Barnich-Troessaert Bracket as a Dirac Bracket on the Covariant Phase Space

TL;DR

The paper shows that the Barnich–Troessaert bracket can be realized as a Dirac bracket on a reduced covariant phase space obtained by imposing second-class constraints that remove radiative data, leaving an edge-mode sector. This yields BT-like charges on a cross-section of null infinity, with a complementary radiative flux term accounting for evolution toward i^+. The framework clarifies the relationship among ADM, radiative, and edge-mode phase spaces and offers a holographic viewpoint on asymptotic gravitational symmetries with potential implications for quantum gravity and edge-mode dynamics.

Abstract

The Barnich--Troessaert bracket is a proposal for a modified Poisson bracket on the covariant phase space for general relativity. The new bracket allows us to compute charges, which are otherwise not integrable. Yet there is a catch. There is a clear prescription for how to evaluate the new bracket for any such charge, but little is known how to extend the bracket to the entire phase space. This is a problem, because not every gravitational observable is also a charge. In this paper, we propose such an extension. The basic idea is to remove the radiative data from the covariant phase space. This requires second-class constraints. Given a few basic assumptions, we show that the resulting Dirac bracket on the constraint surface is nothing but the BT bracket. A heuristic argument is given to show that the resulting constraint surface can only contain gravitational edge modes.

Figures (1)

• Figure : Figure 1: Setup of the problem. We consider an asymptotically flat spacetime. The three-manifolds $M$ and $M_+$ are partial Cauchy hypersurfaces, which are bounded by consecutive cross sections $\mathcal{C}=\partial M$ and $\mathcal{C}_+=\partial M_+$ of future null infinity $\mathcal{I}^+$. The null surface $\mathcal{N}$ is the portion of $\mathcal{I}^+$ between $\mathcal{C}$ and $\mathcal{C}_+$. We restrict ourselves to regions in phase space where $\mathcal{C}_+$ lies far enough ahead such that all radiation at $\mathcal{I}^+$ vanishes at and beyond the cross section $\mathcal{C}_+$. Care needs to be taken with orientations. Our conventions are as follows. The orientation of $\mathcal{N}$ is induced from the bulk, which is $\mathcal{M}$, whereas the orientation of the cross sections $\{\mathcal{C},\mathcal{C}_+\}$ is induced from $M$ and $M_+$. The boundary of $\mathcal{M}$ is $\partial\mathcal{M}=M\cup M_+^{-1}\cup\mathcal{N}$.