Modern classical physics: optics, fluids, plasmas, elasticity, relativity, and statistical physics

Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
CONTENTS......Page 8
List of Boxes......Page 28
Preface......Page 32
Acknowledgments......Page 40
PART I FOUNDATIONS......Page 42
1.1.1 The Geometric Viewpoint on the Laws of Physics......Page 46
1.1.3 Overview of This Chapter......Page 48
1.2 Foundational Concepts......Page 49
1.3 Tensor Algebra without a Coordinate System......Page 51
1.4 Particle Kinetics and Lorentz Force in Geometric Language......Page 54
1.5 Component Representation of Tensor Algebra......Page 57
1.5.1 Slot-Naming Index Notation......Page 58
1.5.2 Particle Kinetics in Index Notation......Page 60
1.6 Orthogonal Transformations of Bases......Page 61
1.7 Differentiation of Scalars, Vectors, and Tensors; Cross Product and Curl......Page 63
1.8 Volumes, Integration, and Integral Conservation Laws......Page 67
1.8.1 Gauss’s and Stokes’ Theorems......Page 68
1.9 The Stress Tensor and Momentum Conservation......Page 70
1.9.1 Examples: Electromagnetic Field and Perfect Fluid......Page 71
1.9.2 Conservation of Momentum......Page 72
1.10.1 Geometrized Units......Page 74
1.10.2 Energy and Momentum of a Moving Particle......Page 75
Bibliographic Note......Page 76
2.1 Overview......Page 78
2.2.1 Inertial Frames, Inertial Coordinates, Events, Vectors, and Spacetime Diagrams......Page 79
2.2.2 The Principle of Relativity and Constancy of Light Speed......Page 83
2.2.3 The Interval and Its Invariance......Page 86
2.3 Tensor Algebra without a Coordinate System......Page 89
2.4.1 Relativistic Particle Kinetics: World Lines, 4-Velocity, 4-Momentum and Its Conservation, 4-Force......Page 90
2.4.2 Geometric Derivation of the Lorentz Force Law......Page 93
2.5.2 Index Gymnastics......Page 95
2.5.3 Slot-Naming Notation......Page 97
2.6 Particle Kinetics in Index Notation and in a Lorentz Frame......Page 98
2.7 Lorentz Transformations......Page 104
2.8 Spacetime Diagrams for Boosts......Page 106
2.9.1 Measurement of Time; Twins Paradox......Page 108
2.9.2 Wormholes......Page 109
2.9.3 Wormhole as Time Machine......Page 110
2.10 Directional Derivatives, Gradients, and the Levi-Civita Tensor......Page 111
2.11 Nature of Electric and Magnetic Fields; Maxwell’s Equations......Page 112
2.12.1 Spacetime Volumes and Integration......Page 116
2.12.2 Conservation of Charge in Spacetime......Page 119
2.12.3 Conservation of Particles, Baryon Number, and Rest Mass......Page 120
2.13.1 Stress-Energy Tensor......Page 123
2.13.2 4-Momentum Conservation......Page 125
2.13.3 Stress-Energy Tensors for Perfect Fluids and Electromagnetic Fields......Page 126
Bibliographic Note......Page 129
PART II STATISTICAL PHYSICS......Page 132
3.1 Overview......Page 136
3.2.1 Newtonian Number Density in Phase Space, N......Page 138
3.2.2 Relativistic Number Density in Phase Space, N......Page 140
3.2.3 Distribution Function f (x, v, t) for Particles in a Plasma......Page 146
3.2.4 Distribution Function Iv/v^3 for Photons......Page 147
3.2.5 Mean Occupation Number n......Page 149
3.3 Thermal-Equilibrium Distribution Functions......Page 152
3.4.1 Particle Density n, Flux S, and Stress Tensor T......Page 158
3.4.2 Relativistic Number-Flux 4-Vector S and Stress-Energy Tensor T......Page 159
3.5.1 Newtonian Density, Pressure, Energy Density, and Equation of State......Page 161
3.5.2 Equations of State for a Nonrelativistic Hydrogen Gas......Page 163
3.5.3 Relativistic Density, Pressure, Energy Density, and Equation of State......Page 166
3.5.4 Equation of State for a Relativistic Degenerate Hydrogen Gas......Page 167
3.5.5 Equation of State for Radiation......Page 169
3.6 Evolution of the Distribution Function: Liouville’s Theorem, the Collisionless Boltzmann Equation, and the Boltzmann Transport Equation......Page 173
3.7 Transport Coefficients......Page 180
3.7.1 Diffusive Heat Conduction inside a Star......Page 183
3.7.2 Order-of-Magnitude Analysis......Page 184
3.7.3 Analysis Using the Boltzmann Transport Equation......Page 185
Bibliographic Note......Page 194
4.1 Overview......Page 196
4.2.1 Systems......Page 198
4.2.2 Ensembles......Page 201
4.2.3 Distribution Function......Page 202
