Abstract
In our previous work, we proposed an algorithm to transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation in harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this algorithm at linear order and for specific quadratic interactions known as quadratic tail terms. In the present work, we extend this analysis to quadratic interactions associated with the coupling of two mass quadrupole moments, including both instantaneous and hereditary terms. Our main result is the derivation of the metric in Newman-Unti and Bondi gauges with complete quadrupole-quadrupole interactions. We rederive the displacement memory effect and provide expressions for all Bondi aspects and dressed Bondi aspects relevant to the study of leading and subleading memory effects. Then we obtain the Newman-Penrose charges, the BMS charges as well as the second and third order celestial charges defined from the known second order and novel third order dressed Bondi aspects for mass monopole-quadrupole and quadrupole-quadrupole interactions.
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References
L. Blanchet, Contribution à l’étude du rayonnement gravitationnel émis par un système isolé (in French), habilitation thesis, Université Pierre et Marie Curie, Paris VI, Paris, France (1990).
D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].
A.G. Wiseman and C.M. Will, Christodoulou’s nonlinear gravitational wave memory: evaluation in the quadrupole approximation, Phys. Rev. D 44 (1991) R2945 [INSPIRE].
K.S. Thorne, Gravitational-wave bursts with memory: the Christodoulou effect, Phys. Rev. D 45 (1992) 520 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [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 [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].
E.T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs group, J. Math. Phys. 7 (1966) 863 [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
L. Blanchet, G. Faye, B.R. Iyer and S. Sinha, The third post-Newtonian gravitational wave polarisations and associated spherical harmonic modes for inspiralling compact binaries in quasi-circular orbits, Class. Quant. Grav. 25 (2008) 165003 [Erratum ibid. 29 (2012) 239501] [arXiv:0802.1249] [INSPIRE].
L. Blanchet and T. Damour, Radiative gravitational fields in general relativity 1. General structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A 320 (1986) 379 [INSPIRE].
L. Blanchet, Radiative gravitational fields in general relativity 2. Asymptotic behaviour at future null infinity, Proc. Roy. Soc. Lond. A 409 (1987) 383 [INSPIRE].
L. Blanchet and T. Damour, Tail transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37 (1988) 1410 [INSPIRE].
K. Mitman et al., Computation of displacement and spin gravitational memory in numerical relativity, Phys. Rev. D 102 (2020) 104007 [arXiv:2007.11562] [INSPIRE].
K. Mitman et al., Fixing the BMS frame of numerical relativity waveforms, Phys. Rev. D 104 (2021) 024051 [arXiv:2105.02300] [INSPIRE].
M. Favata, Nonlinear gravitational-wave memory from binary black hole mergers, Astrophys. J. Lett. 696 (2009) L159 [arXiv:0902.3660] [INSPIRE].
P.D. Lasky et al., Detecting gravitational-wave memory with LIGO: implications of GW150914, Phys. Rev. Lett. 117 (2016) 061102 [arXiv:1605.01415] [INSPIRE].
L.O. McNeill, E. Thrane and P.D. Lasky, Detecting gravitational wave memory without parent signals, Phys. Rev. Lett. 118 (2017) 181103 [arXiv:1702.01759] [INSPIRE].
J.B. Wang et al., Searching for gravitational wave memory bursts with the Parkes Pulsar Timing Array, Mon. Not. Roy. Astron. Soc. 446 (2015) 1657 [arXiv:1410.3323] [INSPIRE].
NANOGrav collaboration, NANOGrav constraints on gravitational wave bursts with memory, Astrophys. J. 810 (2015) 150 [arXiv:1501.05343] [INSPIRE].
Y.B. Zel’dovich and A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars, Sov. Astron. 18 (1974) 17 [INSPIRE].
V.B. Braginsky and L.P. Grishchuk, Kinematic resonance and memory effect in free mass gravitational antennas, Sov. Phys. JETP 62 (1985) 427 [INSPIRE].
M. Turner, Gravitational radiation from point-masses in unbound orbits — Newtonian results, Astrophys. J. 216 (1977) 610.
R. Epstein, The generation of gravitational radiation by escaping supernova neutrinos, Astrophys. J. 223 (1978) 1037 [INSPIRE].
L. Bieri, P.N. Chen and S.-T. Yau, Null asymptotics of solutions of the Einstein-Maxwell equations in general relativity and gravitational radiation, Adv. Theor. Math. Phys. 15 (2011) 1085 [arXiv:1011.2267] [INSPIRE].
L. Bieri and D. Garfinkle, Perturbative and gauge invariant treatment of gravitational wave memory, Phys. Rev. D 89 (2014) 084039 [arXiv:1312.6871] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New gravitational memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [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].
É.É. Flanagan, A.M. Grant, A.I. Harte and D.A. Nichols, Persistent gravitational wave observables: general framework, Phys. Rev. D 99 (2019) 084044 [arXiv:1901.00021] [INSPIRE].
