Cover
Title
Copyright
Dedication
CONTENTS
List of Boxes
Preface
Acknowledgments
PART I FOUNDATIONS
1 Newtonian Physics: Geometric Viewpoint
1.1 Introduction
1.1.1 The Geometric Viewpoint on the Laws of Physics
1.1.2 Purposes of This Chapter
1.1.3 Overview of This Chapter
1.2 Foundational Concepts
1.3 Tensor Algebra without a Coordinate System
1.4 Particle Kinetics and Lorentz Force in Geometric Language
1.5 Component Representation of Tensor Algebra
1.5.1 Slot-Naming Index Notation
1.5.2 Particle Kinetics in Index Notation
1.6 Orthogonal Transformations of Bases
1.7 Differentiation of Scalars, Vectors, and Tensors; Cross Product and Curl
1.8 Volumes, Integration, and Integral Conservation Laws
1.8.1 Gauss’s and Stokes’ Theorems
1.9 The Stress Tensor and Momentum Conservation
1.9.1 Examples: Electromagnetic Field and Perfect Fluid
1.9.2 Conservation of Momentum
1.10 Geometrized Units and Relativistic Particles for Newtonian Readers
1.10.1 Geometrized Units
1.10.2 Energy and Momentum of a Moving Particle
Bibliographic Note
2 Special Relativity: Geometric Viewpoint
2.1 Overview
2.2 Foundational Concepts
2.2.1 Inertial Frames, Inertial Coordinates, Events, Vectors, and Spacetime Diagrams
2.2.2 The Principle of Relativity and Constancy of Light Speed
2.2.3 The Interval and Its Invariance
2.3 Tensor Algebra without a Coordinate System
2.4 Particle Kinetics and Lorentz Force without a Reference Frame
2.4.1 Relativistic Particle Kinetics: World Lines, 4-Velocity, 4-Momentum and Its Conservation, 4-Force
2.4.2 Geometric Derivation of the Lorentz Force Law
2.5 Component Representation of Tensor Algebra
2.5.1 Lorentz Coordinates
2.5.2 Index Gymnastics
2.5.3 Slot-Naming Notation
2.6 Particle Kinetics in Index Notation and in a Lorentz Frame
2.7 Lorentz Transformations
2.8 Spacetime Diagrams for Boosts
2.9 Time Travel
2.9.1 Measurement of Time; Twins Paradox
2.9.2 Wormholes
2.9.3 Wormhole as Time Machine
2.10 Directional Derivatives, Gradients, and the Levi-Civita Tensor
2.11 Nature of Electric and Magnetic Fields; Maxwell’s Equations
2.12 Volumes, Integration, and Conservation Laws
2.12.1 Spacetime Volumes and Integration
2.12.2 Conservation of Charge in Spacetime
2.12.3 Conservation of Particles, Baryon Number, and Rest Mass
2.13 Stress-Energy Tensor and Conservation of 4-Momentum
2.13.1 Stress-Energy Tensor
2.13.2 4-Momentum Conservation
2.13.3 Stress-Energy Tensors for Perfect Fluids and Electromagnetic Fields
Bibliographic Note
PART II STATISTICAL PHYSICS
3 Kinetic Theory
3.1 Overview
3.2 Phase Space and Distribution Function
3.2.1 Newtonian Number Density in Phase Space, N
3.2.2 Relativistic Number Density in Phase Space, N
3.2.3 Distribution Function f (x, v, t) for Particles in a Plasma
3.2.4 Distribution Function Iv/v^3 for Photons
3.2.5 Mean Occupation Number n
3.3 Thermal-Equilibrium Distribution Functions
3.4 Macroscopic Properties of Matter as Integrals over Momentum Space
3.4.1 Particle Density n, Flux S, and Stress Tensor T
3.4.2 Relativistic Number-Flux 4-Vector S and Stress-Energy Tensor T
3.5 Isotropic Distribution Functions and Equations of State
3.5.1 Newtonian Density, Pressure, Energy Density, and Equation of State
3.5.2 Equations of State for a Nonrelativistic Hydrogen Gas
3.5.3 Relativistic Density, Pressure, Energy Density, and Equation of State
3.5.4 Equation of State for a Relativistic Degenerate Hydrogen Gas
3.5.5 Equation of State for Radiation
3.6 Evolution of the Distribution Function: Liouville’s Theorem, the Collisionless Boltzmann Equation, and the Boltzmann Transport Equation
