作用角变量方法研究行星进动和广义M(?)ller变换
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摘要
众所周知,爱因斯坦于1915年创立的广义相对论是研究引力和时空结构的理论。本文从爱因斯坦场方程出发,运用作用角变量方法研究行星进动,并对广义Mφller变换进行了讨论.
我们首先介绍了测地线方程和黎曼曲率张量,这是我们求解爱因斯坦场方程的数学基础.其次,我们介绍了经典力学中的正则变换、哈密顿-雅可比理论及作用角变量方法,为后面研究行星进动问题提供了方法。在第三章中,我们介绍了广义相对论的基本原理和基本任务,并介绍了广义相对论的一个新的研究方向——数值广义相对论。
我们具体讨论了运用作用角变量方法研究行星分别在近圆和椭圆轨道下的近日点进动值,并给出其进动值的明确表述.在回顾了爱因斯坦场方程的Schwarzschild真空解之后,我们介绍了质点在Schwarzschild场中的运动方程.从这组完备的质点动力学微分方程出发,我们探讨了质点运动相应的作用变量的一般形式。并且,分别讨论了在近圆轨道和椭圆轨道两种情况下,质点运动的作用变量的共轭变量——角变量的性质,从而研究出,经过一个周期后,质点(行星)的轨道进动值。
我们还研究了广义Mφller变换——由惯性参考系到任意变速参考系的一种变换。重点讨论了广义Mφller变换的微分形式.我们首先考虑一个沿某一方向变速运动的参考系中的引力场。从爱因斯坦场方程出发,求出该引力场的时空度规。通过分析计算,我们寻求到通过精确对钟的方法来确定某一参考系是否是惯性系,并研究出广义Mφller变换的微分形式。
It is well known that, general relativity is the theory of space, time, and gravitation formulated by Einstein in 1915. In this dissertation, based on the Einstein's field equation, the action-angle variables method was used to study the precession of planets, and the general M(?)ller transformation was discussed.
Firstly, we gave an introduction of Riemann curvature tensor and the geodesic equation, which are very important for comprehending general relativity and solving Einstein's field equation. Secondly, canonical transformations, Hamilton-Jacobi theory and action-angle variables were investigated, which made a foundation to study the precession of planets. Thirdly, the fundamental principles and the basic mission of general relativity were introduced, and a brief introduction to a rising direction for general relativity - Numerical Relativity was presented.
We concretely gave a detailed study of the perihelion precession of planets, in nearly-circular orbital and elliptical orbital respectively, with action-angle variables method. And, the explicit expressions of orbital precession were presented. After reviewed the Schwarzschild vacuum solution of Einstein's field equation, the motion equations of a particle in this field were investigated. Based on this set of self-contained equations, we presented the general integral form of action variables and discussed the properties of the angle variables in cases of nearly-circular orbital and elliptical orbital respectively. Then, the orbital precessions of the particle (a planet) in a cycle were come out by some analysis.
On the other hand, we also gave a detailed discussion on the general M(?)ller transformation, which is a transformation from inertial reference frame to arbitrary time-varied reference system. And its differential form was em- phatically discussed. We considered a gravitational field in a reference system moving variably along a certain direction, the metric of which was derived later based on Einstein's equation. After the analysis and calculation, it came out that an inertial reference system can be determined by precisely checking the distantly separated watches, and the differential form of general M(?)ller transformation was presented.
我们首先介绍了测地线方程和黎曼曲率张量,这是我们求解爱因斯坦场方程的数学基础.其次,我们介绍了经典力学中的正则变换、哈密顿-雅可比理论及作用角变量方法,为后面研究行星进动问题提供了方法。在第三章中,我们介绍了广义相对论的基本原理和基本任务,并介绍了广义相对论的一个新的研究方向——数值广义相对论。
我们具体讨论了运用作用角变量方法研究行星分别在近圆和椭圆轨道下的近日点进动值,并给出其进动值的明确表述.在回顾了爱因斯坦场方程的Schwarzschild真空解之后,我们介绍了质点在Schwarzschild场中的运动方程.从这组完备的质点动力学微分方程出发,我们探讨了质点运动相应的作用变量的一般形式。并且,分别讨论了在近圆轨道和椭圆轨道两种情况下,质点运动的作用变量的共轭变量——角变量的性质,从而研究出,经过一个周期后,质点(行星)的轨道进动值。
我们还研究了广义Mφller变换——由惯性参考系到任意变速参考系的一种变换。重点讨论了广义Mφller变换的微分形式.我们首先考虑一个沿某一方向变速运动的参考系中的引力场。从爱因斯坦场方程出发,求出该引力场的时空度规。通过分析计算,我们寻求到通过精确对钟的方法来确定某一参考系是否是惯性系,并研究出广义Mφller变换的微分形式。
It is well known that, general relativity is the theory of space, time, and gravitation formulated by Einstein in 1915. In this dissertation, based on the Einstein's field equation, the action-angle variables method was used to study the precession of planets, and the general M(?)ller transformation was discussed.
