Chapter
2.2 Equilibrium structure of a spherical body
2.2.1 Equations of body structure
2.2.2 Incompressible fluid
2.2.3 Polytropes and the Lane–Emden equation
2.3 Rotating self-gravitating bodies
2.3.1 Foundations of the theory of rotating bodies
2.3.2 Rotating bodies of uniform density
2.4 General theory of deformed bodies
2.4.2 Unperturbed configuration
2.4.3 Fluid perturbations
2.4.4 Perturbed equilibrium
2.4.5 Rotational deformations
2.5 Tidally deformed bodies
2.6 Bibliographical notes
3 Newtonian orbital dynamics
3.1 Celestial mechanics from Newton to Einstein
3.2 Two bodies: Kepler's problem
3.2.1 Effective one-body description
3.2.4 Solution to Kepler's problem
3.2.5 Keplerian orbits in space
3.3 Perturbed Kepler problem
3.4 Case studies of perturbed Keplerian motion
3.4.1 Perturbations by a third body
3.4.2 The Kozai mechanism
3.4.3 Effects of oblateness
3.4.4 Tidally interacting bodies
3.4.5 Luni-solar precession of the Earth
3.5.1 The three-body problem
3.6 Lagrangian formulation of Newtonian dynamics
3.6.1 Lagrangian and action principle
3.6.2 Lagrangian mechanics of a two-body system
3.6.3 Lagrangian mechanics of a test mass
3.7 Bibliographical notes
4.1.1 Lorentz transformation and spacetime
4.1.3 Kinematics of particles
4.1.4 Momentum and energy
4.1.5 Particle rest-frame
4.1.8 Free particle motion and maximum proper time
4.2 Relativistic hydrodynamics
4.2.3 Energy-momentum tensor
4.3.1 Maxwell's equations
4.3.3 Energy-momentum tensor
4.4 Point particles in spacetime
4.5 Bibliographical notes
5.1 Gravitation as curved spacetime
5.1.1 Principle of equivalence
5.1.2 Metric theory of gravitation
5.1.3 Newtonian gravity as warped time
5.2 Mathematics of curved spacetime
5.2.3 Parallel transport and geodesic equation
5.2.5 Curvature and the local inertial frame
5.3 Physics in curved spacetime
5.3.1 From flat to curved spacetime
5.3.2 Hydrodynamics in curved spacetime
5.3.3 Electrodynamics in curved spacetime
5.3.4 Point particles in curved spacetime
5.4 Einstein field equations
5.5.1 Metric and coordinate freedom
5.5.2 Curvature and field equations
5.5.4 Decomposition of the metric into irreducible pieces
5.5.5 Coulomb gauge and gauge-invariant potentials
5.5.6 Curvature and field equations (revisited)
5.6 Spherical bodies and Schwarzschild spacetime
5.6.1 Spherically symmetric spacetimes
5.6.2 The vacuum Schwarzschild metric
5.6.3 Motion of a test mass
5.6.5 Spherical bodies in hydrostatic equilibrium
5.7 Bibliographical notes
6 Post-Minkowskian theory: Formulation
6.1 Landau–Lifshitz formulation of general relativity
6.1.1 New formulation of the field equations
6.1.3 Integral conservation identities
6.1.4 Total mass, momentum, and angular momentum
6.2 Relaxed Einstein equations
6.2.1 Harmonic coordinates and a wave equation
6.2.2 Formal solution to the wave equation
6.2.3 Iteration of the relaxed field equations
6.3 Integration of the wave equation
6.3.1 Retarded Green's function
6.3.2 Near zone and wave zone: slow-motion condition
6.3.3 Integration domains
6.3.4 Integration over the near zone
6.3.5 Integration over the wave zone
6.4 Bibliographical notes
7 Post-Minkowskian theory: Implementation
7.1.2 General structure of the potentials: Near zone
7.1.4 General structure of the potentials: Wave zone
7.1.5 Toward two iterations of the field equations
7.2.1 Energy-momentum tensor
7.3 Second iteration: Near zone
