Surface Charges in Gravitation

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


The main purpose of this first lecture is to introduce the concept of canonical surface charges in a generally covariant theory of gravity, whose General Relativity is the most famous representative. We will first show that a conserved stress tensor can be generated for any classical field theory. We will then motivate why we cannot define conserved currents and charges in Noether’s fashion for generally covariant theories, and more globally, for theories that include gauge transformations. This will lead us to extend Noether’s first theorem to formulate lower degree conservation laws, which will be exploitable for theories such as Einstein’s gravity. On the way, we will discuss about the symplectic structure of abstract spaces of fields, and use the covariant phase space formalism to derive a powerful result linking this structure and the lower degree conserved forms. We will then be able to compute surface charges associated to these quantities and study their properties and their algebra. Along the text, some pedagogical examples will be provided, namely for pure Einstein’s gravity, and Maxwells electrodynamics. Finally, we will present another possible definition of these surface charges and use the latter definition as an efficient tool to derive the conserved charges of Chern-Simons theory.


  1. 1.
    L.F. Abbott, S. Deser, Stability of gravity with a cosmological constant. Nucl. Phys. B195, 76–96 (1982). ADSCrossRefGoogle Scholar
  2. 2.
    R.L. Arnowitt, S. Deser, C.W. Misner, Dynamical structure and definition of energy in general relativity. Phys. Rev. 116, 1322–1330 (1959). ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    T. Azeyanagi, G. Compère, N. Ogawa, Y. Tachikawa, S. Terashima, Higher-derivative corrections to the asymptotic virasoro symmetry of 4d extremal black holes. Prog. Theor. Phys. 122, 355–384 (2009). arXiv:0903.4176 [hep-th]; ADSCrossRefGoogle Scholar
  4. 4.
    G. Barnich, F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges. Nucl. Phys. B633, 3–82 (2002). arXiv:hep-th/0111246 [hep-th];
  5. 5.
    G. Barnich, G. Compère, Surface charge algebra in gauge theories and thermodynamic integrability. J. Math. Phys. 49, 042901 (2008). arXiv:0708.2378 [gr-qc]; ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Barnich, F. Brandt, M. Henneaux, Local BRST cohomology in gauge theories. Phys. Rep. 338, 439–569 (2000) . arXiv:hep-th/0002245 [hep-th];
  7. 7.
    J.D. Brown, M. Henneaux, On the Poisson brackets of differentiable generators in classical field theory. J. Math. Phys. 27, 489–491 (1986). ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Compère, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions. PhD thesis, Brussels U., 2007. arXiv:0708.3153 [hep-th];
  9. 9.
    G. Compère, F. Dehouck, Relaxing the parity conditions of asymptotically flat gravity. Class. Quant. Grav. 28, 245016 (2011). arXiv:1106.4045 [hep-th];; [Erratum: Class. Quant. Grav. 30, 039501 (2013)]ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Compère, D. Marolf, Setting the boundary free in AdS/CFT. Class. Quant. Grav. 25, 195014 (2008). arXiv:0805.1902 [hep-th];
  11. 11.
    G. Compère, K. Murata, T. Nishioka, Central charges in extreme black Hole/CFT correspondence. J. High Energy Phys. 05, 077 (2009). arXiv:0902.1001 [hep-th]; ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    G. Compère, L. Donnay, P.-H. Lambert, W. Schulgin, Liouville theory beyond the cosmological horizon. J. High Energy Phys. 1503, 158 (2015). arXiv:1411.7873 [hep-th];
  13. 13.
    G. Compère, K. Hajian, A. Seraj, M.M. Sheikh-Jabbari, Wiggling throat of extremal black holes. J. High Energy Phys. 10, 093 (2015). arXiv:1506.07181 [hep-th];
  14. 14.
    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];
  15. 15.
    S. Deser, B. Tekin, Gravitational energy in quadratic curvature gravities. Phys. Rev. Lett. 89, 101101 (2002). arXiv:hep-th/0205318 [hep-th];
  16. 16.
    S. Deser, B. Tekin, Energy in generic higher curvature gravity theories. Phys. Rev. D67, 084009 (2003). arXiv:hep-th/0212292 [hep-th];
  17. 17.
    P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory. Graduate Texts in Contemporary Physics (Springer, New York, 1997).;
  18. 18.
    V. Iyer, R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D50, 846–864 (1994). arXiv:gr-qc/9403028 [gr-qc];
  19. 19.
    J. Lee, R.M. Wald, Local symmetries and constraints. J. Math. Phys. 31, 725–743 (1990). ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    T. Regge, C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. 88, 286 (1974). ADSMathSciNetCrossRefGoogle Scholar

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© 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

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