4.3 Liouville’s Theorem and the Evolution of the Distribution Function......Page 207
4.4 Statistical Equilibrium......Page 209
4.4.1 Canonical Ensemble and Distribution......Page 210
4.4.2 General Equilibrium Ensemble and Distribution; Gibbs Ensemble; Grand Canonical Ensemble......Page 213
4.4.3 Fermi-Dirac and Bose-Einstein Distributions......Page 215
4.4.4 Equipartition Theorem for Quadratic, Classical Degrees of Freedom......Page 218
4.5 The Microcanonical Ensemble......Page 219
4.6 The Ergodic Hypothesis......Page 221
4.7.1 Entropy and the Second Law of Thermodynamics......Page 222
4.7.2 What Causes the Entropy to Increase?......Page 224
4.8 Entropy per Particle......Page 232
4.9 Bose-Einstein Condensate......Page 234
4.10.1 Galaxies......Page 242
4.10.2 Black Holes......Page 245
4.10.3 The Universe......Page 250
4.10.4 Structure Formation in the Expanding Universe: Violent Relaxation and Phase Mixing......Page 251
4.11.1 Information Gained When Measuring the State of a System in a Microcanonical Ensemble......Page 252
4.11.2 Information in Communication Theory......Page 253
4.11.3 Examples of Information Content......Page 255
4.11.5 Capacity of Communication Channels; Erasing Information from Computer Memories......Page 257
Bibliographic Note......Page 259
5.1 Overview......Page 260
5.2.1 Extensive and Intensive Variables; Fundamental Potential......Page 262
5.2.2 Energy as a Fundamental Potential......Page 263
5.2.3 Intensive Variables Identified Using Measuring Devices; First Law of Thermodynamics......Page 264
5.2.4 Euler’s Equation and Form of the Fundamental Potential......Page 267
5.2.5 Everything Deducible from First Law; Maxwell Relations......Page 268
5.2.6 Representations of Thermodynamics......Page 269
5.3.1 The Grand-Potential Representation, and Computation of Thermodynamic Properties as a Grand Canonical Sum......Page 270
5.3.2 Nonrelativistic van der Waals Gas......Page 273
5.4 Canonical Ensemble and the Physical-Free-Energy Representation of Thermodynamics......Page 280
5.4.1 Experimental Meaning of Physical Free Energy......Page 282
5.4.2 Ideal Gas with Internal Degrees of Freedom......Page 283
5.5 Gibbs Ensemble and Representation of Thermodynamics; Phase Transitions and Chemical Reactions......Page 287
5.5.1 Out-of-Equilibrium Ensembles and Their Fundamental Thermodynamic Potentials and Minimum Principles......Page 289
5.5.2 Phase Transitions......Page 292
5.5.3 Chemical Reactions......Page 297
5.6 Fluctuations away from Statistical Equilibrium......Page 301
5.7 Van der Waals Gas: Volume Fluctuations and Gas-to-Liquid Phase Transition......Page 307
5.8 Magnetic Materials......Page 311
5.8.1 Paramagnetism; The Curie Law......Page 312
5.8.2 Ferromagnetism: The Ising Model......Page 313
5.8.3 Renormalization Group Methods for the Ising Model......Page 314
5.8.4 Monte Carlo Methods for the Ising Model......Page 320
Bibliographic Note......Page 323
6.1 Overview......Page 324
6.2.1 Random Variables and Random Processes......Page 326
6.2.2 Probability Distributions......Page 327
6.2.3 Ergodic Hypothesis......Page 329
6.3.1 Markov Processes; Random Walk......Page 330
6.3.2 Gaussian Processes and the Central Limit Theorem; Random Walk......Page 333
6.3.3 Doob’s Theorem for Gaussian-Markov Processes, and Brownian Motion......Page 336
6.4.1 Correlation Functions; Proof of Doob’s Theorem......Page 338
6.4.2 Spectral Densities......Page 340
6.4.3 Physical Meaning of Spectral Density, Light Spectra, and Noise in a Gravitational Wave Detector......Page 342
6.4.4 The Wiener-Khintchine Theorem; Cosmological Density Fluctuations......Page 344
6.5.1 Cross Correlation and Correlation Matrix......Page 347
6.5.2 Spectral Densities and the Wiener-Khintchine Theorem......Page 348
6.6.1 Shot Noise, Flicker Noise, and Random-Walk Noise; Cesium Atomic Clock......Page 349
6.6.2 Information Missing from Spectral Density......Page 351
6.7.1 Filters, Their Kernels, and the Filtered Spectral Density......Page 352
6.7.2 Brownian Motion and Random Walks......Page 354
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......Page 356
6.7.4 Shot Noise......Page 362
