Relativistic Celestial Mechanics of the Solar System

• Main Authors: Kopeikin, Sergei, Efroimsky, Michael, Kaplan, George
• Format: eBook Book
• Language: English
• Published: Weinheim Wiley-VCH 2011; WILEY; John Wiley & Sons, Incorporated
• Edition: 1. Aufl.
• Subjects: Astronomy; Celestial mechanics; Relativity (Physics); Science; Technik; Wissen Astronomie

This authoritative book presents the theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations. In so doing, it fills the need for a textbook that teaches modern dynamical astronomy with a strong emphasis...

Table of Contents:

• Relativistic celestial mechanics of the solar system -- Related titles -- Contents -- Preface -- Symbols and abbreviations -- 1. Newtonian celestial mechanics -- 2. Introduction to special relativity -- 3. General relativity -- 4. Relativistic reference frames -- 5. Post-newtonian coordinate transformations -- 6. Relativistic celestial mechanics -- 7. Relativistic astrometry -- 8. Relativistic geodesy -- 9. Relativity in IAU resolutions -- Appendix a. Fundamental solution of the laplace equation -- Appendix b. Astronomical constants -- Appendix c. Text of IAU resolutions -- Index.
• 6.2.2 Equations of Rotational Motion in the Local Coordinates
• Relativistic Celestial Mechanics of the Solar System -- Contents -- Preface -- Symbols and Abbreviations -- References -- 1 Newtonian Celestial Mechanics -- 1.1 Prolegomena - Classical Mechanics in a Nutshell -- 1.1.1 Kepler's Laws -- 1.1.2 Fundamental Laws of Motion - from Descartes, Newton, and Leibniz to Poincaré and Einstein -- 1.1.3 Newton's Law of Gravity -- 1.2 The N-body Problem -- 1.2.1 Gravitational Potential -- 1.2.2 Gravitational Multipoles -- 1.2.3 Equations of Motion -- 1.2.4 The Integrals of Motion -- 1.2.5 The Equations of Relative Motion with Perturbing Potential -- 1.2.6 The Tidal Potential and Force -- 1.3 The Reduced Two-Body Problem -- 1.3.1 Integrals of Motion and Kepler's Second Law -- 1.3.2 The Equations of Motion and Kepler's First Law -- 1.3.3 The Mean and Eccentric Anomalies - Kepler's Third Law -- 1.3.4 The Laplace-Runge-Lenz Vector -- 1.3.5 Parameterizations of the Reduced Two-Body Problem -- 1.3.6 The Freedom of Choice of the Anomaly -- 1.4 A Perturbed Two-Body Problem -- 1.4.1 Prefatory Notes -- 1.4.2 Variation of Constants - Osculating Conics -- 1.4.3 The Lagrange and Poisson Brackets -- 1.4.4 Equations of Perturbed Motion for Osculating Elements -- 1.4.5 Equations for Osculating Elements in the Euler-Gauss Form -- 1.4.6 The Planetary Equations in the Form of Lagrange -- 1.4.7 The Planetary Equations in the Form of Delaunay -- 1.4.8 Marking a Minefield -- 1.5 Re-examining the Obvious -- 1.5.1 Why Did Lagrange Impose His Constraint? Can It Be Relaxed? -- 1.5.2 Example - the Gauge Freedom of a Harmonic Oscillator -- 1.5.3 Relaxing the Lagrange Constraint in Celestial Mechanics -- 1.5.4 The Gauge-Invariant Perturbation Equation in Terms of the Disturbing Force -- 1.5.5 The Gauge-Invariant Perturbation Equation in Terms of the Disturbing Function -- 1.5.6 The Delaunay Equations without the Lagrange Constraint
