广义相对论和Brans-Dicke引力理论下的行星进动与星光偏折
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摘要
作为两种重要的候选引力理论,广义相对论以及Brans-Dicke引力理论对于理解天体及宇宙的形成与演化具有重要的意义.本文在广义相对论和Brans-Dicke引力理论下,考察恒星附近的行星进动以及星光偏折效应.首先推导出恒星周围的静态球对称背景时空下粒子轨道方程的普适形式;然后分别针对两种引力理论,进一步利用相应的真空场方程的静态球对称精确解推导出描述行星及星光轨道方程,其为非线性的二阶常微分方程;利用微扰法求解轨道方程,得到了含高阶修正的近似解,进而给出了对应的行星进动角和星光偏折角.本文对利用更高精度的实验观测来检验、甄别引力理论具有重要的意义.
As two important alternative theories of gravitation,general relativity and Brans-Dicke's theory,play important roles in our understanding the formation and evolution of celestial bodies and cosmology. In this paper,we study the planet precession and light deflection in general relativity and Brans-Dicke's theory. Firstly,we derive the general orbital equation for a free particle moving in the static,spherically symmetric spacetime by a spherical star. Then,we write down the corresponding equations,which are nonlinear ordinary differential equations of motion for the planet and light ray by using the spherically symmetric solutions to these vacuum field equations in these two theories. Finally,the approximation solutions to the equations of motion can be obtained by the perturbation method,which yield the angles corresponding to the planet precession and light deflection. These results presented in this paper will help us to check and distinguish these two theories of gravity using high-precision experimental observation.
As two important alternative theories of gravitation,general relativity and Brans-Dicke's theory,play important roles in our understanding the formation and evolution of celestial bodies and cosmology. In this paper,we study the planet precession and light deflection in general relativity and Brans-Dicke's theory. Firstly,we derive the general orbital equation for a free particle moving in the static,spherically symmetric spacetime by a spherical star. Then,we write down the corresponding equations,which are nonlinear ordinary differential equations of motion for the planet and light ray by using the spherically symmetric solutions to these vacuum field equations in these two theories. Finally,the approximation solutions to the equations of motion can be obtained by the perturbation method,which yield the angles corresponding to the planet precession and light deflection. These results presented in this paper will help us to check and distinguish these two theories of gravity using high-precision experimental observation.
引文
[1]梁灿彬,周彬.微分几何入门与广义相对论(上册)[M].北京:科学出版社,2006.
[2]刘辽,赵峥.广义相对论[M].北京:高等教育出版社,2010.
[3]赵峥,刘文彪.广义相对论基础[M].北京:清华大学出版社,2010.
[4]Weinberg S.Gravitation and Cosmology[M].A WileyInterscience publication,1972.
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[6]LIGO Scientific Collaboration,Virgo Collaboration.Observation of gravitational waves from a binary black hole merger[J].Physical Review Letters,2016,116:061102.
[7]Brans C,Dicke R H.Mach's principle and a relativistic theory of gravitation[J].Physical Review,1961,124:925.
[8]Schwarz J H.Superstring theory[J].Physics Reports,1982,89:223.
[9]Bertolami O,Martins P J.Nonminimal coupling and quintessence[J].Physical Review D,2000,61:064007.
[10]Qiang L E,Ma Y G,Han M X,et al.5-dimensional Brans-Dicke theory and cosmic acceleration[J].Physical Review D,2005,71:061501.
[11]Bisabr Y.Cosmic acceleration in Brans-Dicke cosmology[J].General Relativity and Gravitation,2012,44:427.
[12]Virbhadra,K S,Narasimha D,Chitre S M.Role of the scalar field in gravitational lensing[J].Astronomy and Astrophysics,1998,337:1.
[13]Bodenner J,Will C M.Deflection of light to second order:A tool for illustrating principles of general relativity[J].American Journal of Physics,2003,71:770.
[14]Keeton C R,Petters A O.Formalism for testing theories of gravity using lensing by compact objects:Static,spherically symmetric case[J].Physical Review D,2000,72:104006.
[15]Rodriguez C,Marin C A.Higher-order corrections for the deflection of light around a massive object.2017,ar Xiv:1701.04434.
[16]Hayfeh A H.Perturbation Methods[M].A Wiley-Interscience publication,1973.
[17]Brans C H.Mach's principle and a relativistic theory of gravitation.II[J].Physical Review,1962,125:2194.
[18]Bhadra A,Sarkar K.On static spherically symmetric solutions of the vacuum Brans-Dicke theory[J].General Relativity and Gravitation,2005,37:2189.
[19]Bertotti B,Iess L.A test of general relativity using radio links with the Cassini spacecraft[J].Nature,2003,425:374.
[20]数据来源于http://pdg.lbl.gov/2014/reviews/rpp2014-rev-astrophysical-constants.pdf.
[21]Barricelli N A.Preferential perihelion and aphelion distances,and planetary formation[J].Earth,Moon,and Planets,1986,34:1.
[2]刘辽,赵峥.广义相对论[M].北京:高等教育出版社,2010.
[3]赵峥,刘文彪.广义相对论基础[M].北京:清华大学出版社,2010.
[4]Weinberg S.Gravitation and Cosmology[M].A WileyInterscience publication,1972.
[5]Wald R M.General Relativity[M].The University of Chicago Press,1984.
[6]LIGO Scientific Collaboration,Virgo Collaboration.Observation of gravitational waves from a binary black hole merger[J].Physical Review Letters,2016,116:061102.
[7]Brans C,Dicke R H.Mach's principle and a relativistic theory of gravitation[J].Physical Review,1961,124:925.
[8]Schwarz J H.Superstring theory[J].Physics Reports,1982,89:223.
[9]Bertolami O,Martins P J.Nonminimal coupling and quintessence[J].Physical Review D,2000,61:064007.
[10]Qiang L E,Ma Y G,Han M X,et al.5-dimensional Brans-Dicke theory and cosmic acceleration[J].Physical Review D,2005,71:061501.
[11]Bisabr Y.Cosmic acceleration in Brans-Dicke cosmology[J].General Relativity and Gravitation,2012,44:427.
[12]Virbhadra,K S,Narasimha D,Chitre S M.Role of the scalar field in gravitational lensing[J].Astronomy and Astrophysics,1998,337:1.
[13]Bodenner J,Will C M.Deflection of light to second order:A tool for illustrating principles of general relativity[J].American Journal of Physics,2003,71:770.
[14]Keeton C R,Petters A O.Formalism for testing theories of gravity using lensing by compact objects:Static,spherically symmetric case[J].Physical Review D,2000,72:104006.
[15]Rodriguez C,Marin C A.Higher-order corrections for the deflection of light around a massive object.2017,ar Xiv:1701.04434.
[16]Hayfeh A H.Perturbation Methods[M].A Wiley-Interscience publication,1973.
[17]Brans C H.Mach's principle and a relativistic theory of gravitation.II[J].Physical Review,1962,125:2194.
[18]Bhadra A,Sarkar K.On static spherically symmetric solutions of the vacuum Brans-Dicke theory[J].General Relativity and Gravitation,2005,37:2189.
[19]Bertotti B,Iess L.A test of general relativity using radio links with the Cassini spacecraft[J].Nature,2003,425:374.
[20]数据来源于http://pdg.lbl.gov/2014/reviews/rpp2014-rev-astrophysical-constants.pdf.
[21]Barricelli N A.Preferential perihelion and aphelion distances,and planetary formation[J].Earth,Moon,and Planets,1986,34:1.