In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini-Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the SL(2, ℝ) symmetries of the boundary fields are manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental SL(2, ℝ) variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.
L.B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Living Rev. Rel. 7 (2004) 4 [INSPIRE].
B. Dittrich, C. Goeller, E. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity I – Convergence of multiple approaches and examples of Ponzano-Regge statistical duals, Nucl. Phys. B 938 (2019) 807 [arXiv:1710.04202] [INSPIRE].
R.E. Peierls, The commutation laws of relativistic field theory, Proc. Roy. Soc. Lond. A 214 (1952) 143.
A. Ashtekar, L. Bombelli and O. Reula, The Covariant Phase Space Of Asymptotically Flat Gravitational Fields, in Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia and D. Holm eds., Amsterdam North Holland (1990).
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge University Press, Cambridge U.K. (2003).
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
J. Frauendiener, Note on the memory effect, Class. Quant. Grav. 9 (1992) 1639.
A. Ashtekar, New Variables for Classical and Quantum Gravity, Phys. Rev. Lett. 57 (1986) 2244 [INSPIRE].
R. Penrose and W. Rindler, Spinors and Space-Time, Two-Spinor Calculus and Relativistic Fields. Vol. 1. Cambridge University Press, Cambridge U.K. (1984).
R. Penrose and W. Rindler, Spinors and Space-Time, Two-Spinor Calculus and Relativistic Fields. Vol. 2. Cambridge University Press, Cambridge U.K. (1986).
C.-N. Yang, Charge quantization, compactness of the gauge group, and flux quantization, Phys. Rev. D 1 (1970) 2360 [INSPIRE].
C. Rovelli, Area is the length of Ashtekar’s triad field, Phys. Rev. D 47 (1993) 1703 [Erratum ibid. 87 (2013) 089902] [INSPIRE].
B. Dittrich, The continuum limit of loop quantum gravity - a framework for solving the theory, in Loop Quantum Gravity: The First 30 Years, A. Ashtekar and J. Pullin, eds. (2017), DOI [arXiv:1409.1450] [INSPIRE].
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2104.05803