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The Barnich--Troessaert bracket is a proposal for a modified Poisson bracketon the covariant phase space for general relativity. The new bracket allows usto 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 suchcharge, but little is known how to extend the bracket to the entire phasespace. This is a problem, because not every gravitational observable is also acharge. In this paper, we propose such an extension. The basic idea is toremove the radiative data from the covariant phase space. This requiressecond-class constraints. Given a few basic assumptions, we show that theresulting Dirac bracket on the constraint surface is nothing but the BTbracket. A heuristic argument is given to show that the resulting constraintsurface can only contain gravitational edge modes.

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