引力N体系统的后牛顿力学
摘要
爱因斯坦提出广义相对论已近一百年,但时至今日相对论引力N体问题仍然是一个远未被深入研究和有待解决的棘手问题。为了进一步认识物质和时空之间的相互作用特性,即给出一个全面而自洽的后牛顿参考系和运动方程理论,同时也为了服务于未来的高精度天体测量和实验,尤其是未来1毫米精度的激光测月,作者以太阳系和地月系统为实例,研究引力N体系统的后牛顿力学。
本文采用标量—张量理论做为引力理论的框架,它相对于广义相对论包含了两个参数化后牛顿参数γ和β。针对太阳系和地月系统,作者使用并拓展了国际天文学联合会2000年有关相对论参考系决议的概念,建立了一系列的后牛顿参考系。假设太阳系是孤立的且时空在无穷远处渐近平坦,可以首先建立太阳系质心参考系。它是一个全局参考系,覆盖整个太阳系的时空,原点位于太阳系质心,空间轴延伸至无穷远。太阳系质心参考系相对于遥远的类星体不存在任何转动,即相对于国际天球参考架保持静止。接着建立的是地月质心参考系。它是一个局部的惯性系,涵盖地月系统,坐标轴延伸到金星和火星的轨道。地月质心参考系是一个动力学上不转动的参考系,即在地月质心参考系中检验粒子的运动方程不包含科里奥利力和惯性离心力。这一建立在引力N系统中某个子系统上的局部参考系拓展了国际天文学联合会2000年相对论参考系决议中建立在某一个天体上的局部参考系的概念。除此之外,作者还建立了其他两个局部参考系：地球质心参考系和月球质心参考系。它们的原点分别位于地球和月球的质心,在动力学上也都是不转动的。相对于其他的局部参考系和国际天球参考架,每一个局部参考系都存在相对论进动。选择动力学无转动局部参考系的理论优点是其数学描述更为简单。在修正了相对论进动之后,每个局部参考系的坐标轴可以和国际天球参考架保持静止。引入这样一个全局参考系和三个局部参考系的目的是为了能够把月球相对于地球的运动和地月质心本身的轨道运动分解开,同时为高精度激光测月实验建立起坐标描述(月球的运动、地球上的观测者和月面上的后向反射镜)和可直接测量量(激光往返原时间隔)之间的联系。作者在所有的参考系中求解了引力场方程,给出了度规张量和标量场的解,其中包含了各个天体的后牛顿多极矩。随后作者推导了参考系之间的后牛顿坐标变换关系,分析了标量一张量引力理论在度规张量和运动方程中所具有的剩余规范自由度,目的是剔除地球和月球运动方程中依赖于坐标的后牛顿效应。
在此基础上,作者还推导了有关的后牛顿运动方程。根据由不同参考系间匹配过程所得到的局部参考系原点在全局参考系中的运动规律,加上局部引力体质心和局部参考系原点时时都重合的这一限制,作者给出了地月质心在太阳系质心参考系下的运动方程(三维坐标加速度),其中的所有项都表达成了引力体局部质量多极矩和自转多极矩的形式。通过把笛卡尔对称无迹外部引力多极矩和质量以及自转多极矩扩充成四维协变形式,作者又给出了地月质心在太阳系质心参考系下运动方程的四维协变形式,并由此证明在一阶后牛顿近似下其世界线与背景引力场中测地线的偏离是由于其自身的高阶多极矩(l≥1)与外部引力场的耦合以及强等效原理破缺所造成的。使用类似的方法,作者还推导了月球和地球在地月质心参考系中的运动方程以及月球相对于地球的运动方程。这一在局部参考系下所给出的相对运动方程可以用于下一代的激光测月实验。
虽然上述所有的结果都是针对太阳系和地月系统而给出的,但它们都可以直接推广到一般的引力N体系统中。
It has been almost a century since Einstein published the general relativity. But the relativistic N-body problem is still an unsolved problem and far from well-studied. Focusing on the solar system and the Earth-Moon system, this thesis tries to establish the post-Newtonian (PN) mechanics of a gravitational N-body system for future high-precision astrometry and experiments, especially the lunar laser ranging (LLR).
