Abstract
We identify in Einstein gravity an asymptotic spin-2 charge aspect whose conservation equation gives rise, after quantization, to the sub-subleading soft theorem. Our treatment reveals that this spin-2 charge generates a non-local spacetime symmetry represented at null infinity by pseudo-vector fields. Moreover, we demonstrate that the non-linear nature of Einstein’s equations is reflected in the Ward identity through collinear corrections to the sub-subleading soft theorem. Our analysis also provides a unified treatment of the universal soft theorems as conservation equations for the spin-0,-1,-2 canonical generators, while highlighting the important role played by the dual mass.
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References
H. Bondi, Gravitational Waves in General Relativity, Nature 186 (1960) 535 [INSPIRE].
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.
R.K. Sachs, On the Characteristic Initial Value Problem in Gravitational Theory, J. Math. Phys. 3 (1962) 908 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
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].
K.S.T. Braginsky, Gravitational-wave bursts with memory and experimental prospects, Nature 327 (1987) 123.
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
K.S. Thorne, Gravitational-wave bursts with memory: The Christodoulou effect, Phys. Rev. D 45 (1992) 520 [INSPIRE].
M. Favata, The gravitational-wave memory effect, Class. Quant. Grav. 27 (2010) 084036 [arXiv:1003.3486] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [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, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
G. Barnich and C. Troessaert, Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity, JHEP 11 (2013) 003 [arXiv:1309.0794] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
C.D. White, Diagrammatic insights into next-to-soft corrections, Phys. Lett. B 737 (2014) 216 [arXiv:1406.7184] [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, JHEP 12 (2016) 053 [arXiv:1502.06120] [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, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].
J. de Boer and S.N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [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.
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
E.E. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP 11 (2018) 200 [Erratum ibid. 04 (2020) 172] [arXiv:1810.00377] [INSPIRE].
M. Campiglia and J. Peraza, Generalized BMS charge algebra, Phys. Rev. D 101 (2020) 104039 [arXiv:2002.06691] [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, The Weyl BMS group and Einstein’s equations, JHEP 07 (2021) 170 [arXiv:2104.05793] [INSPIRE].
A. Guevara, Notes on Conformal Soft Theorems and Recursion Relations in Gravity, arXiv:1906.07810 [INSPIRE].
A. Guevara, E. Himwich, M. Pate and A. Strominger, Holographic symmetry algebras for gauge theory and gravity, JHEP 11 (2021) 152 [arXiv:2103.03961] [INSPIRE].
A. Strominger, w(1 + ∞) and the Celestial Sphere, arXiv:2105.14346 [INSPIRE].
A. Ball, S.A. Narayanan, J. Salzer and A. Strominger, Perturbatively exact w1+∞ asymptotic symmetry of quantum self-dual gravity, JHEP 01 (2022) 114 [arXiv:2111.10392] [INSPIRE].
T. Adamo, L. Mason and A. Sharma, Celestial w1+∞ Symmetries from Twistor Space, SIGMA 18 (2022) 016 [arXiv:2110.06066] [INSPIRE].
T. Adamo, W. Bu, E. Casali and A. Sharma, Celestial operator products from the worldsheet, arXiv:2111.02279 [INSPIRE].
M. Zlotnikov, Sub-sub-leading soft-graviton theorem in arbitrary dimension, JHEP 10 (2014) 148 [arXiv:1407.5936] [INSPIRE].
C. Kalousios and F. Rojas, Next to subleading soft-graviton theorem in arbitrary dimensions, JHEP 01 (2015) 107 [arXiv:1407.5982] [INSPIRE].
A. Luna, S. Melville, S.G. Naculich and C.D. White, Next-to-soft corrections to high energy scattering in QCD and gravity, JHEP 01 (2017) 052 [arXiv:1611.02172] [INSPIRE].
S. Banerjee, S. Ghosh and S.S. Samal, Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes, JHEP 08 (2021) 067 [arXiv:2104.02546] [INSPIRE].
M. Campiglia and A. Laddha, Sub-subleading soft gravitons: New symmetries of quantum gravity?, Phys. Lett. B 764 (2017) 218 [arXiv:1605.09094] [INSPIRE].
M. Campiglia and A. Laddha, Sub-subleading soft gravitons and large diffeomorphisms, JHEP 01 (2017) 036 [arXiv:1608.00685] [INSPIRE].
E. Conde and P. Mao, BMS Supertranslations and Not So Soft Gravitons, JHEP 05 (2017) 060 [arXiv:1612.08294] [INSPIRE].
L. Freidel and D. Pranzetti, Gravity from symmetry: duality and impulsive waves, JHEP 04 (2022) 125 [arXiv:2109.06342] [INSPIRE].
E. Newman and R. Penrose, An Approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566 [INSPIRE].
E.T. Newman and T.W.J. Unti, Behavior of Asymptotically Flat Empty Spaces, J. Math. Phys. 3 (1962) 891 [INSPIRE].
T.M. Adamo, C.N. Kozameh and E.T. Newman, Null Geodesic Congruences, Asymptotically Flat Space-Times and Their Physical Interpretation, Living Rev. Rel. 12 (2009) 6 [arXiv:0906.2155] [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, Adv. Math. Phys. 2012 (2012) 197385 [arXiv:1102.0589] [INSPIRE].
G. Barnich and C. Troessaert, Finite BMS transformations, JHEP 03 (2016) 167 [arXiv:1601.04090] [INSPIRE].
G. Barnich, P. Mao and R. Ruzziconi, BMS current algebra in the context of the Newman–Penrose formalism, Class. Quant. Grav. 37 (2020) 095010 [arXiv:1910.14588] [INSPIRE].
