Hamiltonian analysis of 4-dimensional spacetime inBondi-like coordinates

Abstract：

We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates { v,r,x ^a, 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 ( e _μ ^I) and connection ( ω _μ ^IJ) 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 e ², e ³. 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 π _IJ ^μ. The consistency conditions of the primary constraints π _IJ ⁰=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 _IJ ⁰=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 ₀ ^Aare Ricci identities. The equations of motion of the canonical variables have also been shown.