Abstract
The Poincaré invariant vacuum is not unique in quantum gravity. The BMS supertranslation symmetry originally defined at null infinity is spontaneously broken and results in inequivalent Poincaré vacua. In this paper we construct the unique vacua which interpolate between past and future null infinity in BMS gauge and which are entirely characterized by an arbitrary Goldstone boson defined on the sphere which breaks BMS invariance. We show that these vacua contain a defect which carries no Poincaré charges but which generically carries superrotation charges. We argue that there is a huge degeneracy of vacua with multiple defects. We also present the single defect vacua with its canonically conjugated source which can be constructed from a Liouville boson on the stereographic plane. We show that positivity of the energy forces the stress-tensor of the boson to vanish as a boundary condition. Finite superrotations, which turn on the sources, are therefore physically ruled out as canonical transformations around the vacua. Yet, infinitesimal superrotations are external symplectic symmetries which are associated with conserved charges which characterize the Goldstone boson.
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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].
L.B.A. Ashtekar and O. Reula, The Covariant Phase Space Of Asymptotically Flat Gravitational Fields, Analysis, Geometry and Mechanics: 200 Years After Lagrange, M. Francaviglia and D. Holm eds., North-Holland, Amsterdam (1991).
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
C. Fefferman and C. Robin Graham, Conformal Invariants, in Elie Cartan et les Mathématiques d’aujourd’hui (Astérisque) (1985), pg. 95.
M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].
M. Rooman and P. Spindel, Uniqueness of the asymptotic AdS 3 geometry, Class. Quant. Grav. 18 (2001) 2117 [gr-qc/0011005] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich, A. Gomberoff and H.A. Gonzlez, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].
G. Barnich and H.A. Gonzalez, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity, JHEP 05 (2013) 016 [arXiv:1303.1075] [INSPIRE].
A. Garbarz and M. Leston, Classification of Boundary Gravitons in AdS 3 Gravity, JHEP 05 (2014) 141 [arXiv:1403.3367] [INSPIRE].
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
G. Compère, P.-J. Mao, A. Seraj and M.M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS 3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080 [arXiv:1511.06079] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
S. Deser and R. Jackiw, Three-Dimensional Cosmological Gravity: Dynamics of Constant Curvature, Annals Phys. 153 (1984) 405 [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
G. Barnich and P.-H. Lambert, A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients, J. Phys. Conf. Ser. 410 (2013) 012142 [INSPIRE].
G. Compère, L. Donnay, P.-H. Lambert and W. Schulgin, Liouville theory beyond the cosmological horizon, JHEP 03 (2015) 158 [arXiv:1411.7873] [INSPIRE].
G. Compère, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, Wiggling Throat of Extremal Black Holes, JHEP 10 (2015) 093 [arXiv:1506.07181] [INSPIRE].
E.E. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, arXiv:1510.03386 [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].
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].
T. Banks, A Critique of pure string theory: Heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity \( \mathcal{S} \) -matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, arXiv:1502.06120 [INSPIRE].
C. Troessaert, Hamiltonian surface charges using external sources, J. Math. Phys. 57 (2016) 053507 [arXiv:1509.09094] [INSPIRE].
G. Compère and J. Long, Classical static final state of collapse with supertranslation memory, arXiv:1602.05197 [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS (CNCFG2010) 010 [Ann. U. Craiova Phys. 21(2011) S11] [arXiv:1102.4632] [INSPIRE].
J. Balog, L. Feher and L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Int. J. Mod. Phys. A 13 (1998) 315 [hep-th/9703045] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton (1993).
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in Einstein Yang-Mills theory, Nucl. Phys. B 455 (1995) 357 [hep-th/9505173] [INSPIRE].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [INSPIRE].
G. Barnich and G. Compère, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
A. Dickenstein, M.S. Iriondo and T.A. Rojas, Integrating singular functions on the sphere, J. Math. Phys. 38 (1997) 5361 [gr-qc/9902013] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
G. Barnich, Boundary charges in gauge theories: Using Stokes theorem in the bulk, Class. Quant. Grav. 20 (2003) 3685 [hep-th/0301039] [INSPIRE].
S. Deser, R. Jackiw and G. ’t Hooft, Three-Dimensional Einstein Gravity: Dynamics of Flat Space, Annals Phys. 152 (1984) 220 [INSPIRE].
A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
G. Barnich and G. Compère, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press (2000).
E. Witten, Cosmic Superstrings, Phys. Lett. B 153 (1985) 243 [INSPIRE].
E.J. Copeland, R.C. Myers and J. Polchinski, Cosmic F and D strings, JHEP 06 (2004) 013 [hep-th/0312067] [INSPIRE].
S. Hossenfelder, Theory and Phenomenology of Spacetime Defects, Adv. High Energy Phys. 2014 (2014) 950672 [arXiv:1401.0276] [INSPIRE].
R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].
B. Oblak, Characters of the BMS Group in Three Dimensions, Commun. Math. Phys. 340 (2015) 413 [arXiv:1502.03108] [INSPIRE].
G. Barnich, H.A. Gonzalez, A. Maloney and B. Oblak, One-loop partition function of three-dimensional flat gravity, JHEP 04 (2015) 178 [arXiv:1502.06185] [INSPIRE].
A. Garbarz and M. Leston, Quantization of BMS 3 orbits: a perturbative approach, Nucl. Phys. B 906 (2016) 133 [arXiv:1507.00339] [INSPIRE].
N. Banerjee, D.P. Jatkar, S. Mukhi and T. Neogi, Free-field realisations of the BMS 3 algebra and its extensions, JHEP 06 (2016) 024 [arXiv:1512.06240] [INSPIRE].
B. Zeldovich and A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars, Ya. Sov. Astron. Lett. 18 (1974) 17.
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, Print-86-1309, Princeton, (1986).
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
G. Compère, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions, Ph.D. Thesis, Vrije University, Brussels (2007) [arXiv:0708.3153] [INSPIRE].
L.F. Abbott and S. Deser, Stability of Gravity with a Cosmological Constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].
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Compère, G., Long, J. Vacua of the gravitational field. J. High Energ. Phys. 2016, 137 (2016). https://doi.org/10.1007/JHEP07(2016)137
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DOI: https://doi.org/10.1007/JHEP07(2016)137