Firstly, we gave an introduction of Riemann curvature tensor and the geodesic equation, which are very important for comprehending general relativity and solving Einstein's field equation. Secondly, canonical transformations, Hamilton-Jacobi theory and action-angle variables were investigated, which made a foundation to study the precession of planets. Thirdly, the fundamental principles and the basic mission of general relativity were introduced, and a brief introduction to a rising direction for general relativity - Numerical Relativity was presented.
We concretely gave a detailed study of the perihelion precession of planets, in nearly-circular orbital and elliptical orbital respectively, with action-angle variables method. And, the explicit expressions of orbital precession were presented. After reviewed the Schwarzschild vacuum solution of Einstein's field equation, the motion equations of a particle in this field were investigated. Based on this set of self-contained equations, we presented the general integral form of action variables and discussed the properties of the angle variables in cases of nearly-circular orbital and elliptical orbital respectively. Then, the orbital precessions of the particle (a planet) in a cycle were come out by some analysis.
On the other hand, we also gave a detailed discussion on the general M(?)ller transformation, which is a transformation from inertial reference frame to arbitrary time-varied reference system. And its differential form was em- phatically discussed. We considered a gravitational field in a reference system moving variably along a certain direction, the metric of which was derived later based on Einstein's equation. After the analysis and calculation, it came out that an inertial reference system can be determined by precisely checking the distantly separated watches, and the differential form of general M(?)ller transformation was presented.
引文
[1]王永久,《引力论和宇宙论》,长沙:湖南师范大学出版社,(2004).
[2]须重明 吴雪君,《广义相对论与现代宇宙学》,南京:南京师范大学出版社,(1999).
[3]梁灿彬 周彬,《微分几何入门与广义相对论》(上册.第二版),北京:科学出版社,(2006).
[4]刘辽 赵峥,《广义相对论》(第二版),北京:高等教育出版社,(2004).
[5]俞允强,《广义相对论引论》(第二版),北京:北京大学出版社,(1997).
[6]H.J.Hay,J.P.Schiffer,T.E.Cranshaw,and P.A.Egelstaff,Phys.Rev.Lett.4,165-166(1960).
[7]L.I.Schiff,Phys.Rev.Lett.4,215-217(1960).
[8]Y.Mino,M.Sasaki and T.Tanaka,Phys.Rev.D 55,3457(1997).
[9]W.Hu and A.Cooray,Phys.Rev.D 63,023504(2001).
[10]P.Martinean and R.H.Brandenberger,Phys.Rev.D 72,023507(2005).
[11]http://baike.baidu.com/view/34995.htm
[12]A.Ⅱ.马尔契夫,《理论力学》(第3版),北京:高等教育出版社,(2006).
[13]H.戈德斯坦,《经典力学》(第二版),北京:科学出版社,(1986).
[14]周衍柏,《理论力学教程》(第二版),北京:高等教育出版社,(1986).
[15]W.Dittrich and M.Reuter,Classical and Quantum Dynamics:from Classical Paths to Path Integrals,2nd edn,(Springer-Verlag,Berlin/Heidelberg/New York,1994).
[16]段一士,《广义相对论讲义》,(2006).
[17]刘辽 费保俊 张允中,《狭义相对论》(第二版),北京:科学出版社,(2008).
[18]赵峥,《黑洞与弯曲的时空》,太原:山西科学技术出版社,(2000).
[19]A.Palatini,Rend.Circ.Mat.Palermo.43,203(1919).
[20]H.Stephani,D.Kramer,M.A.H.MacCallum,C.Hoenselaers,E.Herlt,Exact Solutions of Einstein's Field Equations,2nd Edn,(Cambridge University Press,New York,2003).
[21]M.Aleubierre,Introduction to 3+1 Numerical Relativity,(Oxford University Press Inc.,New York,2008).
[22]S.G.Hahn and R.W.Lindquist,Ann.Phys.29,304(1964).
