Gravitation in Astrophysics

I. Gravitation in Localized Systems.- An Introduction to the Theory of Gravitational Radiation.- 1. Introduction.- 1.1 Scope of these lectures.- 1.2 Conventions and Notation.- 2. What is a Gravitational Wave?.- 2.1 First viewpoint: propagation of discontinuities.- 2.2 Second viewpoint: high-frequency waves.- 2.3 Third viewpoint: weak gravitational waves on a flat background.- 2.4 Other viewpoints.- 3. Basic Problems of Gravitational Radiation Theory.- 3.1 A catalogue of problems.- 3.2 A catalogue of approximation methods.- 4. Quadrupole Moment Formalisms.- 4.1 “Quadrupole laws” versus “quadrupole equations”.- 4.2 The three “quadrupole laws”.- 4.3 General discussion of “quadrupole equations”.- 4.4 The “standard,” or “Einstein,” far-field quadrupole equation.- 4.5 The “Landau-Lifshitz” quadrupole equation.- 4.6 The “Fock” quadrupole equation.- 4.7 Recent improvements.- 4.8 Thorne’s generation formalism.- 4.9 Tentative conclusion about the “standard” far-field quadrupole equation.- 4.10 Far-field quadrupole equations for some specific problems.- 4.11 The generalized quadrupole equation of Halpern-Desbrandes and Press.- 4.12 The far-field multipole law.- 4.13 On the definition of the asymptotic outgoing radiation field.- 4.14 Energy-loss quadrupole equations.- 4.15 Radiation-reaction quadrupole equations à la Burke-Thorne.- 4.16 Post-Newtonian radiation-reaction quadrupole equations.- 4.17 Radiation-reaction quadrupole equations in the N-body problem.- 4.18 Conclusion.- 5. Multipolar-Post-Minkowskian Formalisms.- 5.1 Introduction.- 5.2 Formal framework.- 5.3 The hierarchy of equations to be solved.- 5.4 The first step of the hierarchy.- 5.5 On the meaning of the “algorithmic multipole moments”.- 5.6 Recursive algorithm for constructing the higher steps of the hierarchy.- 5.7 Partial results on the asymptotic problem.- 5.8 Preliminary results on the propagation problem.- 5.9 Partial results on the generation problem.- 5.10 Preliminary results on the radiation-reaction problem.- 6. Gravitational Radiation and Binary Systems of Condensed Objects.- 6.1 One method for two questions.- 6.2 The two-condensed-body problem in General Relativity.- 6.3 The internal problems.- 6.4 The matching: internal ? external.- 6.5 The external problem.- 6.6 A convenient auxiliary mathematical technique.- 6.7 Answer to the first question of §6.1 (generation).- 6.8 Equations of motion of a binary system of condensed bodies.- 6.9 Radiation-reaction force versus the relativistic Laplace effect.- 6.10 Poincaré on gravitational waves.- 6.11 Answer to the second question of §6.1 (radiation reaction).- 6.12 Application to the binary pulsar PSR 1913+.- 6.13 Conclusion.- Acknowledgements.- Some Books Fully Devoted to Gravitational Radiation.- Bibliographical References.- Mathematical Foundations of the Theory of Relativistic Stellar and Black Hole Configurations.- 1. Introduction.- 1.1 Background.- 1.2 Purpose and Plan.- 2. Notions of General (Dynamic) Black Hole Theory.- 2.1 Definition.- 2.2 Kinematics of Characteristic (Null) Boundaries.- 2.3 Generalised Raychaudhuri Equation for Timelike and Null Geodesic Congruences.- 2.4 The Horizon of a Black Hole.- 2.5 Asymptotic Predictability, Closed Trapped Surfaces, and Apparent Horizons.- 2.6 Cosmic Censorship and the Existence of an Asymptotic Equilibrium.- 2.7 Approximate Equilibrium.- 3. Stationary and Static Equilibrium.- 3.1 Overview.- 3.2 Elementary Local Properties of Killing Horizons.- 3.3 Uniformity of the Corotating Potential on a Killing Horizon.- 3.4 Uniformity of K (the “zeroth law”) on a Killing Horizon.- 3.5 The Mass of a Stationary System.- 3.6 Globally Bradyonic Character of Generators of Stationary D.O.C..- 3.7 The Staticity Theorem for Non-Rotating Electromagnetic Black Holes.- 3.8 Local Characterisation of Simply Connected Static Domain of Outer Communications.- 4. Axisymmetric Equilibrium States.- 4.1 Mass and Angular Momentum of Stationary Axisymmetric Systems.- 4.2 Circularity Theorem for Stationary Axisymmetric Systems.- 4.3 The Ergoregion and ZAMOs in Papapetrou Coordinates.