6.8.1 Elementary Version of the Fluctuation-Dissipation Theorem; Langevin Equation, Johnson Noise in a Resistor, and Relaxation Time for Brownian Motion......Page 364
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......Page 372
6.9 Fokker-Planck Equation......Page 376
6.9.1 Fokker-Planck for a 1-Dimensional Markov Process......Page 377
6.9.2 Optical Molasses: Doppler Cooling of Atoms......Page 381
6.9.3 Fokker-Planck for a Multidimensional Markov Process; Thermal Noise in an Oscillator......Page 384
Bibliographic Note......Page 386
PART III OPTICS......Page 388
7.1 Overview......Page 392
7.2.1 Monochromatic Plane Waves; Dispersion Relation......Page 393
7.2.2 Wave Packets......Page 395
7.3 Waves in an Inhomogeneous, Time-Varying Medium: The Eikonal Approximation and Geometric Optics......Page 398
7.3.1 Geometric Optics for a Prototypical Wave Equation......Page 399
7.3.2 Connection of Geometric Optics to Quantum Theory......Page 403
7.3.3 Geometric Optics for a General Wave......Page 407
7.3.4 Examples of Geometric-Optics Wave Propagation......Page 409
7.3.5 Relation to Wave Packets; Limitations of the Eikonal Approximation and Geometric Optics......Page 410
7.3.6 Fermat’s Principle......Page 412
7.4 Paraxial Optics......Page 416
7.4.1 Axisymmetric, Paraxial Systems: Lenses, Mirrors, Telescopes, Microscopes, and Optical Cavities......Page 418
7.4.2 Converging Magnetic Lens for Charged Particle Beam......Page 422
7.5.1 Image Formation......Page 425
7.5.2 Aberrations of Optical Instruments......Page 436
7.6.1 Gravitational Deflection of Light......Page 437
7.6.2 Optical Configuration......Page 438
7.6.3 Microlensing......Page 439
7.6.4 Lensing by Galaxies......Page 442
7.7.1 Polarization Vector and Its Geometric-Optics Propagation Law......Page 446
7.7.2 Geometric Phase......Page 447
Bibliographic Note......Page 450
8.1 Overview......Page 452
8.2 Helmholtz-Kirchhoff Integral......Page 454
8.2.1 Diffraction by an Aperture......Page 455
8.2.2 Spreading of the Wavefront: Fresnel and Fraunhofer Regions......Page 458
8.3 Fraunhofer Diffraction......Page 461
8.3.1 Diffraction Grating......Page 463
8.3.2 Airy Pattern of a Circular Aperture: Hubble Space Telescope......Page 466
8.3.3 Babinet’s Principle......Page 469
8.4 Fresnel Diffraction......Page 470
8.4.1 Rectangular Aperture, Fresnel Integrals, and the Cornu Spiral......Page 471
8.4.3 Fresnel Diffraction by a Straight Edge: Lunar Occultation of a Radio Source......Page 473
8.4.4 Circular Apertures: Fresnel Zones and Zone Plates......Page 475
8.5 Paraxial Fourier Optics......Page 477
8.5.1 Coherent Illumination......Page 478
8.5.2 Point-Spread Functions......Page 479
8.5.3 Abbé’s Description of Image Formation by a Thin Lens......Page 480
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......Page 482
8.5.5 Gaussian Beams: Optical Cavities and Interferometric Gravitational-Wave Detectors......Page 486
8.6 Diffraction at a Caustic......Page 492
Bibliographic Note......Page 495
9.1 Overview......Page 496
9.2.1 Young’s Slits......Page 497
9.2.2 Interference with an Extended Source: Van Cittert-Zernike Theorem......Page 500
9.2.3 More General Formulation of Spatial Coherence; Lateral Coherence Length......Page 503
9.2.4 Generalization to 2 Dimensions......Page 504
9.2.5 Michelson Stellar Interferometer; Astronomical Seeing......Page 505
9.2.6 Temporal Coherence......Page 513
9.2.7 Michelson Interferometer and Fourier-Transform Spectroscopy......Page 515
9.2.8 Degree of Coherence; Relation to Theory of Random Processes......Page 518
9.3.1 Two-Element Radio Interferometer......Page 520
9.3.2 Multiple-Element Radio Interferometers......Page 521
9.3.3 Closure Phase......Page 522
9.3.4 Angular Resolution......Page 523
9.4.1 Multiple-Beam Interferometry; Etalons......Page 524
9.4.2 Fabry-Perot Interferometer and Modes of a Fabry-Perot Cavity with Spherical Mirrors......Page 531
9.4.3 Fabry-Perot Applications: Spectrometer, Laser, Mode-Cleaning Cavity, Beam-Shaping Cavity, PDH Laser Stabilization, Optical Frequency Comb......Page 537
9.5 Laser Interferometer Gravitational-Wave Detectors......Page 543
9.6 Power Correlations and Photon Statistics: Hanbury Brown and Twiss Intensity Interferometer......Page 550
Bibliographic Note......Page 553
10.1 Overview......Page 554