[23]L.Smart,A.(?)ade(?),B.DeWitt and K.R.Eppley,Phys.Rev.D 14,2443(1976).
[24]K.R.Eppley,The numerical evolution of the collision of two black holes.PhD thesis,(Princeton University,Princeton,New Jersey,1975).
[25]K.R.Eppley,Phys.Rev.D 16,1609(1977).
[26]L.Lehner,Class.Quantum Grav.18,R25(2001).
[27]Astronomy and Astrophysics Survey Committee,Astronomy and Astrophysics in the New Millennium,National Academies Press,(2001).
[28]http://www.ligo.caltech.edu/.
[29]http://lisa.nasa.gov/.
[30]T.Cokelaer,Phys.Rev.D 76,102004(2007).
[31]B.Abbott et al.(LIGO Scientific Collaboration),Phys.Rev.D 76,082001(2007).
[32]M.A.Gasparini and F.Dubath,Phys.Rev.D 74,122003(2006).
[33]B.Vaishnav,I.Hinder,F.Herrmann and D.Shoemaker,Phys.Rev.D 76,084020(2007).
[34]F.Pretorius,Phys.Rev.Lett.95,121101(2005).
[35]M.Shibata and K.Ury(?),Phys.Rev.D74,121503(R)(2006);Classical Quantum Gravity 24,S125(2007).
[36]M.Shibata and K.Taniguchi,Phys.Rev.D 77,084015(2008).
[37]Z.B.Etienne,J.A.Faber,Y.T.Liu,S.L.Shapiro,K.Taniguchi,and T.W.Baumgarte,Phys.Rev.D77,084002(2008).
[38]T.Yamamoto,M.Shibata,and K.Taniguchi,Phys.Rev.D 78,064054(2008).
[39]M.D.Duez,F.Fourcart,L.E.Kidder,H.P.Pfeiffer,M.A.Scheel,and S.A.Teukolsky,Phys.Rev.D 78,104015(2008).
[40]M.Shibata,K.Kyutoku,T.Yamamoto,and K.Taniguchi,Phys.Rev.D 79,044030(2009).
[41]Zhoujian Cao,Hwei-Jang Yo and Jui-Ping Yu,Phys.Rev.D 78,124011(2008).
[42]U.J.Leverrier,Ann.Obs.Paris 5,1(1859).
[43]S.Newcomb,Tables of Mercury,Astr.Pap.Am.Ephem.,vol 6,part 2,Washington,(1895-1898).
[44]A.Einstein,Erkl(a｜¨)rung der Perihelbewegung des Merkur aus der allgemeinen Relativit(a｜¨)tstheorie,Sitzungsberichte der Preussischen Akademie der Wissenschaften,pp.831-9(1915).
[45]I.I.Shapiro,W.B.Smith,M.E.Ash,et al.,Astron.J.76,588(1971).
[46]G.S.Adkins and J.McDonnell,Phys.Rev.D 75,082001(2007).
[47]N.Straumann,General Relativity and Relativistic Astrophysics,Springer,Berlin,(1984).
[48]M.Carmeli,Classical Fields:General Relativity and Gauge Theory,John Wiley,New York,(1982).
[49]S.Weinberg,Gravitation and Cosmology:Principles and Applications of the General Theory of Relativity,John Wiley,New York,(1972).
[50]S.Chandrasekhar,The mathematical theory of black holes,Oxford University Press,New York,(1992).
[51]J.W.Moffat,J.Cosmol.Astropart.Phys.,JCAP03(2006)004(2006).
[52]R.Kerner,J.W.van Holten,and R.Colistete,Class.Quant.Grav.18,4725(2001).
[53]T.Damour and G.Sch(a｜¨)fer,I1 Nuovo Cimento B 101,127(1988).
[54]G.V.Kraniotis and S.B.Whitehouse,Class.Quantum Grav.20,4817-4835(2003).
[55]K.Schwarzsehild,(U｜¨)ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,Sitzber.Deut.Akad.Wiss.Berlin,K1.Math.-Phys.Tech.,189-196(1916).
[56]C.W.Misner,K.S.Thorne,and J.A.Wheeler,Gravitation,Freeman,San Francisco(1973).
[57]R.M.Wald,General Relativity,The University of Chicago Press,Chicago,(1984).
[58]S.M.Carroll,Lecture Notes on General Relativity,Preprint gr-qc/9712019,(1997).
[59]A.Einstein,Ann.Phys.Lpz.17,891(1905).