3.7 Transport Coefficients
3.7.1 Diffusive Heat Conduction inside a Star
3.7.2 Order-of-Magnitude Analysis
3.7.3 Analysis Using the Boltzmann Transport Equation
Bibliographic Note
4 Statistical Mechanics
4.1 Overview
4.2 Systems, Ensembles, and Distribution Functions
4.2.1 Systems
4.2.2 Ensembles
4.2.3 Distribution Function
4.3 Liouville’s Theorem and the Evolution of the Distribution Function
4.4 Statistical Equilibrium
4.4.1 Canonical Ensemble and Distribution
4.4.2 General Equilibrium Ensemble and Distribution; Gibbs Ensemble; Grand Canonical Ensemble
4.4.3 Fermi-Dirac and Bose-Einstein Distributions
4.4.4 Equipartition Theorem for Quadratic, Classical Degrees of Freedom
4.5 The Microcanonical Ensemble
4.6 The Ergodic Hypothesis
4.7 Entropy and Evolution toward Statistical Equilibrium
4.7.1 Entropy and the Second Law of Thermodynamics
4.7.2 What Causes the Entropy to Increase?
4.8 Entropy per Particle
4.9 Bose-Einstein Condensate
4.10 Statistical Mechanics in the Presence of Gravity
4.10.1 Galaxies
4.10.2 Black Holes
4.10.3 The Universe
4.10.4 Structure Formation in the Expanding Universe: Violent Relaxation and Phase Mixing
4.11 Entropy and Information
4.11.1 Information Gained When Measuring the State of a System in a Microcanonical Ensemble
4.11.2 Information in Communication Theory
4.11.3 Examples of Information Content
4.11.4 Some Properties of Information
4.11.5 Capacity of Communication Channels; Erasing Information from Computer Memories
Bibliographic Note
5 Statistical Thermodynamics
5.1 Overview
5.2 Microcanonical Ensemble and the Energy Representation of Thermodynamics
5.2.1 Extensive and Intensive Variables; Fundamental Potential
5.2.2 Energy as a Fundamental Potential
5.2.3 Intensive Variables Identified Using Measuring Devices; First Law of Thermodynamics
5.2.4 Euler’s Equation and Form of the Fundamental Potential
5.2.5 Everything Deducible from First Law; Maxwell Relations
5.2.6 Representations of Thermodynamics
5.3 Grand Canonical Ensemble and the Grand-Potential Representation of Thermodynamics
5.3.1 The Grand-Potential Representation, and Computation of Thermodynamic Properties as a Grand Canonical Sum
5.3.2 Nonrelativistic van der Waals Gas
5.4 Canonical Ensemble and the Physical-Free-Energy Representation of Thermodynamics
5.4.1 Experimental Meaning of Physical Free Energy
5.4.2 Ideal Gas with Internal Degrees of Freedom
5.5 Gibbs Ensemble and Representation of Thermodynamics; Phase Transitions and Chemical Reactions
5.5.1 Out-of-Equilibrium Ensembles and Their Fundamental Thermodynamic Potentials and Minimum Principles
5.5.2 Phase Transitions
5.5.3 Chemical Reactions
5.6 Fluctuations away from Statistical Equilibrium
5.7 Van der Waals Gas: Volume Fluctuations and Gas-to-Liquid Phase Transition
5.8 Magnetic Materials
5.8.1 Paramagnetism; The Curie Law
5.8.2 Ferromagnetism: The Ising Model
5.8.3 Renormalization Group Methods for the Ising Model
5.8.4 Monte Carlo Methods for the Ising Model
Bibliographic Note
6 Random Processes
6.1 Overview
6.2 Fundamental Concepts
6.2.1 Random Variables and Random Processes
6.2.2 Probability Distributions
6.2.3 Ergodic Hypothesis
6.3 Markov Processes and Gaussian Processes
6.3.1 Markov Processes; Random Walk
6.3.2 Gaussian Processes and the Central Limit Theorem; Random Walk
6.3.3 Doob’s Theorem for Gaussian-Markov Processes, and Brownian Motion
6.4 Correlation Functions and Spectral Densities
6.4.1 Correlation Functions; Proof of Doob’s Theorem
6.4.2 Spectral Densities
6.4.3 Physical Meaning of Spectral Density, Light Spectra, and Noise in a Gravitational Wave Detector
6.4.4 The Wiener-Khintchine Theorem; Cosmological Density Fluctuations
6.5 2-Dimensional Random Processes
6.5.1 Cross Correlation and Correlation Matrix
6.5.2 Spectral Densities and the Wiener-Khintchine Theorem
6.6 Noise and Its Types of Spectra
