Elements of Astrophysics

Elements of Astrophysics

Nick Kaiser

April 21, 2002

2Preface

These are the notes that have grown out of a introductory graduate course I have given for thepast few years at the IfA. They are meant to be a primer for students embarking on a Ph.D. inastronomy. The level is somewhat shallower than standard textbook courses, but quite a broadrange of material is covered. The goal is to get the student to the point of being able to makemeaningful order-of-magnitude calculations and a number of problems are included and togive the students a fairly uniform base in the relevant physics that they can use as a starting pointand introduction to the more detailed textbooks they will need to use when they come to addressserious problems. The books that I have drawn upon extensively in devising this course are Rybickiand Lightman; Shus two-volume series; Longairs two-volume series; various sections of Landau andLifshitz (Classical theory of fields; Mechanics and Fluid Mechanics in particular); Huangs statisticalmechanics, and Binney and Tremaine. Some sections here are rather terse overviews of the relevantparts of these texts, but there are some other areas which I felt were not adequately covered, where Ihave tried to give more elaborate coverage. The reader is strongly encouraged to consult these textsalong with the present work and particularly to attempt the relevant problems contained in manyof these.

The book is organised in the following sections:

PreliminariesWe review aspects special relativity, Lagrangian and Hamiltonian dynamics,and the mathematics of random processes.

Radiation The course follows quite closely the first few chapters of Rybicki and Lightman.We review the macroscopic properties of electromagnetic radiation; we briefly review the con-cepts of radiative transfer and then consider the properties of thermal radiation and show therelation connection between the Planck spectrum and Einsteins discovery of stimulated emis-sion. The treatment of polarization in chapter is done somewhat differently and more attentionis given to radiation propagation both in the geometric optics limit and via diffraction theory.This section concludes with a general discussion of radiation from moving charges, followedby specific chapters for the important radiation mechanisms of bremsstrahlung, synchrotronradiation and Compton scattering.

Field Theory Initially an informal introduction to the matter section, this has now expandedto become a major part of the book.

Matter Starting with the reaction cross sections as computed from field theory we developkinetic theory and the Boltzmann transport equation, which in turn forms the basis for fluiddynamics. The goal is to show how the approximate, macroscopic theory is based on funda-mental physics. We then consider ideal fluids; viscous fluids; fluid instabilities and supersonicflows and shocks. Also covered here is the propagation of electromagnetic waves in a plasma.

Gravity We start with a brief review of Newtonian gravity and review properties of simplespherical model systems. We then consider collisionless dynamics, with particular emphasison their use for determining masses of astronomical systems.

Cosmology We then consider cosmology, cosmological fluctuations and gravitational lensing(TBD).

Appendices In an attempt to make the course self-contained I have included some basicresults from vector calculus and Fourier transform theory. There is a brief and simple reviewof the Boltzmann formula, and appendices on dispersive waves, the relativistic covariance ofelectromagnetism, and complex analysis.

There are still some major holes in the syllabus. Little attention is given to neutron stars andblack holes, for instance, nor to accretion disk theory or MHD. These shortcoming reflect the interestsof the author.

I am continually finding errors in the text, and am grateful to the students who have sufferedthrough the course who have pointed out many other errors.

Contents

I Preliminaries 17

1 Special Relativity 191.1 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 The 4-velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.6 The 4-acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7 The 4-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9 Relativistic Beaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.10 Relativistic Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.11 Invariant Volumes and Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.11.1 Space-Time Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.11.2 Momentum-Space Volume Element . . . . . . . . . . . . . . . . . . . . . . . . 321.11.3 Momentum-Space Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.11.4 Spatial Volume and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.11.5 Phase-Space Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.11.6 Specific Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.12 Emission from Relativistic Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.13.1 Speed and Velocity Transformation . . . . . . . . . . . . . . . . . . . . . . . . 351.13.2 Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.13.3 Geometry of Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.13.4 Relativistic decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Dynamics 372.1 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.2 The Lagrangian and the Action . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.3 The Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1.4 The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.5 Example Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4 Hamiltons Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5 Adiabatic Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 Extremal paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6.2 Schwarzschild Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3

4 CONTENTS

2.6.3 Lagrangian electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Random Fields 453.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 N -Point Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1.2 N -Point Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 Two-point Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Measuring the Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.5 Moments of the Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Variance of Smoothed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.7 Power Law Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.8 Projections of Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.9 Gaussian Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.9.1 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.9.2 Multi-Variate Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 563.9.3 Gaussian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.10 Gaussian N -point Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . 573.11 Gaussian Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.12 Ricean Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.13 Variance of the Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.14.1 Radiation autocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . 60

II Radiation 61

4 Properties of Electromagnetic Radiation 634.1 Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Macroscopic Description of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 The Specific Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.2 Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2.3 Momentum Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.4 Inverse Square Law for Energy Flux . . . . . . . . . . . . . . . . . . . . . . . 654.2.5 Specific Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.6 Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Constancy of Specific Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Thermal Radiation 715.1 Thermodynamics of Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . 71

5.1.1 Stefan-Boltzmann Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Entropy of Black-Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.3 Adiabatic Expansion Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Planck Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.2 Mean Energy per State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.3 Occupation Number in the Planck Spectrum . . . . . . . . . . . . . . . . . . 755.2.4 Specific Energy Density and Brightness . . . . . . . . . . . . . . . . . . . . . 76

5.3 Properties of the Planck Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.1 Raleigh-Jeans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.2 Wien Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.3 Monotonicity with Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.4 Wien Displacement Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.5 Radiation Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

CONTENTS 5

5.4 Characteristic Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4.1 Brightness Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4.2 Color Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.4.3 Effective Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5 Bose-Einstein Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.6.1 Thermodynamics of Black-Body Radiation . . . . . . . . . . . . . . . . . . . 785.6.2 Black body radiation and adiabatic invariance . . . . . . . . . . . . . . . . . 785.6.3 Black-body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6.4 Planck Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Radiative Transfer 816.1 Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 The Equation of Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Kirchoffs Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5 Mean Free Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.6 Radiation Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.7 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.8 Combined Scattering and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.9 Rosseland Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.10 The Eddington Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.11 Einstein A, B Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.11.1 Einstein Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.11.2 Emission and Absorption Coefficients . . . . . . . . . . . . . . . . . . . . . . 89

