Hamiltonian Analysis of 4-dimensional Spacetime in Bondi-like Coordinates

TL;DR

The authors develop a Hamiltonian formulation of 4D gravity on a null Bondi-like foliation using a coframe/connection (Palatini) approach, and perform a full Dirac constraint analysis. By decomposing $SO(1,3)$ into $SO(1,1)\oplus SO(2)\oplus T^{\pm}(2)$, they obtain a null coframe with two null and two spacelike directions, derive 32 primary and 14 secondary constraints (46 total), and show that 6 are first class while 40 are second class, yielding 2 local gravitational degrees of freedom. The torsion-free conditions appear as consistency requirements, and the integrability conditions reduce to Ricci identities, with the equations of motion for the canonical variables fully displayed. The results provide a concrete Hamiltonian framework for gravity in Bondi-like coordinates, potentially connecting to BF-type formulations and pure-connection dynamics in constrained null geometries. Overall, the work advances canonical gravity on null surfaces and clarifies the constraint structure and physical content of 4D Bondi-like spacetimes.

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 $T^\pm(2)$, whose Lie algebra $so(1,3)$ is decomposed into $so(1,1)$, $so(2)$, $t^\pm(2)$ 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 $n_0$, $l_0$, and $e^A_0$ are Ricci identities. The equations of motion of the canonical variables have also been shown.