- 4.4 Slowly Rotating (but Strongly Gravitating) Stellar Equilibrium Configurations.- 4.5 Killing Horizon (Nullity and Rigidity) Property of Locus Where ZAMOs Go Null.- 4.6 Black Hole Mass and Angular Momentum and their Variations.- 4.7 Superradiance.- 4.8 Local Characterisation of Simply Connected Stationary (Circularity) Axisymmetric D.O.C..- 5. The Source Free Equilibrium State Problem for Axisymmetric Black Holes.- 5.1 Canonical Global Coordinate System for the D.O.C. of a Stationary Axisymmetric Black Hole.- 5.2 Reduction to a 2-dimensional Boundary Problem.- 5.3 The Final Step in the Uniqueness Theorem.- 5.4 Killing-Maxwell-Yano System.- 5.5 The Canonical Tetrad.- Acknowledgements.- References.- Relativistic Gravitational Instabilities.- Spherical Pulsation of Spherical Stars.- Newtonian stars.- Relativistic stars.- The turning point criterion for white dwarfs and neutron stars.- Star clusters.- Nonspherical Pulsation of Spherical Stars.- Newtonian stars.- Relativistic stars.- Strongly damped modes.- Quadrupole gravitational radiation.- Nonspherical Perturbations of Spherical Black Holes.- Formulation as a scattering problem.- Calculations of the normal modes.- Stability of Rotating Stars: General Remarks.- The Maclaurin Spheroids.- The nonaxisymmetric modes.- The secular instabilities.- The T/W criterion for instability.- A Relativistic Approach to Stability.- Perfect fluids in general relativity.- Definition of a perturbation in terms of a sequence of solutions.- Two preferred perturbations; Eulerian and Lagrangian.- Perturbations of Einstein’s Equations.- A stability criterion.- A Simple Approach to the Radiation Instability.- Conserved quantities for wave fields.- Mechanism for the gravitational wave instability.- Gravitational wave instability as a two-stream instability.- Other ways of exciting the instability.- The instability due to viscosity.- The Perturbed Energy of a Rotating System.- Orbiting particle: an elementary example.- The second-order energy of a rotating fluid.- Maximum Rotation Rate of Neutron Stars.- Stability of the Kerr Black Hole.- References.- Accretion and Collapse.- I. The Gravothermal Catastrophe.- 1. Specific Heats.- 2. A Thought Experiment.- 3. Why Self-Similar Solutions Occur in Science.- 4. Evolution After Core Collapse.- II. Spherical Accretion.- 5. Bondi Accretion.- 6. Relativistic Accretion.- 7. Cold Self-Similar Gravitational Collapse.- III. Disk Accretion.- 8. Energy, Angular Momentum and Dissipation.- 9. Viscous Newtonian Accretion Disks.- 10. Relativistic Accretion Disks.- IV. Optically Thick Accretion.- 11. Self-Similar Solutions.- 12. Enthalpy Theorem and Jet Production.- Accretion Disk Electrodynamics.- 1. The Standard Thin Disk.- 2. Turbulent Dynamo in Accretion Disks.- 3. Electrodynamic Coupling of Accretion-Disk Coronae.- 4. The Interaction of a Neutron Star with an Accretion Disk.- 4.1 Spin up of neutron stars.- 4.2 Quasiperiodic oscillations.- References.- Special Topics I.- The Membrane Paradigm for Black-Hole Astrophysics.- Tidal Disruption.- 1. Introduction.- 2. Tidal Tensor.- 3. Ellipsoidal Deformations of Homogeneous Bodies.- 3.1 The stationary rotational problem.- 3.2 The stationary tidal problem.- 3.3 The dynamical tidal problem.- 4. Tidal Deformations of a Compressible Body.- 4.1 The tidal rolling mill effect.- 4.2 Tidal versus collisional disruption of stars.- 4.3 The affine star model.- 4.4 Motion in relativistic tidal field.- 4.5 Pancake nucleosynthesis and the fate of debris.- References.- Naked Singularities in Spherical Gravitational Collapse.- 1. Introduction.- 2. Dust Collapse and Shell Focusing Singularities.- 2.1 Shell-crossing.- 2.2 Tolman-Bondi solutions.- 2.3 Causal structure of the Tolman-Bondi solutions.- 2.4 Interpretation of shell-focusing singularities.- 2.5 Collapse of null fluid.- 3. Collapse of Scalar Field Configurations.- 3.1 Self-similar collapse.- 3.2 Generic spherical collapse.- Acknowledgements.- References.- II. Gravitation in Cosmology.- Some Topics in Relativistic Cosmology.- Orientation.