6.6.1 Shot Noise, Flicker Noise, and Random-Walk Noise; Cesium Atomic Clock
6.6.2 Information Missing from Spectral Density
6.7 Filtering Random Processes
6.7.1 Filters, Their Kernels, and the Filtered Spectral Density
6.7.2 Brownian Motion and Random Walks
6.7.3 Extracting a Weak Signal from Noise: Band-Pass Filter, Wiener’s Optimal Filter, Signal-to-Noise Ratio, and Allan Variance of Clock Noise
6.7.4 Shot Noise
6.8 Fluctuation-Dissipation Theorem
6.8.1 Elementary Version of the Fluctuation-Dissipation Theorem; Langevin Equation, Johnson Noise in a Resistor, and Relaxation Time for Brownian Motion
6.8.2 Generalized Fluctuation-Dissipation Theorem; Thermal Noise in a Laser Beam’s Measurement of Mirror Motions; Standard Quantum Limit for Measurement Accuracy and How to Evade It
6.9 Fokker-Planck Equation
6.9.1 Fokker-Planck for a 1-Dimensional Markov Process
6.9.2 Optical Molasses: Doppler Cooling of Atoms
6.9.3 Fokker-Planck for a Multidimensional Markov Process; Thermal Noise in an Oscillator
Bibliographic Note
PART III OPTICS
7 Geometric Optics
7.1 Overview
7.2 Waves in a Homogeneous Medium
7.2.1 Monochromatic Plane Waves; Dispersion Relation
7.2.2 Wave Packets
7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approximation and Geometric Optics
7.3.1 Geometric Optics for a Prototypical Wave Equation
7.3.2 Connection of Geometric Optics to Quantum Theory
7.3.3 Geometric Optics for a General Wave
7.3.4 Examples of Geometric-Optics Wave Propagation
7.3.5 Relation to Wave Packets; Limitations of the Eikonal Approximation and Geometric Optics
7.3.6 Fermat’s Principle
7.4 Paraxial Optics
7.4.1 Axisymmetric, Paraxial Systems: Lenses, Mirrors, Telescopes, Microscopes, and Optical Cavities
7.4.2 Converging Magnetic Lens for Charged Particle Beam
7.5 Catastrophe Optics
7.5.1 Image Formation
7.5.2 Aberrations of Optical Instruments
7.6 Gravitational Lenses
7.6.1 Gravitational Deflection of Light
7.6.2 Optical Configuration
7.6.3 Microlensing
7.6.4 Lensing by Galaxies
7.7 Polarization
7.7.1 Polarization Vector and Its Geometric-Optics Propagation Law
7.7.2 Geometric Phase
Bibliographic Note
8 Diffraction
8.1 Overview
8.2 Helmholtz-Kirchhoff Integral
8.2.1 Diffraction by an Aperture
8.2.2 Spreading of the Wavefront: Fresnel and Fraunhofer Regions
8.3 Fraunhofer Diffraction
8.3.1 Diffraction Grating
8.3.2 Airy Pattern of a Circular Aperture: Hubble Space Telescope
8.3.3 Babinet’s Principle
8.4 Fresnel Diffraction
8.4.1 Rectangular Aperture, Fresnel Integrals, and the Cornu Spiral
8.4.2 Unobscured Plane Wave
8.4.3 Fresnel Diffraction by a Straight Edge: Lunar Occultation of a Radio Source
8.4.4 Circular Apertures: Fresnel Zones and Zone Plates
8.5 Paraxial Fourier Optics
8.5.1 Coherent Illumination
8.5.2 Point-Spread Functions
8.5.3 Abbé’s Description of Image Formation by a Thin Lens
8.5.4 Image Processing by a Spatial Filter in the Focal Plane of a Lens: High-Pass, Low-Pass, and Notch Filters; Phase-Contrast Microscopy
8.5.5 Gaussian Beams: Optical Cavities and Interferometric Gravitational-Wave Detectors
8.6 Diffraction at a Caustic
Bibliographic Note
9 Interference and Coherence
9.1 Overview
9.2 Coherence
9.2.1 Young’s Slits
9.2.2 Interference with an Extended Source: Van Cittert-Zernike Theorem
9.2.3 More General Formulation of Spatial Coherence; Lateral Coherence Length
9.2.4 Generalization to 2 Dimensions
9.2.5 Michelson Stellar Interferometer; Astronomical Seeing
9.2.6 Temporal Coherence
9.2.7 Michelson Interferometer and Fourier-Transform Spectroscopy
9.2.8 Degree of Coherence; Relation to Theory of Random Processes
9.3 Radio Telescopes
9.3.1 Two-Element Radio Interferometer
9.3.2 Multiple-Element Radio Interferometers
9.3.3 Closure Phase
9.3.4 Angular Resolution
9.4 Etalons and Fabry-Perot Interferometers
9.4.1 Multiple-Beam Interferometry; Etalons
9.4.2 Fabry-Perot Interferometer and Modes of a Fabry-Perot Cavity with Spherical Mirrors
