Hamiltonian derivation of dual gravitational charges
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Springer
Received: July 21, 2020 Accepted: August 11, 2020 Published: September 11, 2020
Hadi Godazgar,a Mahdi Godazgarb and Malcolm J. Perryc,d,e a
Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), M¨ uuhlenberg 1, D-14476 Potsdam, Germany b School of Mathematical Sciences, Queen Mary University of London, Mile End Road, E1 4NS, U.K. c DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge CB3 OWA, U.K. d Department of Physics, Queen Mary University of London, Mile End Road, E1 4NS. U.K. e Trinity College, Cambridge University, Cambridge, CB2 1TQ. U.K.
E-mail: [email protected], [email protected], [email protected] Abstract: We provide a Hamiltonian derivation of recently discovered dual BMS charges. In order to do so, we work in the first order formalism and add to the usual Palatini action, the Holst term, which does not contribute to the equations of motion. We give a method for finding the leading order integrable dual charges ` a la Wald-Zoupas and construct the corresponding charge algebra. We argue that in the presence of fermions, the relevant term that leads to dual charges is the topological Nieh-Yan term. Keywords: Black Holes in String Theory, Classical Theories of Gravity, Space-Time Symmetries ArXiv ePrint: 2007.07144v2
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)084
JHEP09(2020)084
Hamiltonian derivation of dual gravitational charges
Contents 1
2 Review of the covariant phase space formalism
3
3 Gravitational theory in first order formalism
6
4 Asymptotic flatness and symmetries 4.1 Boundary conditions 4.2 Asymptotic symmetry generators
7 7 9
5 Asymptotic charges 5.1 Diffeomorphisms: standard and dual BMS charges 5.1.1 Standard BMS charges 5.1.2 Dual BMS charges 5.2 Lorentz transformations
9 10 11 12 13
6 Identifying the integrable charge
15
7 Charge algebra for leading order dual charges
17
8 Fermions
18
9 Discussion
21
A The metric and spin connection
23
B Twisting on the 2-sphere
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C Derivation of the leading dual charge algebra
25
1
Introduction
The intimate relation between symmetries and charges, as manifested in the Noether theorem, is a fundamental result of mathematical physics. The application of these ideas in a gravitational setting is intricate, yet fundamental to almost any investigation involving gravity, from gravitational wave astrophysics to quantum gravity. In this paper, we apply the prescription set out in ref. [1], which uses the covariant phase space formalism [2–9] to propose a systematic method for determining, in principle, all possible gravitational charges, to give a Hamiltonian derivation of a recently discovered tower of dual BMS charges [10, 11]. One can think of dual BMS charges as generalisations of the Taub-NUT
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JHEP09(2020)084
1 Introduction
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charge [12–15] in the same way that standard BMS charges [8, 9, 16–19] generalise the notion of the Bondi linear four-momentum [20, 21]. The
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