2.5.1 Static tides Tidal dissipation Dynamical tides Bibliographical notes Exercises 118

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3 Contents Preface xiii 1 Foundations of Newtonian gravity Newtonian gravity Equations of Newtonian gravity Newtonian field equation Equations of hydrodynamics Motion of fluid elements Thermodynamics of fluid elements Global conservation laws Mass-momentum tensor Spherical and nearly spherical bodies Spherical bodies Nonspherical bodies Symmetric tracefree tensors Motion of extended fluid bodies From fluid configurations to isolated bodies Center-of-mass variables Internal and external potential Taylor expansion of the external potential Equations of motion for isolated bodies Conserved quantities Equations of motion for binary systems Spin dynamics Bibliographical notes Exercises 53 2 Structure of self-gravitating bodies Equations of internal structure Equilibrium structure of a spherical body Equations of body structure Incompressible fluid Polytropes and the Lane-Emden equation Isothermal spheres White dwarfs Rotating self-gravitating bodies Foundations of the theory of rotating bodies Rotating bodies of uniform density General theory of deformed bodies Fluid equations Unperturbed configuration Fluid perturbations Perturbed equilibrium Rotational deformations Tidally deformed bodies 104 iii

4 iv Contents Static tides Tidal dissipation Dynamical tides Bibliographical notes Exercises Newtonian orbital dynamics Celestial mechanics from Newton to Einstein Two bodies: Kepler s problem E ective one-body description Orbital plane First integrals Solution to Kepler s problem Keplerian orbits in space Perturbed Kepler problem Perturbing force Osculating orbits Case studies of perturbed Keplerian motion Perturbations by a third body The Kozai mechanism E ects of oblateness Tidally interacting bodies Luni-solar precession of the Earth More bodies The 3-body problem The N-body problem Lagrangian formulation of Newtonian dynamics Lagrangian and action principle Lagrangian mechanics of a two-body system Lagrangian mechanics of a test mass Bibliographical notes Exercises Minkowski spacetime Spacetime Lorentz transformation and spacetime Metric tensor Kinematics of particles Momentum and energy Particle rest-frame Photons Particle dynamics Free particle motion and maximum proper time Relativistic hydrodynamics Fluid variables Mass current Energy-momentum tensor Fluid dynamics Electrodynamics Maxwell s equations Vector potential Energy-momentum tensor Point particles in spacetime Bibliographical notes 189

5 Contents v 4.6 Exercises Curved spacetime Gravitation as curved spacetime Principle of equivalence Metric theory of gravitation Newtonian gravity as warped time Mathematics of curved spacetime Metric Tensor calculus Parallel transport and geodesic equation Curvature tensors Curvature and the local inertial frame Physics in curved spacetime From flat to curved spacetime Hydrodynamics in curved spacetime Electrodynamics in curved spacetime Point particles in curved spacetime Einstein field equations Linearized theory Metric and coordinate freedom Curvature and field equations Lorenz gauge Decomposition of the metric into irreducible pieces Coulomb gauge and gauge-invariant potentials Curvature and field equations (revisited) Newtonian limit Spherical bodies and Schwarzschild spacetime Spherically symmetric spacetimes The vacuum Schwarzschild metric Motion of a test mass Motion of light Spherical bodies in hydrostatic equilibrium Bibliographical notes Exercises Post-Minkowskian theory: Formulation Landau-Lifshitz formulation of general relativity New formulation of the field equations Coordinate freedom Integral conservation identities Total mass, momentum, and angular momentum Relaxed Einstein equations Harmonic coordinates and a wave equation Formal solution to the wave equation Iteration of the relaxed field equations Integration of the wave equation Retarded Green s function Near zone and wave zone; slow-motion condition Integration domains Integration over the near zone Integration over the wave zone Bibliographical notes Exercises 290

6 vi Contents 7 Post-Minkowskian theory: Implementation Assembling the tools Fluid variables General structure of the potentials: Near zone Near-zone metric General structure of the potentials: Wave zone Toward two iterations of the field equations First iteration Energy-momentum tensor Near zone Wave zone Second iteration: Near zone E ective energy momentum pseudotensor Energy-momentum conservation Near-zone contribution to potentials Wave-zone contribution to potentials Near-zone potentials: Final answer Second iteration: Wave zone Near-zone contribution to potentials Wave-zone contribution to potentials Bibliographical notes Exercises Post-Newtonian theory: Fundamentals Equations of post-newtonian theory Post-Newtonian metric Energy-momentum tensor Auxiliary potentials Geodesic equations Classic approach to post-newtonian theory Coordinate transformations Introduction Newtonian transformations Post-Newtonian transformations Harmonic transformations Comoving frame of a moving body Post-Galilean transformations Pure-gauge transformations Post-Newtonian hydrodynamics Introduction Energy-momentum conservation Post-Newtonian Euler equation Interlude: Integral identities Conservation of mass-energy Conservation of momentum Center-of-mass Bibliographical remarks Exercises 365