7.3.1 Effective energy momentum pseudotensor
7.3.2 Energy-momentum conservation
7.3.3 Near-zone contribution to potentials
7.3.4 Wave-zone contribution to potentials
7.3.5 Near-zone potentials: Final answer
7.4 Second iteration: Wave zone
7.4.1 Near-zone contribution to potentials
7.4.2 Wave-zone contribution to potentials
7.5 Bibliographical notes
8 Post-Newtonian theory: Fundamentals
8.1 Equations of post-Newtonian theory
8.1.1 Post-Newtonian metric
8.1.2 Energy-momentum tensor
8.1.3 Auxiliary potentials
8.2 Classic approach to post-Newtonian theory
8.3 Coordinate transformations
8.3.2 Newtonian transformations
8.3.3 Post-Newtonian transformations
8.3.4 Harmonic transformations
8.3.5 Comoving frame of a moving body
8.3.6 Post-Galilean transformations
8.3.7 Pure-gauge transformations
8.4 Post-Newtonian hydrodynamics
8.4.2 Energy-momentum conservation
8.4.3 Post-Newtonian Euler equation
8.4.4 Interlude: Integral identities
8.4.5 Conservation of mass-energy
8.4.6 Conservation of momentum
8.5 Bibliographical notes
9 Post-Newtonian theory: System of isolated bodies
9.1 From fluid configurations to isolated bodies
9.1.1 Center-of-mass variables
9.1.2 Relative variables; reflection symmetry
9.1.3 Structure integrals; equilibrium conditions
9.1.4 Multipole structure
9.1.5 Internal and external potentials
9.2.3 At long last, the metric
9.3 Motion of isolated bodies
9.3.2 Results and sample computations
9.3.3 Equations of motion (in terms of external potentials)
9.3.4 Evaluation of the external potentials
9.3.5 Equations of motion (final form)
9.3.6 Conserved quantities
9.4 Motion of compact bodies
9.4.1 Zones and matching strategy
9.4.3 Post-Newtonian metric
9.4.4 Transformation to the comoving frame
9.4.6 Equations of motion
9.5 Motion of spinning bodies
9.5.1 Definitions of spin
9.5.2 Equilibrium conditions
9.5.3 Inter-body metric of spinning bodies
9.5.4 Spin–orbit and spin–spin accelerations
9.5.5 Conserved quantities
9.5.7 Comoving frame and proper spin
9.5.8 Choice of representative world line
9.6.1 Energy-momentum tensor
9.7 Bibliographical notes
10 Post-Newtonian celestial mechanics, astrometry and navigation
10.1 Post-Newtonian two-body problem
10.1.1 Equations of motion
10.1.3 Perturbed Keplerian orbits
10.1.4 Pericenter advance
10.1.5 Integration of the equations of motion
10.1.6 de Sitter precession
10.2 Motion of light in post-Newtonian gravity
10.2.1 Motion of a photon
10.2.2 Deflection by a spherical body
10.2.3 Measurement of light deflection
10.2.4 Gravitational lenses
10.2.5 Shapiro time delay
10.3 Post-Newtonian gravity in timekeeping and navigation
10.3.1 "A brief history of time''
10.3.4 Temps Atomique International
10.3.6 Timing of binary pulsars
10.4.1 Frame dragging and Gravity Probe B
10.4.2 Frame dragging and LAGEOS satellites
10.4.3 Binary systems of spinning bodies
10.5 Bibliographical notes
11.1 Gravitational-wave field and polarizations
11.1.1 Far-away wave zone
11.1.2 Gravitational potentials in the far-away wave zone
11.1.3 Decomposition into irreducible components
11.1.4 Harmonic gauge conditions
11.1.5 Transformation to the TT gauge
11.1.6 Geodesic deviation
11.1.7 Extraction of the TT part
11.1.8 Distortion of a ring of particles by a gravitational wave
11.2 The quadrupole formula
11.2.2 Application: Binary system
11.2.3 Application: Rotating neutron star