9.4.3 Fabry-Perot Applications: Spectrometer, Laser, Mode-Cleaning Cavity, Beam-Shaping Cavity, PDH Laser Stabilization, Optical Frequency Comb
9.5 Laser Interferometer Gravitational-Wave Detectors
9.6 Power Correlations and Photon Statistics: Hanbury Brown and Twiss Intensity Interferometer
Bibliographic Note
10 Nonlinear Optics
10.1 Overview
10.2 Lasers
10.2.1 Basic Principles of the Laser
10.2.2 Types of Lasers and Their Performances and Applications
10.2.3 Ti:Sapphire Mode-Locked Laser
10.2.4 Free Electron Laser
10.3 Holography
10.3.1 Recording a Hologram
10.3.2 Reconstructing the 3-Dimensional Image from a Hologram
10.3.3 Other Types of Holography; Applications
10.4 Phase-Conjugate Optics
10.5 Maxwell’s Equations in a Nonlinear Medium; Nonlinear Dielectric Susceptibilities; Electro-Optic Effects
10.6 Three-Wave Mixing in Nonlinear Crystals
10.6.1 Resonance Conditions for Three-Wave Mixing
10.6.2 Three-Wave-Mixing Evolution Equations in a Medium That Is Dispersion-Free and Isotropic at Linear Order
10.6.3 Three-Wave Mixing in a Birefringent Crystal: Phase Matching and Evolution Equations
10.7 Applications of Three-Wave Mixing: Frequency Doubling, Optical Parametric Amplification, and Squeezed Light
10.7.1 Frequency Doubling
10.7.2 Optical Parametric Amplification
10.7.3 Degenerate Optical Parametric Amplification: Squeezed Light
10.8 Four-Wave Mixing in Isotropic Media
10.8.1 Third-Order Susceptibilities and Field Strengths
10.8.2 Phase Conjugation via Four-Wave Mixing in CS2 Fluid
10.8.3 Optical Kerr Effect and Four-Wave Mixing in an Optical Fiber
Bibliographic Note
PART IV ELASTICITY
11 Elastostatics
11.1 Overview
11.2 Displacement and Strain
11.2.1 Displacement Vector and Its Gradient
11.2.2 Expansion, Rotation, Shear, and Strain
11.3 Stress, Elastic Moduli, and Elastostatic Equilibrium
11.3.1 Stress Tensor
11.3.2 Realm of Validity for Hooke’s Law
11.3.3 Elastic Moduli and Elastostatic Stress Tensor
11.3.4 Energy of Deformation
11.3.5 Thermoelasticity
11.3.6 Molecular Origin of Elastic Stress; Estimate of Moduli
11.3.7 Elastostatic Equilibrium: Navier-Cauchy Equation
11.4 Young’s Modulus and Poisson’s Ratio for an Isotropic Material: A Simple Elastostatics Problem
11.5 Reducing the Elastostatic Equations to 1 Dimension for a Bent Beam: Cantilever Bridge, Foucault Pendulum, DNA Molecule, Elastica
11.6 Buckling and Bifurcation of Equilibria
11.6.1 Elementary Theory of Buckling and Bifurcation
11.6.2 Collapse of the World Trade Center Buildings
11.6.3 Buckling with Lateral Force; Connection to Catastrophe Theory
11.6.4 Other Bifurcations: Venus Fly Trap, Whirling Shaft, Triaxial Stars, and Onset of Turbulence
11.7 Reducing the Elastostatic Equations to 2 Dimensions for a Deformed Thin Plate: Stress Polishing a Telescope Mirror
11.8 Cylindrical and Spherical Coordinates: Connection Coefficients and Components of the Gradient of the Displacement Vector
11.9 Solving the 3-Dimensional Navier-Cauchy Equation in Cylindrical Coordinates
11.9.1 Simple Methods: Pipe Fracture and Torsion Pendulum
11.9.2 Separation of Variables and Green’s Functions: Thermoelastic Noise in Mirrors
Bibliographic Note
12 Elastodynamics
12.1 Overview
12.2 Basic Equations of Elastodynamics; Waves in a Homogeneous Medium
12.2.1 Equation of Motion for a Strained Elastic Medium
12.2.2 Elastodynamic Waves
12.2.3 Longitudinal Sound Waves
12.2.4 Transverse Shear Waves
12.2.5 Energy of Elastodynamic Waves
12.3 Waves in Rods, Strings, and Beams
12.3.1 Compression Waves in a Rod
12.3.2 Torsion Waves in a Rod
12.3.3 Waves on Strings
12.3.4 Flexural Waves on a Beam
12.3.5 Bifurcation of Equilibria and Buckling (Once More)
12.4 Body Waves and Surface Waves—Seismology and Ultrasound
12.4.1 Body Waves
12.4.2 Edge Waves
12.4.3 Green’s Function for a Homogeneous Half-Space
12.4.4 Free Oscillations of Solid Bodies
12.4.5 Seismic Tomography
