Abstract
Asymptotically flat spacetimes admit both supertranslations and Lorentz transformations as asymptotic symmetries. Furthermore, they admit super-Lorentz transformations, namely superrotations and superboosts, as outer symmetries associated with super-angular momentum and super-center-of-mass charges. In this paper, we present comprehensively the flux-balance laws for all such BMS charges. We distinguish the Poincaré flux-balance laws from the proper BMS flux-balance laws associated with the three relevant memory effects defined from the shear, namely, the displacement, spin and center-of-mass memory effects. We scrutinize the prescriptions used to define the angular momentum and center-of-mass. In addition, we provide the exact form of all Poincaré and proper BMS flux-balance laws in terms of radiative symmetric tracefree multipoles. Fluxes of energy, angular momentum and octupole super-angular momentum arise at 2.5PN, fluxes of quadrupole supermomentum arise at 3PN and fluxes of momentum, center-of-mass and octupole super-center-of-mass arise at 3.5PN. We also show that the BMS flux-balance laws lead to integro-differential consistency constraints on the radiation-reaction forces acting on the sources. Finally, we derive the exact form of all BMS charges for both an initial Kerr binary and a final Kerr black hole in an arbitrary Lorentz and supertranslation frame, which allows to derive exact constraints on gravitational waveforms produced by binary black hole mergers from each BMS flux-balance law.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London Ser. A 269 (1962) 21.
R.K. Sachs, Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London Ser. A 270 (1962) 103.
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 and J. Long, Classical static final state of collapse with supertranslation memory, Class. Quant. Grav. 33 (2016) 195001 [arXiv:1602.05197] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Superrotation charge and supertranslation hair on black holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].
G. Compère and J. Long, Vacua of the gravitational field, JHEP 07 (2016) 137 [arXiv:1601.04958] [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].
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].
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].
J. Podolsky and R. Steinbauer, Geodesics in space-times with expanding impulsive gravitational waves, Phys. Rev. D 67 (2003) 064013 [gr-qc/0210007] [INSPIRE].
J. Podolsky and R. Svarc, Refraction of geodesics by impulsive spherical gravitational waves in constant-curvature spacetimes with a cosmological constant, Phys. Rev. D 81 (2010) 124035 [arXiv:1005.4613] [INSPIRE].
J. Podolsky, C. Sämann, R. Steinbauer and R. Svarc, The global uniqueness and C1-regularity of geodesics in expanding impulsive gravitational waves, Class. Quant. Grav. 33 (2016) 195010 [arXiv:1602.05020] [INSPIRE].
J.B. Griffiths and P. Docherty, A disintegrating cosmic string, Class. Quant. Grav. 19 (2002) L109 [gr-qc/0204085] [INSPIRE].
J.B. Griffiths, J. Podolsky and P. Docherty, An Interpretation of Robinson-Trautman type N solutions, Class. Quant. Grav. 19 (2002) 4649 [gr-qc/0208022] [INSPIRE].
G. Barnich and C. Troessaert, Finite BMS transformations, JHEP 03 (2016) 167 [arXiv:1601.04090] [INSPIRE].
A. Strominger and A. Zhiboedov, Superrotations and Black Hole Pair Creation, Class. Quant. Grav. 34 (2017) 064002 [arXiv:1610.00639] [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].
E.E. Flanagan, K. Prabhu and I. Shehzad, Extensions of the asymptotic symmetry algebra of general relativity, JHEP 01 (2020) 002 [arXiv:1910.04557] [INSPIRE].
S. Bhattacharjee, S. Kumar and A. Bhattacharyya, Memory effect and BMS-like symmetries for impulsive gravitational waves, Phys. Rev. D 100 (2019) 084010 [arXiv:1905.12905] [INSPIRE].
M.G.J. Van der Burg, Gravitational waves in general relativity IX. Conserved quantities, Proc. Roy. Soc. London Ser. A 294 (1966) 112.
P. N. Payne, Smarr’s zero-frequency-limit calculation, Phys. Rev. D 28 (1982) 8.
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [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].
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].
B. Bonga and E. Poisson, Coulombic contribution to angular momentum flux in general relativity, Phys. Rev. D 99 (2019) 064024 [arXiv:1808.01288] [INSPIRE].