[60]E.A.Desloge,and R.J.Philpott,Am.J.Phys.55,252(1987).
[61]C.MΦller,The Theory of Relativity,Clarendon,Oxford(1952).
[62]H.Lass,Am.J.Phys.31,274(1963).
[63]J.E.Romain,Am.J.Phys.32,279(1964).
[64]L.M.Marsh,Am.J.Phys.33,934(1965).
[65]W.Rindler,Am.J.Phy.34,1174(1976).
[66]W.Rindler,Phys.Rev.119,2082(1960);W.Rindler,Essential Relativity,2nd ed.,Springer,New York,(1977).
[67]C.-G.Huang and H.-Y.Guo,Int.J.Mod.Phys.D 15,1035(2006).
[68]C.-G.Huang,Int.J.Mod.Phys.D 17,1071(2008).
[69]Huang C.-G.and Sun J.-R.Commun.Theor.Phys.49,928(2008).
[70]R.Perrin,Am.J.Phy.47,317(1979).
[71]J.D.Jackson Classical Electrodynamics,John Wiley,New York,(1999).
[2]须重明 吴雪君,《广义相对论与现代宇宙学》,南京:南京师范大学出版社,(1999).
[3]梁灿彬 周彬,《微分几何入门与广义相对论》(上册.第二版),北京:科学出版社,(2006).
[4]刘辽 赵峥,《广义相对论》(第二版),北京:高等教育出版社,(2004).
[5]俞允强,《广义相对论引论》(第二版),北京:北京大学出版社,(1997).
[6]H.J.Hay,J.P.Schiffer,T.E.Cranshaw,and P.A.Egelstaff,Phys.Rev.Lett.4,165-166(1960).
[7]L.I.Schiff,Phys.Rev.Lett.4,215-217(1960).
[8]Y.Mino,M.Sasaki and T.Tanaka,Phys.Rev.D 55,3457(1997).
[9]W.Hu and A.Cooray,Phys.Rev.D 63,023504(2001).
[10]P.Martinean and R.H.Brandenberger,Phys.Rev.D 72,023507(2005).
[11]http://baike.baidu.com/view/34995.htm
[12]A.Ⅱ.马尔契夫,《理论力学》(第3版),北京:高等教育出版社,(2006).
[13]H.戈德斯坦,《经典力学》(第二版),北京:科学出版社,(1986).
[14]周衍柏,《理论力学教程》(第二版),北京:高等教育出版社,(1986).
[15]W.Dittrich and M.Reuter,Classical and Quantum Dynamics:from Classical Paths to Path Integrals,2nd edn,(Springer-Verlag,Berlin/Heidelberg/New York,1994).
[16]段一士,《广义相对论讲义》,(2006).
[17]刘辽 费保俊 张允中,《狭义相对论》(第二版),北京:科学出版社,(2008).
[18]赵峥,《黑洞与弯曲的时空》,太原:山西科学技术出版社,(2000).
[19]A.Palatini,Rend.Circ.Mat.Palermo.43,203(1919).
[20]H.Stephani,D.Kramer,M.A.H.MacCallum,C.Hoenselaers,E.Herlt,Exact Solutions of Einstein's Field Equations,2nd Edn,(Cambridge University Press,New York,2003).
[21]M.Aleubierre,Introduction to 3+1 Numerical Relativity,(Oxford University Press Inc.,New York,2008).
[22]S.G.Hahn and R.W.Lindquist,Ann.Phys.29,304(1964).
[23]L.Smart,A.(?)ade(?),B.DeWitt and K.R.Eppley,Phys.Rev.D 14,2443(1976).
[24]K.R.Eppley,The numerical evolution of the collision of two black holes.PhD thesis,(Princeton University,Princeton,New Jersey,1975).
[25]K.R.Eppley,Phys.Rev.D 16,1609(1977).
[26]L.Lehner,Class.Quantum Grav.18,R25(2001).
[27]Astronomy and Astrophysics Survey Committee,Astronomy and Astrophysics in the New Millennium,National Academies Press,(2001).
[28]http://www.ligo.caltech.edu/.
[29]http://lisa.nasa.gov/.
[30]T.Cokelaer,Phys.Rev.D 76,102004(2007).
[31]B.Abbott et al.(LIGO Scientific Collaboration),Phys.Rev.D 76,082001(2007).
[32]M.A.Gasparini and F.Dubath,Phys.Rev.D 74,122003(2006).
[33]B.Vaishnav,I.Hinder,F.Herrmann and D.Shoemaker,Phys.Rev.D 76,084020(2007).