12.4.6 Ultrasound; Shock Waves in Solids
12.5 The Relationship of Classical Waves to Quantum Mechanical Excitations
Bibliographic Note
PART V FLUID DYNAMICS
13 Foundations of Fluid Dynamics
13.1 Overview
13.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow Velocity; Liquids versus Gases
13.3 Hydrostatics
13.3.1 Archimedes’ Law
13.3.2 Nonrotating Stars and Planets
13.3.3 Rotating Fluids
13.4 Conservation Laws
13.5 The Dynamics of an Ideal Fluid
13.5.1 Mass Conservation
13.5.2 Momentum Conservation
13.5.3 Euler Equation
13.5.4 Bernoulli’s Theorem
13.5.5 Conservation of Energy
13.6 Incompressible Flows
13.7 Viscous Flows with Heat Conduction
13.7.1 Decomposition of the Velocity Gradient into Expansion, Vorticity, and Shear
13.7.2 Navier-Stokes Equation
13.7.3 Molecular Origin of Viscosity
13.7.4 Energy Conservation and Entropy Production
13.7.5 Reynolds Number
13.7.6 Pipe Flow
13.8 Relativistic Dynamics of a Perfect Fluid
13.8.1 Stress-Energy Tensor and Equations of Relativistic Fluid Mechanics
13.8.2 Relativistic Bernoulli Equation and Ultrarelativistic Astrophysical Jets
13.8.3 Nonrelativistic Limit of the Stress-Energy Tensor
Bibliographic Note
14 Vorticity
14.1 Overview
14.2 Vorticity, Circulation, and Their Evolution
14.2.1 Vorticity Evolution
14.2.2 Barotropic, Inviscid, Compressible Flows: Vortex Lines Frozen into Fluid
14.2.3 Tornados
14.2.4 Circulation and Kelvin’s Theorem
14.2.5 Diffusion of Vortex Lines
14.2.6 Sources of Vorticity
14.3 Low-Reynolds-Number Flow—Stokes Flow and Sedimentation
14.3.1 Motivation: Climate Change
14.3.2 Stokes Flow
14.3.3 Sedimentation Rate
14.4 High-Reynolds-Number Flow—Laminar Boundary Layers
14.4.1 Blasius Velocity Profile Near a Flat Plate: Stream Function and Similarity Solution
14.4.2 Blasius Vorticity Profile
14.4.3 Viscous Drag Force on a Flat Plate
14.4.4 Boundary Layer Near a Curved Surface: Separation
14.5 Nearly Rigidly Rotating Flows—Earth’s Atmosphere and Oceans
14.5.1 Equations of Fluid Dynamics in a Rotating Reference Frame
14.5.2 Geostrophic Flows
14.5.3 Taylor-Proudman Theorem
14.5.4 Ekman Boundary Layers
14.6 Instabilities of Shear Flows—Billow Clouds and Turbulence in the Stratosphere
14.6.1 Discontinuous Flow: Kelvin-Helmholtz Instability
14.6.2 Discontinuous Flow with Gravity
14.6.3 Smoothly Stratified Flows: Rayleigh and Richardson Criteria for Instability
Bibliographic Note
15 Turbulence
15.1 Overview
15.2 The Transition to Turbulence—Flow Past a Cylinder
15.3 Empirical Description of Turbulence
15.3.1 The Role of Vorticity in Turbulence
15.4 Semiquantitative Analysis of Turbulence
15.4.1 Weak-Turbulence Formalism
15.4.2 Turbulent Viscosity
15.4.3 Turbulent Wakes and Jets; Entrainment; the Coanda Effect
15.4.4 Kolmogorov Spectrum for Fully Developed, Homogeneous, Isotropic Turbulence
15.5 Turbulent Boundary Layers
15.5.1 Profile of a Turbulent Boundary Layer
15.5.2 Coanda Effect and Separation in a Turbulent Boundary Layer
15.5.3 Instability of a Laminar Boundary Layer
15.5.4 Flight of a Ball
15.6 The Route to Turbulence—Onset of Chaos
15.6.1 Rotating Couette Flow
15.6.2 Feigenbaum Sequence, Poincaré Maps, and the Period-Doubling Route to Turbulence in Convection
15.6.3 Other Routes to Turbulent Convection
15.6.4 Extreme Sensitivity to Initial Conditions
Bibliographic Note
16 Waves
16.1 Overview
16.2 Gravity Waves on and beneath the Surface of a Fluid
16.2.1 Deep-Water Waves and Their Excitation and Damping
16.2.2 Shallow-Water Waves
16.2.3 Capillary Waves and Surface Tension
16.2.4 Helioseismology
16.3 Nonlinear Shallow-Water Waves and Solitons
16.3.1 Korteweg–de Vries (KdV) Equation
16.3.2 Physical Effects in the KdV Equation
16.3.3 Single-Soliton Solution
16.3.4 Two-Soliton Solution
16.3.5 Solitons in Contemporary Physics
16.4 Rossby Waves in a Rotating Fluid
16.5 Sound Waves
16.5.1 Wave Energy
16.5.2 Sound Generation
