Hamiltonian analysis of 4-dimensional spacetime inBondi-like coordinates

  • We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,xa,a=2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T±(2), whose Lie algebra so(1,3) is decomposed into so(1,1), so(2), and t±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in so(1,1) BF theory. General relativity can be reformulated as the four-dimensional coframe (eμI) and connection (ωμIJ) dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e-, e+ and two spacelike 1-forms e2, e3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about πIJμ. The consistency conditions of the primary constraints πIJ0=0 can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints πIJ0=0 and 40 second-class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n0, l0, and e0A are Ricci identities. The equations of motion of the canonical variables have also been shown.
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  • [1] R. Arnowitt, S. Deser and C. Misner, Phys. Rev., 116:1322 (1959)
    [2] R. A. d'Inverno and J. Smallwood, Phys. Rev. D, 22:1233 (1980)
    [3] J. Goldberg, Found. Phys., 14:1211 (1984)
    [4] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Proc. R. Soc. Lond. A, 269:21 (1962)
    [5] R. K. Sachs, Proc. R. Soc. Lond. A, 270:103 (1962)
    [6] A. Ashtakar, C. Beetle, and S. Fairhurst, Class. Quant. Grav., 16:L1 (1999)
    [7] A. Ashtekar and B. Krishnan, Living Rev. Relativ., 7:10 (2004)
    [8] B. Krishnan (2014) pp 527-555. In:Ashtekar A., Petkov V. (eds) Springer Handbook of Spacetime, (Springer, Berlin, Heidelberg)
    [9] A. Ashtekar, Phys. Rev. Lett., 57:2244 (1986)
    [10] R. d'Inverno and J. Vickers, Class. Quant. Grav., 12:753 (1995)
    [11] R. d'Inverno, P. Lambert, and J. Vickers, Class. Quant. Grav., 23:3747 (2006)
    [12] R. d'Inverno, P. Lambert, and J. Vickers, Class. Quant. Grav., 23:4511 (2006)
    [13] J. Goldberg, D. Robinson, and C. Soteriou, Class. Quant. Grav., 9:1309 (1992)
    [14] J. Goldberg and C. Soteriou, Class. Quant. Grav., 12:2779 (1995)
    [15] N. Bodendorfer, T. Thiemann, and A. Thurn, Class. Quant. Grav., 30:045001, 045002, 045003, 045004 (2013)
    [16] G. S. Hall, Symmetries and Curvature Structure in General Relativity, (World Scientific, Singapore, 2004)
    [17] J. Wang, Y. Ma, and X.-A. Zhao, Phys. Rev. D, 89:084065 (2014)
    [18] J. Wang and C.-G. Huang, Int. J. Mod. Phys. D, 25:1650100 (2016)
    [19] C.-G. Huang and J. Wang, Gen. Rel. Grav., 45:(2016)
    [20] J. Wang, C.-G. Huang, and L. Li, Chin. Phys. C, 40:(2016)
    [21] E. T. Newman and R. Penrose, J. Math. Phys., 3:566 (1962)
    [22] R. Basu, A. Chatterjee, and A. Ghosh, Class. Quant. Grav., 29:235010 (2012)
    [23] C. G. Huang and S. B. Kong, Commun. Theor. Phys., 68:227 (2017)
    [24] P. A. M. Dirac, Lectures on Quantum Mechanics, (Yeshiva University, New York, 1964)
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Chao-Guang Huang and Shi-Bei Kong. Hamiltonian analysis of 4-dimensional spacetime inBondi-like coordinates[J]. Chinese Physics C, 2018, 42(10): 105101. doi: 10.1088/1674-1137/42/10/105101
Chao-Guang Huang and Shi-Bei Kong. Hamiltonian analysis of 4-dimensional spacetime inBondi-like coordinates[J]. Chinese Physics C, 2018, 42(10): 105101.  doi: 10.1088/1674-1137/42/10/105101 shu
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Received: 2018-05-17
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    Supported by the National Natural Science Foundation of China (11690022).

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Hamiltonian analysis of 4-dimensional spacetime inBondi-like coordinates

  • 1. Theoretical Physics Division, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
  • 2. University of Chinese Academy of Sciences, Beijing 100049, China
Fund Project:  Supported by the National Natural Science Foundation of China (11690022).

Abstract: We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,xa,a=2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T±(2), whose Lie algebra so(1,3) is decomposed into so(1,1), so(2), and t±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in so(1,1) BF theory. General relativity can be reformulated as the four-dimensional coframe (eμI) and connection (ωμIJ) dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e-, e+ and two spacelike 1-forms e2, e3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about πIJμ. The consistency conditions of the primary constraints πIJ0=0 can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints πIJ0=0 and 40 second-class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n0, l0, and e0A are Ricci identities. The equations of motion of the canonical variables have also been shown.

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