Coadjoint representation of the BMS group on celestial Riemann surfaces. (English) Zbl 1466.81028
Summary: The coadjoint representation of the BMS group in four dimensions is constructed in a formulation that covers both the sphere and the punctured plane. The structure constants are worked out for different choices of bases. The conserved current algebra of non-radiative asymptotically flat spacetimes is explicitly interpreted in these terms.
MSC:
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 53Z05 | Applications of differential geometry to physics |
Keywords:
classical theories of gravity; differential and algebraic geometry; gauge-gravity correspondence; space-time symmetriesReferences:
| [1] | 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. A269 (1962) 21 [INSPIRE]. · Zbl 0106.41903 |
| [2] | R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A270 (1962) 103 [INSPIRE]. · Zbl 0101.43605 |
| [3] | Sachs, R., Asymptotic symmetries in gravitational theory, Phys. Rev., 128, 2851 (1962) · Zbl 0114.21202 · doi:10.1103/PhysRev.128.2851 |
| [4] | Penrose, R., Asymptotic properties of fields and space-times, Phys. Rev. Lett., 10, 66 (1963) · doi:10.1103/PhysRevLett.10.66 |
| [5] | Newman, ET; Penrose, R., Note on the Bondi-Metzner-Sachs group, J. Math. Phys., 7, 863 (1966) · doi:10.1063/1.1931221 |
| [6] | Foster, J., Conformal structure of ı^+and asymptotic symmetry. I. Definitions and local theory, J. Phys. A, 11, 93 (1978) · doi:10.1088/0305-4470/11/1/012 |
| [7] | Foster, J., Asymptotic symmetry and the global structure of future null infinity, Int. J. Theor. Phys., 26, 1107 (1987) · Zbl 0631.53063 · doi:10.1007/BF00669365 |
| [8] | Newman, ET, A possible connexion between the gravitational field and elementary particle physics, Nature, 206, 811 (1965) · Zbl 0125.45401 · doi:10.1038/206811a0 |
| [9] | Komar, A., Quantized gravitational theory and internal symmetries, Phys. Rev. Lett., 15, 76 (1965) · Zbl 0125.45306 · doi:10.1103/PhysRevLett.15.76 |
| [10] | McCarthy, PJ, The Bondi-Metzner-Sachs group in the nuclear topology, Proc. Roy. Soc. Lond. A, 343, 489 (1975) · Zbl 0303.22011 · doi:10.1098/rspa.1975.0083 |
| [11] | Rawnsley, JH, Representations of a semi-direct product by quantization, Math. Proc. Cambridge Phil. Soc., 78, 345 (1975) · Zbl 0313.22014 · doi:10.1017/S0305004100051793 |
| [12] | Fulton, T.; Rohrlich, F.; Witten, L., Conformal invariance in physics, Rev. Mod. Phys., 34, 442 (1962) · Zbl 0107.40804 · doi:10.1103/RevModPhys.34.442 |
| [13] | D’Hoker, E.; Phong, DH, The geometry of string perturbation theory, Rev. Mod. Phys., 60, 917 (1988) · doi:10.1103/RevModPhys.60.917 |
| [14] | Goldberg, JN; MacFarlane, AJ; Newman, ET; Rohrlich, F.; Sudarshan, ECG, Spin s spherical harmonics and edth, J. Math. Phys., 8, 2155 (1967) · Zbl 0155.57402 · doi:10.1063/1.1705135 |
| [15] | Held, A.; Newman, ET; Posadas, R., The Lorentz group and the sphere, J. Math. Phys., 11, 3145 (1970) · Zbl 0202.27401 · doi:10.1063/1.1665105 |
| [16] | Geroch, RP; Held, A.; Penrose, R., A space-time calculus based on pairs of null directions, J. Math. Phys., 14, 874 (1973) · Zbl 0875.53014 · doi:10.1063/1.1666410 |
| [17] | Penrose, R.; Rindler, W., Spinors and space-time, volume 1: two-spinor calculus and relativistic fields (1984), Cambridge, U.K.: Cambridge University Press, Cambridge, U.K. · Zbl 0538.53024 · doi:10.1017/CBO9780511564048 |