11.2.4 Application: Tidally deformed star
11.3 Beyond the quadrupole formula: Waves at 1.5pn order
11.3.1 Requirements and strategy
11.3.2 Integration techniques for field integrals
11.3.3 Radiative quadrupole moment
11.3.4 Radiative octupole moment
11.3.5 Radiative 4-pole and 5-pole moments
11.3.7 Tails: Wave-zone contribution to the gravitational waves
11.3.8 Summary: Gravitational-wave field
11.4 Gravitational waves emitted by a two-body system
11.4.1 Motion in the barycentric frame
11.4.2 Radiative multipole moments
11.4.3 Computation of retarded-time derivatives
11.4.4 Gravitational-wave field
11.4.6 Specialization to circular orbits
11.4.7 Beyond 1.5 pn order
11.5 Gravitational waves and laser interferometers
11.6 Bibliographical notes
12 Radiative losses and radiation reaction
12.1 Radiation reaction in electromagnetism
12.1.1 System of charged bodies
12.1.2 Motion of charged bodies
12.1.4 Radiation reaction
12.1.6 Looking ahead: gravity
12.2 Radiative losses in gravitating systems
12.2.2 The shortwave approximation
12.2.3 Energy and momentum fluxes
12.2.4 Angular-momentum flux
12.2.5 Isaacson's effective energy-momentum tensor
12.3 Radiative losses in slowly-moving systems
12.3.1 Leading-order multipole radiation
12.3.2 Leading-order fluxes
12.3.3 Application: Newtonian binary system
12.4 Astrophysical implications of radiative losses
12.4.2 Inspiralling compact binaries
12.4.3 "How black holes get their kicks''
12.5 Radiation-reaction potentials
12.5.1 Near-zone potentials
12.5.2 Odd terms in the potentials
12.5.3 Odd terms in the effective energy-momentum tensor
12.5.4 Radiation-reaction potentials: Final expressions
12.6 Radiation reaction of fluid systems
12.6.1 Metric, Christoffel symbols, and matter variables
12.6.2 Radiation-reaction force density
12.6.5 Angular-momentum balance
12.7 Radiation reaction of N-body systems
12.8 Radiation reaction in alternative gauges
12.8.1 Coordinate transformation
12.8.2 Two-parameter family of radiation-reaction gauges
12.8.3 Radiation-reaction force
12.9 Orbital evolution under radiation reaction
12.9.1 Evolution of orbital elements
12.9.2 Multi-scale analysis of orbital evolution
12.10 Bibliographical notes
13 Alternative theories of gravity
13.1 Metric theories and the strong equivalence principle
13.2 Parameterized post-Newtonian framework
13.2.1 A class of post-Newtonian theories
13.2.2 Parameterized post-Newtonian metric
13.2.3 Equations of hydrodynamics
13.2.4 Motion of isolated bodies
13.2.6 Metric near a moving body and local gravitational constant
13.3 Experimental tests of gravitational theories
13.3.1 Two-body problem and pericenter advance
13.3.2 Light deflection and Shapiro time delay
13.3.3 Tests of the strong equivalence principle: Nordtvedt effect
13.3.4 Tests of the strong equivalence principle: preferred-frame and preferred-location effects
13.4 Gravitational radiation in alternative theories of gravity
13.4.1 Gravitational potentials in the far-away wave zone
13.4.3 Interaction with a laser interferometer
13.4.4 Multipolar structure of gravitational waves
13.5 Scalar–tensor gravity
13.5.2 Post-Minkowskian formulation
13.5.3 Slow-motion condition
13.5.4 Near-zone solution: PPN metric
13.5.5 Wave-zone solution: gravitational waves
13.6 Bibliographical notes
Gravity
：Newtonian, Post-Newtonian, Relativistic
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