Astronomical units and constants in the general relativity framework

An attempt is made to analyze the existing system of astronomical constants within the general relativity theory (GRT) framework. The general conclusion is that, to avoid any confusion in the GRT compatible interpretation of units and constants, one should give precisely, with full post-Newtonian accuracy, the expressions of the metric forms describing the astronomical barycentric and geocentric reference systems used, for example, in IERS analysis of observations.Institute of Applied Astronomy, St. Petersburg, 197042, Russia     

The General Method for Fixing the Gauges of Relativistic Astronomical Reference Systems

This article applies a new scheme of the first post-Newtonian theory (Damour et al., 1991–1994) to the problem of gauge in relativistic reference systems. Choosing and fixing gauge are necessary when the precision of time measurement and application needs to reach the 2PN level (10⁻¹⁶ or better). We present a general method for fixing the gauges of both the global and local coordinate systems, and for determining the expressions of gravitational potentials and coordinate transformations. The results relevant are consistent with the newest IAU resolutions, therefore they can be applied to astronomical practice.     

Relativistic Celestial Mechanics on the verge of its 100 year anniversary

As we are now approaching 2015, both the General Relativity Theory (GRT) and the relativistic Celestial Mechanics based on it will soon arrive at their 100 year anniversaries. There is no border between Newtonian and relativistic Celestial Mechanics. The five-decade period of intensive development of Celestial Mechanics in the second half of the 20th century left many interesting techniques and problems uncompleted. This lecture reviews some problems of Newtonian and relativistic Celestial Mechanics worthy of further investigation. Concerning Newtonian mechanics, these problems include general solution of the three-body problem by means of the series of polynomials, construction of the short-term and long-term theories of motion using the fast converging elliptic function expansions, and representation of the rotation of the planets in the form compatible with the General Planetary Theory reducing the problem to the combined secular system for translatory motion and rotation. Relativistic problems considered here include the determination of the main relativistic effects in the motion of a satellite, e.g. the Moon, and in the rotation of the primary planet using the Newtonian theories of motion and rotation combined with the relativistic transformation of the reference systems, the use of the linearized weak-field GRT metric as a basis of relativistic Celestial Mechanics in the post-Newtonian approximation, and the motion of the Solar System bodies at the cosmological background in the framework of the basic cosmological models. The exposition of the chosen relativistic problems is preceded by reminding the basic features of relativistic Celestial Mechanics with discussing some present tendencies concerning the Parametrized Post-Newtonian formalism, International Astronomical Union resolutions, and standardization of the GRT routines.     

Second post-Minkowskian order harmonic metric for a moving Kerr-Newman black hole

We apply the Lorentz boosting method to the Kerr-Newman metric in harmonic coordinates, and obtain the second post-Minkowskian order harmonic metric for a moving Kerr-Newman black hole with an arbitrary constant speed. This metric may be useful for investigating observable relativistic effects due to the motion of the moving source. As an application, the post-Newtonian equations of motion for a particle and a photon in the far field of this black hole are calculated.     

Why Newtonian gravity is reliable in large-scale cosmological simulations

Until now, it has been common to use Newtonian gravity to study the non-linear clustering properties of large-scale structures. Without confirmation from Einstein's theory, however, it has been unclear whether we can rely on the analysis (e.g. near the horizon scale). In this work we will provide confirmation of the use of Newtonian gravity in cosmology, based on the relativistic analysis of weakly non-linear situations to third order in perturbations. We will show that, except for the gravitational-wave contribution, the relativistic zero-pressure fluid equations perturbed to second order in a flat Friedmann background coincide exactly with the Newtonian results. We will also present the pure relativistic correction terms appearing in the third order. The third-order correction terms show that these terms are the linear-order curvature perturbation times the second-order relativistic/Newtonian terms. Thus, the pure general relativistic corrections in the third order are independent of the horizon scale and are small when considering the large-scale structure of the Universe because of the low-level temperature anisotropy of the cosmic microwave background radiation. Since we include the cosmological constant, our results are relevant to currently favoured cosmology. As we prove that the Newtonian hydrodynamic equations are valid in all cosmological scales to second order, and that the third-order correction terms are small, our result has the important practical implication that one can now use the large-scale Newtonian numerical simulation more reliably as the simulation scale approaches and even goes beyond the horizon. In a complementary situation, where the system is weakly relativistic (i.e. far inside the horizon) but fully non-linear, we can employ the post-Newtonian approximation. We also show that in large-scale structures, the post-Newtonian effects are quite small.     

