Abstract
Dual gravitational charges have been recently computed from the Holst term in tetrad variables using covariant phase space methods. We highlight that they originate from an exact 3-form in the tetrad symplectic potential that has no analogue in metric variables. Hence there exists a choice of the tetrad symplectic potential that sets the dual charges to zero. This observation relies on the ambiguity of the covariant phase space methods. To shed more light on the dual contributions, we use the Kosmann variation to compute (quasi-local) Hamiltonian charges for arbitrary diffeomorphisms. We obtain a formula that illustrates comprehensively why the dual contribution to the Hamiltonian charges: (i) vanishes for exact isometries and asymptotic symmetries at spatial infinity; (ii) persists for asymptotic symmetries at future null infinity, in addition to the usual BMS contribution. Finally, we point out that dual gravitational charges can be equally derived using the Barnich-Brandt prescription based on cohomological methods, and that the same considerations on asymptotic symmetries apply.
Highlights
Our second goal is to better understand this state of affairs: why shifting from the Lie transformation to the Kosmann transformation is needed for exact isometries but not for asymptotic symmetries, and why the dual contribution survives at null infinity but not at spatial infinity
Our analysis shows that the situation is different for the case of asymptotic symmetries: the presence of a dual contribution depends on the order of the Lagrangian
We drew attention to the fact that the dual contribution to the Hamiltonian charges arises from an exact 3-form in the tetrad symplectic potential originating from the Holst sector
Summary
The tetrad and metric Lagrangians in the first-order formalism have independent connection variables, and are given, respectively, by. Taking arbitrary variations of the Lagrangian over the independent variables, one gets the field equations and a boundary term dθ(δ) The latter is used to read the symplectic potential θ(δ) in covariant phase space methods. The Lagrangian prescribes the symplectic potential up to the addition of an exact 3-form To stress this fact, we refer to the choices (2.3) and (2.4) as ‘bare’. Using the defining relations between the tetrad and the metric, and the corresponding one for the connections, (2.3) and (2.4) differ by an exact 3-form [20, 25], θ(g,γ)(δ) = θ(e,γ)(δ)+dα(δ), α(δ) = −PIJKL eI ∧eJ eρK δeLρ We conclude that the second-order tetrad Lagrangian will always produce the same charges as the metric theory with vanishing torsion
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