6.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.12.1 Main sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.12.2 Radiative Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.12.3 Eddington luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.12.4 Poissonian statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 Radiation Fields 937.1 Lorentz Force Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.2 Field Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.3 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.4 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.5 Radiation Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.5.1 Polarization of Planar Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.5.2 Polarization of Quasi-Monochromatic Waves . . . . . . . . . . . . . . . . . . 977.5.3 Power Spectrum Tensor and Stokes Parameters . . . . . . . . . . . . . . . . . 97

7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.6.1 Maxwells equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.6.2 Energy and momentum of radiation field . . . . . . . . . . . . . . . . . . . . . 100

8 Geometric Optics 1018.1 Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.2 Random Caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.2.1 Probability for Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.2.2 Caustics from Gaussian Deflections . . . . . . . . . . . . . . . . . . . . . . . . 105

6 CONTENTS

9 Diffraction Theory 1079.1 Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089.2 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

9.2.1 Babinets Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.3 Telescope Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9.3.1 The Optical Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.3.2 Properties of the Telescope PSF and OTF . . . . . . . . . . . . . . . . . . . . 1139.3.3 Random Phase Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

9.4 Image Wander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189.5 Occultation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.6 Scintillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199.7 Transition to Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.8.1 Diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.8.2 Fraunhofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.8.3 Telescope PSF from Fresnel Integral . . . . . . . . . . . . . . . . . . . . . . . 124

10 Radiation from Moving Charges 12710.1 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.2 Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.3 Lienard-Wiechart Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910.4 Dipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13210.5 Larmors Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.6 General Multi-pole Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.7 Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13510.8 Radiation Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13710.9 Radiation from Harmonically Bound Particles . . . . . . . . . . . . . . . . . . . . . . 13810.10Scattering by Bound Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13810.11Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

10.11.1Antenna beam pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.11.2Multipole radiation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.11.3Multipole radiation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.11.4Electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110.11.5Thomson drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14110.11.6Polarisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

11 Cerenkov Radiation 14311.1 Retarded Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14411.2 LW Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

12 Bremsstrahlung 14912.1 Radiation from a Single Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14912.2 Photon Discreteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15012.3 Single-Speed Electron Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15112.4 Thermal Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15112.5 Thermal Bremsstrahlung Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 15212.6 Relativistic Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15312.7 Applications of Thermal Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . 154

12.7.1 Low Frequency Emission from Ionized Gas Clouds . . . . . . . . . . . . . . . 15412.7.2 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15412.7.3 Bremsstrahlung from High Energy Electrons . . . . . . . . . . . . . . . . . . 154

12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15512.8.1 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

CONTENTS 7

13 Synchrotron Radiation 15713.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15713.2 Total Power Radiated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15813.3 Synchrotron Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15813.4 Spectrum of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

13.4.1 Pulse Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16013.4.2 Low-Frequency Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 163

13.5 Power-Law Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

14 Compton Scattering 16714.1 Kinematics of Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16714.2 Inverse Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16914.3 Inverse Compton Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16914.4 Compton vs Inverse Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 17214.5 The Compton y-Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17214.6 Repeated Scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

14.6.1 Non-Relativistic, High Optical Depth . . . . . . . . . . . . . . . . . . . . . . 17314.6.2 Highly-Relativistic, Low Optical Depth . . . . . . . . . . . . . . . . . . . . . 173

14.7 The Sunyaev-Zeldovich Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17314.8 Compton Cooling and Compton Drag . . . . . . . . . . . . . . . . . . . . . . . . . . 17414.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

14.9.1 Compton scattering 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17414.9.2 Inverse Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17514.9.3 Compton y-parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

III Field Theory 177

15 Field Theory Overview 179

16 Classical Field Theory 18316.1 The BRS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18316.2 The Continuum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18516.3 Conservation of Wave-Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18916.4 Energy and Momentum in the BRS Model . . . . . . . . . . . . . . . . . . . . . . . . 19116.5 Covariance of the BRS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19116.6 Interactions in Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 19216.7 Wave-Momentum Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19516.8 Conservation of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19616.9 Conservation of Particle Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19916.10Particle Number Conservation at Low Energies . . . . . . . . . . . . . . . . . . . . . 20116.11Ideal Fluid Limit of Wave Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

16.11.1Local Average Stress-Energy Tensor . . . . . . . . . . . . . . . . . . . . . . . 20516.11.2Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

16.12Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

17 Quantum Fields 21117.1 The Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21117.2 The Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

17.2.1 The S-Matrix Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21417.2.2 Example: A Forced Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

17.3 Free Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21517.3.1 Discrete 1-Dimensional Lattice Model . . . . . . . . . . . . . . . . . . . . . . 21517.3.2 Continuous 3-Dimensional Field . . . . . . . . . . . . . . . . . . . . . . . . . 217

17.4 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8 CONTENTS

17.4.1 Scattering off an Impurity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21817.4.2 Self Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21917.4.3 Second Order Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22117.4.4 Contour Integral Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22617.4.5 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22717.4.6 Kinematics of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22717.4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

17.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22817.5.1 Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

18 Relativistic Field Theory 22918.1 The Klein-Gordon Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22918.2 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23118.3 Connection to Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23218.4 The Scalar Field in an Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . 23318.5 Non-Relativistic Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23518.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

18.6.1 Stress-Energy Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23618.6.2 Klein-Gordon Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23618.6.3 Scalar Field Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23718.6.4 Domain Walls and Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

IV Matter 239

19 Kinetic Theory 24119.1 The Collisionless Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 24119.2 The Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24219.3 Applications of the Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 244

19.3.1 Equilibrium Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24419.3.2 Boltzmanns H-Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

19.4 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24619.4.1 Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24619.4.2 Momentum Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24719.4.3 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

19.5 Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24819.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

19.6.1 Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24819.6.2 Kinetic theory and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24919.6.3 Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24919.6.4 Massive neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

20 Ideal Fluids 25120.1 Adiabatic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25120.2 Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25220.3 Convective Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25220.4 Bernoullis Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25220.5 Kelvins Circulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25320.6 Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25520.7 Incompressible Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25520.8 Gravity Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25620.9 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25820.10Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

20.10.1 Ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

CONTENTS 9

20.10.2Potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25920.10.3Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

21 Viscous Fluids 26121.1 Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26121.2 Damping of Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26321.3 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26421.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

21.4.1 Viscous hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26421.4.2 Damping of Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26421.4.3 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