J. Distler, R. Flauger and B. Horn, Double-soft graviton amplitudes and the extended BMS charge algebra, JHEP 08 (2019) 021 [arXiv:1808.09965] [INSPIRE].
A. Ashtekar, T. De Lorenzo and N. Khera, Compact binary coalescences: constraints on waveforms, arXiv:1906.00913 [INSPIRE].
A. Ashtekar, T. De Lorenzo and N. Khera, Compact binary coalescences: the subtle issue of angular momentum, Phys. Rev. D 101 (2020) 044005 [arXiv:1910.02907] [INSPIRE].
K.S. Thorne, Multipole expansions of gravitational radiation, Rev. Mod. Phys. 52 (1980) 299 [INSPIRE].
A. Ashtekar and M. Streubel, Symplectic geometry of radiative modes and conserved quantities at null infinity, Proc. Roy. Soc. London Ser. A 376 (1981) 585.
T. Dray and M. Streubel, Angular momentum at null infinity, Class. Quant. Grav. 1 (1984) 15 [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].
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].
Y.B. Zel’dovich and A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars, Sov. Astron. 18 (1974) [Astron. Zh. 51 (1974) 30] [INSPIRE].
M. Turner, Gravitational radiation from point-masses in unbound orbits: Newtonian results, Astrophys. J. 216 (1977) 610.
M. Turner and C.M. Will, Post-Newtonian gravitational bremsstrahlung, Astrophys. J. 220 (1978) 1107 [INSPIRE].
M.S. Turner, Gravitational radiation from supernova neutrino bursts, Nature 274 (1978) 565 [INSPIRE].
S.J. Kovacs and K.S. Thorne, The generation of gravitational waves. 4. Bremsstrahlung, Astrophys. J. 224 (1978) 62 [INSPIRE].
V.B. Braginskii and L.P. Grishchuk, Kinematic resonance and the memory effect in free mass gravitational antennas, Zh. Eksp. Teor. Fiz. 89 (1985) 744.
V.B. Braginskii and K.S. Thorne, Gravitational-wave bursts with memory and experimental prospects, Nature 327 (1987) 123.
L. Blanchet and T. Damour, Tail transported temporal correlations in the dynamics of a gravitating system, Phys. Rev. D 37 (1988) 1410 [INSPIRE].
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) 2945 [INSPIRE].
L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
J. Frauendiener, Note on the memory effect, Class. Quant. Grav. 9 (1992) 1639.
M. Favata, Post-Newtonian corrections to the gravitational-wave memory for quasi-circular, inspiralling compact binaries, Phys. Rev. D 80 (2009) 024002 [arXiv:0812.0069] [INSPIRE].
D. Pollney and C. Reisswig, Gravitational memory in binary black hole mergers, Astrophys. J. Lett. 732 (2011) L13 [arXiv:1004.4209] [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].
P.D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen, Detecting gravitational-wave memory with LIGO: implications of GW150914, Phys. Rev. Lett. 117 (2016) 061102 [arXiv:1605.01415] [INSPIRE].
T. Mädler and J. Winicour, The sky pattern of the linearized gravitational memory effect, Class. Quant. Grav. 33 (2016) 175006 [arXiv:1605.01273] [INSPIRE].
M. Hübner, C. Talbot, P.D. Lasky and E. Thrane, Measuring gravitational-wave memory in the first LIGO/Virgo gravitational-wave transient catalog, Phys. Rev. D 101 (2020) 023011 [arXiv:1911.12496] [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].
S. Pasterski, A. Strominger and A. Zhiboedov, New gravitational memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [INSPIRE].
E.E. 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, arXiv:1902.01840 [INSPIRE].
L. Blanchet, T. Damour and B.R. Iyer, Surface-integral expressions for the multipole moments of post-Newtonian sources and the boosted Schwarzschild solution, Class. Quant. Grav. 22 (2005) 155 [gr-qc/0410021] [INSPIRE].
T. Mädler and J. Winicour, Kerr black holes and nonlinear radiation memory, Class. Quant. Grav. 36 (2019) 095009 [arXiv:1811.04711] [INSPIRE].
