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

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.

💡 Research Summary

The paper presents a complete Hamiltonian formulation of four‑dimensional general relativity in Bondi‑like coordinates ({v,r,x^{a}}) with (a=2,3). In this setting the hypersurfaces (v=\text{const}) are null and the evolution parameter is the advanced time (v). The authors first decompose the internal Lorentz group (SO(1,3)) into the product (SO(1,1)\times SO(2)\times T^{-}(2)\times T^{+}(2)). Correspondingly the Lie algebra splits into (\mathfrak{so}(1,1)\oplus\mathfrak{so}(2)\oplus\mathfrak{t}^{-}(2)\oplus\mathfrak{t}^{+}(2)). This decomposition is naturally realized by a co‑frame consisting of two null one‑forms (e^{-},e^{+}) and two spacelike one‑forms (e^{A};(A=2,3)). The null co‑frame is chosen so that (e^{-}=n_{0},dv), making the (SO(1,1)) boost symmetry manifest.

Using the Palatini action \