A set of PN reference frames are introduced for a comprehensive study of the or-bital dynamics and rotational motion of Moon and Earth by LLR with the precision of 1 millimeter. We employ a framework of a scalar-tensor theory of gravity depending on two parameters,βandγ, of the parameterized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. We assume that the solar system is isolated and space-time is asymptotically fiat at infinity. The primary reference frame covers the entire space-time, has its origin at the solar-system barycenter (SSB) and spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The secondary reference frame has its origin at the Earth-Moon barycenter (EMB). The EMB frame is locally-inertial with its spatial axes spreading out up to the orbits of Venus and Mars, and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames-geocentric (GRF) and selenocentric (SRF)-have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is sub-ject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of their relative motion with respect to each other. Theoretical advantage of the dynamically non-rotating local frames is in a more simple mathemat-ical description. Each local frame can be aligned with the axes of ICRF after applying the matrix of the relativistic precession. The set of one global and three local frames is introduced in order to fully decouple the relative motion of Moon with respect to Earth from the orbital.motion of the Earth-Moon barycenter as well as to connect the coordinate description of the lunar motion, an observer on Earth, and a retro-reflector on Moon to directly measurable quantities such as the proper time and the round-trip laser-light distance. We solve the gravitational field equations and find out the metric tensor and the scalar field in all frames, which description includes the PN definition of the multipole moments of the gravitational field of Earth and Moon. We also de-rive the PN coordinate transformations between the frames and analyze the residual gauge freedom imposed by the scalar-tensor theory of gravity on the metric tensor and equations of motion. The residual gauge freedom is used for removing the spurious, coordinate-dependent PN effects from the equations of motion of Earth and Moon.
Based on the previous reference frames, the equations of motion are derived. With the law of motion of the origin of the EMB frame in the SSB frame given by the matching procedure and the condition that the origin of the EMB frame coincides with the center of mass of the Earth-Moon system at any instant, the equations of the motion of the center of mass of the Earth-Moon system in the SSB frame are obtained. The 3D coordinate accelerations in these equations are expressed by the local multipole moments. Through extension of multipole moments of external gravitational fields, masses and spins from Cartesian symmetry-trace-free tensors to 4D covariant tensors, it is shown that the equations of motion of the center of mass of the Earth-Moon system in the SSB frame can be written in a covariant form and the higher moments (l> 1) of the Earth-Moon system and the violation of the strong equivalence principle cause the world line of its center of mass to deviate from the geodesics in the background gravitational field. The equations of motion of the Earth or the Moon in the EMB frame are given, and then they lead to the equations of motion of the Moon with respect to the Earth, which could be used in the next generation of LLR. All of the results could be easily extended and applied to a gravitational N-body system.
本文采用标量—张量理论做为引力理论的框架,它相对于广义相对论包含了两个参数化后牛顿参数γ和β。针对太阳系和地月系统,作者使用并拓展了国际天文学联合会2000年有关相对论参考系决议的概念,建立了一系列的后牛顿参考系。假设太阳系是孤立的且时空在无穷远处渐近平坦,可以首先建立太阳系质心参考系。它是一个全局参考系,覆盖整个太阳系的时空,原点位于太阳系质心,空间轴延伸至无穷远。太阳系质心参考系相对于遥远的类星体不存在任何转动,即相对于国际天球参考架保持静止。接着建立的是地月质心参考系。它是一个局部的惯性系,涵盖地月系统,坐标轴延伸到金星和火星的轨道。地月质心参考系是一个动力学上不转动的参考系,即在地月质心参考系中检验粒子的运动方程不包含科里奥利力和惯性离心力。这一建立在引力N系统中某个子系统上的局部参考系拓展了国际天文学联合会2000年相对论参考系决议中建立在某一个天体上的局部参考系的概念。除此之外,作者还建立了其他两个局部参考系：地球质心参考系和月球质心参考系。它们的原点分别位于地球和月球的质心,在动力学上也都是不转动的。相对于其他的局部参考系和国际天球参考架,每一个局部参考系都存在相对论进动。选择动力学无转动局部参考系的理论优点是其数学描述更为简单。在修正了相对论进动之后,每个局部参考系的坐标轴可以和国际天球参考架保持静止。引入这样一个全局参考系和三个局部参考系的目的是为了能够把月球相对于地球的运动和地月质心本身的轨道运动分解开,同时为高精度激光测月实验建立起坐标描述(月球的运动、地球上的观测者和月面上的后向反射镜)和可直接测量量(激光往返原时间隔)之间的联系。作者在所有的参考系中求解了引力场方程,给出了度规张量和标量场的解,其中包含了各个天体的后牛顿多极矩。随后作者推导了参考系之间的后牛顿坐标变换关系,分析了标量一张量引力理论在度规张量和运动方程中所具有的剩余规范自由度,目的是剔除地球和月球运动方程中依赖于坐标的后牛顿效应。