广义相对论框架中的IAU时间尺度和参考系

简要地回顾和介绍了IAU时间尺度和参考系的历史和进展，其主要内容：（1）牛顿时空观和相对论时空观，（2）IAU各种时间尺度的历史演变和相互关系；（3）IAU的天文参考系，有关的最新决议,相对论框架下度规及其规范问题，四维时空中的空间1PN坐标变换，也介绍了一些有关工作，阐明了与IAU最新决议稍有不同的观点，指出目前IAU有关决议可能仍存在的某种程度上的不完善。     

Post-Newtonian satellite orbits

The first post-Newtonian approximation of general relativity is used to account for the motion of solar system bodies and near-Earth objects which are slow moving and produce weak gravitational fields. The \(n\)-body relativistic equations of motion are given by the Einstein-Infeld-Hoffmann equations. For \(n=2\), we investigate the associated dynamics of two-body systems in the first post-Newtonian approximation. By direct integration of the associated planar equations of motion, we deduce a new expression that characterises the orbit of test particles in the first post-Newtonian regime generalising the well-known Binet equation for Newtonian mechanics. The expression so obtained does not appear to have been given in the literature and is consistent with classical orbiting theory in the Newtonian limit. Further, the accuracy of the post-Newtonian Binet equation is numerically verified by comparing secular variations of known expression with the full general relativistic orbit equation.     

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.     

The Time Equation of Light Propagation in XNAV

We discuss in detail the general relativistic effect in the X-ray sourcebased navigation for autonomous position determination program (XNAV). By using the post-Newtonian approximate method of the DSX scheme, we calculate the bending of light and the gravitational time delay under the 1PN metric, as well as the gravitational time delay under the 2PN metric, and finally obtain the high-accuracy time equation of light propagation in XNAV.     

天文常数中的相对论问题

现代天文观测技术的日新月异、广义相对论的１ＰＮ近似方法在天体力学和天体测量中的广泛应用,使得有必要在１ＰＮ框架中严格而细致地重新审查天文常数系统。在相对论框架里,太阳系天体的质量应当定义为ＢＤ质量,它们的相对变化不超过１０＾－１９,可视为守恒量;引力势满足的方程不再是Ｐｏｉｓｓｏｎ方程而与坐标规范的选择有关,引力势也不再能用传统的球谐函数展开。应当选定一种规范,并且以ＢＤ多极矩作为天文常数。黄赤交     

Relativistic and the first sectorial harmonics corrections in the critical inclination

The problem of the critical inclination is treated in the Hamiltonian framework taking into consideration post-Newtonian corrections as well as the main correction term of sectorial harmonics for an earth-like planet. The Hamiltonian is expressed in terms of Delaunay canonical variables. A canonical transformation is applied to eliminate short period terms. A modified critical inclination is obtained due to relativistic and the first sectorial harmonics corrections.     

Geodynamic implications of astrometry

The transition from local horizon and terrestrial BIH-systems to celestial reference frames is well known to be affected by various geodetic parameters such as polar motion (x_p(t), y_p(t)), UT1-TUC (where UT1 is basically dependent on variations in UT0 and t=time), plumb line deflections (, ) of observation stations, global and local tidal deformations etc. Variations of such quantities with (relative) resolution of the order of 0.001 and better, such as VLBI, demand the application of continuous high-precision (world-wide) geodynamic surveys whenever global theories and sufficient models are not available and the introduction of improved local and global models (geophysical and relativistic) is needed in order to match astrometric observations related to different reference frames. Prediction of parameters for immediate transformation from one system of reference into the other is sometimes of interest.The paperreviews recent results of different observations,points out a number of still open and unresolved problems in observations and modeling, anddiscusses related consequences. Conclusions for geodynamics drawn from comparison of observed data with models based on astronomical and geophysical observations give way to new understanding of basic phenomena of relevance for various disciplines.     