| [18] | Penrose, R.; Rindler, W., Spinors and space-time, volume 2: spinor and twistor methods in space-time geometry (1986), Cambridge, U.K.: Cambridge University Press, Cambridge, U.K. · Zbl 0591.53002 · doi:10.1017/CBO9780511524486 |
| [19] | Barnich, G.; Troessaert, C., Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett., 105, 111103 (2010) · doi:10.1103/PhysRevLett.105.111103 |
| [20] | Barnich, G.; Troessaert, C., Aspects of the BMS/CFT correspondence, JHEP, 05, 062 (2010) · Zbl 1287.83043 · doi:10.1007/JHEP05(2010)062 |
| [21] | G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 (2010) [arXiv:1102.4632] [INSPIRE]. |
| [22] | Barnich, G.; Troessaert, C., Finite BMS transformations, JHEP, 03, 167 (2016) · Zbl 1388.83514 · doi:10.1007/JHEP03(2016)167 |
| [23] | Ashtekar, A.; Bičák, J.; Schmidt, BG, Asymptotic structure of symmetry-reduced general relativity, Phys. Rev. D, 55, 669 (1997) · doi:10.1103/PhysRevD.55.669 |
| [24] | 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]. · Zbl 1111.83045 |
| [25] | G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP06 (2014) 129 [arXiv:1403.5803] [INSPIRE]. · Zbl 1388.83006 |
| [26] | G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP03 (2015) 033 [arXiv:1502.00010] [INSPIRE]. · Zbl 1388.83006 |
| [27] | Barnich, G., Centrally extended BMS4 Lie algebroid, JHEP, 06, 007 (2017) · Zbl 1380.83244 · doi:10.1007/JHEP06(2017)007 |
| [28] | Antoniou, IE; Misra, B., Characterization of semidirect sum Lie algebras, J. Math. Phys., 32, 864 (1991) · Zbl 0760.22027 · doi:10.1063/1.529344 |
| [29] | Carmeli, M., Group theory and general relativity: representations of the Lorentz group and their applications to the gravitational field (2000), London, U.K.: Imperial College Press, London, U.K. · Zbl 0970.83001 · doi:10.1142/p199 |
| [30] | Alessio, F.; Esposito, G., On the structure and applications of the Bondi-Metzner-Sachs group, Int. J. Geom. Meth. Mod. Phys., 15, 1830002 (2018) · Zbl 1387.83017 · doi:10.1142/S0219887818300027 |
| [31] | R. Geroch, Asymptotic structure of space-time, in Symposium on the asymptotic structure of space-time, P. Esposito and L. Witten eds., Plenum, New York, NY, U.S.A. (1977), pg. 1. |
| [32] | Duval, C.; Gibbons, GW; Horvathy, PA, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav., 31, 092001 (2014) · Zbl 1291.83084 · doi:10.1088/0264-9381/31/9/092001 |
| [33] | Duval, C.; Gibbons, GW; Horvathy, PA, Conformal Carroll groups, J. Phys. A, 47, 335204 (2014) · Zbl 1297.83028 · doi:10.1088/1751-8113/47/33/335204 |
| [34] | I. Gradshteyn and I. Ryzhik, 8-9 special functions, in Table of integrals, series, and products, Elsevier, The Netherlands (1980), pg. 904. |
| [35] | Porter, JR, Green’s functions associated with the edth operators, Gen. Rel. Grav., 13, 531 (1981) · Zbl 0536.53034 · doi:10.1007/BF00757239 |
| [36] | V. Kac, Vertex algebras for beginners, 2^nd edition, University Lecture Series, volume 10, American Mathematical Society, U.S.A. (1997). · Zbl 0861.17017 |
| [37] | Barnich, G.; Lambert, P-H, A note on the Newman-Unti group and the BMS charge algebra in terms of Newman-Penrose coefficients, Adv. Math. Phys., 2012, 197385 (2012) · Zbl 1266.83132 · doi:10.1155/2012/197385 |
| [38] | Barnich, G.; Troessaert, C., Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity, JHEP, 11, 003 (2013) · Zbl 1342.83228 · doi:10.1007/JHEP11(2013)003 |
