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 ± (2), whose Lie algebra so(1,3) is decomposed into so(1, 1), so(2),t ± (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 µ ) and connection (ω IJ µ ) 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 π µ IJ . The consistency conditions of the primary constraints π 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 π 0 IJ = 0 and 40 second class constraints. The local physical degrees of freedom is 2. The integrability conditions of Lagrange multipliers n0, l0, and e A 0 are Ricci identities. The equations of motion of the canonical variables have also been shown.