Bondi-Sachs度量与引力波
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摘要
广义相对论中,引力波是具波的形式并且辐射能量的真空时空度量,而Bondi-Sachs度量是这类度量的自然描述之一.本文将概述零宇宙常数时Bondi-Sachs度量的Bondi能量动量非负性定理、Bondi能量动量和ADM(Arnowitt-Deser-Misner)能量动量的关系,以及非零宇宙常数时BondiSachs度量的一类自然边值条件及其在该边值条件下的peeling性质、B-引力波的一些新特性.
In general relativity, gravitational waves are wave-like metrics which radiate energy, and Bondi-Sachs metrics are one of natural metrics describing them. In this paper, we provide a short review on positivity of the Bondi energy-momentum and the relation between the Bondi energy-momentum and the ADM energy-momentum for Bondi-Sachs metrics when the cosmological constant is zero. We also review a natural boundary condition of the Bondi-Sachs metrics when the cosmological constant is nonzero, and present the peeling property as well as some new features of gravitational B-waves under this boundary condition.
In general relativity, gravitational waves are wave-like metrics which radiate energy, and Bondi-Sachs metrics are one of natural metrics describing them. In this paper, we provide a short review on positivity of the Bondi energy-momentum and the relation between the Bondi energy-momentum and the ADM energy-momentum for Bondi-Sachs metrics when the cosmological constant is zero. We also review a natural boundary condition of the Bondi-Sachs metrics when the cosmological constant is nonzero, and present the peeling property as well as some new features of gravitational B-waves under this boundary condition.
引文
1胡和生.球对称引力场方程的严格解.复旦学报,1974,1:92-98(Hu H S.Exact solutions of the spherically symmetric gravitational field equations.Front Math China,2006,1:169-177)
2谷超豪.非均匀的球对称引力坍缩问题.复旦学报,1973,3:73-78(Gu C H.Gravitation collapse of spherical symmetry with non-uniform density.Front Math China,2006,1:161-168)
3 Choquet-Bruhat Y.General Relativity and the Einstein Equations.Oxford:Oxford University Press,2009
4 Einstein A.N¨aherungsweise integration der feldgleichungen der gravitation.In:Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissenschaften.Berlin:Wiley,1916,688-696
5 Abbott B P,Abbott R,Abbott T D,et al.Observation of gravitational waves from a binary black hole merger.Phys Rev Lett,2016,116:061102
6 Bondi H,van der Burg M,Metzner A.Gravitational waves in general relativity,VII:Waves from axi-symmetric isolated systems.Proc R Soc Lond Ser A Math Phys Eng Sci,1962,269:21-52
7 Sachs R.Gravitational waves in general relativity,VIII:Waves in asymptotically flat space-time.Proc R Soc Lond Ser A Math Phys Eng Sci,1962,270:103-126
8 Ge H,Luo M,Su Q,et al.Bondi-Sachs metrics and photon rockets.Gen Relativity Gravitation,2011,43:2729-2742
9 Huang W L,Yau S T,Zhang X.Positivity of the Bondi mass in Bondi’s radiating spacetimes.Rend Lincei Mat Appl,2006,17:335-349
10 Chru′sciel P T,Jezierski J,l?eski S.The Trautman-Bondi mass of hyperboloidal initial data sets.Adv Theor Math Phys,2004,8:83-139
11 Zhang X.A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds,I.Comm Math Phys,2004,249:529-548
12 Zhang X.Recent progress on the positive energy theorem.Internat J Modern Phys A,2015,30:1545018
13张晓.广义相对论正能量定理.中国科学:数学,2017,47:673-688
14 Arnowitt R,Deser S,Misner C.The dynamics of general relativity.In:Gravitation:An Introduction to Current Research.New York:Wiley,1962,227-264.Gen Relativity Gravitation,2008,40:1997-2027
15 Ashtekar A,Magnon-Ashtekar A.Energy-momentum in general relativity.Phys Rev Lett,1979,43:181-184
16 Penrose R,Rindler W.Spinors and Space-Time,I;II.Cambridge:Cambridge University Press,1984;1988
17 Zhang X.On the relation between ADM and Bondi energy-momenta.Adv Theor Math Phys,2006,10:261-282
18 Huang W,Zhang X.On the relation between ADM and Bondi energymomenta III-perturbed radiative spatial infinity.Sci China Ser A,2007,50:1316-1324
19 Lax P D,Phillips R S.Scattering Theory.Cambridge:Academic Press,1967