10.2.1 Basic Principles of the Laser......Page 556
10.2.2 Types of Lasers and Their Performances and Applications......Page 560
10.2.3 Ti:Sapphire Mode-Locked Laser......Page 561
10.3 Holography......Page 562
10.3.1 Recording a Hologram......Page 563
10.3.2 Reconstructing the 3-Dimensional Image from a Hologram......Page 566
10.3.3 Other Types of Holography; Applications......Page 568
10.4 Phase-Conjugate Optics......Page 572
10.5 Maxwell’s Equations in a Nonlinear Medium; Nonlinear Dielectric Susceptibilities; Electro-Optic Effects......Page 577
10.6.1 Resonance Conditions for Three-Wave Mixing......Page 581
10.6.2 Three-Wave-Mixing Evolution Equations in a Medium That Is Dispersion-Free and Isotropic at Linear Order......Page 585
10.6.3 Three-Wave Mixing in a Birefringent Crystal: Phase Matching and Evolution Equations......Page 587
10.7.1 Frequency Doubling......Page 594
10.7.2 Optical Parametric Amplification......Page 596
10.7.3 Degenerate Optical Parametric Amplification: Squeezed Light......Page 597
10.8.1 Third-Order Susceptibilities and Field Strengths......Page 599
10.8.2 Phase Conjugation via Four-Wave Mixing in CS2 Fluid......Page 600
10.8.3 Optical Kerr Effect and Four-Wave Mixing in an Optical Fiber......Page 603
Bibliographic Note......Page 605
PART IV ELASTICITY......Page 606
11.1 Overview......Page 608
11.2.1 Displacement Vector and Its Gradient......Page 611
11.2.2 Expansion, Rotation, Shear, and Strain......Page 612
11.3.1 Stress Tensor......Page 618
11.3.3 Elastic Moduli and Elastostatic Stress Tensor......Page 621
11.3.4 Energy of Deformation......Page 623
11.3.5 Thermoelasticity......Page 625
11.3.6 Molecular Origin of Elastic Stress; Estimate of Moduli......Page 626
11.3.7 Elastostatic Equilibrium: Navier-Cauchy Equation......Page 628
11.4 Young’s Modulus and Poisson’s Ratio for an Isotropic Material: A Simple Elastostatics Problem......Page 630
11.5 Reducing the Elastostatic Equations to 1 Dimension for a Bent Beam: Cantilever Bridge, Foucault Pendulum, DNA Molecule, Elastica......Page 633
11.6.1 Elementary Theory of Buckling and Bifurcation......Page 643
11.6.2 Collapse of the World Trade Center Buildings......Page 646
11.6.3 Buckling with Lateral Force; Connection to Catastrophe Theory......Page 647
11.6.4 Other Bifurcations: Venus Fly Trap, Whirling Shaft, Triaxial Stars, and Onset of Turbulence......Page 648
11.7 Reducing the Elastostatic Equations to 2 Dimensions for a Deformed Thin Plate: Stress Polishing a Telescope Mirror......Page 650
11.8 Cylindrical and Spherical Coordinates: Connection Coefficients and Components of the Gradient of the Displacement Vector......Page 655
11.9.1 Simple Methods: Pipe Fracture and Torsion Pendulum......Page 660
11.9.2 Separation of Variables and Green’s Functions: Thermoelastic Noise in Mirrors......Page 663
Bibliographic Note......Page 668
12.1 Overview......Page 670
12.2.1 Equation of Motion for a Strained Elastic Medium......Page 671
12.2.2 Elastodynamic Waves......Page 677
12.2.3 Longitudinal Sound Waves......Page 678
12.2.4 Transverse Shear Waves......Page 679
12.2.5 Energy of Elastodynamic Waves......Page 681
12.3 Waves in Rods, Strings, and Beams......Page 683
12.3.2 Torsion Waves in a Rod......Page 684
12.3.3 Waves on Strings......Page 685
12.3.4 Flexural Waves on a Beam......Page 686
12.3.5 Bifurcation of Equilibria and Buckling (Once More)......Page 688
12.4 Body Waves and Surface Waves—Seismology and Ultrasound......Page 689
12.4.1 Body Waves......Page 691
12.4.2 Edge Waves......Page 695
12.4.3 Green’s Function for a Homogeneous Half-Space......Page 699
12.4.4 Free Oscillations of Solid Bodies......Page 702
12.4.6 Ultrasound; Shock Waves in Solids......Page 704
12.5 The Relationship of Classical Waves to Quantum Mechanical Excitations......Page 708
Bibliographic Note......Page 711
PART V FLUID DYNAMICS......Page 712
13.1 Overview......Page 716
13.2 The Macroscopic Nature of a Fluid: Density, Pressure, Flow Velocity; Liquids versus Gases......Page 718
13.3 Hydrostatics......Page 722
13.3.1 Archimedes’ Law......Page 725
13.3.2 Nonrotating Stars and Planets......Page 727
13.3.3 Rotating Fluids......Page 730
13.4 Conservation Laws......Page 732