E. Himwich, Z. Mirzaiyan and S. Pasterski, A note on the subleading soft graviton, JHEP 04 (2021) 172 [arXiv:1902.01840] [INSPIRE].
A.M. Grant and D.A. Nichols, Persistent gravitational wave observables: curve deviation in asymptotically flat spacetimes, Phys. Rev. D 105 (2022) 024056 [Erratum ibid. 107 (2023) 109902] [arXiv:2109.03832] [INSPIRE].
A. Seraj, Gravitational breathing memory and dual symmetries, JHEP 05 (2021) 283 [arXiv:2103.12185] [INSPIRE].
A. Seraj and B. Oblak, Gyroscopic gravitational memory, arXiv:2112.04535 [INSPIRE].
A. Seraj and B. Oblak, Precession caused by gravitational waves, Phys. Rev. Lett. 129 (2022) 061101 [arXiv:2203.16216] [INSPIRE].
M. Godazgar, G. Macaulay, G. Long and A. Seraj, Gravitational memory effects and higher derivative actions, JHEP 09 (2022) 150 [arXiv:2206.12339] [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].
M. Campiglia and A. Laddha, New symmetries for the gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, Gravitational multipole moments from Noether charges, JHEP 05 (2018) 054 [arXiv:1711.08806] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, New dual gravitational charges, Phys. Rev. D 99 (2019) 024013 [arXiv:1812.01641] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, Tower of subleading dual BMS charges, JHEP 03 (2019) 057 [arXiv:1812.06935] [INSPIRE].
G. Compère, Infinite towers of supertranslation and superrotation memories, Phys. Rev. Lett. 123 (2019) 021101 [arXiv:1904.00280] [INSPIRE].
A. Strominger, w1+∞ and the celestial sphere, arXiv:2105.14346 [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].
H. Godazgar, M. Godazgar and C.N. Pope, Dual gravitational charges and soft theorems, JHEP 10 (2019) 123 [arXiv:1908.01164] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, Higher spin dynamics in gravity and w1+∞ celestial symmetries, Phys. Rev. D 106 (2022) 086013 [arXiv:2112.15573] [INSPIRE].
A. Papapetrou, Coordonnées radiatives cartésiennes (in French), Ann. Inst. Henri Poincaré A XI (1969) 251.
J. Madore, Gravitational radiation from a bounded source. I, Ann. Inst. Henri Poincaré 12 (1970) 285.
J. Madore, Gravitational radiation from a bounded source. II, Ann. Inst. Henri Poincaré 12 (1970) 365.
E.T. Newman and T.W.J. Unti, A class of null flat-space coordinate systems, J. Math. Phys. 4 (1963) 1467 [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].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
M.G.J. van der Burg, Gravitational waves in general relativity IX. Conserved quantities, Proc. Roy. Soc. Lond. A 294 (1966) 112.
L.A. Tamburino and J.H. Winicour, Gravitational fields in finite and conformal Bondi frames, Phys. Rev. 150 (1966) 1039 [INSPIRE].
J. Winicour, Logarithmic asymptotic flatness, Found. Phys. 15 (1985) 605.
L. Freidel, D. Pranzetti and A.-M. Raclariu, Sub-subleading soft graviton theorem from asymptotic Einstein’s equations, JHEP 05 (2022) 186 [arXiv:2111.15607] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
É.É. 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. 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].
G. Compère, R. Oliveri and A. Seraj, The Poincaré and BMS flux-balance laws with application to binary systems, JHEP 10 (2020) 116 [arXiv:1912.03164] [INSPIRE].
L. Blanchet et al., Multipole expansion of gravitational waves: from harmonic to Bondi coordinates, JHEP 02 (2021) 029 [arXiv:2011.10000] [INSPIRE].
L. Blanchet, B.R. Iyer, C.M. Will and A.G. Wiseman, Gravitational wave forms from inspiralling compact binaries to second post-Newtonian order, Class. Quant. Grav. 13 (1996) 575 [gr-qc/9602024] [INSPIRE].
K.G. Arun, L. Blanchet, B.R. Iyer and M.S.S. Qusailah, The 2.5PN gravitational wave polarisations from inspiralling compact binaries in circular orbits, Class. Quant. Grav. 21 (2004) 3771 [Erratum ibid. 22 (2005) 3115] [gr-qc/0404085] [INSPIRE].
L.E. Kidder, L. Blanchet and B.R. Iyer, Radiation reaction in the 2.5PN waveform from inspiralling binaries in circular orbits, Class. Quant. Grav. 24 (2007) 5307 [arXiv:0706.0726] [INSPIRE].
D.A. Nichols, Spin memory effect for compact binaries in the post-Newtonian approximation, Phys. Rev. D 95 (2017) 084048 [arXiv:1702.03300] [INSPIRE].
