Relativistic Celestial Mechanics of the Solar System

Transcription

1 Sergei Kopeikin, Michael Efroimsky, and George Kaplan Relativistic Celestial Mechanics of the Solar System WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA

2 Contents Preface XVII Symbols and Abbreviations References XXXI XXIII 1 Newtonian Celestial Mechanics Prolegomena - Classical Mechanics in a Nutshell Kepler's Laws Fundamental Laws of Motion - from Descartes, Newton, and Leibniz to Poincare and Einstein Newton's Law of Gravity The N-body Problem Gravitational Potential Gravitational Multipoles Equations of Motion The Integrals of Motion The Equations of Relative Motion with Perturbing Potential The Tidal Potential and Force The Reduced Two-Body Problem Integrals of Motion and Kepler's Second Law The Equations of Motion and Kepler's First Law The Mean and Eccentric Anomalies - Kepler's Third Law The Laplace-Runge-Lenz Vector Parameterizations of the Reduced Two-Body Problem A Keplerian Orbit in the Euclidean Space A Keplerian Orbit in the Projective Space The Freedom of Choice of the Anomaly A Perturbed Two-Body Problem Prefatory Notes Variation of Constants - Osculating Conies The Lagrange and Poisson Brackets Equations of Perturbed Motion for Osculating Elements Equations for Osculating Elements in the Euler-Gauss Form The Planetary Equations in the Form of Lagrange 55

3 VI I Contents The Planetary Equations in the Form of Delaunay Marking a Minefield Re-examining the Obvious Why Did Lagrange Impose His Constraint? Can It Be Relaxed? Example-the Gauge Freedom of a Harmonic Oscillator Relaxing the Lagrange Constraint in Celestial Mechanics The Gauge Freedom The Gauge Transformations The Gauge-Invariant Perturbation Equation in Terms of the Disturbing Force The Gauge-Invariant Perturbation Equation in Terms of the Disturbing Function The Delaunay Equations without the Lagrange Constraint Contact Orbital Elements Osculation and Nonosculation in Rotational Dynamics Epilogue to the Chapter 76 References 77 2 Introduction to Special Relativity From Newtonian Mechanics to Special Relativity The Newtonian Spacetime The Newtonian Transformations The Galilean Transformations Form-Invariance of the Newtonian Equations of Motion The Maxwell Equations and the Lorentz Transformations Building the Special Relativity Basic Requirements to a New Theory of Space and Time On the "Single-Postulate" Approach to Special Relativity The Difference in the Interpretation of Special Relativity by Einstein, Poincare and Lorentz From Einstein's Postulates to Minkowski's Spacetime of Events Dimension of the Minkowski Spacetime Homogeneity and Isotropy of the Minkowski Spacetime Coordinates and Reference Frames Spacetime Interval The Null Cone The Proper Time The Proper Distance Causal Relationship Minkowski Spacetime as a Pseudo-Euclidean Vector Space Axioms of Vector Space Dot-Products and Norms Euclidean Space Pseudo-Euclidean Space The Vector Basis 108

4 Contents I VII The Metric Tensor The Lorentz Group General Properties Parametrization of the Lorentz Group The Poincare Group Tensor Algebra Warming up in Three Dimensions - Scalars, Vectors, What Next? Covectors Axioms of Covector Space The Basis in the Covector Space Duality of Covectors and Vectors The Transformation Law of Covectors Bilinear Forms A Tensors A A.I Definition of Tensors as Linear Mappings Transformations of Tensors Under a Change of the Basis Rising and Lowering Indices of Tensors Contraction of Tensor Indices Tensor Equations Kinematics The Proper Frame of Observer Four-Velocity and Four-Acceleration Transformation of Velocity Transformation of Acceleration Dilation of Time Simultaneity and Synchronization of Clocks Contraction of Length Aberration of Light The Doppler Effect Accelerated Frames Worldline of a Uniformly-Accelerated Observer A Tetrad Comoving with a Uniformly-Accelerated Observer The Rindler Coordinates The Radar Coordinates Relativistic Dynamics Linear Momentum and Energy Relativistic Force and Equations of Motion The Relativistic Transformation of the Minkowski Force A The Lorentz Force and Transformation of Electromagnetic Field The Aberration of the Minkowski Force The Center-of-Momentum Frame The Center-of-Mass Frame Energy-Momentum Tensor Noninteracting Particles Perfect Fluid 188