13.5 The Dynamics of an Ideal Fluid......Page 736
13.5.2 Momentum Conservation......Page 737
13.5.4 Bernoulli’s Theorem......Page 738
13.5.5 Conservation of Energy......Page 745
13.6 Incompressible Flows......Page 750
13.7.1 Decomposition of the Velocity Gradient into Expansion, Vorticity, and Shear......Page 751
13.7.2 Navier-Stokes Equation......Page 752
13.7.3 Molecular Origin of Viscosity......Page 754
13.7.4 Energy Conservation and Entropy Production......Page 755
13.7.6 Pipe Flow......Page 757
13.8.1 Stress-Energy Tensor and Equations of Relativistic Fluid Mechanics......Page 760
13.8.2 Relativistic Bernoulli Equation and Ultrarelativistic Astrophysical Jets......Page 762
13.8.3 Nonrelativistic Limit of the Stress-Energy Tensor......Page 764
Bibliographic Note......Page 767
14.1 Overview......Page 770
14.2 Vorticity, Circulation, and Their Evolution......Page 772
14.2.1 Vorticity Evolution......Page 775
14.2.2 Barotropic, Inviscid, Compressible Flows: Vortex Lines Frozen into Fluid......Page 777
14.2.3 Tornados......Page 779
14.2.4 Circulation and Kelvin’s Theorem......Page 780
14.2.5 Diffusion of Vortex Lines......Page 782
14.2.6 Sources of Vorticity......Page 785
14.3 Low-Reynolds-Number Flow—Stokes Flow and Sedimentation......Page 787
14.3.1 Motivation: Climate Change......Page 789
14.3.2 Stokes Flow......Page 790
14.3.3 Sedimentation Rate......Page 795
14.4 High-Reynolds-Number Flow—Laminar Boundary Layers......Page 798
14.4.1 Blasius Velocity Profile Near a Flat Plate: Stream Function and Similarity Solution......Page 799
14.4.3 Viscous Drag Force on a Flat Plate......Page 804
14.4.4 Boundary Layer Near a Curved Surface: Separation......Page 805
14.5 Nearly Rigidly Rotating Flows—Earth’s Atmosphere and Oceans......Page 807
14.5.1 Equations of Fluid Dynamics in a Rotating Reference Frame......Page 808
14.5.2 Geostrophic Flows......Page 811
14.5.3 Taylor-Proudman Theorem......Page 812
14.5.4 Ekman Boundary Layers......Page 813
14.6.1 Discontinuous Flow: Kelvin-Helmholtz Instability......Page 819
14.6.2 Discontinuous Flow with Gravity......Page 823
14.6.3 Smoothly Stratified Flows: Rayleigh and Richardson Criteria for Instability......Page 825
Bibliographic Note......Page 827
15.1 Overview......Page 828
15.2 The Transition to Turbulence—Flow Past a Cylinder......Page 830
15.3 Empirical Description of Turbulence......Page 839
15.3.1 The Role of Vorticity in Turbulence......Page 840
15.4.1 Weak-Turbulence Formalism......Page 841
15.4.2 Turbulent Viscosity......Page 845
15.4.3 Turbulent Wakes and Jets; Entrainment; the Coanda Effect......Page 846
15.4.4 Kolmogorov Spectrum for Fully Developed, Homogeneous, Isotropic Turbulence......Page 851
15.5 Turbulent Boundary Layers......Page 858
15.5.1 Profile of a Turbulent Boundary Layer......Page 859
15.5.2 Coanda Effect and Separation in a Turbulent Boundary Layer......Page 861
15.5.3 Instability of a Laminar Boundary Layer......Page 863
15.5.4 Flight of a Ball......Page 864
15.6.1 Rotating Couette Flow......Page 866
15.6.2 Feigenbaum Sequence, Poincaré Maps, and the Period-Doubling Route to Turbulence in Convection......Page 869
15.6.3 Other Routes to Turbulent Convection......Page 872
15.6.4 Extreme Sensitivity to Initial Conditions......Page 873
Bibliographic Note......Page 875
16.1 Overview......Page 876
16.2 Gravity Waves on and beneath the Surface of a Fluid......Page 878
16.2.2 Shallow-Water Waves......Page 881
16.2.3 Capillary Waves and Surface Tension......Page 885
16.2.4 Helioseismology......Page 889
16.3.1 Korteweg–de Vries (KdV) Equation......Page 891
16.3.2 Physical Effects in the KdV Equation......Page 894
16.3.3 Single-Soliton Solution......Page 895
16.3.4 Two-Soliton Solution......Page 896
16.3.5 Solitons in Contemporary Physics......Page 897
16.4 Rossby Waves in a Rotating Fluid......Page 899
16.5 Sound Waves......Page 903
16.5.1 Wave Energy......Page 904
16.5.2 Sound Generation......Page 906
16.5.3 Radiation Reaction, Runaway Solutions, and Matched Asymptotic Expansions......Page 910
Bibliographic Note......Page 915
17.1 Overview......Page 916
17.2 Equations of Compressible Flow......Page 918
17.3.1 Basic Equations; Transition from Subsonic to Supersonic Flow......Page 921