[34]F.Pretorius,Phys.Rev.Lett.95,121101(2005).
[35]M.Shibata and K.Ury(?),Phys.Rev.D74,121503(R)(2006);Classical Quantum Gravity 24,S125(2007).
[36]M.Shibata and K.Taniguchi,Phys.Rev.D 77,084015(2008).
[37]Z.B.Etienne,J.A.Faber,Y.T.Liu,S.L.Shapiro,K.Taniguchi,and T.W.Baumgarte,Phys.Rev.D77,084002(2008).
[38]T.Yamamoto,M.Shibata,and K.Taniguchi,Phys.Rev.D 78,064054(2008).
[39]M.D.Duez,F.Fourcart,L.E.Kidder,H.P.Pfeiffer,M.A.Scheel,and S.A.Teukolsky,Phys.Rev.D 78,104015(2008).
[40]M.Shibata,K.Kyutoku,T.Yamamoto,and K.Taniguchi,Phys.Rev.D 79,044030(2009).
[41]Zhoujian Cao,Hwei-Jang Yo and Jui-Ping Yu,Phys.Rev.D 78,124011(2008).
[42]U.J.Leverrier,Ann.Obs.Paris 5,1(1859).
[43]S.Newcomb,Tables of Mercury,Astr.Pap.Am.Ephem.,vol 6,part 2,Washington,(1895-1898).
[44]A.Einstein,Erkl(a｜¨)rung der Perihelbewegung des Merkur aus der allgemeinen Relativit(a｜¨)tstheorie,Sitzungsberichte der Preussischen Akademie der Wissenschaften,pp.831-9(1915).
[45]I.I.Shapiro,W.B.Smith,M.E.Ash,et al.,Astron.J.76,588(1971).
[46]G.S.Adkins and J.McDonnell,Phys.Rev.D 75,082001(2007).
[47]N.Straumann,General Relativity and Relativistic Astrophysics,Springer,Berlin,(1984).
[48]M.Carmeli,Classical Fields:General Relativity and Gauge Theory,John Wiley,New York,(1982).
[49]S.Weinberg,Gravitation and Cosmology:Principles and Applications of the General Theory of Relativity,John Wiley,New York,(1972).
[50]S.Chandrasekhar,The mathematical theory of black holes,Oxford University Press,New York,(1992).
[51]J.W.Moffat,J.Cosmol.Astropart.Phys.,JCAP03(2006)004(2006).
[52]R.Kerner,J.W.van Holten,and R.Colistete,Class.Quant.Grav.18,4725(2001).
[53]T.Damour and G.Sch(a｜¨)fer,I1 Nuovo Cimento B 101,127(1988).
[54]G.V.Kraniotis and S.B.Whitehouse,Class.Quantum Grav.20,4817-4835(2003).
[55]K.Schwarzsehild,(U｜¨)ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,Sitzber.Deut.Akad.Wiss.Berlin,K1.Math.-Phys.Tech.,189-196(1916).
[56]C.W.Misner,K.S.Thorne,and J.A.Wheeler,Gravitation,Freeman,San Francisco(1973).
[57]R.M.Wald,General Relativity,The University of Chicago Press,Chicago,(1984).
[58]S.M.Carroll,Lecture Notes on General Relativity,Preprint gr-qc/9712019,(1997).
[59]A.Einstein,Ann.Phys.Lpz.17,891(1905).
[60]E.A.Desloge,and R.J.Philpott,Am.J.Phys.55,252(1987).
[61]C.MΦller,The Theory of Relativity,Clarendon,Oxford(1952).
[62]H.Lass,Am.J.Phys.31,274(1963).
[63]J.E.Romain,Am.J.Phys.32,279(1964).
[64]L.M.Marsh,Am.J.Phys.33,934(1965).
[65]W.Rindler,Am.J.Phy.34,1174(1976).
[66]W.Rindler,Phys.Rev.119,2082(1960);W.Rindler,Essential Relativity,2nd ed.,Springer,New York,(1977).
[67]C.-G.Huang and H.-Y.Guo,Int.J.Mod.Phys.D 15,1035(2006).
[68]C.-G.Huang,Int.J.Mod.Phys.D 17,1071(2008).
[69]Huang C.-G.and Sun J.-R.Commun.Theor.Phys.49,928(2008).
[70]R.Perrin,Am.J.Phy.47,317(1979).
[71]J.D.Jackson Classical Electrodynamics,John Wiley,New York,(1999).