• 3.6.5 Lie Transport of Tensors -- 3.7 The Riemann Tensor and Curvature of Manifold -- 3.7.1 Noncommutation of Covariant Derivatives -- 3.7.2 The Dependence of the Parallel Transport on the Path -- 3.7.3 The Holonomy of a Connection -- 3.7.4 The Riemann Tensor as a Measure of Flatness -- 3.7.5 The Jacobi Equation and the Geodesics Deviation -- 3.7.6 Properties of the Riemann Tensor -- 3.8 Mathematical and Physical Foundations of General Relativity -- 3.8.1 General Covariance on Curved Manifolds -- 3.8.2 General Relativity Principle Links Gravity to Geometry -- 3.8.3 The Equations of Motion of Test Particles -- 3.8.4 The Correspondence Principle - the Interaction of Matter and Geometry -- 3.8.5 The Principle of the Gauge Invariance -- 3.8.6 Principles of Measurement of Gravitational Field -- 3.8.7 Experimental Testing of General Relativity -- 3.9 Variational Principle in General Relativity -- 3.9.1 The Action Functional -- 3.9.2 Variational Equations -- 3.9.3 The Hilbert Action and the Einstein Equations -- 3.9.4 The Noether Theorem and Conserved Currents -- 3.9.5 The Metrical Energy-Momentum Tensor -- 3.9.6 The Canonical Energy-Momentum Tensor -- 3.9.7 Pseudotensor of Landau and Lifshitz -- 3.10 Gravitational Waves -- 3.10.1 The Post-Minkowskian Approximations -- 3.10.2 Multipolar Expansion of a Retarded Potential -- 3.10.3 Multipolar Expansion of Gravitational Field -- 3.10.4 Gravitational Field in Transverse-Traceless Gauge -- 3.10.5 Gravitational Radiation and Detection of Gravitational Waves -- References -- 4 Relativistic Reference Frames -- 4.1 Historical Background -- 4.2 Isolated Astronomical Systems -- 4.2.1 Field Equations in the Scalar-Tensor Theory of Gravity -- 4.2.2 The Energy-Momentum Tensor -- 4.2.3 Basic Principles of the Post-Newtonian Approximations -- 4.2.4 Gauge Conditions and Residual Gauge Freedom
• 4.2.5 The Reduced Field Equations -- 4.3 Global Astronomical Coordinates -- 4.3.1 Dynamic and Kinematic Properties of the Global Coordinates -- 4.3.2 The Metric Tensor and Scalar Field in the Global Coordinates -- 4.4 Gravitational Multipoles in the Global Coordinates -- 4.4.1 General Description of Multipole Moments -- 4.4.2 Active Multipole Moments -- 4.4.3 Scalar Multipole Moments -- 4.4.4 Conformal Multipole Moments -- 4.4.5 Post-Newtonian Conservation Laws -- 4.5 Local Astronomical Coordinates -- 4.5.1 Dynamic and Kinematic Properties of the Local Coordinates -- 4.5.2 The Metric Tensor and Scalar Field in the Local Coordinates -- 4.5.3 Multipolar Expansion of Gravitational Field in the Local Coordinates -- References -- 5 Post-Newtonian Coordinate Transformations -- 5.1 The Transformation from the Local to Global Coordinates -- 5.1.1 Preliminaries -- 5.1.2 General Structure of the Coordinate Transformation -- 5.1.3 Transformation of the Coordinate Basis -- 5.2 Matching Transformation of the Metric Tensor and Scalar Field -- 5.2.1 Historical Background -- 5.2.2 Method of the Matched Asymptotic Expansions in the PPN Formalism -- 5.2.3 Transformation of Gravitational Potentials from the Local to Global Coordinates -- 5.2.4 Matching for the Scalar Field -- 5.2.5 Matching for the Metric Tensor -- 5.2.6 Final Form of the PPN Coordinate Transformation -- References -- 6 Relativistic Celestial Mechanics -- 6.1 Post-Newtonian Equations of Orbital Motion -- 6.1.1 Introduction -- 6.1.2 Macroscopic Post-Newtonian Equations of Motion -- 6.1.3 Mass and the Linear Momentum of a Self-Gravitating Body -- 6.1.4 Translational Equation of Motion in the Local Coordinates -- 6.1.5 Orbital Equation of Motion in the Global Coordinates -- 6.2 Rotational Equations of Motion of Extended Bodies -- 6.2.1 The Angular Momentum of a Self-Gravitating Body