| [39] | Barnich, G.; Mao, P.; Ruzziconi, R., BMS current algebra in the context of the Newman-Penrose formalism, Class. Quant. Grav., 37, 095010 (2020) · Zbl 1479.83179 · doi:10.1088/1361-6382/ab7c01 |
| [40] | Campiglia, M.; Laddha, A., Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D, 90, 124028 (2014) · doi:10.1103/PhysRevD.90.124028 |
| [41] | G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP11 (2018) 200 [Erratum ibid.04 (2020) 172] [arXiv:1810.00377] [INSPIRE]. · Zbl 1405.83004 |
| [42] | Flanagan, EE; Prabhu, K.; Shehzad, I., Extensions of the asymptotic symmetry algebra of general relativity, JHEP, 01, 002 (2020) · Zbl 1442.83007 · doi:10.1007/JHEP01(2020)002 |
| [43] | Campiglia, M.; Peraza, J., Generalized BMS charge algebra, Phys. Rev. D, 101, 104039 (2020) · doi:10.1103/PhysRevD.101.104039 |
| [44] | Compère, G.; Fiorucci, A.; Ruzziconi, R., The Λ-BMS_4charge algebra, JHEP, 10, 205 (2020) · Zbl 1456.81215 · doi:10.1007/JHEP10(2020)205 |
| [45] | B. Oblak, BMS particles in three dimensions, Ph.D. thesis, Brussels U., Brussels, Belgium (2016) [arXiv:1610.08526] [INSPIRE]. · Zbl 1379.81011 |
| [46] | W. Donnelly, L. Freidel, S.F. Moosavian and A.J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, arXiv:2012.10367 [INSPIRE]. |
| [47] | Safari, HR; Sheikh-Jabbari, MM, BMS_4algebra, its stability and deformations, JHEP, 04, 068 (2019) · doi:10.1007/JHEP04(2019)068 |
| [48] | Cantoni, V., A class of representations of the generalized Bondi-Metzner group, J. Math. Phys., 7, 1361 (1966) · Zbl 0147.27404 · doi:10.1063/1.1705045 |
| [49] | P.J. McCarthy, Representations of the Bondi-Metzner-Sachs group I. Determination of the representations, Proc. Roy. Soc. Lond. A330 (1972) 517. · Zbl 0351.22012 |
| [50] | P.J. McCarthy, Representations of the Bondi-Metzner-Sachs group II. Properties and classification of the representations, Proc. Roy. Soc. Lond. A333 (1973) 317. · Zbl 0351.22013 |
| [51] | P.J. McCarthy and M. Crampin, Representations of the Bondi-Metzner-Sachs group III. Poincaré spin multiplicities and irreducibility, Proc. Roy. Soc. Lond. A335 (1973) 301. · Zbl 0354.22020 |
| [52] | Girardello, L.; Parravicini, G., Continuous spins in the Bondi-Metzner-Sachs group of asymptotic symmetry in general relativity, Phys. Rev. Lett., 32, 565 (1974) · doi:10.1103/PhysRevLett.32.565 |
| [53] | Kirillov, AA, Lectures on the orbit method (2004), U.S.A.: American Mathematical Society, U.S.A. · Zbl 1229.22003 · doi:10.1090/gsm/064 |
| [54] | Alekseev, A.; Faddeev, LD; Shatashvili, SL, Quantization of symplectic orbits of compact Lie groups by means of the functional integral, J. Geom. Phys., 5, 391 (1988) · Zbl 0698.58025 · doi:10.1016/0393-0440(88)90031-9 |
| [55] | Alekseev, A.; Shatashvili, SL, Path integral quantization of the coadjoint orbits of the Virasoro group and 2D gravity, Nucl. Phys. B, 323, 719 (1989) · doi:10.1016/0550-3213(89)90130-2 |
| [56] | Barnich, G.; Gonzalez, HA; Salgado-ReboLledó, P., Geometric actions for three-dimensional gravity, Class. Quant. Grav., 35, 014003 (2018) · Zbl 1382.83073 · doi:10.1088/1361-6382/aa9806 |
| [57] | Nguyen, K.; Salzer, J., The effective action of superrotation modes, JHEP, 02, 108 (2021) · Zbl 1460.83008 · doi:10.1007/JHEP02(2021)108 |
| [58] | Israel, W.; Nester, JM, Positivity of the Bondi gravitational mass, Phys. Lett. A, 85, 259 (1981) · doi:10.1016/0375-9601(81)90951-8 |
| [59] | Ludvigsen, M.; Vickers, JAG, The positivity of the Bondi mass, J. Phys. A, 14, L389 (1981) · doi:10.1088/0305-4470/14/10/002 |