20 Alexakis S,Shao A.Global uniqueness theorems for linear and nonlinear waves.J Funct Anal,2015,269:3458-3499
21 He X,Cao Z.New Bondi-type outgoing boundary condition for the Einstein equations with cosmological constant.Internat J Modern Phys D,2015,24:1550081
22 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,I:Basic framework.Classical Quantum Gravity,2015,32:025004
23 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,II:Linear fields on de Sitter spacetime.Phys Rev D,2015,92:044011
24 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,III:The quadrupole formula.Phys Rev D,2015,92:104032
25 Ashtekar A,Bonga B,Kesavan A.Gravitational waves from isolated systems:Surprising consequences of a positive cosmological constant.Phys Rev Lett,2016,116:051101
26 Bishop N T.Gravitational waves in a de Sitter universe.Phys Rev D,2016,93:044025
27 Date G,Hoque S J.Gravitational waves from compact sources in a de Sitter background.Phys Rev D,2016,94:064039
28 Date G,Hoque S J.Cosmological horizon and the quadrupole formula in de Sitter background.Phys Rev D,2017,96:044026
29 Saw V L.Mass-loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant.Phys Rev D,2016,94:104004
30 Saw V L.Behaviour of asymptotically electro-Λspacetimes.Phys Rev D,2017,95:084038
31 Bieri L,Garfinkle D,Yau S T.Gravitational wave memory in de Sitter spacetime.Phys Rev D,2016,94:064040
32 Xie F,Zhang X.Peeling property of Bondi-Sachs metrics for nonzero cosmological constant.Ar Xiv:1704.06015,2017
33 Penrose R.Zero rest mass fields including gravitation:Asymptotic behavior.Proc R Soc Lond Ser A Math Phys Eng Sci,1965,284:159-203
34 Luo M,Xie N,Zhang X.Positive mass theorems for asymptotically de Sitter spacetimes.Nuclear Phys B,2010,825:98-118
35 Liang Z,Zhang X.Spacelike hypersurfaces with negative total energy in de Sitter spacetime.J Math Phys,2012,53:022502
36 Henneaux M,Teitelboim C.Asymptotically anti-de Sitter spaces.Comm Math Phys,1985,98:391-424
37 Wang Y,Xie N,Zhang X.The positive energy theorem for asymptotically anti-de Sitter spacetimes.Commun Contemp Math,2015,17:1550015
1)It is believed that when a star has exhausted all thermonuclear sources of energy,it will collapse under its own gravitational field.Oppenheimer and Snyder10have given a rigorous study of the process for a spherically symmetric dust cloud by using a special solution given by Tolman11,in the restricted case of a spatially constant matter density starting from rest.This study was taken anew by Gu Chao Hao12,in a study of the Tolman solution,but not restricted to spatially constant matter.Gu Chao Hao gave an analysis of shell crossing singularities as well as the essential singularity at the centre of symmetry.He gave a detailed mathematical discussion of the formation of horizons and of what is now called the nakedness of these singularities.Hu Hesheng13constructed the complete solutions of the fully general spherically symmetric solution for dust,not restricted to the Tolman class,and discussed their properties.The work done by Gu Chao Hao and Hu Hesheng for non-homogeneous dust clouds was done again,unaware of these papers published in China,by Mu¨ller zum Hagen,Seifert,and Yodzis14(Tolman class,shell crossing singularities),Christodoulou15(starting from rest,central singularities)and Richard Newman16(general solution).These western papers were motivated by the cosmic censorship conjecture,which is violated both at“shell crossing”singularities and at the centre,as was already shown in Gu’s paper.10 Oppenheimer,J.R.and Snyder,H.(1939)Phys.Rev.56,455.11 Tolman,R.(1934)Proc.Natl.Acad.Sci.USA,20,169-76,Bondi,H.(1947)Mon.Nat.R.Astron.Soc.,107,410-25.12 Gu Chao Hao(1973)J.Fudan University,1,73-8.13 Hu Hesheng(1974)J.Fudan University,2,92-8.14 Müller zum Hagen,H.,Seifert,H.J.,and Yodzis,P.(1973)Comm.Math.Phys.,34,134-48).15 Christodoulou,D.(1984)Commun.Math.Phys.,93,171-95.16 Newman,R.(1986)Class.Quantum Grav.,3,527-39.
