A note on dual gravitational charges
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Springer
Received: October 17, 2020 Accepted: November 3, 2020 Published: December 11, 2020
Roberto Oliveria and Simone Spezialeb a
CEICO, Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic b Aix Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France
E-mail: [email protected], [email protected] 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. Keywords: Classical Theories of Gravity, Space-Time Symmetries ArXiv ePrint: 2010.01111
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2020)079
JHEP12(2020)079
A note on dual gravitational charges
Contents 1
2 Tetrad and metric covariant phase spaces 2.1 Brief review of first-order formalism 2.2 The origin of dual charges
3 3 5
3 Dressing the symplectic potential
7
4 Isometries and the Kosmann derivative 4.1 Isometries 4.2 Asymptotic symmetries at spatial infinity 4.3 Asymptotic symmetries at future null infinity
7 9 9 9
5 Dual charges from cohomological methods
11
6 Conclusions
12
A The Lie vs. Kosmann discrepancy
14
B Limit at future null infinity
14
1
Introduction
Recent work [1, 2] has shown that, if one starts from the first-order tetrad Lagrangian including a dual term (often called Holst term), the Hamiltonian charges at future null infinity contain contributions from both the standard BMS terms and the new dual gravitational charges defined in [3]. These were further studied in [4–6]. For previous related work, see e.g. [7–10]. The calculation in [1, 2] uses covariant phase space methods [11–13]. These dual contributions to the charges, or dual charges for short, are present only in tetrad variables, and not in metric variables. The first goal of this brief note is to highlight that the origin of this difference lies in the more general discrepancy between the metric and tetrad symplectic potentials, which differ by a certain exact 3-form. In covariant phase space methods one is free to add exact 3-forms to
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