Extensions of the asymptotic symmetry algebra of general relativity
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Springer
Received: October 18, Revised: December 7, Accepted: December 17, Published: January 2,
2019 2019 2019 2020
´ ´ Flanagan,a Kartik Prabhub and Ibrahim Shehzada Eanna E. a
Department of Physics, Cornell University, Ithaca, NY 14853, U.S.A. b Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Cornell University, Ithaca, NY 14853, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We consider a recently proposed extension of the Bondi-Metzner-Sachs algebra to include arbitrary infinitesimal diffeomorphisms on a 2-sphere. To realize this extended algebra as asymptotic symmetries, we work with an extended class of spacetimes in which the unphysical metric at null infinity is not universal. We show that the symplectic current evaluated on these extended symmetries is divergent in the limit to null infinity. We also show that this divergence cannot be removed by a local and covariant redefinition of the symplectic current. This suggests that such an extended symmetry algebra cannot be realized as symmetries on the phase space of vacuum general relativity at null infinity, and that the corresponding asymptotic charges are ill-defined. However, a possible loophole in the argument is the possibility that symplectic current may not need to be covariant in order to have a covariant symplectic form. We also show that the extended algebra does not have a preferred subalgebra of translations and therefore does not admit a universal definition of Bondi 4-momentum. Keywords: Classical Theories of Gravity, Gauge Symmetry, Space-Time Symmetries ArXiv ePrint: 1910.04557
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)002
JHEP01(2020)002
Extensions of the asymptotic symmetry algebra of general relativity
Contents 1 4
2 Asymptotic flatness at null infinity and the BMS algebra 2.1 Definition and properties of asymptotic flatness at null infinity 2.2 Review of derivation of the Bondi-Metzner-Sachs symmetry algebra
4 4 6
3 An extended field configuration space and extended algebra 3.1 Extended field configuration space 3.2 Extended algebra
8 9 9
4 The 4.1 4.2 4.3
symplectic current of general relativity at null infinity The symplectic current for general perturbations Divergence of the symplectic current on the extended phase space Ambiguities in the symplectic current
10 11 12 14
5 Other issues 5.1 Existence of Bondi four-momentum 5.2 Choice of field configuration space
17 17 18
6 Discussion and conclusions
18
A Metric on I and conformal factor in a neighborhood can be chosen to be universal 19 B The extended BMS algebra does not contain any preferred translation subalgebra 21
1
Introduction and summary
The asymptotic symmetry group at null infinity of asymptotically-flat spacetimes in general relativity is normally considered to be the infinite-dimensional Bondi-Metzner-Sachs (BMS) group. Associated with the Lie algebra of the BMS group there are an infinite number of charges and fluxes (due to gravitati
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