2谷超豪.非均匀的球对称引力坍缩问题.复旦学报,1973,3:73-78(Gu C H.Gravitation collapse of spherical symmetry with non-uniform density.Front Math China,2006,1:161-168)
3 Choquet-Bruhat Y.General Relativity and the Einstein Equations.Oxford:Oxford University Press,2009
4 Einstein A.N¨aherungsweise integration der feldgleichungen der gravitation.In:Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissenschaften.Berlin:Wiley,1916,688-696
5 Abbott B P,Abbott R,Abbott T D,et al.Observation of gravitational waves from a binary black hole merger.Phys Rev Lett,2016,116:061102
6 Bondi H,van der Burg M,Metzner A.Gravitational waves in general relativity,VII:Waves from axi-symmetric isolated systems.Proc R Soc Lond Ser A Math Phys Eng Sci,1962,269:21-52
7 Sachs R.Gravitational waves in general relativity,VIII:Waves in asymptotically flat space-time.Proc R Soc Lond Ser A Math Phys Eng Sci,1962,270:103-126
8 Ge H,Luo M,Su Q,et al.Bondi-Sachs metrics and photon rockets.Gen Relativity Gravitation,2011,43:2729-2742
9 Huang W L,Yau S T,Zhang X.Positivity of the Bondi mass in Bondi’s radiating spacetimes.Rend Lincei Mat Appl,2006,17:335-349
10 Chru′sciel P T,Jezierski J,l?eski S.The Trautman-Bondi mass of hyperboloidal initial data sets.Adv Theor Math Phys,2004,8:83-139
11 Zhang X.A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds,I.Comm Math Phys,2004,249:529-548
12 Zhang X.Recent progress on the positive energy theorem.Internat J Modern Phys A,2015,30:1545018
13张晓.广义相对论正能量定理.中国科学:数学,2017,47:673-688
14 Arnowitt R,Deser S,Misner C.The dynamics of general relativity.In:Gravitation:An Introduction to Current Research.New York:Wiley,1962,227-264.Gen Relativity Gravitation,2008,40:1997-2027
15 Ashtekar A,Magnon-Ashtekar A.Energy-momentum in general relativity.Phys Rev Lett,1979,43:181-184
16 Penrose R,Rindler W.Spinors and Space-Time,I;II.Cambridge:Cambridge University Press,1984;1988
17 Zhang X.On the relation between ADM and Bondi energy-momenta.Adv Theor Math Phys,2006,10:261-282
18 Huang W,Zhang X.On the relation between ADM and Bondi energymomenta III-perturbed radiative spatial infinity.Sci China Ser A,2007,50:1316-1324
19 Lax P D,Phillips R S.Scattering Theory.Cambridge:Academic Press,1967
20 Alexakis S,Shao A.Global uniqueness theorems for linear and nonlinear waves.J Funct Anal,2015,269:3458-3499
21 He X,Cao Z.New Bondi-type outgoing boundary condition for the Einstein equations with cosmological constant.Internat J Modern Phys D,2015,24:1550081
22 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,I:Basic framework.Classical Quantum Gravity,2015,32:025004
23 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,II:Linear fields on de Sitter spacetime.Phys Rev D,2015,92:044011
24 Ashtekar A,Bonga B,Kesavan A.Asymptotics with a positive cosmological constant,III:The quadrupole formula.Phys Rev D,2015,92:104032
25 Ashtekar A,Bonga B,Kesavan A.Gravitational waves from isolated systems:Surprising consequences of a positive cosmological constant.Phys Rev Lett,2016,116:051101
26 Bishop N T.Gravitational waves in a de Sitter universe.Phys Rev D,2016,93:044025
