Einstein’s Physics

Atoms, Quanta, and Relativity Derived, explained, and appraised

Ta-Pei Cheng University of Missouri - St. Louis Portland State University

To be published by Oxford University Press 2013 Contents

I ATOMIC NATURE OF MATTER 1

1 Molecular size from classical fluid 3 1.1 Two relations of molecular size and the Avogadro number . . 4 1.2 The relation for the effective viscosity ...... 6 1.2.1 The equation of motion for a viscous fluid ...... 7 1.2.2 Viscosity and heat loss in a fluid ...... 8 1.2.3 Volume fraction in terms of molecular dimensions . . . 11 1.3 The relation for the diffusion coefficient ...... 11 1.3.1 Osmoticforce...... 12 1.3.2 Frictional drag force — the Stokes law ...... 13 1.4 SuppMat: Basics of fluid mechanics ...... 15 1.4.1 The equation of continuity ...... 15 1.4.2 The Euler equation for an ideal fluid ...... 16 1.5 SuppMat: Calculating the effective viscosity ...... 18 1.5.1 The induced velocity field v′ ...... 18 1.5.2 The induced pressure field p′ ...... 20 1.5.3 Heat dissipation in a fluid with suspended particles . . 20 1.6 SuppMat: The Stokes formula for viscous force ...... 24

2 The Brownian motion 27 2.1 Diffusion and Brownian motion ...... 29 2.1.1 Einstein’s statistical derivation of the diffusion equation 29 2.1.2 The solution of the diffusion equation and the mean- squaredisplacement ...... 32 2.2 Fluctuations of a particle system ...... 32 2.2.1 Randomwalk...... 33 2.2.2 Brownian motion as a random walk ...... 34 2.3 The Einstein-Smoluchowski relation ...... 34 2.3.1 Fluctuation and dissipation ...... 36

iii iv CONTENTS

2.3.2 Mean-square displacement and molecular dimensions . 36 2.4 Perrin’s experimental verification ...... 37

II QUANTUM THEORY 41

3 Blackbody radiation: From Kirchhoff to Planck 43 3.1 Radiation as a collection of oscillators ...... 45 3.2 Thermodynamics of blackbody radiation ...... 47 3.2.1 Radiation energy density is an universal function . . . 47 3.2.2 The Stefan-Boltzmann law ...... 49 3.2.3 Wien’s displacement law ...... 50 3.2.4 Planck’s distribution proposed ...... 54 3.3 Planck’s investigation of cavity oscillator entropy ...... 54 3.3.1 Relating the oscillator energy to the radiation density 55 3.3.2 The mean entropy of an oscillator ...... 55 3.4 Planck’s statistical analysis ...... 57 3.4.1 Calculating the complexion of Planck’s distribution . . 57 3.4.2 The Planck’s constant and the Boltzmann’s constant . 61 3.4.3 Planck’s energy quantization proposal — a summary . 62 3.5 SuppMat: Radiation oscillator energy and frequency . . . . . 63 3.5.1 Ratio of the oscillator energy and frequency is an adi- abaticinvariant...... 64 3.5.2 The thermodynamic derivation of the relation between radiation pressure and energy density ...... 67

4 Einstein’s proposal of light quanta 69 4.1 TheRayleigh-Jeanslaw ...... 71 4.1.1 Einstein’s derivation of the Rayleigh-Jeans law . . . . 71 4.1.2 The history of the Rayleigh-Jeans law and "Planck’s fortunatefailure"...... 73 4.1.3 An excursion to Rayleigh’s calculation of the density ofwavestates...... 74 4.2 Radiation entropy and complexion á la Einstein ...... 76 4.2.1 The entropy and complexion of radiation in the Wien limit...... 77 4.2.2 The entropy and complexion of an ideal gas ...... 79 4.2.3 Radiation as a gas of light quanta ...... 80 4.2.4 Photons as quanta of radiation ...... 81 4.3 Thephotoelectriceffect ...... 81 CONTENTS v

4.4 SuppMat: The equipartition theorem ...... 83

5 Quantum theory of specific heat 85 5.1 The quantum postulate: Einstein vs. Planck...... 86 5.1.1 Einstein’s derivation of Planck’s distribution . . . . . 87 5.2 Specific heat and the equipartition theorem ...... 88 5.2.1 The study of heat capacity in the pre-quantum era . . 88 5.2.2 Einstein’s quantum insight ...... 90 5.3 TheEinsteinsolid ...... 91 5.4 TheDebyesolidandphonons ...... 94

