[Paper Review] Barnich-Troessaert Bracket as a Dirac Bracket on the Covariant Phase Space

Wolfgang Wieland|arXiv (Cornell University)|Apr 16, 2021

Black Holes and Theoretical Physics55 references18 citations

TL;DR

This paper proposes that the Barnich–Troessaert (BT) bracket—used to define integrable charges in general relativity at null infinity—arises naturally as a Dirac bracket on a reduced covariant phase space where radiative degrees of freedom are removed via second-class constraints. By treating radiative data as auxiliary background fields, the resulting phase space describes only gravitational edge modes, and the BT bracket emerges as the Dirac bracket, with an additional flux integral shifting charges to future null infinity, ensuring integrability and closure of the algebra.

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.

Motivation & Objective

• To resolve the non-integrability of BMS supertranslation charges on the covariant phase space due to radiative degrees of freedom.
• To clarify the phase space structure underlying the Barnich–Troessaert bracket, which is not a standard Poisson bracket on the full phase space.
• To demonstrate that the BT bracket can be understood as a Dirac bracket on a reduced phase space where radiative data are constrained to c-numbers.
• To show that the resulting phase space describes only gravitational edge modes, with the BT bracket capturing their symplectic structure.

Proposed method

• Remove radiative data from the covariant phase space by introducing second-class constraints, effectively replacing them with background c-number fields.
• Apply the Dirac bracket formalism to the constrained system, deriving a new symplectic structure on the reduced phase space.
• Use the pre-symplectic two-form ΩM and the radiative symplectic current Jrad to define the flux integral Fξ[M→M+], which shifts charges along null generators.
• Show that the resulting Dirac bracket reproduces the Barnich–Troessaert bracket on a cross section C of null infinity I+, up to a flux correction.
• Demonstrate that the flux integral commutes under the Dirac bracket, as it depends only on radiative modes.
• Establish that the reduced phase space is isomorphic to the phase space of gravitational edge modes (Coulombic modes), with the symplectic structure inherited from the Dirac procedure.

Experimental results

Research questions

• RQ1 How can the Barnich–Troessaert bracket be consistently extended to the entire covariant phase space, given its original definition only for charges?
• RQ2 What is the underlying phase space structure for which the BT bracket defines a non-degenerate symplectic form?
• RQ3 How do generic Dirac observables commute under the BT bracket, and what is their dynamics on the reduced phase space?
• RQ4 Can the BT bracket be derived from a standard Hamiltonian constraint system with second-class constraints?
• RQ5 What is the physical interpretation of the phase space obtained after removing radiative data?

Key findings

• The Barnich–Troessaert bracket is shown to be equivalent to a Dirac bracket on a reduced phase space where radiative data are removed via second-class constraints.
• The resulting phase space describes only gravitational edge modes, with no radiative degrees of freedom.
• The Dirac bracket reproduces the BT bracket on a cross section C of I+, up to a flux integral that shifts charges to future null infinity.
• The flux integral F[ξ,ξ′][M→M+] depends only on radiative modes and commutes under the Dirac bracket, ensuring consistency of the algebra.
• The charges Q+ξ on the reduced phase space satisfy the commutation relation {Q+ξ, Q+ξ′}* = −Q+[ξ,ξ′], confirming integrability and closure of the BMS algebra.
• The symplectic structure on the reduced phase space can be obtained either via the BT bracket with a flux shift or via the Dirac bracket, providing two equivalent descriptions.