22 Fluid Instabilities 26722.1 Rayleigh-Taylor and Kelvin-Helmholtz . . . . . . . . . . . . . . . . . . . . . . . . . . 26722.2 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26722.3 Thermal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26822.4 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

22.4.1 Kolmogorov Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26922.4.2 Passive Additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27022.4.3 Inner Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27022.4.4 Atmospheric Seeing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27022.4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

22.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27222.5.1 Kolmogorov turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

23 Supersonic Flows and Shocks 27323.1 The de Laval Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27323.2 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

23.2.1 The Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27423.2.2 Vorticity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27623.2.3 Taylor-Sedov Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

23.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27623.3.1 Collisional shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

24 Plasma 27724.1 Time and Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

24.1.1 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27724.1.2 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27824.1.3 Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

24.2 Electromagnetic Waves in a Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 27924.2.1 Dispersion in a Cold Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 27924.2.2 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

24.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28324.3.1 Dispersion Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

V Gravity 285

25 The Laws of Gravity 28725.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28725.2 Newtonian Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28725.3 Spherical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

25.3.1 Newtons Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28825.3.2 Circular and Escape Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28825.3.3 Useful Spherical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

10 CONTENTS

26 Collisionless Systems 29126.1 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29126.2 Jeans Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29226.3 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29226.4 Applications of the Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

26.4.1 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29326.4.2 Galaxy Cluster Mass to Light Ratios . . . . . . . . . . . . . . . . . . . . . . . 29326.4.3 Flat Rotation Curve Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

26.5 Masses from Kinematic Tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29426.6 The Oort Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29526.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

26.7.1 Two-body Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

27 Evolution of Gravitating Systems 29727.1 Negative Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29727.2 Phase Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29827.3 Violent Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29927.4 Dynamical Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29927.5 Collisions Between Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30027.6 Tidal Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

VI Cosmology 301

28 Friedmann-Robertson-Walker Models 30328.1 Newtonian Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30328.2 Solution of the Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30428.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30628.4 The Density Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30728.5 The Cosmological Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30828.6 The Horizon Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30828.7 Cosmology with Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30928.8 Radiation Dominated Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31228.9 Number of Quanta per Horizon Volume . . . . . . . . . . . . . . . . . . . . . . . . . 31228.10Curvature of Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31328.11Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

28.11.1Energy of a Uniform Expanding Sphere . . . . . . . . . . . . . . . . . . . . . 31628.11.2Solution of the FRW Energy Equation . . . . . . . . . . . . . . . . . . . . . . 316

29 Inflation 31929.1 Problems with the FRW Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31929.2 The Inflationary Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31929.3 Chaotic Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32129.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32529.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

29.5.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

30 Observations in FRW Cosmologies 32930.1 Distances in FRW Cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

30.1.1 Scale Factor vs Hubble Parameter . . . . . . . . . . . . . . . . . . . . . . . . 32930.1.2 Redshift vs Comoving Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 329

30.2 Angular Diameter and Luminosity Distances . . . . . . . . . . . . . . . . . . . . . . 33130.3 Magnitudes and Distance Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33430.4 K-Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

CONTENTS 11

31 Linear Cosmological Perturbation Theory 33731.1 Perturbations of Zero-Pressure Models . . . . . . . . . . . . . . . . . . . . . . . . . . 337

31.1.1 The Spherical Top-Hat Perturbation . . . . . . . . . . . . . . . . . . . . . . 33731.1.2 General Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

31.2 Non-zero Pressure and the Jeans Length . . . . . . . . . . . . . . . . . . . . . . . . . 34331.2.1 Matter Dominated Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34331.2.2 Radiation Dominated Era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34431.2.3 Super-Horizon Scale Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 34731.2.4 Isocurvature vs Isentropic Perturbations . . . . . . . . . . . . . . . . . . . . . 34731.2.5 Diffusive Damping and Free-Streaming . . . . . . . . . . . . . . . . . . . . . . 348

31.3 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34931.3.1 The Adiabatic-Baryonic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 34931.3.2 The Hot-Dark-Matter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 35131.3.3 The Cold Dark Matter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

32 Origin of Cosmological Structure 35532.1 Spontaneous Generation of Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 35532.2 Fluctuations from Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35832.3 Self-Ordering Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

32.3.1 Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36132.3.2 Cosmic Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

33 Probes of Large-Scale Structure 36933.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36933.2 Galaxy Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

33.2.1 Redshift Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37033.2.2 Poisson Sample Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37033.2.3 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37133.2.4 The Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37333.2.5 Redshift Space Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37433.2.6 Angular Clustering Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

33.3 Bulk-Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37833.3.1 Measuring Bulk-Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

33.4 Microwave Background Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . 38033.4.1 Recombination and the Cosmic Photosphere . . . . . . . . . . . . . . . . . . 38033.4.2 Large-Angle Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38033.4.3 Small-Angle Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38133.4.4 Polarization of the CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

33.5 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

34 Non-Linear Cosmological Structure 38534.1 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38534.2 Gunn-Gott Spherical Accretion Model . . . . . . . . . . . . . . . . . . . . . . . . . . 38734.3 The Zeldovich Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38834.4 Press-Schechter Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39034.5 Biased Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39034.6 Self-Similar Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39134.7 Davis and Peebles Scaling Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39334.8 Cosmic Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

12 CONTENTS

VII Appendices 395

A Vector Calculus 397A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397A.2 Vector Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397A.3 Div, Grad and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397A.4 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398A.5 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398A.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

A.6.1 Vector Calculus Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

B Fourier Transforms 401B.1 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401B.2 Continuous Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402B.3 Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403B.4 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403B.5 Wiener-Khinchin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403B.6 Fourier Transforms of Derivatives and Integrals . . . . . . . . . . . . . . . . . . . . . 403B.7 Fourier Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404B.8 Utility of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404B.9 Commonly Occurring Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404B.10 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405B.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

B.11.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

C The Boltzmann Formula 409

D Dispersive Waves 411D.1 The Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411D.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413D.3 Evolution of Dispersive Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

E Relativistic Covariance of Electromagnetism 421E.1 EM Field of a Rapidly Moving Charge . . . . . . . . . . . . . . . . . . . . . . . . . . 422

F Complex Analysis 425F.1 Complex Numbers and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425F.2 Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425F.3 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426F.4 Contour Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