17.3.2 Setting up a Stationary, Transonic Flow......Page 924
17.3.3 Rocket Engines......Page 928
17.4.1 Riemann Invariants......Page 932
17.4.2 Shock Tube......Page 936
17.5 Shock Fronts......Page 938
17.5.1 Junction Conditions across a Shock; Rankine-Hugoniot Relations......Page 939
17.5.2 Junction Conditions for Ideal Gas with Constant......Page 945
17.5.3 Internal Structure of a Shock......Page 947
17.5.4 Mach Cone......Page 948
17.6 Self-Similar Solutions—Sedov-Taylor Blast Wave......Page 949
17.6.1 The Sedov-Taylor Solution......Page 950
17.6.2 Atomic Bomb......Page 953
17.6.3 Supernovae......Page 955
Bibliographic Note......Page 957
18.1 Overview......Page 958
18.2 Diffusive Heat Conduction—Cooling a Nuclear Reactor; Thermal Boundary Layers......Page 959
18.3 Boussinesq Approximation......Page 964
18.4 Rayleigh-Bénard Convection......Page 966
18.5 Convection in Stars......Page 974
18.6 Double Diffusion—Salt Fingers......Page 978
Bibliographic Note......Page 982
19.1 Overview......Page 984
19.2 Basic Equations of MHD......Page 985
19.2.1 Maxwell’s Equations in the MHD Approximation......Page 987
19.2.2 Momentum and Energy Conservation......Page 991
19.2.3 Boundary Conditions......Page 994
19.2.4 Magnetic Field and Vorticity......Page 998
19.3.1 Controlled Thermonuclear Fusion......Page 999
19.3.2 Z-Pinch......Page 1001
19.3.3 Θ-Pinch......Page 1003
19.3.4 Tokamak......Page 1004
19.4 Hydromagnetic Flows......Page 1006
19.5.1 Linear Perturbation Theory......Page 1012
19.5.2 Z-Pinch: Sausage and Kink Instabilities......Page 1016
19.5.3 The Θ-Pinch and Its Toroidal Analog; Flute Instability; Motivation for Tokamak......Page 1019
19.5.4 Energy Principle and Virial Theorems......Page 1021
19.6.1 Cowling’s Theorem......Page 1025
19.6.2 Kinematic Dynamos......Page 1026
19.6.3 Magnetic Reconnection......Page 1027
19.7.1 Cosmic Rays......Page 1029
19.7.2 Magnetosonic Dispersion Relation......Page 1030
19.7.3 Scattering of Cosmic Rays by Alfvén Waves......Page 1033
Bibliographic Note......Page 1034
PART VI PLASMA PHYSICS......Page 1036
20.1 Overview......Page 1038
20.2.1 Ionization Boundary......Page 1039
20.2.3 Relativistic Boundary......Page 1041
20.2.5 Examples of Natural and Human-Made Plasmas......Page 1042
20.3.1 Debye Shielding......Page 1044
20.3.2 Collective Behavior......Page 1045
20.3.3 Plasma Oscillations and Plasma Frequency......Page 1046
20.4.1 Collision Frequency......Page 1047
20.4.2 The Coulomb Logarithm......Page 1049
20.4.3 Thermal Equilibration Rates in a Plasma......Page 1051
20.4.4 Discussion......Page 1053
20.5.1 Coulomb Collisions......Page 1056
20.5.2 Anomalous Resistivity and Anomalous Equilibration......Page 1057
20.6.1 Cyclotron Frequency and Larmor Radius......Page 1060
20.6.2 Validity of the Fluid Approximation......Page 1061
20.6.3 Conductivity Tensor......Page 1063
20.7 Particle Motion and Adiabatic Invariants......Page 1065
20.7.2 Homogeneous, Time-Independent Electric and Magnetic Fields......Page 1066
20.7.3 Inhomogeneous, Time-Independent Magnetic Field......Page 1067
20.7.4 A Slowly Time-Varying Magnetic Field......Page 1070
20.7.5 Failure of Adiabatic Invariants; Chaotic Orbits......Page 1071
Bibliographic Note......Page 1073
21.1 Overview......Page 1074
21.2 Dielectric Tensor, Wave Equation, and General Dispersion Relation......Page 1076
21.3 Two-Fluid Formalism......Page 1078
21.4.1 Dielectric Tensor and Dispersion Relation for a Cold, Unmagnetized Plasma......Page 1081
21.4.2 Plasma Electromagnetic Modes......Page 1083
21.4.3 Langmuir Waves and Ion-Acoustic Waves in Warm Plasmas......Page 1085
21.4.4 Cutoffs and Resonances......Page 1090
21.5.1 Dielectric Tensor and Dispersion Relation......Page 1091
21.5.2 Parallel Propagation......Page 1093
21.5.3 Perpendicular Propagation......Page 1098
21.5.4 Propagation of Radio Waves in the Ionosphere; Magnetoionic Theory......Page 1099
21.5.5 CMA Diagram for Wave Modes in a Cold, Magnetized Plasma......Page 1103
21.6 Two-Stream Instability......Page 1106
Bibliographic Note......Page 1109
22.1 Overview......Page 1110
22.2.1 Distribution Function and Vlasov Equation......Page 1111
22.2.2 Relation of Kinetic Theory to Two-Fluid Theory......Page 1114
22.2.3 Jeans’ Theorem......Page 1115
