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

TL;DR

This paper develops a Hamiltonian formulation for 3D gravity using an so(1,2) connection, analyzed in Bondi-like null coordinates where spatial slices are degenerate while the full spacetime geometry remains nondegenerate. By reorganizing the internal Lorentz algebra as $\mathfrak{so}(1,1) \oplus \mathfrak{so}(1)$ plus translation sectors, the authors construct a Palatini action, derive canonical momenta, and identify 12 primary and 12 secondary constraints, all of which are second-class. The analysis shows zero local degrees of freedom in the bulk, with torsion-free conditions arising as secondary constraints and equations of motion, and provides a concrete BTZ spacetime check to confirm consistency. These results illuminate a path toward connecting bulk Hamiltonian dynamics to a boundary $\mathfrak{so}(1,1)$-BF theory on horizons, and suggest a framework that generalizes to higher dimensions via the proposed decomposition. Overall, the work offers a self-consistent, null-foliated Hamiltonian perspective on 3D gravity with potential implications for boundary holography and quantum gravity formalisms in degenerate-slice geometries.

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 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.