在此基础上,作者还推导了有关的后牛顿运动方程。根据由不同参考系间匹配过程所得到的局部参考系原点在全局参考系中的运动规律,加上局部引力体质心和局部参考系原点时时都重合的这一限制,作者给出了地月质心在太阳系质心参考系下的运动方程(三维坐标加速度),其中的所有项都表达成了引力体局部质量多极矩和自转多极矩的形式。通过把笛卡尔对称无迹外部引力多极矩和质量以及自转多极矩扩充成四维协变形式,作者又给出了地月质心在太阳系质心参考系下运动方程的四维协变形式,并由此证明在一阶后牛顿近似下其世界线与背景引力场中测地线的偏离是由于其自身的高阶多极矩(l≥1)与外部引力场的耦合以及强等效原理破缺所造成的。使用类似的方法,作者还推导了月球和地球在地月质心参考系中的运动方程以及月球相对于地球的运动方程。这一在局部参考系下所给出的相对运动方程可以用于下一代的激光测月实验。
虽然上述所有的结果都是针对太阳系和地月系统而给出的,但它们都可以直接推广到一般的引力N体系统中。
It has been almost a century since Einstein published the general relativity. But the relativistic N-body problem is still an unsolved problem and far from well-studied. Focusing on the solar system and the Earth-Moon system, this thesis tries to establish the post-Newtonian (PN) mechanics of a gravitational N-body system for future high-precision astrometry and experiments, especially the lunar laser ranging (LLR).
A set of PN reference frames are introduced for a comprehensive study of the or-bital dynamics and rotational motion of Moon and Earth by LLR with the precision of 1 millimeter. We employ a framework of a scalar-tensor theory of gravity depending on two parameters,βandγ, of the parameterized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. We assume that the solar system is isolated and space-time is asymptotically fiat at infinity. The primary reference frame covers the entire space-time, has its origin at the solar-system barycenter (SSB) and spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The secondary reference frame has its origin at the Earth-Moon barycenter (EMB). The EMB frame is locally-inertial with its spatial axes spreading out up to the orbits of Venus and Mars, and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames-geocentric (GRF) and selenocentric (SRF)-have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is sub-ject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of their relative motion with respect to each other. Theoretical advantage of the dynamically non-rotating local frames is in a more simple mathemat-ical description. Each local frame can be aligned with the axes of ICRF after applying the matrix of the relativistic precession. The set of one global and three local frames is introduced in order to fully decouple the relative motion of Moon with respect to Earth from the orbital.motion of the Earth-Moon barycenter as well as to connect the coordinate description of the lunar motion, an observer on Earth, and a retro-reflector on Moon to directly measurable quantities such as the proper time and the round-trip laser-light distance. We solve the gravitational field equations and find out the metric tensor and the scalar field in all frames, which description includes the PN definition of the multipole moments of the gravitational field of Earth and Moon. We also de-rive the PN coordinate transformations between the frames and analyze the residual gauge freedom imposed by the scalar-tensor theory of gravity on the metric tensor and equations of motion. The residual gauge freedom is used for removing the spurious, coordinate-dependent PN effects from the equations of motion of Earth and Moon.
Based on the previous reference frames, the equations of motion are derived. With the law of motion of the origin of the EMB frame in the SSB frame given by the matching procedure and the condition that the origin of the EMB frame coincides with the center of mass of the Earth-Moon system at any instant, the equations of the motion of the center of mass of the Earth-Moon system in the SSB frame are obtained. The 3D coordinate accelerations in these equations are expressed by the local multipole moments. Through extension of multipole moments of external gravitational fields, masses and spins from Cartesian symmetry-trace-free tensors to 4D covariant tensors, it is shown that the equations of motion of the center of mass of the Earth-Moon system in the SSB frame can be written in a covariant form and the higher moments (l> 1) of the Earth-Moon system and the violation of the strong equivalence principle cause the world line of its center of mass to deviate from the geodesics in the background gravitational field. The equations of motion of the Earth or the Moon in the EMB frame are given, and then they lead to the equations of motion of the Moon with respect to the Earth, which could be used in the next generation of LLR. All of the results could be easily extended and applied to a gravitational N-body system.
引文
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