L. Blanchet and G. Faye, Flux-balance equations for linear momentum and center-of-mass position of self-gravitating post-Newtonian systems, Class. Quant. Grav. 36 (2019) 085003 [arXiv:1811.08966] [INSPIRE].
E.T. Newman and R. Penrose, 10 exact gravitationally-conserved quantities, Phys. Rev. Lett. 15 (1965) 231 [INSPIRE].
E.T. Newman and R. Penrose, New conservation laws for zero rest-mass fields in asymptotically flat space-time, Proc. Roy. Soc. Lond. A 305 (1968) 175 [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].
R. Sachs and P.G. Bergmann, Structure of particles in linearized gravitational theory, Phys. Rev. 112 (1958) 674 [INSPIRE].
F.A.E. Pirani, Introduction to gravitational radiation theory, volume 1 of Brandeis summer institute in theoretical physics, Prentice-Hall, Englewood Cliffs, NJ, U.S.A. (1964), p. 249.
K.S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52 (1980) 299 [INSPIRE].
L. Blanchet and T. Damour, Radiative gravitational fields in general relativity I. General structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A 320 (1986) 379 [INSPIRE].
R. Epstein and R.V. Wagoner, Post-Newtonian generation of gravitational waves, Astrophys. J. 197 (1975) 717.
C.N. Kozameh and G.D. Quiroga, Center of mass and spin for isolated sources of gravitational radiation, Phys. Rev. D 93 (2016) 064050 [arXiv:1311.5854] [INSPIRE].
C.N. Kozameh, J.I. Nieva and G.D. Quiroga, Spin and center of mass comparison between the post-Newtonian approach and the asymptotic formulation, Phys. Rev. D 98 (2018) 064005 [arXiv:1711.11375] [INSPIRE].
L. Blanchet, Quadrupole-quadrupole gravitational waves, Class. Quant. Grav. 15 (1998) 89 [gr-qc/9710037] [INSPIRE].
V.B. Braginsky and K.S. Thorne, Gravitational-wave bursts with memory and experimental prospects, Nature 327 (1987) 123.
P.N. Payne, Smarr’s zero frequency limit calculation, Phys. Rev. D 28 (1983) 1894 [INSPIRE].
M. Ludvigsen, Geodesic deviation at null infinity and the physical effects of very long wave gravitational radiation, Gen. Rel. Grav. 21 (1989) 1205 [INSPIRE].
J.M. Martín-García et al., xAct: efficient tensor computer algebra for Mathematica, http://www.xact.es/, GPL (2002)–(2012).
L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Rel. 17 (2014) 2 [arXiv:1310.1528] [INSPIRE].
G. Compère, R. Oliveri and A. Seraj, Metric reconstruction from celestial multipoles, JHEP 11 (2022) 001 [arXiv:2206.12597] [INSPIRE].
A.R. Exton, E.T. Newman and R. Penrose, Conserved quantities in the Einstein-Maxwell theory, J. Math. Phys. 10 (1969) 1566 [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].
L. Freidel and D. Pranzetti, Gravity from symmetry: duality and impulsive waves, JHEP 04 (2022) 125 [arXiv:2109.06342] [INSPIRE].
H. Godazgar, M. Godazgar and C.N. Pope, Subleading BMS charges and fake news near null infinity, JHEP 01 (2019) 143 [arXiv:1809.09076] [INSPIRE].
E. Whittaker and G. Watson, A course of modern analysis, Cambridge University Press, Cambridge, U.K. (1990).
L. Blanchet, G. Faye and F. Larrouturou, The quadrupole moment of compact binaries to the fourth post-Newtonian order: from source to canonical moment, Class. Quant. Grav. 39 (2022) 195003 [arXiv:2204.11293] [INSPIRE].
D. Trestini, F. Larrouturou and L. Blanchet, The quadrupole moment of compact binaries to the fourth post-Newtonian order: relating the harmonic and radiative metrics, Class. Quant. Grav. 40 (2023) 055006 [arXiv:2209.02719] [INSPIRE].
Acknowledgments
G.C. is Senior Research Associate of the F.R.S.-FNRS and acknowledges support from the FNRS research credit J.0036.20F and the IISN convention 4.4503.15. The work of R.O. is supported by the Région Île-de-France within the DIM ACAV+ project SYMONGRAV (Symétries asymptotiques et ondes gravitationnelles). A.S. is supported by a Royal Society University Research Fellowship. L.B. and R.O. ackowledge support from the Partenariat Hubert Curien within the Barrande mobility programme (project number 46771VC).
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Blanchet, L., Compère, G., Faye, G. et al. Multipole expansion of gravitational waves: memory effects and Bondi aspects. J. High Energ. Phys. 2023, 123 (2023). https://doi.org/10.1007/JHEP07(2023)123
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DOI: https://doi.org/10.1007/JHEP07(2023)123