5 VIM I Contents Nonperfect Fluid and Solids Electromagnetic Field Scalar Field 191 References General Relativity The Principle of Equivalence The Inertial and Gravitational Masses The Weak Equivalence Principle The Einstein Equivalence Principle The Strong Equivalence Principle The Mach Principle The Principle of Covariance Lorentz Covariance in Special Relativity Lorentz Covariance in Arbitrary Coordinates Covariant Derivative and the Christoffel Symbols in Special Relativity Relationship Between the Christoffel Symbols and the Metric Tensor Covariant Derivative of the Metric Tensor From Lorentz to General Covariance Two Approaches to Gravitation in General Relativity A Differentiable Manifold Topology of Manifold Local Charts and Atlas Functions Tangent Vectors Tangent Space Covectors and Cotangent Space Tensors The Metric Tensor Operation of Rising and Lowering Indices Magnitude of a Vector and an Angle Between Vectors The Riemann Normal Coordinates Affine Connection on Manifold Axiomatic Definition of the Affine Connection Components of the Connection Covariant Derivative of Tensors Parallel Transport of Tensors Equation of the Parallel Transport Geodesies Transformation Law for Connection Components The Levi-Civita Connection Commutator of Two Vector Fields Torsion Tensor 240

6 Contents IX Nonmetricity Tensor Linking the Connection with the Metric Structure Lie Derivative A Vector Flow The Directional Derivative of a Function Geometric Interpretation of the Commutator of Two Vector Fields Definition of the Lie Derivative Lie Transport of Tensors The Riemann Tensor and Curvature of Manifold Noncommutation of Covariant Derivatives The Dependence of the Parallel Transport on the Path The Holonomy of a Connection A The Riemann Tensor as a Measure of Flatness The Jacobi Equation and the Geodesies Deviation Properties of the Riemann Tensor Algebraic Symmetries The Weyl Tensor and the Ricci Decomposition The Bianchi Identities Mathematical and Physical Foundations of General Relativity General Covariance on Curved Manifolds General Relativity Principle Links Gravity to Geometry The Equations of Motion of Test Particles The Correspondence Principle - the Interaction of Matter and Geometry The Newtonian Gravitational Potential and the Metric Tensor The Newtonian Gravity and the Einstein Field Equations The Principle of the Gauge Invariance Principles of Measurement of Gravitational Field Clocks and Rulers Time Measurements Space Measurements Are Coordinates Measurable? Experimental Testing of General Relativity Variational Principle in General Relativity The Action Functional Variational Equations Variational Equations for Matter Variational Equations for Gravitational Field The Hilbert Action and the Einstein Equations The Hilbert Lagrangian The Einstein Lagrangian The Einstein Tensor The Generalizations of the Hilbert Lagrangian The Noether Theorem and Conserved Currents The Anatomy of the Infinitesimal Variation 316