• 2.7.3 The Relativistic Transformation of the Minkowski Force -- 2.7.4 The Lorentz Force and Transformation of Electromagnetic Field -- 2.7.5 The Aberration of the Minkowski Force -- 2.7.6 The Center-of-Momentum Frame -- 2.7.7 The Center-of-Mass Frame -- 2.8 Energy-Momentum Tensor -- 2.8.1 Noninteracting Particles -- 2.8.2 Perfect Fluid -- 2.8.3 Nonperfect Fluid and Solids -- 2.8.4 Electromagnetic Field -- 2.8.5 Scalar Field -- References -- 3 General Relativity -- 3.1 The Principle of Equivalence -- 3.1.1 The Inertial and Gravitational Masses -- 3.1.2 The Weak Equivalence Principle -- 3.1.3 The Einstein Equivalence Principle -- 3.1.4 The Strong Equivalence Principle -- 3.1.5 The Mach Principle -- 3.2 The Principle of Covariance -- 3.2.1 Lorentz Covariance in Special Relativity -- 3.2.2 Lorentz Covariance in Arbitrary Coordinates -- 3.2.3 From Lorentz to General Covariance -- 3.2.4 Two Approaches to Gravitation in General Relativity -- 3.3 A Differentiable Manifold -- 3.3.1 Topology of Manifold -- 3.3.2 Local Charts and Atlas -- 3.3.3 Functions -- 3.3.4 Tangent Vectors -- 3.3.5 Tangent Space -- 3.3.6 Covectors and Cotangent Space -- 3.3.7 Tensors -- 3.3.8 The Metric Tensor -- 3.4 Affine Connection on Manifold -- 3.4.1 Axiomatic Definition of the Affine Connection -- 3.4.2 Components of the Connection -- 3.4.3 Covariant Derivative of Tensors -- 3.4.4 Parallel Transport of Tensors -- 3.4.5 Transformation Law for Connection Components -- 3.5 The Levi-Civita Connection -- 3.5.1 Commutator of Two Vector Fields -- 3.5.2 Torsion Tensor -- 3.5.3 Nonmetricity Tensor -- 3.5.4 Linking the Connection with the Metric Structure -- 3.6 Lie Derivative -- 3.6.1 A Vector Flow -- 3.6.2 The Directional Derivative of a Function -- 3.6.3 Geometric Interpretation of the Commutator of Two Vector Fields -- 3.6.4 Definition of the Lie Derivative
• 1.5.7 Contact Orbital Elements -- 1.5.8 Osculation and Nonosculation in Rotational Dynamics -- 1.6 Epilogue to the Chapter -- References -- 2 Introduction to Special Relativity -- 2.1 From Newtonian Mechanics to Special Relativity -- 2.1.1 The Newtonian Spacetime -- 2.1.2 The Newtonian Transformations -- 2.1.3 The Galilean Transformations -- 2.1.4 Form-Invariance of the Newtonian Equations of Motion -- 2.1.5 The Maxwell Equations and the Lorentz Transformations -- 2.2 Building the Special Relativity -- 2.2.1 Basic Requirements to a New Theory of Space and Time -- 2.2.2 On the "Single-Postulate" Approach to Special Relativity -- 2.2.3 The Difference in the Interpretation of Special Relativity by Einstein, Poincaré and Lorentz -- 2.2.4 From Einstein's Postulates to Minkowski's Spacetime of Events -- 2.3 Minkowski Spacetime as a Pseudo-Euclidean Vector Space -- 2.3.1 Axioms of Vector Space -- 2.3.2 Dot-Products and Norms -- 2.3.3 The Vector Basis -- 2.3.4 The Metric Tensor -- 2.3.5 The Lorentz Group -- 2.3.6 The Poincaré Group -- 2.4 Tensor Algebra -- 2.4.1 Warming up in Three Dimensions - Scalars, Vectors, What Next? -- 2.4.2 Covectors -- 2.4.3 Bilinear Forms -- 2.4.4 Tensors -- 2.5 Kinematics -- 2.5.1 The Proper Frame of Observer -- 2.5.2 Four-Velocity and Four-Acceleration -- 2.5.3 Transformation of Velocity -- 2.5.4 Transformation of Acceleration -- 2.5.5 Dilation of Time -- 2.5.6 Simultaneity and Synchronization of Clocks -- 2.5.7 Contraction of Length -- 2.5.8 Aberration of Light -- 2.5.9 The Doppler Effect -- 2.6 Accelerated Frames -- 2.6.1 Worldline of a Uniformly-Accelerated Observer -- 2.6.2 A Tetrad Comoving with a Uniformly-Accelerated Observer -- 2.6.3 The Rindler Coordinates -- 2.6.4 The Radar Coordinates -- 2.7 Relativistic Dynamics -- 2.7.1 Linear Momentum and Energy -- 2.7.2 Relativistic Force and Equations of Motion