16.5.3 Radiation Reaction, Runaway Solutions, and Matched Asymptotic Expansions
Bibliographic Note
17 Compressible and Supersonic Flow
17.1 Overview
17.2 Equations of Compressible Flow
17.3 Stationary, Irrotational, Quasi-1-Dimensional Flow
17.3.1 Basic Equations; Transition from Subsonic to Supersonic Flow
17.3.2 Setting up a Stationary, Transonic Flow
17.3.3 Rocket Engines
17.4 1-Dimensional, Time-Dependent Flow
17.4.1 Riemann Invariants
17.4.2 Shock Tube
17.5 Shock Fronts
17.5.1 Junction Conditions across a Shock; Rankine-Hugoniot Relations
17.5.2 Junction Conditions for Ideal Gas with Constant
17.5.3 Internal Structure of a Shock
17.5.4 Mach Cone
17.6 Self-Similar Solutions—Sedov-Taylor Blast Wave
17.6.1 The Sedov-Taylor Solution
17.6.2 Atomic Bomb
17.6.3 Supernovae
Bibliographic Note
18 Convection
18.1 Overview
18.2 Diffusive Heat Conduction—Cooling a Nuclear Reactor; Thermal Boundary Layers
18.3 Boussinesq Approximation
18.4 Rayleigh-Bénard Convection
18.5 Convection in Stars
18.6 Double Diffusion—Salt Fingers
Bibliographic Note
19 Magnetohydrodynamics
19.1 Overview
19.2 Basic Equations of MHD
19.2.1 Maxwell’s Equations in the MHD Approximation
19.2.2 Momentum and Energy Conservation
19.2.3 Boundary Conditions
19.2.4 Magnetic Field and Vorticity
19.3 Magnetostatic Equilibria
19.3.1 Controlled Thermonuclear Fusion
19.3.2 Z-Pinch
19.3.3 Θ-Pinch
19.3.4 Tokamak
19.4 Hydromagnetic Flows
19.5 Stability of Magnetostatic Equilibria
19.5.1 Linear Perturbation Theory
19.5.2 Z-Pinch: Sausage and Kink Instabilities
19.5.3 The Θ-Pinch and Its Toroidal Analog; Flute Instability; Motivation for Tokamak
19.5.4 Energy Principle and Virial Theorems
19.6 Dynamos and Reconnection of Magnetic Field Lines
19.6.1 Cowling’s Theorem
19.6.2 Kinematic Dynamos
19.6.3 Magnetic Reconnection
19.7 Magnetosonic Waves and the Scattering of Cosmic Rays
19.7.1 Cosmic Rays
19.7.2 Magnetosonic Dispersion Relation
19.7.3 Scattering of Cosmic Rays by Alfvén Waves
Bibliographic Note
PART VI PLASMA PHYSICS
20 The Particle Kinetics of Plasma
20.1 Overview
20.2 Examples of Plasmas and Their Density-Temperature Regimes
20.2.1 Ionization Boundary
20.2.2 Degeneracy Boundary
20.2.3 Relativistic Boundary
20.2.4 Pair-Production Boundary
20.2.5 Examples of Natural and Human-Made Plasmas
20.3 Collective Effects in Plasmas—Debye Shielding and Plasma Oscillations
20.3.1 Debye Shielding
20.3.2 Collective Behavior
20.3.3 Plasma Oscillations and Plasma Frequency
20.4 Coulomb Collisions
20.4.1 Collision Frequency
20.4.2 The Coulomb Logarithm
20.4.3 Thermal Equilibration Rates in a Plasma
20.4.4 Discussion
20.5 Transport Coefficients
20.5.1 Coulomb Collisions
20.5.2 Anomalous Resistivity and Anomalous Equilibration
20.6 Magnetic Field
20.6.1 Cyclotron Frequency and Larmor Radius
20.6.2 Validity of the Fluid Approximation
20.6.3 Conductivity Tensor
20.7 Particle Motion and Adiabatic Invariants
20.7.1 Homogeneous, Time-Independent Magnetic Field and No Electric Field
20.7.2 Homogeneous, Time-Independent Electric and Magnetic Fields
20.7.3 Inhomogeneous, Time-Independent Magnetic Field
20.7.4 A Slowly Time-Varying Magnetic Field
20.7.5 Failure of Adiabatic Invariants; Chaotic Orbits
Bibliographic Note
21 Waves in Cold Plasmas: Two-Fluid Formalism
21.1 Overview
21.2 Dielectric Tensor, Wave Equation, and General Dispersion Relation
21.3 Two-Fluid Formalism
21.4 Wave Modes in an Unmagnetized Plasma
21.4.1 Dielectric Tensor and Dispersion Relation for a Cold, Unmagnetized Plasma
21.4.2 Plasma Electromagnetic Modes
21.4.3 Langmuir Waves and Ion-Acoustic Waves in Warm Plasmas
21.4.4 Cutoffs and Resonances
21.5 Wave Modes in a Cold, Magnetized Plasma
21.5.1 Dielectric Tensor and Dispersion Relation
21.5.2 Parallel Propagation
21.5.3 Perpendicular Propagation
21.5.4 Propagation of Radio Waves in the Ionosphere; Magnetoionic Theory
21.5.5 CMA Diagram for Wave Modes in a Cold, Magnetized Plasma
