scholarly journals Coordinate system in general relativity

1988 ◽  
Vol 128 ◽  
pp. 105-106
Author(s):  
Toshio Fukushima
Keyword(s):  
Solar System ◽  
Coordinate System ◽  
Reference Frame ◽  
Coordinate Transformation ◽  
Time Coordinate ◽  
Coordinate Axis ◽  
The Earth ◽  
Natural Coordinate System ◽  
Natural Coordinate ◽  
Barycentric Coordinate

The proper reference frame comoving with a system of mass-points is defined as a general relativistic extension of the relative coordinate system in the Newtonian mechanics. The coordinate transformation connecting this and the background coordinate systems is presented explicitly in the post-Newtonian formalism. The conversion formulas of some physical quantities caused by this coordinate transformation are discussed. The concept of the rotating coordinate system is reexamined within the relativistic framework. A modification of the introduced proper reference frame named the Natural Coordinate System (NCS) is proposed as the basic coordinate system in the astrometry. By means of the concept of the natural coordinate system, the relation between the solar system barycentric coordinate system and the terrestrial coordinate system is given explicitly. To illustrate the concept of NCS, we quote in the following the definition of the non-rotating NCS comoving with the Earth, i.e. the Terrestrial Coordinate System (TCS) (Fukushima et al., 1986a, 1986b):1) Consider a fictitious spacetime with the metric obtained by subtracting the direct terms due to the Earth from the true metric in the solar system Barycentric Coordinate System (BCS).2) The time coordinate axis of the TCS is defined as the worldline of the geocenter, i.e. the timelike geodesic of the geocenter in the above ficititious spacetime.3) The unit of time in the TCS, terrestrial second sT, is defined as the unit of time in the BCS, barycentric second, multiplied by a certain factor so that there exist periodic differences only between the time coordinate of any event in the TCS, i.e. TDT, and the corresponding time coordinate in the BCS, i.e. TDB.4) The space coordinate axes of the TCS are defined as three geometrically straight lines satisfying that they and the time coordinate axis of the TCS are orthogonal to each other at the geocenter in the above fictitious spacetime, and that the coordinate triad constructed by them is symmetric.5) The unit of length in the TCS, terrestrial meter mT, is defined as the length so that c = 299792458 mT/sT.

Self-Adaptation of Natural-Coordinate System

2013 ◽  
Vol 367 ◽  
pp. 286-291
Author(s):  
Ke Wei Zhang ◽  
Yun Qing Zhang
Keyword(s):  
Coordinate System ◽  
Equal Weight ◽  
Natural Coordinates ◽  
Model Quality ◽  
Step Method ◽  
Natural Coordinate System ◽  
Self Adaptation ◽  
Natural Coordinate ◽  
Analysis Platform ◽  
Adaptation Method

A self-adaptation method for natural-coordinate systems is proposed, in order to automate the selection of natural coordinates for each rigid element of a multibody system. The four-step method includes: First, find out all empty positions, which come from the feature points or vectors of the joints attached to the element, and give equal weight to them; second, delete redundant empty positions and add their weight to the unique one; third, select at most four empty positions which have a maximum total weight and can be occupied by a natural-coordinate system at the same time; fourth, the standard natural-coordinate system on the element can adapt itself to the selected empty positions, leading to an actual natural-coordinate system, which contains twelve rational natural coordinates for the element. The implementation of the method has been achieved on a multibody dynamics and motion analysis platform, InteDyna, with the result that modeling efficiency is enhanced and model quality improved.

scholarly journals Post-Newtonian Reference Frames for Advanced Theory of the Lunar Motion and for a New Generation of Lunar Laser Ranging

2010 ◽  
Vol 60 (4) ◽  
Author(s):  
Yi Xie ◽  
Sergei Kopeikin
Keyword(s):  
Solar System ◽  
Reference Frame ◽  
Reference Frames ◽  
Laser Ranging ◽  
Lunar Laser Ranging ◽  
Local Frame ◽  
The Earth ◽  
Lunar Laser ◽  
Lunar Motion ◽  
New Generation

Post-Newtonian Reference Frames for Advanced Theory of the Lunar Motion and for a New Generation of Lunar Laser RangingWe overview a set of post-Newtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of Moon and Earth by means of lunar laser ranging (LLR). We employ a scalar-tensor theory of gravity depending on two post-Newtonian parameters, β and γ, and utilize 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 flat 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 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 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 subject 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 mathematical 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 gravity field equations and find out the metric tensor and the scalar field in all frames which description includes the post-Newtonian multipole moments of the gravitational field of Earth and Moon. We also derive the post-Newtonian coordinate transformations between the frames and analyze the residual gauge freedom.

scholarly journals Symmetry types of the piezoelectric tensor and their identification

2013 ◽  
Vol 469 (2155) ◽  
pp. 20120755 ◽  
Author(s):  
W.-N. Zou ◽  
C.-X. Tang ◽  
E. Pan
Keyword(s):  
Coordinate System ◽  
Mirror Symmetry ◽  
Symmetry Type ◽  
Third Order ◽  
Order Linear ◽  
The Third ◽  
Natural Coordinate System ◽  
Linear Piezoelectricity ◽  
Rotation Transformation ◽  
Natural Coordinate

The third-order linear piezoelectricity tensor seems to be simpler than the fourth-order linear elasticity one, yet its total number of symmetry types is larger than the latter and the exact number is still inconclusive. In this paper, by means of the irreducible decomposition of the linear piezoelectricity tensor and the multipole representation of the corresponding four deviators, we conclude that there are 15 irreducible piezoelectric symmetry types, and thus further establish their characteristic web tree. By virtue of the notion of mirror symmetry and antisymmetry, we define three indicators with respect to two Euler angles and plot them on a unit disk in order to identify the symmetry type of a linear piezoelectricity tensor measured in an arbitrarily oriented coordinate system. Furthermore, an analytic procedure based on the solved axis-direction sets is also proposed to precisely determine the symmetry type of a linear piezoelectricity tensor and to trace the rotation transformation back to its natural coordinate system.

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