R.V. Wagoner and C.M. Will, PostNewtonian gravitational radiation from orbiting point masses, Astrophys. J. 210 (1976) 764 [Erratum ibid. 215 (1977) 984] [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. London Ser. A 320 (1986) 379.
C.M. Will and A.G. Wiseman, Gravitational radiation from compact binary systems: Gravitational wave forms and energy loss to second postNewtonian order, Phys. Rev. D 54 (1996) 4813 [gr-qc/9608012] [INSPIRE].
W.D. Goldberger and I.Z. Rothstein, An effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156] [INSPIRE].
R.A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [arXiv:1601.04914] [INSPIRE].
M. Ruiz, R. Takahashi, M. Alcubierre and D. Núñez, Multipole expansions for energy and momenta carried by gravitational waves, Gen. Rel. Grav. 40 (2008) 2467 [arXiv:0707.4654] [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].
T. Mädler and J. Winicour, Bondi-Sachs formalism, Scholarpedia 11 (2016) 33528 [arXiv:1609.01731] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [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].
G. Compère, Advanced lectures on general relativity, Lect. Notes Phys. 952 (2019) 150.
G. Compère, R. Oliveri and A. Seraj, Gravitational multipole moments from Noether charges, JHEP 05 (2018) 054 [arXiv:1711.08806] [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a new soft graviton theorem, arXiv:1404.4091 [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].
M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP 11 (2016) 012 [arXiv:1605.09677] [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].
M. Mirbabayi and M. Porrati, Dressed hard states and black hole soft hair, Phys. Rev. Lett. 117 (2016) 211301 [arXiv:1607.03120] [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, Princeton University Press, Princeton U.S.A. (1986).
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
W.T. Shaw, Symplectic geometry of null infinity and two-surface twistors, Class. Quant. Grav. 1 (1984) L33 [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [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 and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
R. Javadinezhad, U. Kol and M. Porrati, Comments on Lorentz transformations, dressed asymptotic states and Hawking radiation, JHEP 01 (2019) 089 [arXiv:1808.02987] [INSPIRE].
O.M. Moreschi, Intrinsic angular momentum in general relativity, Class. Quant. Grav. 21 (2004) 5409 [gr-qc/0209097] [INSPIRE].
E. Gallo and O.M. Moreschi, Intrinsic angular momentum for radiating spacetimes which agrees with the Komar integral in the axisymmetric case, Phys. Rev. D 89 (2014) 084009 [arXiv:1404.2475] [INSPIRE].
R. Epstein and R.V. Wagoner, Post-Newtonian generation of gravitational waves, Astrophys. J. 197 (1975) 717.
L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Rev. Rel. 17 (2014) 2 [arXiv:1310.1528] [INSPIRE].
A. Peres, Classical radiation recoil, Phys. Rev. 128 (1962) 2471 [INSPIRE].
J.D. Bekenstein, Gravitational-radiation recoil and runaway black holes, Astrophys. J. 183 (1973) 657 [INSPIRE].
P.C. Peters and J. Mathews, Gravitational radiation from point masses in a Keplerian orbit, Phys. Rev. 131 (1963) 435 [INSPIRE].
P.C. Peters, Gravitational radiation and the motion of two point masses, Phys. Rev. 136 (1964) B1224.
K.S. Thorne, Nonradial pulsation of general-relativistic stellar models IV. The Weakfield limit, Astrophys. J. 158 (1969) 997 [INSPIRE].
S. Chandrasekhar, Conservation Laws in General Relativity and in the Post-Newtonian Approximations, Astrophys. J. 158 (1969) 45.
S. Chandrasekhar and Y. Nutku, The second post-Newtonian equations of hydrodynamics in general relativity, Astrophys. J. 158 (1969) 55.
S. Chandrasekhar and F.P. Esposito, The 2.5-post-Newtonian equations of hydrodynamics and radiation reaction in general relativity, Astrophys. J. 160 (1970) 153.
J.L. Anderson and T.C. Decanio, Equations of hydrodynamics in general relativity in the slow motion approximation., Gen. Rel. Grav. 6 (1975) 197.
