Abstract
We identify a symplectic potential for general relativity in tetrad and connection variables that is fully gauge-invariant, using the freedom to add surface terms. When torsion vanishes, it does not lead to surface charges associated with the internal Lorentz transformations, and reduces exactly to the symplectic potential given by the Einstein-Hilbert action. In particular, it reproduces the Komar form when the variation is a Li derivative, and the geometric expression in terms of extrinsic curvature and 2d corner data for a general variation. The additional surface term vanishes at spatial infinity for asymptotically flat spacetimes, thus the usual Poincaré charges are obtained. We prove that the first law of black hole mechanics follows from the Noether identity associated with the covariant Lie derivative, and that it is independent of the ambiguities in the symplectic potential provided one takes into account the presence of non-trivial Lorentz charges that these ambiguities can introduce.
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References
A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Analysis, geometry and mechanics: 200 years after Lagrange, M. Francaviglia and D. Holm eds., North-Holland (1991) [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation S. Hawking and W. Israel eds., Princeton (1986) [INSPIRE].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
A. Ashtekar, S. Fairhurst and B. Krishnan, Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D 62 (2000) 104025 [gr-qc/0005083] [INSPIRE].
A. Ashtekar, J. Engle and D. Sloan, Asymptotics and Hamiltonians in a First order formalism, Class. Quant. Grav. 25 (2008) 095020 [arXiv:0802.2527] [INSPIRE].
A. Corichi, I. Rubalcava and T. Vukasinac, Hamiltonian and Noether charges in first order gravity, Gen. Rel. Grav. 46 (2014) 1813 [arXiv:1312.7828] [INSPIRE].
A. Corichi, I. Rubalcava-García and T. Vukašinac, Actions, topological terms and boundaries in first-order gravity: A review, Int. J. Mod. Phys. D 25 (2016) 1630011 [arXiv:1604.07764] [INSPIRE].
T. Jacobson and A. Mohd, Black hole entropy and Lorentz-diffeomorphism Noether charge, Phys. Rev. D 92 (2015) 124010 [arXiv:1507.01054] [INSPIRE].
K. Prabhu, The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class. Quant. Grav. 34 (2017) 035011 [arXiv:1511.00388] [INSPIRE].
G.A. Burnett and R.M. Wald, A conserved current for perturbations of einstein-maxwell space-times, Proc. Roy. Soc. Lond. A 430 (1990) 57.
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
N. Bodendorfer, T. Thiemann and A. Thurn, New Variables for Classical and Quantum Gravity in all Dimensions V. Isolated Horizon Boundary Degrees of Freedom, Class. Quant. Grav. 31 (2014) 055002 [arXiv:1304.2679] [INSPIRE].
G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne’eman, Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors and breaking of dilation invariance, Phys. Rept. 258 (1995) 1 [gr-qc/9402012] [INSPIRE].
Y.N. Obukhov, The Palatini principle for manifold with boundary, Class. Quant. Grav. 4 (1987) 1085.
N. Bodendorfer and Y. Neiman, Imaginary action, spinfoam asymptotics and the ‘transplanckian’ regime of loop quantum gravity, Class. Quant. Grav. 30 (2013) 195018 [arXiv:1303.4752] [INSPIRE].
I. Jubb, J. Samuel, R. Sorkin and S. Surya, Boundary and Corner Terms in the Action for General Relativity, Class. Quant. Grav. 34 (2017) 065006 [arXiv:1612.00149] [INSPIRE].
T. Thiemann, Modern canonical quantum general relativity, Cambridge University Press (2001).
J.E. Daum and M. Reuter, Renormalization Group Flow of the Holst Action, Phys. Lett. B 710 (2012) 215 [arXiv:1012.4280] [INSPIRE].
D. Benedetti and S. Speziale, Perturbative quantum gravity with the Immirzi parameter, JHEP 06 (2011) 107 [arXiv:1104.4028] [INSPIRE].
D. Benedetti and S. Speziale, Perturbative running of the Immirzi parameter, J. Phys. Conf. Ser. 360 (2012) 012011 [arXiv:1111.0884] [INSPIRE].
W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].
A. Ashtekar and W.M. Wieland, work in progress (2018).
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [INSPIRE].
G. Barnich, P. Mao and R. Ruzziconi, Conserved currents in the Cartan formulation of general relativity, in About Various Kinds of Interactions: Workshop in honour of Professor Philippe Spindel, Mons, Belgium, June 4-5, 2015 (2016) [arXiv:1611.01777] [INSPIRE].
M. Montesinos, D. González, M. Celada and B. Díaz, Reformulation of the symmetries of first-order general relativity, Class. Quant. Grav. 34 (2017) 205002 [arXiv:1704.04248] [INSPIRE].
E. Frodden and D. Hidalgo, Surface Charges for Gravity and Electromagnetism in the First Order Formalism, Class. Quant. Grav. 35 (2018) 035002 [arXiv:1703.10120] [INSPIRE].
M. Blau, private communication (2018).
W.M. Wieland, The Chiral Structure of Loop Quantum Gravity, Ph.D. Thesis, Aix-Marseille Université (2013) [INSPIRE].
M. Henneaux and C. Troessaert, BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach, JHEP 03 (2018) 147 [arXiv:1801.03718] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
H. Gomes and A. Riello, A Unified Geometric Framework for Boundary Charges and Particle Dressings, arXiv:1804.01919 [INSPIRE].
L. Freidel, A. Perez and D. Pranzetti, Loop gravity string, Phys. Rev. D 95 (2017) 106002 [arXiv:1611.03668] [INSPIRE].
W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincaré 18 (2017) 3695 [arXiv:1706.00479] [INSPIRE].
M. Geiller, Lorentz-diffeomorphism edge modes in 3d gravity, JHEP 02 (2018) 029 [arXiv:1712.05269] [INSPIRE].
A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Rel. 7 (2004) 10 [gr-qc/0407042] [INSPIRE].
N. Bodendorfer, A note on entanglement entropy and quantum geometry, Class. Quant. Grav. 31 (2014) 214004 [arXiv:1402.1038] [INSPIRE].
A.S. Cattaneo and M. Schiavina, BV-BFV approach to General Relativity: Palatini-Cartan-Holst action, arXiv:1707.06328 [INSPIRE].
A. Ashtekar and M. Streubel, Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity, Proc. Roy. Soc. Lond. A 376 (1981) 585 [INSPIRE].
M.P. Reisenberger, The symplectic 2-form for gravity in terms of free null initial data, Class. Quant. Grav. 30 (2013) 155022 [arXiv:1211.3880] [INSPIRE].
F. Hopfmüller and L. Freidel, Null Conservation Laws for Gravity, Phys. Rev. D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
S. Alexandrov and S. Speziale, First order gravity on the light front, Phys. Rev. D 91 (2015) 064043 [arXiv:1412.6057] [INSPIRE].
E. De Paoli and S. Speziale, Sachs’ free data in real connection variables, JHEP 11 (2017) 205 [arXiv:1707.00667] [INSPIRE].
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De Paoli, E., Speziale, S. A gauge-invariant symplectic potential for tetrad general relativity. J. High Energ. Phys. 2018, 40 (2018). https://doi.org/10.1007/JHEP07(2018)040
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DOI: https://doi.org/10.1007/JHEP07(2018)040