- The Universe is Unique.- Non-local Influences.- Horizons.- How Many Spatial Dimensions are There?.- Variation of Fundamental “Constants”.- Unknown Physics.- Selection Effects.- Unknown Matter Fields.- How Little Could We Know?.- Newtonian Gravitation.- Newtonian Cosmology.- General Relativistic Cosmology.- The Friedman Cosmological Models.- Observable Parameters.- When Do Closed Universes Recollapse?.- Spatially Homogeneous Universes.- The Microwave Background and the Density of the Universe.- Microwave Background Observations.- Characteristic Microwave Background Patterns.- Quadrupole.- Hotspot.- Spirals.- Observational Limits.- Isotropy and Homogeneity.- The Cosmological Principle(s).- Can We Prove a Cosmological Principle?.- Is Isotropy a Stable Property of Cosmological Models?.- Is Isotropy Really Unstable and Does it Matter Anyway?.- Approach to a Family of Plane Waves.- No Hair Theorems.- Inflation and the Initial Value Problem.- Inflation and the Strong Energy Condition.- The Deflationary Universe.- Resumé.- Acknowledgements.- References.- Cosmic Strings and the Origin of Structure in the Universe.- 1. Introduction.- 1.1 Origin of Perturbations in the Universe.- 2. Quantum Particle Creation in an Inflationary Universe.- 2.1 A Simple Model.- 2.2 More Realistic Models.- 3. Topological Defects in Field Theories.- 3.1 Cosmological Constant.- 3.2 Domain Walls.- 3.3 Cosmic Strings.- 3.4 Monopoles.- 3.5 Instantons.- 4. Cosmological Evolution of Topological Defects.- 4.1 Bounds on Evolution of Defect Density.- 4.2 Evolution of Monopoles and Domain Walls.- 4.3 Evolution of Cosmic Strings.- 5. Motion and Evolution of Cosmic Strings.- 5.1 Closed Loops.- 5.2 Gravitational Waves from Closed Loops.- 5.3 Galaxy Formation by Cosmic Strings.- 5.4 Fluctuations in the Cosmic Background Radiation Due to Cosmic Strings.- 5.5 Light Bending Due to Cosmic Strings.- Acknowledgements.- References.- Cosmological Phase Transitions.- 1. The Evolution of the Vacuum.- 1.1 High Temperature Symmetry Restoration.- 1.2 Domain Walls.- 1.3 Cosmic Strings.- 1.4 Magnetic Monopoles.- 1.5 The Kibble Mechanism.- 2. Inflation.- 2.1 Loose Ends of the Standard Cosmology.- 2.2 Inflation - The Basic Picture.- 2.3 Dynamics of Inflation.- 2.4 Specific Models.- 2.5 Present Status and Future Directions.- Acknowledgements.- References.- Prediction in Quantum Cosmology.- 1. Introduction.- 2. Predictions from the Wave Function of the Universe.- 2.1 The Wave Function of the Universe.- 2.2 Cosmological Observations and Cosmological Predictions.- 2.3 The Nature of Cosmological Predictions.- 2.4 Quantum Mechanics of Individual Systems.- 2.5 The Problem of Time.- 3. Laws for Initial Conditions.- 3.1 The Sum Over Histories Formulation of Quantum Cosmology.- 3.2 Constraints.- 3.3 A Proposal for a Wave Function of the Universe.- 4. The Limit of Classical Geometry and Quantum Field Theory in Curved Spacetime.- 4.1 The Semiclassical Approximation to Non-Relativistic Particle Quantum Mechanics.- 4.2 The Born-Oppenheimer Approximation for Real Clocks.- 4.3 The Approximation of Quantum Field Theory in Curved Spacetime.- 4.4 The Semiclassical Vacuum.- Acknowledgements.- Problems.- References.- Special Topics II.- The Quasi-Isotropic Universe.- 1. Introduction.- 2. The Microwave Background.- 3. Helium Abundance.- 4. Anisotropic Spatially Homogeneous Cosmologies.- 5. The Wainwright and Anderson Solution.- 6. An Inhomogeneous Model.- 7. Discussion of Results.- 7.1 The Anthropic Principle.- 7.2 Initial Conditions.- 7.3 Gravitational Entropy.- References.- Semiclassical Quantum Gravity in Two and Four Dimensions.- 1. Quantum Effects Near Distorted Black Holes.- 2. Q.F.T and the Antipodal Identification of Black Holes and of Desitter Space.- 3. The Back-Reaction Problem in Two Dimensions: Liouville and Schroedinger Equations.- References.- Towards a Theory for the Quantum Mechanics of Gravitational Collapse.- Abstract.- 1. Introduction.- 2. String Theory in a Nut Shell.- 3. The Black Hole.- 4. The Shifting Horizon.- 5. A Link with String Theory.- 6. Conclusion.- References.