List of Figures

1.1 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.2 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3 Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4 Rotated Lorentz Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5 Four Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.6 Relativistic Beaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.7 Relativistic Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.8 Spatial Volume Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2 Adiabatic Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1 Power-Spectrum and Auto-Correlation Function . . . . . . . . . . . . . . . . . . . . 483.2 Measuring the Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 Fields with Power-Law Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1 Specific Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Net Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Radiation Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Invariance of the Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.6 Invariance of the Intensity in a Telescope . . . . . . . . . . . . . . . . . . . . . . . . 68

5.1 A Cylinder Containing Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Planck Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.1 Einstein Coefficents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.1 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

8.1 Formation of a Caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.2 Generic Form of a Caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.1 Huygens Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.2 Fresnel cos(x2) Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099.3 Fresnel Knife-Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.4 Babinets Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119.5 Refracting Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.6 PSF for a Square Pupil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149.7 Circular Pupil PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.8 Wavefront Deformation from Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 1169.9 Speckly PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179.10 Scintillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

13

14 LIST OF FIGURES

9.11 Telescope with Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229.12 Geometric vs Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

10.1 Source for Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.2 Lienard-Wiechart Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13110.3 Dipole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13210.4 Bounded Charge Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.5 Quadrupole Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13510.6 Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

11.1 Conical Shock Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14311.2 Cerenkov Radiation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14511.3 Light Cone-Particle Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14611.4 Pulse of Cerenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

12.1 Electron-Ion Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15012.2 Thermal Bremsstrahlung Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

13.1 Critical Frequency for Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . 15913.2 Geometry for Synchrotron Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 16013.3 Observer Time vs Retarded Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16113.4 Retarded vs Observer Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16213.5 Potential for a Pulse of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . 16313.6 Field for a Pulse of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . 16313.7 Synchrotron Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

14.1 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16814.2 Inverse Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

16.1 Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18416.2 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18516.3 BRS Phase and Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18816.4 The 4 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19316.5 The 2 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19416.6 Interacting Complex Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

17.1 First Order Phonon-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 22117.2 Chion Decay and Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22117.3 Second Order Phonon-Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . . . 22317.4 Phonon-Phonon Scattering Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 22417.5 Phonon-Phonon Scattering Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 22517.6 Twisted Phonon-Phonon Scattering Diagram . . . . . . . . . . . . . . . . . . . . . . 22517.7 Contour Integral for the Chion Propoagator . . . . . . . . . . . . . . . . . . . . . . . 226

18.1 W-Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23118.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

19.1 Two-Body Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

20.1 Bernoulli Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25320.2 Kelvins Circulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25420.3 Gravity Wave Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

21.1 Shear Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

22.1 Thermal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

LIST OF FIGURES 15

22.2 Atmospheric Wavefront Corrugation . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

23.1 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

24.1 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27724.2 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

27.1 Phase Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

28.1 FRW Scale Factor (linear plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30528.2 FRW Scale Factor (logarithmic plot) . . . . . . . . . . . . . . . . . . . . . . . . . . . 30628.3 Density Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30728.4 Causal Structure of the FRW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30928.5 Microwave Background Photon World-Lines . . . . . . . . . . . . . . . . . . . . . . . 31028.6 Mass of FRW Closed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31628.7 Embedding Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

29.1 Horizon-Scale in Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32229.2 Chaotic Inflation Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

30.1 Comoving Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33130.2 Angular Diameter Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33230.3 Angular Diameter Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33230.4 Luminosity Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

31.1 Spherical Top-Hat Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33831.2 Isocurvature and Isentropic Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 34831.3 Adiabatic-Baryonic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35031.4 Adiabatic-Baryonic Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 35031.5 Hot Dark Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35231.6 Cold Dark Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

32.1 Monopole, Dipole and Quadrupole Perturbations . . . . . . . . . . . . . . . . . . . . 35732.2 Multiple Quadrupole Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35832.3 Scalar Field Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36232.4 Domain Wall Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36332.5 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36432.6 2 Dimensional Scalar Field Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

33.1 The Tully-Fisher Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37933.2 Large-Angle CMB Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38133.3 Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

34.1 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38634.2 Gunn-Gott Accretion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38734.3 Biased Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39234.4 Self-Similar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

B.1 Common Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405B.2 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

C.1 Boltzmann Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

D.1 Beating of 2 Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412D.2 Evolution of a Wave Packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414D.3 Fourier Transform of a Dispersive Wave . . . . . . . . . . . . . . . . . . . . . . . . . 415

16 LIST OF FIGURES

D.4 Gravity Wave Chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418D.5 Gravity Wave Swell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

E.1 Electric Field of a Moving Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

F.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426F.2 Contour Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Part I

Preliminaries

17

Chapter 1

Special Relativity

A remarkable feature of Maxwells equations is that they support waves with a unique velocity c,yet there is no underlying medium with respect to which this velocity is defined (in contrast to saysound waves in a physical medium). An equally remarkable observational fact is that the velocity ofpropagation of light is indeed independent of the frame of reference of the observer or of the source(Michelson and Morley experiment). Searches for the expected aether drift proved unsuccessful.These results would seem to conflict with Galilean relativity in which there is a universal time, anduniversal Cartesian spatial coordinates such that each event can be assigned coordinates on whichall observers can agree. Einsteins special theory of relativity makes sense of these results. Theresult is a consistent framework in which events in space-time are assigned coordinates, but wherethe coordinates depend on the state of motion of the observer. The situation is rather analogousto that in planar geometry, where the coordinates of a point depend on the origin and rotationof ones chosen frame of reference. However, one can also use vector notation to express relationsbetween lines and point e.g. a + b = c which are valid for all frames of reference. In specialrelativity the fundamental quantities are points, or events, which are vectors in a 4-dimensionalspace-time. We will see how these 4-vectors transform under changes in the observers frame ofreference, and how particle velocities, momenta and other physical quantities can be expressed inthe language of 4-vector notation. Indeed the fundamental principle of relativity is that all of thelaws of physics can be expressed in a frame invariant manner. The last part of the chapter dealswith the transformation properties of distribution functions (e.g. the density of particles in space,or the distribution of particles over energy, velocity etc).

1.1 Time Dilation

An immediate consequence of the frame-independence of the speed of light is that observers inrelative motion with respect to one another must assign different time separations to events.

Consider an observer A with a simple gedanken clock consisting of a photon bouncing betweenmirrors attached to the ends of a standard rod of length l0 as illustrated in figure 1.1. One roundtrip of the photon takes an interval t0 = 2l0/c in the rest-frame of the clock. Now consider thesame round trip as seen from the point of view of an observer B moving with some relative velocityv in a direction perpendicular to the rod.

