Hamiltonian Analysis of 3-dimensional Spacetime in Bondi-like Coordinates

TL;DR

This paper performs a Hamiltonian analysis of 3D spacetime in Bondi-like coordinates, revealing the system's symmetries, constraints, and the absence of physical degrees of freedom, with applications to BTZ spacetime.

Contribution

It introduces a Hamiltonian framework for 3D connection dynamics in Bondi-like coordinates, emphasizing degenerate geometries and system symmetries, distinct from ADM and Ashtekar formalisms.

Findings

No physical degrees of freedom in the system.

Constraints and equations of motion are explicitly derived.

Application to BTZ spacetime confirms the analysis.

Abstract

The Hamiltonian analysis for a 3-dimensional connection dynamics of $\frak{so}(1,2)$, spanned by $\{L_{-+},L_{-2},L_{+2}\}$ instead of $\{L_{01}, L_{02}, L_{12}\}$, 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 Ba\~nados-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.