21.6 Two-Stream Instability
Bibliographic Note
22 Kinetic Theory of Warm Plasmas
22.1 Overview
22.2 Basic Concepts of Kinetic Theory and Its Relationship to Two-Fluid Theory
22.2.1 Distribution Function and Vlasov Equation
22.2.2 Relation of Kinetic Theory to Two-Fluid Theory
22.2.3 Jeans’ Theorem
22.3 Electrostatic Waves in an Unmagnetized Plasma: Landau Damping
22.3.1 Formal Dispersion Relation
22.3.2 Two-Stream Instability
22.3.3 The Landau Contour
22.3.4 Dispersion Relation for Weakly Damped or Growing Waves
22.3.5 Langmuir Waves and Their Landau Damping
22.3.6 Ion-Acoustic Waves and Conditions for Their Landau Damping to Be Weak
22.4 Stability of Electrostatic Waves in Unmagnetized Plasmas
22.4.1 Nyquist’s Method
22.4.2 Penrose’s Instability Criterion
22.5 Particle Trapping
22.6 N-Particle Distribution Function
22.6.1 BBGKY Hierarchy
22.6.2 Two-Point Correlation Function
22.6.3 Coulomb Correction to Plasma Pressure
Bibliographic Note
23 Nonlinear Dynamics of Plasmas
23.1 Overview
23.2 Quasilinear Theory in Classical Language
23.2.1 Classical Derivation of the Theory
23.2.2 Summary of Quasilinear Theory
23.2.3 Conservation Laws
23.2.4 Generalization to 3 Dimensions
23.3 Quasilinear Theory in Quantum Mechanical Language
23.3.1 Plasmon Occupation Number n
23.3.2 Evolution of n for Plasmons via Interaction with Electrons
23.3.3 Evolution of f for Electrons via Interaction with Plasmons
23.3.4 Emission of Plasmons by Particles in the Presence of a Magnetic Field
23.3.5 Relationship between Classical and Quantum Mechanical Formalisms
23.3.6 Evolution of n via Three-Wave Mixing
23.4 Quasilinear Evolution of Unstable Distribution Functions—A Bump in the Tail
23.4.1 Instability of Streaming Cosmic Rays
23.5 Parametric Instabilities; Laser Fusion
23.6 Solitons and Collisionless Shock Waves
Bibliographic Note
PART VII GENERAL RELATIVITY
24 From Special to General Relativity
24.1 Overview
24.2 Special Relativity Once Again
24.2.1 Geometric, Frame-Independent Formulation
24.2.2 Inertial Frames and Components of Vectors, Tensors, and Physical Laws
24.2.3 Light Speed, the Interval, and Spacetime Diagrams
24.3 Differential Geometry in General Bases and in Curved Manifolds
24.3.1 Nonorthonormal Bases
24.3.2 Vectors as Directional Derivatives; Tangent Space; Commutators
24.3.3 Differentiation of Vectors and Tensors; Connection Coefficients
24.3.4 Integration
24.4 The Stress-Energy Tensor Revisited
24.5 The Proper Reference Frame of an Accelerated Observer
24.5.1 Relation to Inertial Coordinates; Metric in Proper Reference Frame; Transport Law for Rotating Vectors
24.5.2 Geodesic Equation for a Freely Falling Particle
24.5.3 Uniformly Accelerated Observer
24.5.4 Rindler Coordinates for Minkowski Spacetime
Bibliographic Note
25 Fundamental Concepts of General Relativity
25.1 History and Overview
25.2 Local Lorentz Frames, the Principle of Relativity, and Einstein’s Equivalence Principle
25.3 The Spacetime Metric, and Gravity as a Curvature of Spacetime
25.4 Free-Fall Motion and Geodesics of Spacetime
25.5 Relative Acceleration, Tidal Gravity, and Spacetime Curvature
25.5.1 Newtonian Description of Tidal Gravity
25.5.2 Relativistic Description of Tidal Gravity
25.5.3 Comparison of Newtonian and Relativistic Descriptions
25.6 Properties of the Riemann Curvature Tensor
25.7 Delicacies in the Equivalence Principle, and Some Nongravitational Laws of Physics in Curved Spacetime
25.7.1 Curvature Coupling in the Nongravitational Laws
25.8 The Einstein Field Equation
25.8.1 Geometrized Units
25.9 Weak Gravitational Fields
25.9.1 Newtonian Limit of General Relativity
25.9.2 Linearized Theory
25.9.3 Gravitational Field outside a Stationary, Linearized Source of Gravity
25.9.4 Conservation Laws for Mass, Momentum, and Angular Momentum in Linearized Theory
25.9.5 Conservation Laws for a Strong-Gravity Source
Bibliographic Note