First, note that A and B must assign the same length to the rod. To see this imagine B carriesan identical rod, also perpendicular to his direction of motion, with pencils attached which makemarks on As rod as they pass. Since the situation is completely symmetrical, the marks on Asrod must have the same separation as the pencils on Bs. Thus transverse spatial dimensions areindependent of the frame of state of the observer.

From Bs point of view then the distance traveled by the photon in one round trip must exceed2l0, and consequently the time interval between the photons departure and return is t > t0. Inthis time, As rod has moved a distance vt, so, by Pythagoras theorem the total distance traveled

19

20 CHAPTER 1. SPECIAL RELATIVITY

-

6

x

y6

?

l0

A6

?

-

6

x

y6

?

l0

B

-vt

l20 + (vt/2)

2

CCCCCCCCCCCCCCCCCCCW

Figure 1.1: Illustration of time-dilation. Left panel shows one click of the gedanken clock in As frameof reference. Right panel shows the path of the photon as seen by a moving observer. Evidently, ifthe clock is moving the photon has to travel futher, so the interval between clicks for a moving clockis greater that if the clock is at rest. The time dilation factor is = 1/

1 v2/c2.

by the photon is 2l20 + (vt/2)2 = ct and solving for t gives

t = t0/1 v2/c2 (1.1)

or, defining the Lorentz gamma-factor

1/1 v2/c2 (1.2)

we have the time dilation formulat = t0. (1.3)

Thus moving clocks run slow by the factor . This is a small correction for low velocities, butbecomes very large as the velocity approaches c. This behavior is not paradoxical, since the situationis not symmetrical; the two events (departure and return) occur at the same point of space in Asframe of reference, whereas they occur at different positions in Bs frame. The spatial separation ofthe two events in Bs frame is x = vt, and the temporal separation is t = t0/

1 v2/c2 and

so we havet2 x2/c2 = t20. (1.4)

The quantityt0 =

t2 x2/c2 (1.5)

is called the proper time interval between two events. It is invariant; ie it is the same for all observersin a state of constant relative motion, even though they assign different spatial and temporal intervalsto the separation of the events. It is equal to the temporal separation of the events as measured

1.2. LENGTH CONTRACTION 21

by an observer for whom the two events occur at the same point in space. Any other observer willassign a greater temporal separation.

1.2 Length Contraction

We can derive the Lorentz-Fitzgerald length contraction formula in a very similar fashion. Let usequip A with a pair of clocks mounted back to back so that a pair of photons repeatedly depart fromA, travel equal and opposite distances l0, bounce off mirrors, and then return to A. A space-timediagram of one cycle of the clock as perceived by A is shown on the left hand side of figure 1.2.

The same set of events as perceived by an observer B now moving with constant velocity parallelto the arms of the clock is shown on the right for the case of v = c/2. Set the origin of coordinates atthe emission point, and let the length of the arms in Bs rest frame be l. The two photons propagatealong trajectories x = ct until they reach the mirrors, which have world lines x = l+ vt. Solvingfor the reflection times t and locations gives

ct = x = l1 v/c . (1.6)

The return flight of each photon is the same as the outward flight of the other photon, so the totaltime t elapsed between departure and return to A satisfies

ct = c(t+ + t) = l[

11 v/c +

11 + v/c

]=

2l1 v2/c2 = 2

2l. (1.7)

However, we know that t = t0 = 2l0/c and hence we obtain the Lorentz-Fitzgerald lengthcontraction formula

l = l0/. (1.8)

This is sometimes stated as moving rods appear foreshortened. More precisely, we have shownthat two events which occur at the same time in one frame and have separation l in that frame willhave a spatial separation in a relatively moving frame of l0 = l > l. Consider, for example thereflection events. In As frame these occur at the same time, so t0 = 0, and have spatial separationx0 = 2l0. In Bs frame however, they have temporal separation (times c) of

ct = c(t+ t) = 22lv/c = 2l0v/c (1.9)

and spatial separationx = x+ x = 22l = 2l0. (1.10)

Evidentlyx2 c2t2 = 4l202(1 v2/c2) = (2l0)2 = x20 (1.11)

is also an invariant. The quantity

x0 =x2 c2t2 (1.12)

is known as the proper distance between the two events.Finally, the area of the region enclosed by the photon world lines is, in As frame, A0 = (

2l0)2

whereas in Bs frame

A = (2x+)(

2x) =

2l202(1 v2/c2) = A0. (1.13)

so this area is an invariant. Since transverse dimensions are invariant, this means that the space-time4-volume is an invariant.

22 CHAPTER 1. SPECIAL RELATIVITY

6ct

-x

s

ss

s

se se-l0

6

?

2l0

S M+M

r+ r

ss

ss

se se-l

?

6

2l0

r+

r

S M+M

Figure 1.2: Space-time diagram used to derive length contraction. On the left are shown the photontrajectories (wiggly diagonal lines) departing from the source S (whose world-line is the centralvertical line) and reflecting from mirrors M, M+ (also with vertical world-lines) and returning tothe source S. This is for a stationary clock, and the interval between clicks of the clock the timebetween departure and return of the photons is ct = 2l0. The slanted lines on the right showthe world lines for the source/receiver and mirrors for a clock which is moving at a constant velocity(v = c/2 in this case). Since the speed of light is invariant, the photons still move along 45-degreediagonal lines. Now we have already seen that the interval between clicks for a moving clock islarger than that for a stationary clock by a factor . This means that the time between emissionand return for the moving clock is ct = 2l0. It is then a matter of simple geometry (see text) toshow that the distance between the source and the mirrors l is smaller than that for the stationaryclock by a factor , or l = l0/. This is the phenomenon of relativistic length contraction; if we havea metre rod moving in a direction parallel to its length then at a given time the distance betweenthe ends of the rod is less than 1 metre by a factor 1/. The above description is of two differentclocks, viewed in a single coordinate system. There is a different, and illuminating, alternative wayto view the above figure. We can think of these two pictures as being of the same clock andindeed the very same set of emission, reflection and reception events but as viewed from twodifferent frames of reference. The left hand picture shows the events as recorded by an observer whosees the clock as stationary while the right hand picture is the events as recorded by an observermoving at velocity v = c/2 with respect to the clock. Now consider the reflection events, labelledr and r+. In the clock-frame these events have spatial separation x = 2l0, while in the movingframe simple geometrical analysis shows that the spatial separation is x = 2l0. Now we aresaying that the separation of the mirrors is larger for the moving clock, whereas before we weresaying the moving clocks rods were contracted. This sounds contradictory, or paradoxical, but itisnt really. The resolution of the apparent paradox is that the situation is again non-symmetricalbetween the two frames. The reflection events occur at the same time in the clocks rest frame, andthe separation is the so-called proper-separation x0 = 2l0. In the moving frame the two eventshave a time coordinate difference t 6= 0, and the spatial separation, as we show in the text, is nowx =

x20 + c2t2. In the earlier discussion we were computing the distance between the two

events e, e which occur same time in the moving frame. These events in the rest frame do not occurat the same coordinate time.