6 Waves, particles, and quantum jumps 99 6.1 Wave-particleduality...... 101 6.1.1 Fluctuation theory (Einstein 1904) ...... 101 6.1.2 Energy fluctuation of radiation (Einstein 1909a) . . . 102 6.2 Bohr’satom...... 105 6.2.1 Spectroscopy: Balmer and Rydberg ...... 106 6.2.2 Atomic structure: Thomson and Rutherford ...... 107 6.2.3 Bohr’s quantum model and the hydrogen spectrum . . 107 6.3 Einstein’s A and B coefficients ...... 111 6.3.1 Probability introduced in quantum dynamics . . . . . 112 6.3.2 Stimulated emission and the idea of laser ...... 114 6.4 Looking ahead to quantum field theory ...... 115 6.4.1 Oscillators in matrix mechanics ...... 116 6.4.2 Quantum jumps: from emission and absorption of ra- diation to creation and annihilation of particles . . . . 119 6.4.3 Resolving the riddle of wave-particle duality in radia- tionfluctuation...... 122 6.5 SuppMat: Fluctuations of a wave system ...... 124

7 Bose-Einstein statistics and condensation 127 7.1 PhotonandtheComptoneffect ...... 129 7.2 Towards Bose-Einstein statistics ...... 130 7.2.1 Boltzmann statistics ...... 131 7.2.2 Bose’s counting of photon states ...... 133 7.2.3 Einstein’s elaboration of Bose’s counting ...... 136 7.3 Quantum mechanics and identical particles ...... 139 7.4 Bose-Einstein condensation ...... 142 7.4.1 Condensate occupancy calculated ...... 143 7.4.2 The condensation temperature ...... 144 vi CONTENTS

7.4.3 Laboratory observation of BE condensation ...... 146 7.5 SuppMat: Radiation pressure due to photons ...... 148 7.6 SuppMat: Planck’s analysis and BE statistics ...... 149 7.7 SuppMat: The role of indistinguishability in BEC ...... 150

8 Local reality and the Einstein-Bohr debate 153 8.1 QM basics — superposition and probability ...... 154 8.2 The Copenhagen interpretation ...... 154 8.3 EPR paradox : entanglement and nonlocality ...... 156 8.3.1 The post EPR era and Bell’s inequality ...... 159 8.3.2 Local reality vs. QM — the experimental outcome . . . 162 8.4 SuppMat: QM calculation of spin correlations ...... 164

III SPECIAL RELATIVITY 169

9 Prelude to special relativity 171 9.1 Relativity as a coordinate symmetry ...... 172 9.2 Maxwell’sequations ...... 174 9.2.1 The electromagnetic wave equation ...... 175 9.2.2 Aether as the medium for EM wave propagation . . . 176 9.3 Developments prior to special relativity ...... 177 9.3.1 Stellar aberration and Fizeau’s experiment ...... 177 9.3.2 Lorentz’s corresponding states and local time . . . . . 180 9.3.3 The Michelson-Morley experiment ...... 184 9.3.4 Length contraction and the Lorentz transformation . . 187 9.3.5 Poincaré and special relativity ...... 188 9.4 Reconstructing Einstein’s motivation ...... 188 9.4.1 The magnet and conductor thought experiment . . . . 189 9.4.2 From ‘no absolute time’ to the complete theory in five weeks ...... 191 9.4.3 Influence of prior investigators in physics and philosophy193 9.5 SuppMat: Lorentz transformation à la Lorentz ...... 194 9.5.1 Noncovariance under Galilean transformation . . . . . 195 9.5.2 Lorentz’s local time and noncovariance at O
v2/c2 . 196 9.5.3 Maxwell’s equations are Lorentz covariant ...... 198

10 The new kinematics and E = mc2 201 10.1 Thenewkinematics ...... 203 10.1.1 Einstein’s two postulates ...... 203 CONTENTS vii

