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E-raamat: Applied General Relativity: Theory and Applications in Astronomy, Celestial Mechanics and Metrology

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Applied General Relativity: Theory and Applications in Astronomy, Celestial Mechanics and MetrologySuurem pilt
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In the late 20th and beginning 21st century high-precision astronomy, positioning and metrology strongly rely on general relativity. Supported by exercises and solutions this book offers graduate students and researchers entering those fields a self-contained and exhaustive but accessible treatment of applied general relativity. The book is written in a homogenous (graduate level textbook) style allowing the reader to understand the arguments step by step. It first introduces the mathematical and theoretical foundations of gravity theory and then concentrates on its general relativistic applications: clock rates, clock sychronization, establishment of time scales, astronomical references frames, relativistic astrometry, celestial mechanics and metrology. The authors present up-to-date relativistic models for applied techniques such as Satellite LASER Ranging (SLR), Lunar LASER Ranging (LLR), Globale Navigation Satellite Systems (GNSS), Very Large Baseline Interferometry (VLBI), radar measurements, gyroscopes and pulsar timing. A list of acronyms helps the reader keep an overview and a mathematical appendix provides required functions and terms.

1 Introduction
1(14)
1.1 Time and Reference Systems
2(5)
1.2 Space
7(2)
1.3 Astrometry
9(2)
1.4 Celestial Mechanics
11(2)
1.5 Relativistic Astrophysics and Cosmology
13(2)
2 Elements of Differential Geometry
15(32)
2.1 Space-Time Manifold and Fields
15(1)
2.2 Coordinates, Differentials and Tensors
16(3)
2.2.1 Symmetrization and Antisymmetrization
19(1)
2.3 Tensor Algebra
19(1)
2.4 The Lie-Derivative
20(1)
2.5 The Covariant Derivative
21(4)
2.6 Geodesies
25(1)
2.7 Curvature- and Ricci Tensor
26(4)
2.8 The Metric Tensor
30(2)
2.9 Metric Connections
32(6)
2.9.1 Riemann Tensor and Its Symmetries
36(2)
2.10 The Levi-Civita Symbol and Tensor
38(2)
2.11 Symmetric Spaces
40(3)
2.11.1 Maximally Symmetric Spaces
41(1)
2.11.2 Maximally Symmetric 3-Spaces
42(1)
2.12 GRTensor
43(4)
3 Newtonian Celestial Mechanics
47(68)
3.1 Newtonian Theory of Gravity
47(1)
3.2 The Newtonian Space-Time
48(5)
3.2.1 The Galilean Group
48(1)
3.2.2 Weak Equivalence Principle and Newtonian Theory of Gravity
49(4)
3.3 Gravitational Field of a Body
53(20)
3.3.1 Spherical Multipole-Moments
53(6)
3.3.2 Spherical Mass-Moments of an Oblate Spheroid
59(3)
3.3.3 STF-Tensors
62(6)
3.3.4 Cartesian Multipole-Moments
68(5)
3.4 The Tidal Potential
73(4)
3.4.1 Newtonian Tidal Moments
73(2)
3.4.2 The l = 2 Tidal Potential for External Point-Masses
75(2)
3.5 Translational Equations of Motion
77(2)
3.6 Rotational Equations of Motion
79(3)
3.6.1 The Torque Resulting from an External Mass-Monopole
80(2)
3.7 The Newtonian 2-Body Problem
82(18)
3.7.1 Integrals of Motion
82(5)
3.7.2 Orbital Equation; Kepler's First and Third Law
87(2)
3.7.3 Classification of the Conic Sections
89(2)
3.7.4 Kepler's Equation
91(3)
3.7.5 Fourier-Analysis in the Elliptical Orbit
94(1)
3.7.6 The Elliptical Kepler Orbit in Space
95(5)
3.8 Perturbation Theory
100(15)
3.8.1 Variation of Constants
100(1)
3.8.2 Perturbation Equations, Derived from Vectorial Elements
101(14)
4 Relativity
115(42)
4.1 Relativity
115(1)