1.3. LORENTZ TRANSFORMATION 23

-

6

x

ct

-

6

x

ct

Figure 1.3: The Lorentz transformation causes a shearing in the x t space. This was shown abovefor an area bounded by null curves, but the result is true for arbitrary areas.

1.3 Lorentz Transformation

Figure 1.2 shows that the effect of a boost on the area in the x ct plane bounded by the photonpaths is to squash it along one diagonal direction and to stretch it along the other. The same is truefor any area, as illustrated in figure 1.3. In fact, one can write the transformation for the spatialcoordinates as multiplication by a matrix, whose coefficients are a function of the boost velocity.

In this section it will prove convenient to work in units such that c = 1 (or equivalently let t = ctand drop the prime), so photon world lines are diagonals in x t space.

Lets now determine the form of this transformation matrix for the case of a boost along thex-axis. For such a boost, we know that the y and z-cooordinates are unaffected, so we need onlycompute how the x and t coordinates are changed. Consider first what happens if we take the x tplane and rotate it by 45 degrees. Specifically, lets define new coordinates[

TX

]= R(45)

[tx

]=[cos(45) sin(45)sin(45) cos(45)

] [tx

]=

12

[1 11 1

] [tx

]. (1.14)

Now we saw in the previous section that in this rotated frame the effect of a boost is just a stretch inthe horizontal direction with scale factor S+ = x+/l0 = 1/(1 v) and a contraction in the verticaldirection with scale factor S = x/l0 = 1/(1 + v). The effect of the boost on position vectors inthis 45-degree rotated system is just multiplication by the 2 2 matrix

M =[S+ 00 S

]. (1.15)

or, denoting coordinates in the boosted frame by syperscript,[T

X

]=[S+ 00 S

] [TX

]=[S+TSX

]. (1.16)

The effect of these linear transformations is illustrated in figure 1.4.So far we have obtained the linear transformation matrix for transforming the rotated X T

coordinates. What we really want is the matrix that transforms un-rotated x t coordinates. Thisis readily found since we have[

t

x

]= R1

[T

X

]= R1M

[TX

]= R1MR

[tx

](1.17)

with R = R(45) the rotation matrix for a 45 degree rotation. Evidently, the transformation fromx t to boosted x t coordinates is effected by multiplying by the matrix M = R1MR, or

M =12

[1 11 1

] [S+ 00 S

] [1 11 1

]=

12

[S+ + S S+ + SS+ + S S+ + S

]. (1.18)

24 CHAPTER 1. SPECIAL RELATIVITY

-t

6x

-T

6X

-T

6X

Figure 1.4: A region of 2-dimensional x t space-time bounded by photon world lines is shown inthe left hand panel. The center panel shows the same region in X T coordinates, which are justx t coordinates rotated through 45. The right panel shows the same region after applying a boostalong the x-axis.

but S+ + S = 2 and S+ S = 2 where = v/c. Therefore the transformation of x tcoordinate vectors induced by a boost of dimensionless velocity is[

t

x

]=M

[tx

]=[

] [tx

]=[(t x)(x t)

]. (1.19)

Finally, recalling that transverse dimensions y, z are unaffected by a boost in the x-direction weobtain the full transformation as a 4 4 matrix multiplication

t

x

y

z

=

11

txyz

=(t x)(x t)

yz

. (1.20)This is known as the Lorentz transformation.

1.4 Four-vectors

The prototype 4-vector is the separation between two space-time events

x =

x0

x1

x2

x3

=ctxyz

(1.21)which transforms under a boost v/c = as

x = x (1.22)

where the 4 4 transformation matrix is

=

11

(1.23)Summation of repeated indices is implied, and such summations should generally involve one sub-script index and one superscript index.

1.4. FOUR-VECTORS 25

Alternative notation for four vectors is

x = ~x = (ct,x) = (ct, xi) (1.24)

where i = 1, 2, 3.We have seen that the Lorentz transformation matrix corresponds to a diagonal shearing in the

x t subspace. The determinant of is unity, in accord with the invariance of space-time volumepreviously noted.

This is for a boost along the x-axis. The transformation law for a boost in another direction canbe found by multiplying matrices for a spatial rotation and a boost. There are also generalizationsof (1.20) which allow for reflections of the coordinates (including time). See any standard text fordetails.

The vector x is a contravariant vector. It is also convenient to define a covariant 4-vector whichis equivalent, but is defined as

x =

ctxyz

(1.25)with a subscript index to distinguish it.

The two forms of 4-vector can be transformed into each other by multiplying by a 4 4 matrixcalled the Minkowski metric

= =

1

11

1

. (1.26)since clearly x = x and x = x .

There is a version of the Lorentz matrix that transforms covariant 4-vectors as x =

x

which is related to by =

. (1.27)

The norm of the 4-vector ~x is defined as

s2 = ~x ~x = xx = c2t2 + x2 + y2 + z2 = x x c2t2 (1.28)which we recognize as the invariant proper separation of the events. It can be computed as s2 =x

x etc. If the norm is positive, negative or zero the separation is said to be space-like, time-like and null respectively. Since the norm is invariant, a separation which is space-like in one framewill be space-like in all inertial frames etc.

A 4-component entity ~A is a four-vector if it transforms in the same way as ~x under boosts (aswell as spatial transformations, rotations etc).

The scalar product of two four vectors ~A, ~B is defined as

~A ~B = AB = A0B0 +A B (1.29)and it is easy to show that the scalar product is invariant under Lorentz transformations.

