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Index

.0, 423–428, 444

Aberration, 168Absolute time, 47, 48Accelerated observers, 174Accuracy of clocks, 2Addition of velocities, 128ADM mass, 251Advanced PN solution, 244Affine connection, 23, 24Affine parameter, 25ALGOS, 304Ampere’s law, 117Anomalous perihelion precession, 1Astrometric observable, 167Astrometry, 488Asymptotic flatness, 224, 254Atomic clock, 446, 448

Baryon number density, 154Basic angle, 493BCRS, 8BCRS - GCRS transformation, 352BD-moments, 247BEIDOU, 450Bianchi identities, 163

first, 36second, 29

Birkhoff’s Theorem, 210Body moments, 224Body sphere, 254Boyer-Lindquist coordinates, 213Brumberg representation, 382

Canonical gauge, 270CAS, see Computer Algebra System (CAS)Causality, 135Cesium clock, 449Chandler wobble, 484Charge-current, 116Christoffel-symbols, 33, 161, 238Climate changes, 5Clock rates, 293Clock synchronization, 293, 296Computer Algebra System (CAS), 43Conic sections, 89Constant redshift surface, 293Continuity equation, 116Contraction of indices, 19Contravariant vector, 18Coordinate time synchronization, 297Coordinate transformation, 17Cosmological principle, 215Coulomb potential, 117Coulomb’s law, 117Coupling constant, 163Covariance, 15Covariant derivative, 21, 25Covariant vector, 18Curvature scalar, 37Curvature tensor, 26

D’Alembert condition, 375D’Alembert’s principle, 78Damour-Deruelle representation, 387Damour-Iyer moments, 272Damour-Iyer Theorem, 267Delta function, 135

© Springer Nature Switzerland AG 2019M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and AstrophysicsLibrary, https://doi.org/10.1007/978-3-030-19673-8

535

536 Index

De Sitter effect, 2De Sitter metric, 221Differentials, 17DI-mass moments, 270Dispersion measure, 432DI-spin moments, 270Dopller-effect, 167Doppler measurements, 478DORIS, 7DSX framework, vi

Earth orientation parameters (EOP), 8, 459,460

Ec, 3, 5Eccentric anomaly, 91Echelle Atomique Libre, 6EEP, see Einstein’s equivalence principle

(EEP)EIH-equations of motion, 373EIH-Lagrangian, 374Einstein’s equivalence principle (EEP), 158,

159Einstein’s summation convention, 18Einstein tensor, 37, 162Electric field, 116Electric permittivity, 117Electromagnetic field tensor, 117Electromagnetic potential, 119El Niño, 8Energy-momentum tensor, 152, 163Entropy, 155EOP, see Earth orientation parameters (EOP)Ephemeris, 306–308Equations of motion, 377Equipotential surface, 291Equivalence principle, 15Erez-Rosen metric, 204Ernst potential ξ , 191Euclidean space, 16Euler-Lagrange equations, 375Euler’s equation of hydrodynamics, 155Expansion of universe, 364

Fairhead–Bretagnon series, 308Faraday’s law, 117Far zone, 255Fermi-derivative, 172Fiber optic gyroscope, 480Field equations, 162Field equations (static case), 190Field moments, 224Fixed-star oriented tetrad, 321

4-acceleration, 125Frame dragging, 290Front velocity, 145

Gaia, 9, 489, 493Galilean group, 48GALILEO, 449Gauge freedom, 164General Relativity, 1Geodesic, 25

deviation equation, 52precession, 2, 322

Geoid, 6Geopotential, 292Geroch-Hansen moments, 224GLONASS, 300, 448GNSS, 300, 443, 448GNSS system time, 300GOCE, 5Gothic metric, 179GPS, 3, 300, 444GPS, differential, 448Gravitational light deflection, 309Gravitational N-body problem, 367Gravitational redshift, 158, 159Gravitational source term, 183Gravitational time delay, 316GREM, 9Group-velocity, 150GRTensor, 43Gyroscopes, 480

Hansen potentials, 225Harmonic coordinates, 210Harmonic gauge, 245Hartle-Thorne metric, 198Hertzian dipole, 141Hipparcos, 9, 489, 490Hydrogen maser clock, 449, 456, 458Hypersurface orthogonal, 199

ICRF, 8ICRS, 8IERS, 7Interstellar medium, 431Irreducible moments, 264Isochronometric surface, 290, 291ITRS, 8

Jacobi-matrix, 18Julian Date, 6

Index 537

Keplerian two-body problem, 48Kepler orbit in space, 95Kepler’s equation, 91Kepler’s laws, 87Kerr metric, 197, 213Killing equation, 40, 185Killing vector field (KVF), 40, 186Klioner-formalism, 423Kretschmann scalar, 38