4.2 Electrodynamics and Special Theory of Relativity
116(4)
4.2.1 Maxwell's Equations
116(4)
4.3 The Minkowskian Metric, Lorentz-Transformation
120(15)
4.3.1 Addition of Velocities
127(1)
4.3.2 Thomas Precession
128(4)
4.3.3 General Coordinate Transformations and a Derivation of the Lorentz-Transformation
132(3)
4.4 The EM-Field of a Moving Point Charge
135(4)
4.5 The Speed of Propagation in Electromagnetism
139(13)
4.5.1 The Vacuum Case
139(6)
4.5.2 Propagation in a Uniform Dielectric Medium
145(7)
4.6 Energy and Momentum
152(5)
5 Einstein's Theory of Gravity
157(28)
5.1 General Relativity
157(1)
5.2 Einstein's Equivalence Principle
158(3)
5.3 The Motion of Test Bodies
161(1)
5.4 Einstein's Theory of Gravity
162(2)
5.5 The Problem of Observables
164(5)
5.5.1 The Ranging Observable
165(1)
5.5.2 The Spectroscopic Observable
165(2)
5.5.3 The Astrometric Observable
167(2)
5.6 Tetrads and Tetrad Induced Coordinates
169(5)
5.7 Proper Reference Systems of Accelerated Observers
174(5)
5.8 The Landau-Lifshitz Formulation of GR
179(6)
5.8.1 The Landau-Lifshitz Field Equations
179(2)
5.8.2 Harmonic Gauge
181(4)
6 Exact Solutions---Field Moments
185(50)
6.1 Minkowskian Space-Time
185(2)
6.2 Stationary Space-Times
187(26)
6.2.1 Stationary Axially Symmetric Space-Times
192(6)
6.2.2 The Hartle-Thorne Metric
198(1)
6.2.3 Static Axially Symmetric Space-Times
199(10)
6.2.4 Spherically Symmetric Space-Time
209(4)
6.3 The Kerr Metric
213(2)
6.3.1 Boyer-Lindquist Coordinates
213(2)
6.4 Cosmologically Relevant Spacetimes
215(9)
6.4.1 The Cosmological Principle
215(2)
6.4.2 Robertson-Walker Metric
217(4)
6.4.3 De Sitter Space
221(2)
6.4.4 Schwarzschild: De Sitter Solution
223(1)
6.5 Field Moments
224(11)
6.5.1 Geroch-Hansen Moments
224(2)
6.5.2 Thome Moments
226(4)
6.5.3 The FHP Theorem
230(5)
7 The Post-Newtonian and MPM Formalisms
235(54)
7.1 The Post-Newtonian Expansion
235(1)
7.2 The General Form of the Metric
236(5)
7.3 Field Equations and the Gauge Problem
241(5)
7.4 The External Post-Newtonian Field of a Body
246(7)
7.5 The Multi-Polar, Post-Minkowskian (MPM) Formalism
253(1)
7.6 Several Expansions
254(2)
7.7 First Post-Minkowskian Approximation
256(16)
7.8 The MPM Algorithm
272(17)
7.8.1 The First PN Approximation
274(9)
7.8.2 The MPM Iteration Scheme
283(6)
8 First Applications of the PN-Formalism
289(48)
8.1 Equipotential Surfaces and Relativistic Geoid
290(3)
8.1.1 Post-Newtonian Equipotential Surfaces
291(2)
8.2 The Problem of Time in the Vicinity of the Earth
293(13)
8.2.1 Synchronization of Nearby Clocks
293(1)
8.2.2 Rates of Clocks in the Earth's Vicinity
294(2)
8.2.3 Synchronization of Clocks in the Vicinity of the Earth
296(1)
8.2.4 Coordinate Time Synchronization
297(1)
8.2.5 The Relation Between Coordinate and Proper Time
298(2)
8.2.6 Clock Comparisons: Practical Aspects
300(4)
8.2.7 TAI, TT and UTC
304(2)
8.3 Barycentric Timescales TCB, Teph, TDB
306(2)
8.4 Fairhead--Bretagnon Series
308(1)
8.5 Light-Rays in the PN-Field of a Single Body
309(9)
8.5.1 The Celestial Sphere
314(1)
8.5.2 The Astrometric Observable
314(2)
8.5.3 The Gravitational Time Delay
316(2)
8.6 The PN Motion of a Torque-Free Gyroscope
318(5)
8.7 Geodesic Motion in the PN-Schwarzschild Field
323(6)
8.8 Celestial Mechanical Perturbation Theory
329(8)
8.8.1 Post-Newtonian Schwarzschild Effects
329(3)
8.8.2 The Lense-Thirring Effect
332(5)
9 Astronomical Reference Systems
337(30)
9.1 The Problem of Celestial Mechanics
337(1)
9.2 Transformation Between Global and Local Systems
338(5)
9.3 Split of Local Potentials, Multipole-Moments
