Hamiltonian Analysis of 3-Dimensional Connection Dynamics in Bondi-like Coordinates

Abstract

The Hamiltonian analysis for a 3-dimensional connection dynamics of so(1, 2), spanned by {L−+, L−2, L+2} instead of {L01, L02, L12}, is first conducted in a Bondi-like coordinate system. The symmetry of the system is clearly presented. A null coframe with 3 independent variables and 9 connection coefficients are treated as basic configuration variables. All constraints and their consistency conditions, the solutions of Lagrange multipliers as well as the equations of motion are presented. There is no physical degree of freedom in the system. The Bañados-Teitelboim-Zanelli (BTZ) spacetime is discussed as an example to check the analysis. Unlike the ADM formalism, where only non-degenerate geometries on slices are dealt with and the Ashtekar formalism, where non-degenerate geometries on slices are mainly concerned though the degenerate geometries may be studied as well, in the present formalism the geometries on the slices are always degenerate though the geometries for the spacetime are not degenerate.

Hamiltonian analysis of 4-dimensional spacetime in Bondi-like coordinates

“…In our previous study [23], we carried out a Hamiltonian analysis of three-dimensional gravity in Bondilike coordinates, based on Dirac's treatments of a constrained system [24]. In the three-dimensional case, g 01 was fixed to 1; all the three variables e + 0 , e 2 0 , and e 2 2 of the coframe and the connection components ω IJ µ were treated as configuration variables; the Palatini action was used; and the cosmological constant was also included.…”

“…In su(2)-connection dynamics [9], the constraints are classified as the spatial scalar, spatial vector, and su(2) gauge constraints. In comparison, ǫ AB ǫ ab F +A 1a e B b +F 23 23 ≈0 and ǫ AB ǫ ab F −A 1a e B b ≈ 0 are two scalar constraints and…”

“…The aim of the present study is to make a Hamiltonian analysis of four-dimensional gravity in Bondi-like coordinates, by using the same method as in Ref. [23].…”

“…For convenience, we implement some modifications in the treatment. Unlike in the treatment in the threedimensional case [23], g 01 is not fixed, and, therefore, the metric is more general and can be applied to more cases. The other differences are that n 0 , l 0 , and e A 0 are treated as Lagrange multipliers and that the cosmological constant is not included.…”