Numerical simulation of the Moon's rotation in a rigorous relativistic framework

This paper describes a numerical simulation of the rigid rotation of the Moon in a relativistic framework.Following a resolution passed by the International Astronomical Union(IAU) in 2000,we construct a kinematically non-rotating reference system named the Selenocentric Celestial Reference System(SCRS) and give the time transformation between the Selenocentric Coordinate Time(TCS) and Barycentric Coordinate Time(TCB).The post-Newtonian equations of the Moon's rotation are written in the SCRS,and they are integrated numerically.We calculate the correction to the rotation of the Moon due to total relativistic torque which includes post-Newtonian and gravitomagnetic torques as well as geodetic precession.We find two dominant periods associated with this correction:18.6 yr and 80.1 yr.In addition,the precession of the rotating axes caused by fourth-degree and fifth-degree harmonics of the Moon is also analyzed,and we have found that the main periods of this precession are 27.3 d,2.9 yr,18.6 yr and 80.1 yr.     

The Fermi coordinate system in the post-Newtonian framework

The Fermi coordinate system (or the non-rotating proper reference frame) is studied within the framework of the post-Newtonian approximation of general relativity. First its general functional form is found to consist of three parts; (1) the motion of the space origin of the Fermi coordinate system relative to the background one, (2) the Fermi-Walker tetrad transported along the world line of the space origin, and (3) the spacelike geodesic starting from the space origin. Next, the post-Newtonian expressions of the latter two are obtained under the condition that the first is given. Then the full coordinate transformation formula connecting the Fermi coordinate system to the background one is derived explicitly. The effectiveness of the Fermi coordinate system is discussed and the effective region is found to be a cylinder with the radius of about 0.5 kpc for the Fermi coordinate system comoving with the Earth. The mathematical way to derive the generalized Fermi coordinate system which Ashby and Bertotti defined is also shown.     

On derivation of EIH (Einstein–Infeld–Hoffman) equations of motion from the linearized metric of general relativity theory

The half-century old idea of Infeld to use the variational principle of the general relativity field equations is reminded to show that the commonly employed EIH (Einstein–Infeld–Hoffman) equations of motion may be derived from the linearized (weak-field) metric alone. Based on it, the linearized metric might be sufficient for post-Newtonian celestial mechanics and astrometry enabling one to derive the post-Newtonian equations of motion and rotation of celestial bodies as well as the post-Newtonian equations of light propagation within the general relativity framework.     

二阶后牛顿光线方程

近来相继提出一系列的空间天体测量计划，要求考虑在多参考系中二阶后牛顿部分对光线传播的贡献，也就是说，必须讨论在最近完成的扩展的DSX体系下的二阶后牛顿(2PN)光线方程．DSX体系是在20世纪90年代初建立的，用来讨论对N个任意形状和组成、自转可变形物体的一套完整的一阶后牛顿(1PN)天体力学理论．在此建议采用迭代的方法来推导2PN光线方程．从度规和Christoffel记号出发推导太阳系中的2PN光线方程．当忽略掉更高阶的项时，2PN光线方程将回到在很多教科书中广泛出现的1PN光线方程．利用这套方程就可以计算太阳系的光线传播．     

The Internal and External Metrics of a Rotating Ellipsoid Under Post-Newtonianian Approximation

Under the post-Newtonian approximation, the internal and external metrics of a rigidly rotating oblate spheroid filled by a uniform and incompressible perfect fluid are obtained. And the analytic solutions of post-Newtonian metric components are derived by using the series expansion in an ellipsoidal coordinate system. For this specific problem, there are only finite terms remaining in the series expansion, so the obtained results can be used to study particle motion under these metrics.