The gradient operator in space-time is a covariant vector since we require that the differencebetween the values of some scalar quantity f at two neighboring points

df = ~dx ~f = dx x

f (1.30)

should transform as a scalar (ie be invariant). We often write the gradient operator as ~ = /x. We will also use the notation y = y, to denote partial derivatives with respect to spacetime coordinates.

A 4 4 matrix T is a contravariant rank-2 tensor if its components transform in the samemanner as AB . Covariant rank-2 tensors T , or mixed rank-2 tensors T are defined similarly,as are higher rank tensors.

26 CHAPTER 1. SPECIAL RELATIVITY

Example rank-2 tensors are and (the Kronecker -symbol). These are both constant;the components have the same numerical value on all inertial frames. Other examples are theouter product of a pair of vectors AB , and the gradient of a vector field A .

Tensors can be added, so A +B is a tensor. Higher order tensors can be obtained by taking outer products of tensors, such as T =AB.

Indices can be raised and lowered with the Minkowski metric. Pairs of identical indices can be summed over to construct tensors, vectors of lower rankby contraction. For example, on can make a vector by contracting a (mixed) rank-3 tensorA = T . It is important that one contract on one upstairs and one downstairs index. Ifnecessary, one should raise or lower an index with the Minkwski metric.

The fundamental principle of special relativity is that all of the laws of physics can be expressedin terms of 4-vectors and tensors in an invariant manner.

1.5 The 4-velocity

The coordinates of a particle are a 4-vector, as is the difference of the coordinates at two pointsalong its world-line. For two neighboring points or events, we can divide by the proper-time dbetween the events, ie the interval between the events as measured by an observer moving with theparticle and which is a scalar, to obtain the 4-velocity

~U =dx

d(1.31)

which is a contravariant 4-vector.If the particle has 3-velocity u relative to our inertial frame, then the two events in our frame

have temporal coordinate separation dt = ud , with u 1/1 u u/c2 as usual, and hence

the particles 4-velocity is related to its coordinate velocity by U0 = dx0/d = cdt/d = cu andU i = dxi/d = udxi/dt or

~U = u

[cu

]. (1.32)

If we undergo a boost along the x-axis of velocity = v/c into some new inertial frame then thecomponents of the particles 4-velocity transform as

U 0 = (U0 U1)U 1 = (U1 U0)

U 2 = U2

U 3 = U3

(1.33)

These relations can be used to show how speeds and velocities of particles transform underboosts of the observers frame of reference as follows: The first of equations (1.33) with U0 = uc,U1 = uu1 etc and with x-component of the coordinate velocity in the unprimed frame u1 = u cos gives

u = u(1 uv

c2cos

)(1.34)

which allows one to transform the particles Lorentz factor u, and therefore also the particles speed|u|, under changes in inertial frame.

The second of equations (1.33) gives uu1 = (uu1 cu) = u(u1 v) or

u1 =u1 v

1 vu1/c2 (1.35)

1.6. THE 4-ACCELERATION 27

which is the transformation law for the coordinate velocity.If the particle is moving along the x-axis at the speed of light in the unprimed frame (u1 = c)

then then the velocity in the unprimed frame is u1 = (c v)/(1 v/c) = c. This is in accord withthe constancy of the speed of light in all frames.

Finally, in the rest frame of the particle, ~U = (c, 0), so dotting some vector with a particles4-velocity is a useful way to extract the time component of the 4-vector as seen in the particlesframe of reference.

1.6 The 4-acceleration

The four-acceleration is

~A =d~U

d. (1.36)

and is another 4-vector.The scalar product of the 4-acceleration and the 4-velocity is ~A ~U = d(~U ~U)/d which vanishes

because the squared length of a 4-vector is ~U ~U = c2, which is invariant. Thus the four accelerationis always orthogonal to the 4-velocity.

In terms of the coordinate 3-velocity, the 4-acceleration is

~A = (d(c)/dt, d(u)/dt). (1.37)

and a little algebra gives the norm of the 4-acceleration in terms of the particles coordinate accel-eration and coordinate velocity as

~A ~A = 4(a a+ 2(u a/c)2). (1.38)

In the particles rest-frame ~U = (c, 0, 0, 0) so A0 = 0, and therefore the norm is just equal to thesquare of the proper acceleration: ~A ~A = |a0|2 so (1.38) gives the acceleration felt by a particle interms of the coordinate acceleration in the observers frame of reference.

If we decompose the 3-acceleration into components a and a which are perpendicular andparallel to the velocity vector u it is easy to show that

~A ~A = |a0|2 = 4(a2 + 2a2) (1.39)

Of particular interest is the case a = a, as is the case for a particle being accelerated by a staticmagnetic field. In that case, the rest-frame acceleration is larger than in the lab-frame by a factor2. This is easily understood. Observers in different inertial frames agree on the values of transversedistances as these are not affected by the Lorentz boost matrix. The second time derivative of thetransverse position of the particle is larger in the instantaneous rest-frame than in the lab-frame,simply because time runs faster, by a factor , in the rest-frame. This will prove useful when wewant to calculate relativistic synchrotron radiation.

1.7 The 4-momentum

Multiplying the 4-velocity of a particle by its rest mass m (another invariant) gives the four-momentum

~P = m~U = m(c,u). (1.40)

The spatial components of the 4-momentum differ from the non-relativistic form by the factor .To see why this is necessary consider the situation illustrated in figure 1.5.

Some texts use the notation m0 for the rest-mass and set m = m0. The space components ofthe relativistic 4-momentum are then P = mu, just as in non-relativistic mechanics. We do notfollow that convention.

28 CHAPTER 1. SPECIAL RELATIVITY

A A- -

B B

}PPPPPPPPPPPq }

1 }

} PPPPPPPPP

PPi}) }A A- -

B,B

}XXXXXXXXXXXXXXz }

: }

}}6?

Figure 1.5: Illustration of the relativistic form for the momentum. Two observers A and B pass eachother on rapidly moving carriages and as they do so they bounce balls off each other, exchangingmomentum. The upper panel shows the symmetric situation in the center of mass frame. The lowerpanel shows the situation from Bs point of view. Now B assigns a longer time interval to the pairof events A and A than to B, B while transverse distances are invariant so it follows that thetransverse velocity he assigns to As ball is lower than his own by a factor . Thus, in Bs framemux is not conserved, but mux is conserved in the collision.