22.3.1 Formal Dispersion Relation......Page 1118
22.3.2 Two-Stream Instability......Page 1120
22.3.3 The Landau Contour......Page 1121
22.3.4 Dispersion Relation for Weakly Damped or Growing Waves......Page 1126
22.3.5 Langmuir Waves and Their Landau Damping......Page 1127
22.3.6 Ion-Acoustic Waves and Conditions for Their Landau Damping to Be Weak......Page 1129
22.4 Stability of Electrostatic Waves in Unmagnetized Plasmas......Page 1131
22.4.2 Penrose’s Instability Criterion......Page 1132
22.5 Particle Trapping......Page 1139
22.6 N-Particle Distribution Function......Page 1143
22.6.1 BBGKY Hierarchy......Page 1144
22.6.2 Two-Point Correlation Function......Page 1145
22.6.3 Coulomb Correction to Plasma Pressure......Page 1148
Bibliographic Note......Page 1149
23.1 Overview......Page 1152
23.2.1 Classical Derivation of the Theory......Page 1154
23.2.2 Summary of Quasilinear Theory......Page 1161
23.2.3 Conservation Laws......Page 1162
23.2.4 Generalization to 3 Dimensions......Page 1163
23.3.1 Plasmon Occupation Number n......Page 1164
23.3.2 Evolution of n for Plasmons via Interaction with Electrons......Page 1165
23.3.3 Evolution of f for Electrons via Interaction with Plasmons......Page 1170
23.3.5 Relationship between Classical and Quantum Mechanical Formalisms......Page 1172
23.3.6 Evolution of n via Three-Wave Mixing......Page 1173
23.4 Quasilinear Evolution of Unstable Distribution Functions—A Bump in the Tail......Page 1177
23.4.1 Instability of Streaming Cosmic Rays......Page 1179
23.5 Parametric Instabilities; Laser Fusion......Page 1181
23.6 Solitons and Collisionless Shock Waves......Page 1183
Bibliographic Note......Page 1190
PART VII GENERAL RELATIVITY......Page 1192
24.2 Special Relativity Once Again......Page 1194
24.2.1 Geometric, Frame-Independent Formulation......Page 1195
24.2.2 Inertial Frames and Components of Vectors, Tensors, and Physical Laws......Page 1197
24.2.3 Light Speed, the Interval, and Spacetime Diagrams......Page 1200
24.3 Differential Geometry in General Bases and in Curved Manifolds......Page 1201
24.3.1 Nonorthonormal Bases......Page 1202
24.3.2 Vectors as Directional Derivatives; Tangent Space; Commutators......Page 1206
24.3.3 Differentiation of Vectors and Tensors; Connection Coefficients......Page 1210
24.3.4 Integration......Page 1215
24.4 The Stress-Energy Tensor Revisited......Page 1217
24.5 The Proper Reference Frame of an Accelerated Observer......Page 1221
24.5.1 Relation to Inertial Coordinates; Metric in Proper Reference Frame; Transport Law for Rotating Vectors......Page 1224
24.5.2 Geodesic Equation for a Freely Falling Particle......Page 1225
24.5.3 Uniformly Accelerated Observer......Page 1227
24.5.4 Rindler Coordinates for Minkowski Spacetime......Page 1228
Bibliographic Note......Page 1231
25.1 History and Overview......Page 1232
25.2 Local Lorentz Frames, the Principle of Relativity, and Einstein’s Equivalence Principle......Page 1236
25.3 The Spacetime Metric, and Gravity as a Curvature of Spacetime......Page 1237
25.4 Free-Fall Motion and Geodesics of Spacetime......Page 1241
25.5 Relative Acceleration, Tidal Gravity, and Spacetime Curvature......Page 1247
25.5.1 Newtonian Description of Tidal Gravity......Page 1248
25.5.2 Relativistic Description of Tidal Gravity......Page 1249
25.5.3 Comparison of Newtonian and Relativistic Descriptions......Page 1251
25.6 Properties of the Riemann Curvature Tensor......Page 1254
25.7 Delicacies in the Equivalence Principle, and Some Nongravitational Laws of Physics in Curved Spacetime......Page 1258
25.7.1 Curvature Coupling in the Nongravitational Laws......Page 1259
25.8 The Einstein Field Equation......Page 1262
25.9 Weak Gravitational Fields......Page 1265
25.9.1 Newtonian Limit of General Relativity......Page 1266
25.9.2 Linearized Theory......Page 1268
25.9.3 Gravitational Field outside a Stationary, Linearized Source of Gravity......Page 1272
25.9.4 Conservation Laws for Mass, Momentum, and Angular Momentum in Linearized Theory......Page 1278
25.9.5 Conservation Laws for a Strong-Gravity Source......Page 1279
Bibliographic Note......Page 1280
26.1 Overview......Page 1282
26.2.1 The Schwarzschild Metric, Its Connection Coefficients, and Its Curvature Tensors......Page 1283