Landau-Lifshitz complex, 181Landau-Lifshitz field equations, 179Landau-Lifshitz pseudotensor, 180Laplace equation, 200Law of Galileo, 50Legendre polynomials, 54Lense-Thirring orbital precession, 332Lense-Thirring precession, 322Levi-Civita symbol, 38Levi-Civity tensor, 40Lie-derivative, 20Liénard-Wiechert potentials, 136Light-rays

1PN multipoles, 406post-Minkowski, 416

LLR, see Lunar laser ranging (LLR)Local evolution equations, 368Local harmonic proper coordinates, 348LOD-variations, 458, 483, 488Lorentz-boost, 186Lorentz-gauge, 119Lorentz-transformation, 125Lovelock Theorem, 162Lunar laser ranging (LLR), 7, 165, 453

Magnetic field, 116Manifold, 15Mass multipole-moments, 58Mass-moments, 224Matching region, 273Maximally symmetric spaces, 41Maxwell equations, 116Maxwell stress tensor, 154Mean anomaly, 93Metric connections, 32Metric potentials, 237Metric tensor, 30Metric theory, 160Minkowski metric, 125Minkowski space-time, 185Mixed PN solution, 244Motion of a gyroscope, 318

MPM algorithm, 272MPM formalism, vi, 253, 255M-Q-S metric, 206Multipole moments

Cartesian, 68post-Newtonian, 246spherical, 53

Near zone, 255Neutron star, 431Newtonian curvature tensor, 51Newtonian EOM

rotational, 79translational, 78

Newtonian Ricci-tensor, 53Newtonian tidal moments, 73Newtonian tidal potential, 73Newtonian torque, 80Newtonian 2-body problem, 82No incoming radiation condition, 184Norm of a KVF, 187

Observables, 164

Parallel displacement, 22Parallel transport, 26Past null infinity, 314Perihelion precession, 110, 331Perturbation equations (Lagrange), 108Planck-length, 15Plane waves, 121PN, see Post-Newtonian (PN)Poincaré-transformation, 126Poisson equation, 52, 163Post-Galilean transformation, 127Post-Newtonian (PN)

center of mass, 381formalism, 235laws of motion, 375metric, 237rotational motion, 392spin, 394tidal forces, 354two-body problem, 380

Potential coefficients, 58Poynting vector, 154Primary frequency standard, 6Prolate spheroidal coordinates, PS, 194Proper coordinates, 174Proper motion, 490, 492Proper time, 5, 123

538 Index

Pulsar timing, 431Pulsar timing arrays, 442

Quartz clocks, 2

Radio pulsar, 431Relativistic celestial mechanics, vi, 337Relativistic perihelion advance, 332Retarded Green’s function, 135Retarded PN solution, 244Retarded time, 184Ricci rotation coefficients, 170Ricci tensor, 26, 30, 240Riemannian metric, 32Riemann tensor, 36, 239Rigidly rotating multipoles, 397Ring laser, 482, 483Robertson-Walker metric, 217Rotating coordinates, 291Runge-Lenz vector, 87

Satellite motion, 329Scalar, 18Scalar product, 32Schwarzschild metric, 202Schwarzschild—de Sitter metric, 223Sea surface topography, 5Shapiro effect, 316Shapiro time delay, 412Simon-Beig theorem, 230Skeletonized harmonic gauge, 270SLR, 450Spacelike infinity, 225Special Relativity, 1Specific angular momentum, 382Specific energy, 382Spectroscopic observable, 165Spherical harmonics, 53Spin-moments, 224Stability of clocks, 2

Stationary axisymmetric space-times, 192Stationary space-times, 187STF-tensors, 62Symmetric spaces, 40

Teph, 306TAI, 289, 304TCB, 289TCG, 5, 289TDB, 289Tensor, 18Terrestrial Time (TT), 5, 289, 304Tetrad induced coordinates, 169, 170Tetrads, 169Thomas precession, 128, 322Thorne moments, 224, 226Thorne rest-terms, 227Time of arrival, 431Time transfer function, 414Tolman mass, 251Transformation of metric potentials, 342TT, see Terrestrial Time (TT)Twist of a KVF, 187, 189TWSTFT, 300Tycho, 492

UT1, 6UTC, 6, 289, 304

Vectorial elements, 101Virial theorem, 252VLBI, 7, 462VLBI, sources at finite distance, 473VSOP, 308

Wagoner-Will representation, 384Weak equivalence principle, 49Weyl spherical coordinates, 194