7 X I Contents Examples of the Gauge Transformations Proof of the Noether Theorem The Metrical Energy-Momentum Tensor Hardcore of the Metrical Energy-Momentum Tensor Gauge Invariance of the Metrical Energy Momentum Tensor Electromagnetic Energy-Momentum Tensor Energy-Momentum Tensor of a Perfect Fluid Energy-Momentum Tensor of a Scalar Field The Canonical Energy-Momentum Tensor Definition Relationship to the Metrical Energy-Momentum Tensor Killing Vectors and the Global Laws of Conservation The Canonical Energy-Momentum Tensor for Electromagnetic Field The Canonical Energy-Momentum Tensor for Perfect Fluid Pseudotensor of Landau and Lifshitz Gravitational Waves The Post-Minkowskian Approximations Multipolar Expansion of a Retarded Potential Multipolar Expansion of Gravitational Field Gravitational Field in Transverse-Traceless Gauge Gravitational Radiation and Detection of Gravitational Waves 352 References Relativistic Reference Frames Historical Background Isolated Astronomical Systems Field Equations in the Scalar-Tensor Theory of Gravity The Energy-Momentum Tensor Basic Principles of the Post-Newtonian Approximations Gauge Conditions and Residual Gauge Freedom The Reduced Field Equations 389 ; 4.3 Global Astronomical Coordinates Dynamic and Kinematic Properties of the Global Coordinates The Metric Tensor and Scalar Field in the Global Coordinates 395 \ 4.4 Gravitational Multipoles in the Global Coordinates General Description of Multipole Moments 396 ] Active Multipole Moments 399 ; Scalar Multipole Moments Conformal Multipole Moments 402 i Post-Newtonian Conservation Laws 404 j 4.5 Local Astronomical Coordinates Dynamic and Kinematic Properties of the Local Coordinates The Metric Tensor and Scalar Field in the Local Coordinates The Scalar Field: Internal and External Solutions 410

8 Contents XI The Metric Tensor: Internal Solution The Metric Tensor: External Solution The Metric Tensor: The Coupling Terms Multipolar Expansion of Gravitational Field in the Local Coordinates 420 References Post-Newtonian Coordinate Transformations The Transformation from the Local to Global Coordinates Preliminaries General Structure of the Coordinate Transformation Transformation of the Coordinate Basis Matching Transformation of the Metric Tensor and Scalar Field Historical Background Method of the Matched Asymptotic Expansions in the PPN Formalism Transformation of Gravitational Potentials from the Local to Global Coordinates Transformation of the Internal Potentials Transformation of the External Potentials Matching for the Scalar Field Matching for the Metric Tensor Matching goo(t, *) and g a p(u, w) in the Newtonian Approximation Matchinggij(t,x) andg a p(u,w) Matching go>(*.*) and g a p(u,w) Matching goo(t, x) and g a p{u, w) in the Post-Newtonian Approximation Final Form of the PPN Coordinate Transformation 457 References Relativistic Celestial Mechanics Post-Newtonian Equations of Orbital Motion Introduction Macroscopic Post-Newtonian Equations of Motion Mass and the Linear Momentum of a Self-Gravitating Body Translational Equation of Motion in the Local Coordinates Orbital Equation of Motion in the Global Coordinates Rotational Equations of Motion of Extended Bodies The Angular Momentum of a Self-Gravitating Body Equations of Rotational Motion in the Local Coordinates Motion of Spherically-Symmetric and Rigidly-Rotating Bodies Definition of a Spherically-Symmetric and Rigidly-Rotating Body Coordinate Transformation of the Multipole Moments Gravitational Multipoles in the Global Coordinates Orbital Post-Newtonian Equations of Motion Rotational Equations of Motion 500

9 XII I Contents 6.4 Post-Newtonian Two-Body Problem Introduction Perturbing Post-Newtonian Force Orbital Solution in the Two-Body Problem Osculating Elements Parametrization The Damour-Deruelle Parametrization The Epstein-Haugan Parametrization The Brumberg Parametrization 512 References Relativistic Astrometry Introduction Gravitational Lienard-Wiechert Potentials Mathematical Technique for Integrating Equations of Propagation of Photons Gravitational Perturbations of Photon's Trajectory Observable Relativistic Effects Gravitational Time Delay Gravitational Bending and the Deflection Angle of Light Gravitational Shift of Electromagnetic-Wave Frequency Applications to Relativistic Astrophysics and Astrometry Gravitational Time Delay in Binary Pulsars Pulsars - Rotating Radio Beacons The Approximation Scheme Post-Newtonian Versus Post-Minkowski Calculations of Time Delay in Binary Systems Time Delay in the Parameterized Post-Keplerian Formalism Moving Gravitational Lenses Gravitational Lens Equation Gravitational Shift of Frequency by Moving Bodies Relativistic Astrometry in the Solar System Near-Zone and Far-Zone Astrometry Pulsar Timing Very Long Baseline Interferometry A Relativistic Space Astrometry Doppler Tracking of Interplanetary Spacecrafts Definition and Calculation of the Doppler Shift The Null Cone Partial Derivatives Doppler Effect in Spacecraft-Planetary Conjunctions The Doppler Effect Revisited The Explicit Doppler Tracking Formula Astrometric Experiments with the Solar System Planets Motivations The Unperturbed Light-Ray Trajectory The Gravitational Field 626