The time component of the 4-momentum is

cP 0 = mc2 =mc2

1 v2/c2 = mc2 +

12mv2 + . . . (1.41)

which, aside from the constant mc2 coincides with the kinetic energy for low velocities, and we callcP 0 = E the total energy. The 4-momentum is

~P = (E/c,P). (1.42)

The 4-momentum for a massive particle is a time-like vector and its invariant squared length is

E2/c2 P P = m2c2. (1.43)Massive particles are said to live on the mass-shell in 4-momentum space.

All these quantities and relations are well-defined in the limit m 0. For massless particlesE2 = |P|2c2, and with E = h, P = hk the 4-momentum is then

~P = h(/c,k). (1.44)

The total 4-momentum for a composite system is the sum of the 4-momenta for the componentparts, and all 4 components are conserved. Note that the mass of a composite system is not thesum of the masses of the components, since the total mass contains, in addition to the rest mass,any energy associated with internal motions etc.

1.8. DOPPLER EFFECT 29

source frame

z -

6

x

y

z -v

-

6

:

observer frame

x

y

Figure 1.6: Left panel shows a photon emitted from a source as seen in the rest-frame. Right panelshows the situation in the observer frame in which the source has velocity v = vx.

1.8 Doppler Effect

Consider a photon which in the frame of some observer has 4-momentum

~P0 =E0c

1

cos 0sin 00

. (1.45)If the emitter is moving with velocity v = vx with respect to the observer then the 4-momentum

in the emitters frame is

~P Ec

1

cos sin 0

= ~P = E0c(1 cos 0)(cos 0 )

sin 00

. (1.46)The observed energy is therefore related to the energy in the emitters frame by

E0 =E

(1 cos 0) (1.47)

which is the Doppler formula.

1.9 Relativistic Beaming

Consider a source which emits radiation isotropically in its rest frame. What is the angular distri-bution of radiation in some other inertial frame?

Let a particular photon have source-frame 4-momentum

~P =E

c

1

cos sin cossin sin

(1.48)and let the source have velocity v = vx in the observer frame. The observer therefore has velocityv = vx in the source-frame, and so the photon 4-momentum in the observer frame (primed frame)

30 CHAPTER 1. SPECIAL RELATIVITY

is

~P =E

c

1

cos

sin cos

sin sin

= Ec(1 + cos )(cos + )sin cossin sin

. (1.49)Comparing the ratio Py/Pz in the two frames shows that = ; the azimuthal angle is the same inboth frames. Comparing Py/Px reveals that

tan =sin

(cos + ). (1.50)

For large velocities c so 1 and this results in the photon trajectories in the observer framebeing confined to a narrow cone of width 1/ around the direction of motion of the source. Forexample, consider equatorial rays in the source-frame for which = 90. In the observer framethese have

tan =1

' 1

(1.51)

so the width of the beam is on the order of 1/ for 1. This result will be useful when weconsider synchrotron radiation.

It is also interesting to consider the energy flux in the beam. The Doppler formula says that theenergy of the photons are boosted by a factor

h

h=

1(1 cos ) . (1.52)

Now cos ' (1 2/2 + . . .) and = (1 2)1/2 ' (1 2/2 + . . .) so

(1 cos ) ' (1 (1 2/2)(1 2/2) ' 2(2 + 2) 1/, (1.53)

where we have used 1/ for 1. The typical energy boost factor is therefore h/h .These photons are compressed by a factor 2 in angular width so the energy per unit area isincreased by a factor 3. What about the rate at which this energy flows? Consider a finitewave train of N waves. This will be emitted in time t = N/ in the rest frame, but will pass ourobserver in time t = N/ t/, with the net result that the energy flux (ie the energy per unitarea per unit time) is increased by a factor 4.

We will consider the transformation of radiation intensity more rigorously below.

1.10 Relativistic Decays

As an example of the use of 4-momentum conservation, consider a massive particle of massM whichspontaneously decays into two lighter decay products of mass m1 and m2 with energies (as measuredin the rest-frame of the initial particle) E1 and E2 (see figure 1.7). We shall set c = 1 for clarity inthis section.

Conservation of energy and momentum gives

M = E1 + E20 = P1 +P2

(1.54)

where we are using units such that c = 1. The latter tells is that |P1|2 = |P2|2, but |P1|2 = E21 m21and so 4-momentum conservation can also be written as

M = E1 + E2E21 m21 = E22 m22 (1.55)

1.11. INVARIANT VOLUMES AND DENSITIES 31

before

after

M ~P = (M,0)

m1~P1 = (E1,P1)

m2 -~P2 = (E2,P2)

Figure 1.7: Decay of a heavy particle of mass M into two lighter decay products m1, m2. Fourmomenta in the rest frame of the decaying particle are indicated.

and solving this pair of equations for the two unknowns E1, E2 yields

E1 =M2 +m21 m22

2ME2 =

M2 m21 +m222M

(1.56)

which are the energies of the decay products in the center-of-momentum frame.Now consider the inverse process where two energetic particles collide and merge to form a heavier

particle. The sum of the particle energies in the center of momentum frame then sets a threshold;this is the maximum mass particle that can be created. If we fire two equal mass particles at eachother with energy E1 = E2 then the available energy isM = E1+E2. If, on the other hand, we fire aparticle of mass E1 at a stationary target of massm2, then the total energy of the resulting particle isE = E1+m2 and the total momentum of the product is P = P1, or equivalently P 2 = P 21 = E

21m21

and so the mass of the product is

M2 = E2 P 2 = (E1 +m2)2 E21 +m22 = m21 +m22 + 2m2E1. (1.57)In the highly relativistic case where E1 m1, m2 the mass threshold isM '

2m2E1 which is much

less than the mass threshold if one were to collide two particles of energy E1 in a head-on collision.This is because in the stationary target case most of the energy is carried off in the momentum ofthe resulting particle, and the mass threshold is reduced by a factor 1/. This explains why thehighest energy collisions are obtained in particle accelerators which collide counter-rotating beamsof particles and anti-particles.

1.11 Invariant Volumes and Densities

Boosts of the observer induce changes in the spatio-temporal coordinates of events and therebymodify the 3-volume of