 02, 029 (2010). arXiv:0712.0155 [hep-th];
  11. 11.
    G. Compère, W. Song, A. Strominger, New boundary conditions for AdS3. J. High Energy Phys. 05, 152 (2013). arXiv:1303.2662 [hep-th];
  12. 12.
    G. Barnich, G. Compère, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions. Class. Quant. Grav. 24, F15–F23 (2007). arXiv:gr-qc/0610130 [gr-qc];;
  13. 13.
    G. Barnich, C. Troessaert, Aspects of the BMS/CFT correspondence. J. High Energy Phys. 1005, 062 (2010). arXiv:1001.1541 [hep-th];
  14. 14.
    G. Barnich, A. Gomberoff, H.A. Gonzalez, The flat limit of three dimensional asymptotically anti-de Sitter spacetimes. Phys. Rev. D 86, 024020 (2012). arXiv:1204.3288 [gr-qc];
  15. 15.
  16. 16.
    D. Ida, No black hole theorem in three-dimensional gravity. Phys. Rev. Lett. 85, 3758–3760 (2000). arXiv:gr-qc/0005129 [gr-qc]; ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions. J. High Energy Phys. 10, 095 (2012). arXiv:1208.4371 [hep-th];
  18. 18.
    A. Bagchi, S. Detournay, R. Fareghbal, J. Simón, Holography of 3D flat cosmological horizons. Phys. Rev. Lett. 110(14), 141302 (2013). arXiv:1208.4372 [hep-th];
  19. 19.
    S. Deser, R. Jackiw, G. ’t Hooft, Three-dimensional Einstein gravity: dynamics of flat space. Ann. Phys. 152, 220 (1984). ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Deser, R. Jackiw, Three-dimensional cosmological gravity: dynamics of constant curvature. Ann. Phys. 153, 405–416 (1984). ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    J. de Boer, J.I. Jottar, Thermodynamics of higher spin black holes in AdS 3. J. High Energy Phys. 01, 023 (2014). arXiv:1302.0816 [hep-th];
  22. 22.
    G. Compère, P. Mao, A. Seraj, M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons. J. High Energy Phys. 01, 080 (2016). arXiv:1511.06079 [hep-th];
  23. 23.
    A. Ashtekar, J. Bicak, B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity. Phys. Rev. D 55, 669–686 (1997). arXiv:gr-qc/9608042 [gr-qc]; ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Compère, A. Fiorucci, Asymptotically flat spacetimes with BMS3 symmetry. Class. Quant. Grav. 34(20), 204002 (2017). arXiv:1705.06217 [hep-th]; ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Geoffrey Compère
    • 1
  1. 1.Physique théorique mathématiqueUniversité Libre de BruxellesBrusselsBelgium

Personalised recommendations