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

TL;DR

This paper extends the Barnich-Troessaert bracket to the entire covariant phase space in general relativity by using a Dirac bracket on a constrained surface, facilitating the computation of gravitational charges.

Contribution

It introduces a method to extend the BT bracket to the full phase space via a Dirac bracket on a constrained surface, addressing a key limitation.

Findings

The Dirac bracket on the constraint surface equals the BT bracket.

Removing radiative data isolates gravitational edge modes.

The approach enables consistent charge computation on the entire phase space.

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.