| [60] | Schon, R.; Yau, S-T, Proof that the Bondi mass is positive, Phys. Rev. Lett., 48, 369 (1982) · doi:10.1103/PhysRevLett.48.369 |
| [61] | Horowitz, GT; Perry, MJ, Gravitational energy cannot become negative, Phys. Rev. Lett., 48, 371 (1982) · doi:10.1103/PhysRevLett.48.371 |
| [62] | Horowitz, GT; Tod, P., A relation between local and total energy in general relativity, Commun. Math. Phys., 85, 429 (1982) · Zbl 0501.53017 · doi:10.1007/BF01208723 |
| [63] | Ashtekar, A.; Horowitz, GT, Energy-momentum of isolated systems cannot be null, Phys. Lett. A, 89, 181 (1982) · doi:10.1016/0375-9601(82)90203-1 |
| [64] | Reula, O.; Tod, KP, Positivity of the Bondi energy, J. Math. Phys., 25, 1004 (1984) · doi:10.1063/1.526267 |
| [65] | Barnich, G.; Oblak, B., Holographic positive energy theorems in three-dimensional gravity, Class. Quant. Grav., 31, 152001 (2014) · Zbl 1297.83010 · doi:10.1088/0264-9381/31/15/152001 |
| [66] | Ashtekar, A.; Streubel, M., Symplectic geometry of radiative modes and conserved quantities at null infinity, Proc. Roy. Soc. Lond. A, 376, 585 (1981) · doi:10.1098/rspa.1981.0109 |
| [67] | A. Ashtekar, Asymptotic quantization: based on 1984 Naples lectures, Bibliopolis, Naples, Italy (1987). · Zbl 0621.53064 |
| [68] | Adamo, T.; Casali, E.; Skinner, D., Perturbative gravity at null infinity, Class. Quant. Grav., 31, 225008 (2014) · Zbl 1304.81146 · doi:10.1088/0264-9381/31/22/225008 |
| [69] | Troessaert, C., Hamiltonian surface charges using external sources, J. Math. Phys., 57, 053507 (2016) · Zbl 1342.83019 · doi:10.1063/1.4947177 |
| [70] | Wieland, W., Null infinity as an open Hamiltonian system, JHEP, 04, 095 (2021) · Zbl 1462.83010 · doi:10.1007/JHEP04(2021)095 |
| [71] | Troessaert, C., The BMS4 algebra at spatial infinity, Class. Quant. Grav., 35, 074003 (2018) · Zbl 1390.83074 · doi:10.1088/1361-6382/aaae22 |
| [72] | Henneaux, M.; Troessaert, C., BMS group at spatial infinity: the Hamiltonian (ADM) approach, JHEP, 03, 147 (2018) · Zbl 1388.83015 · doi:10.1007/JHEP03(2018)147 |
| [73] | Torre, CG, Null surface geometrodynamics, Class. Quant. Grav., 3, 773 (1986) · Zbl 0596.53065 · doi:10.1088/0264-9381/3/5/008 |
| [74] | Oliveri, R.; Speziale, S., Boundary effects in general relativity with tetrad variables, Gen. Rel. Grav., 52, 83 (2020) · Zbl 1468.83009 · doi:10.1007/s10714-020-02733-8 |
| [75] | Strominger, A., On BMS invariance of gravitational scattering, JHEP, 07, 152 (2014) · Zbl 1392.81215 · doi:10.1007/JHEP07(2014)152 |
| [76] | He, T.; Lysov, V.; Mitra, P.; Strominger, A., BMS supertranslations and Weinberg’s soft graviton theorem, JHEP, 05, 151 (2015) · Zbl 1388.83261 · doi:10.1007/JHEP05(2015)151 |
| [77] | Conde, E.; Mao, P., BMS supertranslations and not so soft gravitons, JHEP, 05, 060 (2017) · Zbl 1380.83199 · doi:10.1007/JHEP05(2017)060 |
| [78] | Cheung, C.; de la Fuente, A.; Sundrum, R., 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP, 01, 112 (2017) · Zbl 1373.81319 · doi:10.1007/JHEP01(2017)112 |
| [79] | Banerjee, S., Symmetries of free massless particles and soft theorems, Gen. Rel. Grav., 51, 128 (2019) · Zbl 1430.83008 · doi:10.1007/s10714-019-2609-z |
| [80] | Distler, J.; Flauger, R.; Horn, B., Double-soft graviton amplitudes and the extended BMS charge algebra, JHEP, 08, 021 (2019) · Zbl 1421.83039 · doi:10.1007/JHEP08(2019)021 |
| [81] | Donnay, L.; Puhm, A.; Strominger, A., Conformally soft photons and gravitons, JHEP, 01, 184 (2019) · Zbl 1409.81116 · doi:10.1007/JHEP01(2019)184 |