10.1.2 The new conception of time and the derivation of the Lorentz transformation ...... 204 10.1.3 Relativity of simultaneity, time dilation and length contraction ...... 207 10.2 The new velocity addition rule ...... 212 10.2.1 The invariant space-time interval ...... 212 10.2.2 Adding velocities but keeping light speed constant . . 212 10.3 Maxwell’s equations are covariant ...... 213 10.3.1 The Lorentz transformation of electromagnetic fields . 214 10.3.2 The Lorentz transformation of radiation energy . . . . 216 10.4 TheLorentzforcelaw ...... 216 10.5 The E = mc2 relation ...... 217 10.5.1 Work-energy theorem in relativity ...... 218 10.5.2 The E = mc2 paper three months later ...... 219 10.6 SuppMat: Relativistic wave motion ...... 220 10.6.1 The Fresnel formula from velocity addition rule . . . . 220 10.6.2 The Doppler effect and aberration of light ...... 221 10.6.3 Derivation of the radiation energy transformation . . . 222 10.7 SuppMat: Relativistic momentum and force ...... 224

11 Geometric formulation of relativity 225 11.1 Minkowskispacetime...... 227 11.1.1 Rotation in 3D space - a review ...... 227 11.1.2 The Lorentz transformation as a rotation in 4D space- time ...... 228 11.2 Tensorsinaflatspacetime...... 229 11.2.1 Tensor contraction and the metric ...... 230 11.2.2 Minkowski spacetime is pseudo-Euclidean ...... 232 11.2.3 Relativistic velocity, momentum, and energy . . . . . 233 11.2.4 The electromagnetic field tensor ...... 234 11.2.5 The energy-momentum-stress tensor for a field system 235 11.3 The spacetime diagram ...... 238 11.3.1 Basic features and invariant regions ...... 239 11.3.2 Lorentz transformation in the spacetime diagram . . . 241 11.4 The geometric formulation — a summary ...... 244

IV GENERAL RELATIVITY 245

12 Towards a general theory of relativity 247 viii CONTENTS

12.1 Einstein’s motivations for GR ...... 248 12.2 Theequivalenceprinciple ...... 249 12.2.1 The inertia mass vs. the gravitational mass ...... 249 12.2.2 ‘Myhappiestthought’ ...... 252 12.3 Implications of the equivalence principle ...... 253 12.3.1 Bendingofalightray ...... 254 12.3.2 Gravitational redshift ...... 255 12.3.3 Gravitational time dilation ...... 258 12.3.4 Gravity-induced index of refraction in free space . . . 259 12.3.5 Light ray deflection calculated ...... 260 12.3.6 From EP to ‘gravity as the structure of spacetime’ . . 262 12.4 Elements of Riemannian geometry ...... 262 12.4.1 Gaussian coordinates and the metric tensor ...... 263 12.4.2 Geodesic equation ...... 265 12.4.3 Flatnesstheorem ...... 267 12.4.4 Curvature...... 268

13 Curved spacetime as gravitational field 273 13.1 EP requires a metric description of gravity ...... 274 13.1.1 What is a geometric theory? ...... 274 13.1.2 Time dilation as a geometric effect ...... 275 13.1.3 Further arguments for warped spacetime as the grav- itationalfield ...... 276 13.2 GRasafieldtheoryofgravitation ...... 279 13.2.1 The geodesic equation as the GR equation of motion . 280 13.2.2 TheNewtonianlimit ...... 281 13.3 Tensors in a curved spacetime ...... 282 13.3.1 General coordinate transformations ...... 283 13.3.2 Covariant differentiation ...... 286 13.4 The principle of general covariance ...... 291 13.4.1 The principle of minimal substitution ...... 291 13.4.2 Geodesic equation from SR equation of motion . . . . 292

14 The Einstein field equation 295 14.1 TheNewtonianfieldequation ...... 296 14.2 SeekingtheGRfieldequation ...... 297 14.3 Curvature tensor and tidal forces ...... 298 14.3.1 Tidal forces – a qualitative discussion ...... 299 14.3.2 Newtonian deviation equation and equation of geo- desicdeviation ...... 301 CONTENTS ix

14.3.3 Symmetries and contractions of the curvature tensor . 303 14.3.4 The Bianchi identities and the Einstein tensor . . . . 304 14.4 TheEinsteinequation ...... 306 14.4.1 The Newtonian limit for a general source ...... 307 14.4.2 Gravitational waves ...... 308 14.5 The Schwarzschild solution ...... 309 14.5.1 Three classical tests ...... 310 14.5.2 Black holes - the full power and glory of GR . . . . . 315