26 Relativistic Stars and Black Holes
26.1 Overview
26.2 Schwarzschild’s Spacetime Geometry
26.2.1 The Schwarzschild Metric, Its Connection Coefficients, and Its Curvature Tensors
26.2.2 The Nature of Schwarzschild’s Coordinate System, and Symmetries of the Schwarzschild Spacetime
26.2.3 Schwarzschild Spacetime at Radii r » M: The Asymptotically Flat Region
26.2.4 Schwarzschild Spacetime at r ~ M
26.3 Static Stars
26.3.1 Birkhoff’s Theorem
26.3.2 Stellar Interior
26.3.3 Local Conservation of Energy and Momentum
26.3.4 The Einstein Field Equation
26.3.5 Stellar Models and Their Properties
26.3.6 Embedding Diagrams
26.4 Gravitational Implosion of a Star to Form a Black Hole
26.4.1 The Implosion Analyzed in Schwarzschild Coordinates
26.4.2 Tidal Forces at the Gravitational Radius
26.4.3 Stellar Implosion in Eddington-Finkelstein Coordinates
26.4.4 Tidal Forces at r = 0—The Central Singularity
26.4.5 Schwarzschild Black Hole
26.5 Spinning Black Holes: The Kerr Spacetime
26.5.1 The Kerr Metric for a Spinning Black Hole
26.5.2 Dragging of Inertial Frames
26.5.3 The Light-Cone Structure, and the Horizon
26.5.4 Evolution of Black Holes—Rotational Energy and Its Extraction
26.6 The Many-Fingered Nature of Time
Bibliographic Note
27 Gravitational Waves and Experimental Tests of General Relativity
27.1 Overview
27.2 Experimental Tests of General Relativity
27.2.1 Equivalence Principle, Gravitational Redshift, and Global Positioning System
27.2.2 Perihelion Advance of Mercury
27.2.3 Gravitational Deflection of Light, Fermat’s Principle, and Gravitational Lenses
27.2.4 Shapiro Time Delay
27.2.5 Geodetic and Lense-Thirring Precession
27.2.6 Gravitational Radiation Reaction
27.3 Gravitational Waves Propagating through Flat Spacetime
27.3.1 Weak, Plane Waves in Linearized Theory
27.3.2 Measuring a Gravitational Wave by Its Tidal Forces
27.3.3 Gravitons and Their Spin and Rest Mass
27.4 Gravitational Waves Propagating through Curved Spacetime
27.4.1 Gravitational Wave Equation in Curved Spacetime
27.4.2 Geometric-Optics Propagation of Gravitational Waves
27.4.3 Energy and Momentum in Gravitational Waves
27.5 The Generation of Gravitational Waves
27.5.1 Multipole-Moment Expansion
27.5.2 Quadrupole-Moment Formalism
27.5.3 Quadrupolar Wave Strength, Energy, Angular Momentum, and Radiation Reaction
27.5.4 Gravitational Waves from a Binary Star System
27.5.5 Gravitational Waves from Binaries Made of Black Holes, Neutron Stars, or Both: Numerical Relativity
27.6 The Detection of Gravitational Waves
27.6.1 Frequency Bands and Detection Techniques
27.6.2 Gravitational-Wave Interferometers: Overview and Elementary Treatment
27.6.3 Interferometer Analyzed in TT Gauge
27.6.4 Interferometer Analyzed in the Proper Reference Frame of the Beam Splitter
27.6.5 Realistic Interferometers
27.6.6 Pulsar Timing Arrays
Bibliographic Note
28 Cosmology
28.1 Overview
28.2 General Relativistic Cosmology
28.2.1 Isotropy and Homogeneity
28.2.2 Geometry
28.2.3 Kinematics
28.2.4 Dynamics
28.3 The Universe Today
28.3.1 Baryons
28.3.2 Dark Matter
28.3.3 Photons
28.3.4 Neutrinos
28.3.5 Cosmological Constant
28.3.6 Standard Cosmology
28.4 Seven Ages of the Universe
28.4.1 Particle Age
28.4.2 Nuclear Age
28.4.3 Photon Age
28.4.4 Plasma Age
28.4.5 Atomic Age
28.4.6 Gravitational Age
28.4.7 Cosmological Age
28.5 Galaxy Formation
28.5.1 Linear Perturbations
28.5.2 Individual Constituents
28.5.3 Solution of the Perturbation Equations
28.5.4 Galaxies
28.6 Cosmological Optics
28.6.1 Cosmic Microwave Background
28.6.2 Weak Gravitational Lensing
28.6.3 Sunyaev-Zel’dovich Effect
28.7 Three Mysteries
28.7.1 Inflation and the Origin of the Universe
28.7.2 Dark Matter and the Growth of Structure
28.7.3 The Cosmological Constant and the Fate of the Universe
Bibliographic Note
References
Name Index
Subject Index
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