7 Contents vii 9 Post-Newtonian theory: System of isolated bodies From fluid configurations to isolated bodies Center-of-mass variables Relative variables; reflection symmetry Structure integrals; equilibrium conditions Multipole structure Internal and external potentials Total mass-energy Virial identities Inter-body metric Introduction Potentials At long last, the metric Motion of isolated bodies Strategy Results and sample computations Equations of motion (in terms of external potentials) Evaluation of the external potentials Equations of motion (final form) Conserved quantities Binary systems Motion of compact bodies Zones and matching strategy Body metric Post-Newtonian metric Transformation to the comoving frame Matching Equations of motion Motion of spinning bodies Definitions of spin Equilibrium conditions Inter-body metric of spinning bodies Spin-orbit and spin-spin accelerations Conserved quantities Spin precession Comoving frame and proper spin Choice of representative world line Binary systems Point particles Energy-momentum tensor Regularization Potentials Bibliographical notes Exercises Post-Newtonian celestial mechanics, astrometry and navigation Post-Newtonian two-body problem Equations of motion Circular orbits Perturbed Keplerian orbits Pericenter advance Integration of the equations of motion de Sitter precession Motion of light in post-newtonian gravity 438

8 viii Contents Motion of a photon Deflection by a spherical body Measurement of light deflection Gravitational lenses Shapiro time delay Post-Newtonian gravity in timekeeping and navigation A brief history of time Reference frames Geoid Temps Atomique International Orbiting clocks Timing of binary pulsars Spinning bodies Frame dragging and Gravity Probe B Frame dragging and LAGEOS satellites Binary systems of spinning bodies Bibliographical notes Exercises Gravitational waves Gravitational-wave field and polarizations Far-away wave zone Gravitational potentials in the far-away wave zone Decomposition into irreducible components Harmonic gauge conditions Transformation to the TT gauge Geodesic deviation Extraction of the TT part Distortion of a ring of particles by a gravitational wave The quadrupole formula Formulation Application: Binary system Application: Rotating neutron star Application: Tidally deformed star Beyond the quadrupole formula: Waves at 1.5pn order Requirements and strategy Integration techniques for field integrals Radiative quadrupole moment Radiative octupole moment Radiative 4-pole and 5-pole moments Surface integrals Tails: Wave-zone contribution to the gravitational waves Summary: Gravitational-wave field Gravitational waves emitted by a two-body system Motion in the barycentric frame Radiative multipole moments Computation of retarded-time derivatives Gravitational-wave field Polarizations Specialization to circular orbits Beyond 1.5pn order Gravitational waves and laser interferometers Bibliographical notes Exercises 551

9 Contents ix 12 Radiative losses and radiation reaction Radiation reaction in electromagnetism System of charged bodies Motion of charged bodies Radiative losses Radiation reaction Energy balance Looking ahead: gravity Radiative losses in gravitating systems Balance equations The shortwave approximation Energy and momentum fluxes Angular-momentum flux Isaacson s e ective energy-momentum tensor Radiative losses in slowly-moving systems Leading-order multipole radiation Leading-order fluxes Application: Newtonian binary system Astrophysical implications of radiative losses Binary pulsars Inspiralling compact binaries How black holes get their kicks Radiation-reaction potentials Near-zone potentials Odd terms in the potentials Odd terms in the e ective energy-momentum tensor Radiation-reaction potentials: Final expressions Radiation reaction of fluid systems Metric, Christo el symbols, and matter variables Radiation-reaction force density Energy balance Momentum balance Angular-momentum balance Radiation reaction of N-body systems N bodies Two bodies Radiation reaction in alternative gauges Coordinate transformation Two-parameter family of radiation-reaction gauges Radiation-reaction force Balance equations Orbital evolution under radiation reaction Evolution of orbital elements Multi-scale analysis of orbital evolution Bibliographical notes Exercises Alternative theories of gravity Metric theories and the strong equivalence principle Parametrized post-newtonian framework A class of post-newtonian theories Parametrized post-newtonian metric Equations of hydrodynamics Motion of isolated bodies 633

10 x Contents Motion of light Metric near a moving body and local gravitational constant Spin dynamics Experimental tests of gravitational theories Two-body problem and pericenter advance Light deflection and Shapiro time delay Tests of the strong equivalence principle: Nordtvedt e ect Tests of the strong equivalence principle: Preferred-frame and preferred-location e ects Gravitational radiation in alternative theories of gravity Gravitational potentials in the far-away wave zone Polarizations Interaction with a laser interferometer Multipolar structure of gravitational waves Scalar-tensor gravity Field equations Post-Minkowskian formulation Slow-motion condition Near-zone solution: PPN metric Wave-zone solution: gravitational waves Bibliographical notes Exercises 674 References 677 Index 688