26.2.2 The Nature of Schwarzschild’s Coordinate System, and Symmetries of the Schwarzschild Spacetime......Page 1285
26.2.3 Schwarzschild Spacetime at Radii r » M: The Asymptotically Flat Region......Page 1286
26.2.4 Schwarzschild Spacetime at r ~ M......Page 1289
26.3.1 Birkhoff’s Theorem......Page 1291
26.3.2 Stellar Interior......Page 1293
26.3.3 Local Conservation of Energy and Momentum......Page 1296
26.3.4 The Einstein Field Equation......Page 1298
26.3.5 Stellar Models and Their Properties......Page 1300
26.3.6 Embedding Diagrams......Page 1302
26.4.1 The Implosion Analyzed in Schwarzschild Coordinates......Page 1305
26.4.2 Tidal Forces at the Gravitational Radius......Page 1307
26.4.3 Stellar Implosion in Eddington-Finkelstein Coordinates......Page 1308
26.4.4 Tidal Forces at r = 0—The Central Singularity......Page 1312
26.4.5 Schwarzschild Black Hole......Page 1313
26.5.1 The Kerr Metric for a Spinning Black Hole......Page 1318
26.5.3 The Light-Cone Structure, and the Horizon......Page 1320
26.5.4 Evolution of Black Holes—Rotational Energy and Its Extraction......Page 1323
26.6 The Many-Fingered Nature of Time......Page 1334
Bibliographic Note......Page 1338
27.1 Overview......Page 1340
27.2.1 Equivalence Principle, Gravitational Redshift, and Global Positioning System......Page 1341
27.2.2 Perihelion Advance of Mercury......Page 1343
27.2.3 Gravitational Deflection of Light, Fermat’s Principle, and Gravitational Lenses......Page 1346
27.2.4 Shapiro Time Delay......Page 1349
27.2.5 Geodetic and Lense-Thirring Precession......Page 1350
27.2.6 Gravitational Radiation Reaction......Page 1351
27.3.1 Weak, Plane Waves in Linearized Theory......Page 1352
27.3.2 Measuring a Gravitational Wave by Its Tidal Forces......Page 1356
27.3.3 Gravitons and Their Spin and Rest Mass......Page 1360
27.4 Gravitational Waves Propagating through Curved Spacetime......Page 1361
27.4.1 Gravitational Wave Equation in Curved Spacetime......Page 1362
27.4.2 Geometric-Optics Propagation of Gravitational Waves......Page 1363
27.4.3 Energy and Momentum in Gravitational Waves......Page 1365
27.5 The Generation of Gravitational Waves......Page 1368
27.5.1 Multipole-Moment Expansion......Page 1369
27.5.2 Quadrupole-Moment Formalism......Page 1371
27.5.3 Quadrupolar Wave Strength, Energy, Angular Momentum, and Radiation Reaction......Page 1373
27.5.4 Gravitational Waves from a Binary Star System......Page 1376
27.5.5 Gravitational Waves from Binaries Made of Black Holes, Neutron Stars, or Both: Numerical Relativity......Page 1382
27.6.1 Frequency Bands and Detection Techniques......Page 1386
27.6.2 Gravitational-Wave Interferometers: Overview and Elementary Treatment......Page 1388
27.6.3 Interferometer Analyzed in TT Gauge......Page 1390
27.6.4 Interferometer Analyzed in the Proper Reference Frame of the Beam Splitter......Page 1393
27.6.6 Pulsar Timing Arrays......Page 1396
Bibliographic Note......Page 1399
28.1 Overview......Page 1402
28.2.1 Isotropy and Homogeneity......Page 1405
28.2.2 Geometry......Page 1407
28.2.3 Kinematics......Page 1414
28.2.4 Dynamics......Page 1417
28.3.1 Baryons......Page 1420
28.3.2 Dark Matter......Page 1421
28.3.3 Photons......Page 1422
28.3.5 Cosmological Constant......Page 1423
28.4 Seven Ages of the Universe......Page 1424
28.4.1 Particle Age......Page 1425
28.4.2 Nuclear Age......Page 1428
28.4.3 Photon Age......Page 1433
28.4.4 Plasma Age......Page 1434
28.4.6 Gravitational Age......Page 1438
28.4.7 Cosmological Age......Page 1441
28.5.1 Linear Perturbations......Page 1442
28.5.2 Individual Constituents......Page 1447
28.5.3 Solution of the Perturbation Equations......Page 1451
28.5.4 Galaxies......Page 1453
28.6.1 Cosmic Microwave Background......Page 1456
28.6.2 Weak Gravitational Lensing......Page 1463
28.6.3 Sunyaev-Zel’dovich Effect......Page 1469
28.7.1 Inflation and the Origin of the Universe......Page 1472
28.7.2 Dark Matter and the Growth of Structure......Page 1481
28.7.3 The Cosmological Constant and the Fate of the Universe......Page 1485
Bibliographic Note......Page 1488
References......Page 1490
Name Index......Page 1514
Subject Index......Page 1518
Blank Page......Page 0