| [82] | Stieberger, S.; Taylor, TR, Symmetries of celestial amplitudes, Phys. Lett. B, 793, 141 (2019) · Zbl 1421.81154 · doi:10.1016/j.physletb.2019.03.063 |
| [83] | Adamo, T.; Mason, L.; Sharma, A., Celestial amplitudes and conformal soft theorems, Class. Quant. Grav., 36, 205018 (2019) · Zbl 1478.81036 · doi:10.1088/1361-6382/ab42ce |
| [84] | Fotopoulos, A.; Taylor, TR, Primary fields in celestial CFT, JHEP, 10, 167 (2019) · Zbl 1427.81127 · doi:10.1007/JHEP10(2019)167 |
| [85] | Fotopoulos, A.; Stieberger, S.; Taylor, TR; Zhu, B., Extended BMS algebra of celestial CFT, JHEP, 03, 130 (2020) · Zbl 1435.83010 · doi:10.1007/JHEP03(2020)130 |
| [86] | Fan, W.; Fotopoulos, A.; Stieberger, S.; Taylor, TR, On Sugawara construction on celestial sphere, JHEP, 09, 139 (2020) · Zbl 1454.81186 · doi:10.1007/JHEP09(2020)139 |
| [87] | Donnay, L.; Pasterski, S.; Puhm, A., Asymptotic symmetries and celestial CFT, JHEP, 09, 176 (2020) · Zbl 1454.81183 · doi:10.1007/JHEP09(2020)176 |
| [88] | Donnay, L.; Giribet, G.; Rosso, F., Quantum BMS transformations in conformally flat space-times and holography, JHEP, 12, 102 (2020) · Zbl 1457.83057 · doi:10.1007/JHEP12(2020)102 |
| [89] | González, HA; Puhm, A.; Rojas, F., Loop corrections to celestial amplitudes, Phys. Rev. D, 102, 126027 (2020) · doi:10.1103/PhysRevD.102.126027 |
| [90] | Banerjee, S.; Ghosh, S.; Gonzo, R., BMS symmetry of celestial OPE, JHEP, 04, 130 (2020) · Zbl 1436.83022 · doi:10.1007/JHEP04(2020)130 |
| [91] | Banerjee, S.; Ghosh, S.; Paul, P., MHV graviton scattering amplitudes and current algebra on the celestial sphere, JHEP, 02, 176 (2021) · doi:10.1007/JHEP02(2021)176 |
| [92] | Himwich, E.; Narayanan, SA; Pate, M.; Paul, N.; Strominger, A., The soft \(\mathcal{S} \)-matrix in gravity, JHEP, 09, 129 (2020) · Zbl 1454.83039 · doi:10.1007/JHEP09(2020)129 |
| [93] | A. Atanasov, A. Ball, W. Melton, A.-M. Raclariu and A. Strominger, (2, 2) scattering and the celestial torus, arXiv:2101.09591 [INSPIRE]. |
| [94] | I.M. Krichever and S.P. Novikov, Algebras of Virasoro type, Riemann surfaces and the structure of soliton theory, Funct. Anal. Appl.21 (1987) 126 [Funkt. Anal. Pril.21 (1987) 47] [INSPIRE]. · Zbl 0634.17010 |
| [95] | Krichever, IM; Novikov, SP, Virasoro-type algebras, Riemann surfaces and strings in Minkowsky space, Funct. Anal. Appl., 21, 294 (1988) · Zbl 0659.17012 · doi:10.1007/BF01077803 |
| [96] | Wu, TT; Yang, CN, Dirac monopole without strings: monopole harmonics, Nucl. Phys. B, 107, 365 (1976) · doi:10.1016/0550-3213(76)90143-7 |
| [97] | Thorne, KS, Multipole expansions of gravitational radiation, Rev. Mod. Phys., 52, 299 (1980) · doi:10.1103/RevModPhys.52.299 |
| [98] | Eastwood, M.; Tod, P., Edth-a differential operator on the sphere, Math. Proc. Cambridge Phil. Soc., 92, 317 (1982) · Zbl 0511.53026 · doi:10.1017/S0305004100059971 |
| [99] | Galperin, AS, Harmonic superspace (2011), New York, U.S.A.: Cambridge University Press, New York, U.S.A. |
| [100] | Beyer, F.; Daszuta, B.; Frauendiener, J.; Whale, B., Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics, Class. Quant. Grav., 31, 075019 (2014) · Zbl 1291.83019 · doi:10.1088/0264-9381/31/7/075019 |
| [101] | Boyle, M., How should spin-weighted spherical functions be defined?, J. Math. Phys., 57, 092504 (2016) · Zbl 1350.83008 · doi:10.1063/1.4962723 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
