Hamiltonian Analysis of 4-dimensional Spacetime in Bondi-like Coordinates

Abstract: We discuss the Hamiltonian formulation of gravity in 4-dimensional spacetime under Bondi-like coordinates{v, r, x^a, a=2, 3}. In Bondi-like coordinates, the 3-dimensional hypersurface is a null hypersurface and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the 4-dimensional spacetime is decomposed into SO(1,1), SO(2), and T^\pm(2), whose Lie algebra so(1,3) is decomposed into so(1,1), so(2), t^\pm(2) correspondingly. The SO(1,1) symmetry is very obvious in this kind of decomposition, which is very useful in so(1,1) BF theory. General relativity can be reformulated as the 4-dimensional coframe (e^I_\mu) and connection ({\omega}^{IJ}_\mu) dynamics of gravity based on this kind of decomposition in the Bondi-like coordinate system. The coframe consists of 2 null 1-forms e^-, e^+ and 2 spacelike 1-forms e^2, e^3. The Palatini action is used. The Hamiltonian analysis is conducted by the Dirac's methods. The consistency analysis of constraints has been done completely. There are 2 scalar constraints and one 2-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about {\pi}^\mu_{IJ}. The consistency conditions of the primary constraints {\pi}^0_{IJ}=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 {\pi}^0_{IJ}=0 and 40 second class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n_0, l_0, and e^A_0 are Ricci identities. The equations of motion of the canonical variables have also been shown.