J. Ehlers, Isolated systems in general relativity, in 9th Texas Symposium on Relativistic Astrophysics , December 14–19, Munich, Germany (1980).
G.D. Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. Part I. First post-Newtonian equations, Gen. Rel. Grav. 12 (1980) 467.
G.D. Kerlick, Finite reduced hydrodynamic equations in the slow-motion approximation to general relativity. Part II. Radiation reaction and higher-order divergent terms, Gen. Rel. Grav. 12 (1980) 521.
A. Papapetrou and B. Linet, Equation of motion including the reaction of gravitational radiation, Gen. Rel. Grav. 13 (1981) 335 [INSPIRE].
R.A. Breuer and E. Rudolph, Radiation reaction and energy loss in the post-Newtonian approximation of general relativity, Gen. Rel. Grav. 13 (1981) 777.
T. Damour and N. Deruelle, Radiation reaction and angular momentum loss in small angle gravitational scattering, Phys. Lett. A 87 (1981) 81 [INSPIRE].
R.A. Breuer and E. Rudolph, The force law for the dynamic two-body problem on the second post-Newtonian approximation of general relativity, Gen. Rel. Grav. 14 (1982) 181.
T. Damour and N. Deruelle, Lagrangien généralisé du système de deux masses ponctuelles, à l’approximation post-post-newtonienne de la relativité générale, C.R. Acad. Sci. Paris 293 (1981) 537.
T. Damour, Problème des deux corps et freinage de rayonnement en relativité générale, C.R. Acad. Sci. Paris 294 (1982) 1355.
T. Damour, Gravitational radiation and the motion of compact bodies, in Gravitational Radiation, N. Deruelle and T. Piran eds., North-Holland publishing Company, The Netherlands (1983).
G. Schaefer, The gravitational quadrupole radiation reaction force and the canonical formalism of ADM, Annals Phys. 161 (1985) 81 [INSPIRE].
B.R. Iyer and C.M. Will, PostNewtonian gravitational radiation reaction for two-body systems, Phys. Rev. Lett. 70 (1993) 113 [INSPIRE].
B.R. Iyer and C.M. Will, PostNewtonian gravitational radiation reaction for two-body systems: nonspinning bodies, Phys. Rev. D 52 (1995) 6882 [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].
M. Boyle et al., The SXS collaboration catalog of binary black hole simulations, Class. Quant. Grav. 36 (2019) 195006 [arXiv:1904.04831] [INSPIRE].
N.W. Taylor et al., Comparing Gravitational Waveform Extrapolation to Cauchy-Characteristic Extraction in Binary Black Hole Simulations, Phys. Rev. D 88 (2013) 124010 [arXiv:1309.3605] [INSPIRE].
V. Varma, D. Gerosa, L.C. Stein, F. Hébert and H. Zhang, High-accuracy mass, spin, and recoil predictions of generic black-hole merger remnants, Phys. Rev. Lett. 122 (2019) 011101 [arXiv:1809.09125] [INSPIRE].
C.J. Woodford, M. Boyle and H.P. Pfeiffer, Compact binary waveform center-of-mass corrections, Phys. Rev. D 100 (2019) 124010 [arXiv:1904.04842] [INSPIRE].
G. Compère, Infinite towers of supertranslation and superrotation memories, Phys. Rev. Lett. 123 (2019) 021101 [arXiv:1904.00280] [INSPIRE].
G. Faye, L. Blanchet and B.R. Iyer, Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order, Class. Quant. Grav. 32 (2015) 045016 [arXiv:1409.3546] [INSPIRE].
F. Beyer, B. Daszuta, J. Frauendiener and B. Whale, Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics, Class. Quant. Grav. 31 (2014) 075019 [arXiv:1308.4729] [INSPIRE].
G. Arfken, Mathematical methods for physicists, third edition, Academic Press, U.S.A. (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1912.03164
Rights and permissions
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.
About this article
Cite this article
Compère, G., Oliveri, R. & Seraj, A. The Poincaré and BMS flux-balance laws with application to binary systems. J. High Energ. Phys. 2020, 116 (2020). https://doi.org/10.1007/JHEP10(2020)116
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)116