27 Date G,Hoque S J.Gravitational waves from compact sources in a de Sitter background.Phys Rev D,2016,94:064039
28 Date G,Hoque S J.Cosmological horizon and the quadrupole formula in de Sitter background.Phys Rev D,2017,96:044026
29 Saw V L.Mass-loss of an isolated gravitating system due to energy carried away by gravitational waves with a cosmological constant.Phys Rev D,2016,94:104004
30 Saw V L.Behaviour of asymptotically electro-Λspacetimes.Phys Rev D,2017,95:084038
31 Bieri L,Garfinkle D,Yau S T.Gravitational wave memory in de Sitter spacetime.Phys Rev D,2016,94:064040
32 Xie F,Zhang X.Peeling property of Bondi-Sachs metrics for nonzero cosmological constant.Ar Xiv:1704.06015,2017
33 Penrose R.Zero rest mass fields including gravitation:Asymptotic behavior.Proc R Soc Lond Ser A Math Phys Eng Sci,1965,284:159-203
34 Luo M,Xie N,Zhang X.Positive mass theorems for asymptotically de Sitter spacetimes.Nuclear Phys B,2010,825:98-118
35 Liang Z,Zhang X.Spacelike hypersurfaces with negative total energy in de Sitter spacetime.J Math Phys,2012,53:022502
36 Henneaux M,Teitelboim C.Asymptotically anti-de Sitter spaces.Comm Math Phys,1985,98:391-424
37 Wang Y,Xie N,Zhang X.The positive energy theorem for asymptotically anti-de Sitter spacetimes.Commun Contemp Math,2015,17:1550015
1)It is believed that when a star has exhausted all thermonuclear sources of energy,it will collapse under its own gravitational field.Oppenheimer and Snyder10have given a rigorous study of the process for a spherically symmetric dust cloud by using a special solution given by Tolman11,in the restricted case of a spatially constant matter density starting from rest.This study was taken anew by Gu Chao Hao12,in a study of the Tolman solution,but not restricted to spatially constant matter.Gu Chao Hao gave an analysis of shell crossing singularities as well as the essential singularity at the centre of symmetry.He gave a detailed mathematical discussion of the formation of horizons and of what is now called the nakedness of these singularities.Hu Hesheng13constructed the complete solutions of the fully general spherically symmetric solution for dust,not restricted to the Tolman class,and discussed their properties.The work done by Gu Chao Hao and Hu Hesheng for non-homogeneous dust clouds was done again,unaware of these papers published in China,by Mu¨ller zum Hagen,Seifert,and Yodzis14(Tolman class,shell crossing singularities),Christodoulou15(starting from rest,central singularities)and Richard Newman16(general solution).These western papers were motivated by the cosmic censorship conjecture,which is violated both at“shell crossing”singularities and at the centre,as was already shown in Gu’s paper.10 Oppenheimer,J.R.and Snyder,H.(1939)Phys.Rev.56,455.11 Tolman,R.(1934)Proc.Natl.Acad.Sci.USA,20,169-76,Bondi,H.(1947)Mon.Nat.R.Astron.Soc.,107,410-25.12 Gu Chao Hao(1973)J.Fudan University,1,73-8.13 Hu Hesheng(1974)J.Fudan University,2,92-8.14 Müller zum Hagen,H.,Seifert,H.J.,and Yodzis,P.(1973)Comm.Math.Phys.,34,134-48).15 Christodoulou,D.(1984)Commun.Math.Phys.,93,171-95.16 Newman,R.(1986)Class.Quantum Grav.,3,527-39.