343(5)
9.4 Local Harmonic Proper Coordinates
348(4)
9.5 The Standard xμ → Xα Transformation
352(2)
9.6 The Description of Tidal Forces
354(10)
9.6.1 Post-Newtonian Tidal Moments
354(10)
9.7 BCRS and the Expansion of the Universe
364(3)
10 The Gravitational N-Body Problem
367(34)
10.1 Local Evolution Equations
368(2)
10.2 The Translational Motion
370(10)
10.2.1 The LD-EIH Lagrangian
374(1)
10.2.2 Laws of Motion
375(2)
10.2.3 Equations of Motion
377(3)
10.3 The PN Two-Body Problem
380(12)
10.3.1 The Brumberg Representation
382(2)
10.3.2 The Wagoner-Will Representation
384(3)
10.3.3 The Damour-Deruelle Representation
387(5)
10.4 The Rotational Motion
392(5)
10.4.1 Landau-Lifshitz and Fock Spin
392(2)
10.4.2 The PN-Spin in the N Body Problem
394(3)
10.5 Rigidly Rotating Multipoles
397(4)
10.5.1 Angular Velocity
397(1)
10.5.2 Rigidly Rotating Multipoles
398(3)
11 Light-Rays
401(30)
11.1 Historical Remarks
401(5)
11.2 Light-Rays for 1PN Stationary Multipoles
406(10)
11.2.1 The Shapiro Time Delay
412(2)
11.2.2 The Time Transfer Function
414(1)
11.2.3 The TTF for a Body Slowly Moving with Constant Velocity
415(1)
11.3 Light-Rays to Post-Minkowskian Order
416(7)
11.3.1 The Shapiro Time Delay
421(2)
11.4 The Klioner-Formalism
423(8)
11.4.1 Relativistic Aberration
424(1)
11.4.2 Gravitational Light Deflection
425(1)
11.4.3 Parallax
425(1)
11.4.4 Proper Motion and Radial Velocity
426(5)
12 Metrology
431(66)
12.1 Pulsar Timing
431(12)
12.1.1 Pulsar Timing Arrays
442(1)
12.2 GNSS
443(7)
12.2.1 Global Positioning System
444(4)
12.2.2 GLONASS
448(1)
12.2.3 GALILEO
449(1)
12.2.4 BEIDOU
450(1)
12.3 SLR--LLR
450(12)
12.3.1 Satellite Laser Ranging
450(3)
12.3.2 Lunar Laser Ranging
453(9)
12.4 VLBI
462(16)
12.4.1 The Gravitational Time Delay
465(4)
12.4.2 The Geometrical Delay
469(4)
12.4.3 Radio Sources at Finite Distance
473(5)
12.5 Doppler Measurements
478(2)
12.6 Gyroscopes
480(8)
12.6.1 Passive Sagnac Interferometers
480(8)
12.7 Astrometry
488(9)
12.7.1 Hipparcos
490(3)
12.7.2 The Astrometric Project Gaia
493(4)
13 Appendix
497(20)
13.1 Legendre-Polynomials
497(2)
13.1.1 Ql(x) for x ≥ 1
498(1)
13.2 Relations for STF-Tensors
499(2)
13.3 Differential Geometry: Formulas
501(1)
13.4 Spherically Symmetric Metric
502(1)
13.5 Spherically Symmetric Static Metric
503(1)
13.6 The Kerr Metric: Geometry
504(6)
13.7 Relations Concerning Multipole-Moments
510(4)
13.7.1 Multipole-Moments Derived from -Moments
510(2)
13.7.2 Multipole-Moments Derived from Spherical Weyl-Moments
512(2)
13.8 Weyl-Moments as Functions of Mass Multipole-Moments
514(3)
Bibliography 517(18)
Index 535
Prof. Dr. phil. nat. Michael Soffel is the director of the Lohrmann Observatory of the Technical University Dresden. He is an internationally known expert in relativistic celestial mechanics, relativistic astronomy and geodesy and experimental gravity research. He is the author of the books Relativity in Astrometry, Celestial Mechanics and Geodesy (Springer, 1989) and Space-Time Reference Systems. Prof. Wen-Biao Han, who receicved his Ph.D. in 2009, is full professor at Shanghai Astronomical Observatory, Chinese Academy of Sciences. His research area focuses on relativistic fundamental astronomy, numerical simulations of gravity waves and low-frequency gravitational astronomy. He is author of many scientific papers and principle investigator of the group 'Gravitational waves and relativistic fundamental astronomy' at Shanghai Astronomical observatory.
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