10 Contents XIII The Field Equations The Planet's Gravitational Multipoles The Light-Ray Gravitational Perturbations The Light-Ray Propagation Equation The Null Cone Integration Technique The Speed of Gravity, Causality, and the Principle of Equivalence Light-Ray Deflection Patterns The Deflection Angle Snapshot Patterns Dynamic Patterns of the Light Deflection Testing Relativity and Reference Frames The Monopolar Deflection The Dipolar Deflection The Quadrupolar Deflection 655 References Relativistic Geodesy Introduction Basic Equations Geocentric Reference Frame Topocentric Reference Frame Relationship Between the Geocentric and Topocentric Frames Post-Newtonian Gravimetry Post-Newtonian Gradiometry Relativistic Geoid Definition of a Geoid in the Post-Newtonian Gravity Post-Newtonian u-geoid Post-Newtonian o-geoid Post-Newtonian Level Surface Post-Newtonian Clairaut's Equation 707 References Relativity in IAU Resolutions Introduction Overview of the Resolutions About this Chapter Other Resources Relativity Background The BCRS and the GCRS Computing Observables Other Considerations Time Scales Different Flavors of Time Time Scales Based on the SI Second Time Scales Based on the Rotation of the Earth 733

11 XIV I Contents Coordinated Universal Time (UTC) To Leap or not to Leap Formulas Formulas for Time Scales Based on the SI Second Formulas for Time Scales Based on the Rotation of the Earth The Fundamental Celestial Reference System The ICRS, ICRF, and the HCRF Background: Reference Systems and Reference Frames The Effect of Catalogue Errors on Reference Frames Late Twentieth Century Developments ICRS Implementation The Defining Extragalactic Frame The Frame at Optical Wavelengths Standard Algorithms Relationship to Other Systems Data in the ICRS Formulas Ephemerides of the Major Solar System Bodies The JPL Ephemerides DE Recent Ephemeris Development Sizes, Shapes, and Rotational Data Precession and Nutation Aspects of Earth Rotation Which Pole? The New Models Formulas Formulas for Precession Formulas for Nutation Alternative Combined Transformation Observational Corrections to Precession-Nutation Sample Nutation Terms Modeling the Earth's Rotation A Messy Business Nonrotating Origins The Path of the CIO on the Sky Transforming Vectors Between Reference Systems Formulas Location of Cardinal Points CIO Location Relative to the Equinox CIO Location from Numerical Integration CIO Location from the Arc-Difference s Geodetic Position Vectors and Polar Motion Complete Terrestrial to Celestial Transformation Hour Angle 802 References 805

12 Contents I XV Appendix A Fundamental Solution of the Laplace Equation 813 References 817 Appendix B Astronomical Constants 819 References 823 Appendix C Text of IAU Resolutions 825 C.I Text of IAU Resolutions of 1997 Adopted at the XXIIIrd General Assembly, Kyoto 825 C.2 Text of IAU Resolutions of 2000 Adopted at the XXIVth General Assembly, Manchester 829 C.3 Text of IAU Resolutions of 2006 Adopted at the XXVIth General Assembly, Prague 841 C.4 Text of IAU Resolutions of 2009 Adopted at the XXVIIth General Assembly, Rio de Janeiro 847 Index 851