A note on dual gravitational charges

TL;DR

This paper analyzes dual gravitational charges that arise in tetrad gravity with the Holst term, showing they originate from an exact 3-form in the tetrad symplectic potential and can be removed by dressing the potential. By deriving a Kosmann-based formula for quasi-local Hamiltonian charges, it clarifies why dual contributions vanish for exact isometries and at spatial infinity but persist for asymptotic BMS symmetries at future null infinity. It furthermore demonstrates, via Barnich-Brandt cohomological methods, that these dual charges are tied to the chosen Lagrangian order and gauge dressing, highlighting ambiguities in covariant phase space approaches. The findings sharpen the understanding of tetrad vs metric phase spaces and illuminate the role of gauge choices, fall-off conditions, and internal Lorentz transformations in gravitational charges. The work thus provides a coherent framework connecting metric and tetrad charges and clarifies the conditions under which dual contributions are physical or gauge artifacts.

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.