W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, JHEP 09 (2016) 102 [arXiv:1601.04744] [INSPIRE].
L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, Extended corner symmetry, charge bracket and Einstein’s equations, JHEP 09 (2021) 083 [arXiv:2104.12881] [INSPIRE].
L. Ciambelli and R.G. Leigh, Isolated surfaces and symmetries of gravity, Phys. Rev. D 104 (2021) 046005 [arXiv:2104.07643] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Celestial diamonds: conformal multiplets in celestial CFT, JHEP 11 (2021) 072 [arXiv:2105.03516] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Revisiting the conformally soft sector with celestial diamonds, JHEP 11 (2021) 143 [arXiv:2105.09792] [INSPIRE].
L. Donnay and R. Ruzziconi, BMS flux algebra in celestial holography, JHEP 11 (2021) 040 [arXiv:2108.11969] [INSPIRE].
L. Ciambelli, R.G. Leigh, C. Marteau and P.M. Petropoulos, Carroll Structures, Null Geometry and Conformal Isometries, Phys. Rev. D 100 (2019) 046010 [arXiv:1905.02221] [INSPIRE].
L. Ciambelli and R.G. Leigh, Weyl Connections and their Role in Holography, Phys. Rev. D 101 (2020) 086020 [arXiv:1905.04339] [INSPIRE].
R. Geroch, Asymptotic structure of space-time, in Asymptotic Structure of Space-Time, F.P. Esposito and L. Witten eds., Springer, Boston, U.S.A. (1977), pg. 1.
S. Pasterski and A. Puhm, Shifting spin on the celestial sphere, Phys. Rev. D 104 (2021) 086020 [arXiv:2012.15694] [INSPIRE].
D.A. Nichols, Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes, Phys. Rev. D 98 (2018) 064032 [arXiv:1807.08767] [INSPIRE].
L. Freidel, E.R. Livine and D. Pranzetti, Gravitational edge modes: from Kac–Moody charges to Poincaré networks, Class. Quant. Grav. 36 (2019) 195014 [arXiv:1906.07876] [INSPIRE].
L. Freidel, E.R. Livine and D. Pranzetti, Kinematical Gravitational Charge Algebra, Phys. Rev. D 101 (2020) 024012 [arXiv:1910.05642] [INSPIRE].
W. Donnelly, L. Freidel, S.F. Moosavian and A.J. Speranza, Gravitational edge modes, coadjoint orbits, and hydrodynamics, JHEP 09 (2021) 008 [arXiv:2012.10367] [INSPIRE].
R.F. Penna, BMS invariance and the membrane paradigm, JHEP 03 (2016) 023 [arXiv:1508.06577] [INSPIRE].
R.F. Penna, Near-horizon BMS symmetries as fluid symmetries, JHEP 10 (2017) 049 [arXiv:1703.07382] [INSPIRE].
G. Barnich and R. Ruzziconi, Coadjoint representation of the BMS group on celestial Riemann surfaces, JHEP 06 (2021) 079 [arXiv:2103.11253] [INSPIRE].
A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [INSPIRE].
A. Laddha and A. Sen, Logarithmic Terms in the Soft Expansion in Four Dimensions, JHEP 10 (2018) 056 [arXiv:1804.09193] [INSPIRE].
I.M. Gel’fand and L.A. Dikii, Fractional powers of operators and hamiltonian systems, Funct. Anal. Appl. 10 (1976) 259.
M. Adler, On a trace functional for formal pseudo differential operators, Invent. Math. 50 (1978) 219.
L.A. Dickey, Soliton Equations And Hamiltonian Systems, second edition, World Scientific, Singapore (2003).
I. Bakas, Higher Spin Fields and the Gelfand-dickey Algebra, Commun. Math. Phys. 123 (1989) 627 [INSPIRE].
L. Bonora, Q.P. Liu and C.S. Xiong, The Integrable hierarchy constructed from a pair of higher KdV hierarchies and its associated W algebra, Commun. Math. Phys. 175 (1996) 177 [hep-th/9408035] [INSPIRE].
A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
A. Ashtekar, Asymptotic Quantization of the Gravitational Field, Phys. Rev. Lett. 46 (1981) 573 [INSPIRE].
A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav. 50 (2018) 140 [arXiv:1808.07093] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, Tower of subleading dual BMS charges, JHEP 03 (2019) 057 [arXiv:1812.06935] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, Dual gravitational charges and soft theorems, JHEP 10 (2019) 123 [arXiv:1908.01164] [INSPIRE].
H. Godazgar, M. Godazgar and M.J. Perry, Hamiltonian derivation of dual gravitational charges, JHEP 09 (2020) 084 [arXiv:2007.07144] [INSPIRE].
M. Pate, A.-M. Raclariu, A. Strominger and E.Y. Yuan, Celestial operator products of gluons and gravitons, Rev. Math. Phys. 33 (2021) 2140003 [arXiv:1910.07424] [INSPIRE].
Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, Higher spin dynamics in gravity and w1+∞ celestial symmetries, arXiv:2112.15573 [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
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Freidel, L., Pranzetti, D. & Raclariu, AM. Sub-subleading soft graviton theorem from asymptotic Einstein’s equations. J. High Energ. Phys. 2022, 186 (2022). https://doi.org/10.1007/JHEP05(2022)186
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DOI: https://doi.org/10.1007/JHEP05(2022)186