Advertisement

Three Dimensional Einstein’s Gravity

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

Abstract

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.

References

  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). http://dx.doi.org/10.1007/BF01211590 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]; http://dx.doi.org/10.1023/A:1026654312961; http://dx.doi.org/10.4310/ATMP.1998.v2.n2.a1 [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]; http://dx.doi.org/10.1016/S0370-1573(99)00083-6 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Witten, (2+1)-Dimensional gravity as an exactly soluble system. Nucl. Phys. B 311, 46 (1988). http://dx.doi.org/10.1016/0550-3213(88)90143-5
  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). http://dx.doi.org/10.1016/0370-2693(86)90140-1 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]; http://dx.doi.org/10.1103/PhysRevLett.69.1849 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]; http://dx.doi.org/10.1103/PhysRevD.48.1506; http://dx.doi.org/10.1103/PhysRevD.88.069902 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]; http://dx.doi.org/10.1088/1126-6708/1998/02/009
  9. 9.
    E. Witten, Three-dimensional gravity revisited (2007, Unpublished). http://arxiv.org/abs/0706.3359
  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]; http://dx.doi.org/10.1007/JHEP02(2010)029
  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]; http://dx.doi.org/10.1007/JHEP05(2013)152
  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]; http://dx.doi.org/10.1088/0264-9381/24/5/F01; http://dx.doi.org/10.1088/0264-9381/24/11/C01
  13. 13.
    G. Barnich, C. Troessaert, Aspects of the BMS/CFT correspondence. J. High Energy Phys. 1005, 062 (2010). arXiv:1001.1541 [hep-th]; http://dx.doi.org/10.1007/JHEP05(2010)062
  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]; http://dx.doi.org/10.1103/PhysRevD.86.024020
  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]; http://dx.doi.org/10.1103/PhysRevLett.85.3758 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]; http://dx.doi.org/10.1007/JHEP10(2012)095
  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]; http://dx.doi.org/10.1103/PhysRevLett.110.141302
  19. 19.
    S. Deser, R. Jackiw, G. ’t Hooft, Three-dimensional Einstein gravity: dynamics of flat space. Ann. Phys. 152, 220 (1984). http://dx.doi.org/10.1016/0003-4916(84)90085-X ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Deser, R. Jackiw, Three-dimensional cosmological gravity: dynamics of constant curvature. Ann. Phys. 153, 405–416 (1984). http://dx.doi.org/10.1016/0003-4916(84)90025-3 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]; http://dx.doi.org/10.1007/JHEP01(2014)023
  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]; http://dx.doi.org/10.1007/JHEP01(2016)080
  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]; http://dx.doi.org/10.1103/PhysRevD.55.669 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]; http://dx.doi.org/10.1088/1361-6382/aa8aad 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