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.
Article PDF
Similar content being viewed by others
References
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond.A 269 (1962) 21 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond.A 270 (1962) 103 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev.128 (1962) 2851 [INSPIRE].
R. Penrose, Zero rest mass fields including gravitation: asymptotic behavior, Proc. Roy. Soc. Lond.A 284 (1965) 159 [INSPIRE].
R. Geroch, Asymptotic structure of space-time, in Asymptotic structure of space-time, F.P. Esposito and L. Witten eds., Plenum Press, New York, NY, U.S.A. (1977) [INSPIRE].
R.P. Geroch and J. Winicour, Linkages in general relativity, J. Math. Phys.22 (1981) 803 [INSPIRE].
A. Ashtekar and M. Streubel, Symplectic geometry of radiative modes and conserved quantities at null infinity, Proc. Roy. Soc. Lond.A 376 (1981) 585 [INSPIRE].
R.M. Wald and A. Zoupas, A general definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev.D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-dimensional supertranslations and Weinberg’s soft graviton theorem, Ann. Math. Sci. Appl.02 (2017) 69 [arXiv:1502.07644] [INSPIRE].
S.G. Avery and B.U.W. Schwab, Burg-Metzner-Sachs symmetry, string theory and soft theorems, Phys. Rev.D 93 (2016) 026003 [arXiv:1506.05789] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of gravity and soft theorems for massive particles, JHEP12 (2015) 094 [arXiv:1509.01406] [INSPIRE].
A. Campoleoni, D. Francia and C. Heissenberg, On higher-spin supertranslations and superrotations, JHEP05 (2017) 120 [arXiv:1703.01351] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New gravitational memories, JHEP12 (2016) 053 [arXiv:1502.06120] [INSPIRE].
S. Hollands, A. Ishibashi and R.M. Wald, BMS supertranslations and memory in four and higher dimensions, Class. Quant. Grav.34 (2017) 155005 [arXiv:1612.03290] [INSPIRE].
P. Mao and H. Ouyang, Note on soft theorems and memories in even dimensions, Phys. Lett.B 774 (2017) 715 [arXiv:1707.07118] [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Gravitational memory in higher dimensions, JHEP06 (2018) 138 [arXiv:1712.01204] [INSPIRE].
A. Chatterjee and D.A. Lowe, BMS symmetry, soft particles and memory, Class. Quant. Grav.35 (2018) 094001 [arXiv:1712.03211] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett.116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
A. Strominger, Black hole information revisited, World Scientific, Singapore (2019), pg. 109 [arXiv:1706.07143] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation charge and supertranslation hair on black holes, JHEP05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett.105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP11 (2018) 200 [arXiv:1810.00377] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev.D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
M. Campiglia and A. Laddha, New symmetries for the gravitational S-matrix, JHEP04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
L. Freidel, F. Hopfmüller and A. Riello, Asymptotic renormalization in flat space: symplectic potential and charges of electromagnetism, JHEP10 (2019) 126 [arXiv:1904.04384] [INSPIRE].
R.M. Wald, General relativity, The University of Chicago Press, Chicago, IL, U.S.A. (1984) [INSPIRE].
L. Bieri, Solutions of the Einstein vacuum equations, in Extensions of the stability theorem of the Minkowski space in general relativity, AMS/IP Stud. Adv. Math.45, (2009).
S. Hollands and A. Ishibashi, Asymptotic flatness and Bondi energy in higher dimensional gravity, J. Math. Phys.46 (2005) 022503 [gr-qc/0304054] [INSPIRE].
S. Hollands and A. Thorne, Bondi mass cannot become negative in higher dimensions, Commun. Math. Phys.333 (2015) 1037 [arXiv:1307.1603] [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, arXiv:1906.08616 [INSPIRE].
R.M. Wald, On identically closed forms locally constructed from a field, J. Math. Phys.31 (1990) 2378.
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys.31 (1990) 725 [INSPIRE].
L.A. Tamburino and J.H. Winicour, Gravitational fields in finite and conformal Bondi frames, Phys. Rev.150 (1966) 1039 [INSPIRE].
F.W.J. Olver et al. eds., NIST digital library of mathematical functions, release 1.0.18, http://dlmf.nist.gov/, 27 March 2018.
J.J. Sakurai, Modern quantum mechanics, Addison-Wesley Publishing Company, U.S.A. (1994).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1910.04557
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Flanagan, É.É., Prabhu, K. & Shehzad, I. Extensions of the asymptotic symmetry algebra of general relativity. J. High Energ. Phys. 2020, 2 (2020). https://doi.org/10.1007/JHEP01(2020)002
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2020)002