15 Cosmology 321 15.1 The cosmological principle ...... 323 15.1.1 The Robertson—Walker spacetime ...... 324 15.1.2 The discovery of the expanding universe ...... 326 15.1.3 Bigbangcosmology ...... 328 15.2 Timeevolutionoftheuniverse ...... 330 15.2.1 The FLRW cosmology ...... 330 15.2.2 Mass/energy content of the universe ...... 332 15.3 The cosmological constant ...... 336 15.3.1 Einstein and the static universe ...... 336 15.3.2 TheInflationaryepoch...... 340 15.3.3 The dark energy leading to an accelerating universe . 342

V WALKING IN EINSTEIN’S STEPS 351

16 Internal symmetry and gauge interactions 353 16.1 Einstein and the symmetry principle ...... 355 16.2 Gauge invariance in classical electromagnetism ...... 356 16.2.1 EM potentials and gauge transformation ...... 357 16.2.2 Hamiltonian of a charged particle in an EM field . . . 359 16.3 Gauge symmetry in quantum mechanics ...... 361 16.3.1 The minimal substitution rule ...... 361 16.3.2 The gauge transformation of wavefuncions ...... 362 16.3.3 Thegaugeprinciple ...... 364 16.4 Electromagnetism as a gauge interaction ...... 367 16.4.1 The 4D spacetime formalism recalled ...... 367 16.4.2 The Maxwell Lagrangian density ...... 370 16.4.3 Maxwell equations from gauge and Lorentz symmetries 371 16.5 Gauge theories: a narrative history ...... 372 x CONTENTS

16.5.1 Einstein’s inspiration, Weyl’s program, and Fock’s dis- covery...... 372 16.5.2 Quantum electrodynamics ...... 374 16.5.3 QCD as a prototype Yang-Mills theory ...... 377 16.5.4 Hidden gauge symmetry and the electroweak interaction381 16.5.5 The Standard Model and beyond ...... 388

17 The Kaluza-Klein theory and extra dimensions 393 17.1 Unification of electrodynamics and gravity ...... 394 17.1.1 Einstein and unified field theory ...... 394 17.1.2 A geometric unification ...... 395 17.1.3 A rapid review of electromagnetic gauge theory . . . . 396 17.1.4 A rapid review of GR gravitational theory ...... 396 17.2 General relativity in 5D spacetime ...... 398 17.2.1 Extra spatial dimension and the KK metric ...... 398 17.2.2 ‘The Kaluza-Klein miracle’ ...... 399 17.3 The physics of the KK spacetime ...... 400 17.3.1 Motivating the KK metric ansatz ...... 400 17.3.2 Gauge transformation as a 5D coordinate change . . . 401 17.3.3 Compactified extra dimension ...... 402 17.3.4 Quantum fields in a compactified space ...... 403 17.4 Further theoretical developments ...... 405 17.4.1 Lessons from Maxwell’s equations ...... 406 17.4.2 Einstein and mathematics ...... 407 17.5 SuppMat: Calculating the 5D tensors ...... 407 17.5.1 The 5D Christoffel symbols ...... 408 17.5.2 The 5D Ricci tensor components ...... 411 17.5.3 From 5D Ricci tensor to 5D Ricci scalar ...... 417

VI APPENDICES 419

A Mathematics supplements 421 A.1 Vectorcalculus ...... 421 A.1.1 The Kronecker delta and Levi-Civita symbols . . . . . 422 A.1.2 Differential calculus of a vector field ...... 423 A.1.3 Vectorintegralcalculus ...... 425 A.1.4 Differential equations of Maxwell electrodynamics . . 428 A.2 TheGaussianintegral ...... 431 A.3 Stirling’s approximation ...... 432 CONTENTS xi

A.3.1 The integral representation for n! ...... 432 A.3.2 Derivation of Stirling’s formula ...... 433 A.4 Lagrangianmultipliers ...... 434 A.4.1 Themethod...... 434 A.4.2 Someexamples ...... 435 A.5 The Euler-Lagrange equation ...... 436 A.5.1 Mechanics of a single particle ...... 436 A.5.2 Lagrangian density of a field system ...... 437

B Einstein’s papers 439 B.1 Cited Einstein’s journal articles ...... 439 B.2 Furtherreading...... 443

C Answers to the 21 Einstein questions 447

VII ENDNOTES 455

D References and bibliography 457

E Glossary of symbols and acronyms 467 E.1 Latinsymbols...... 467 E.2 Greeksymbols ...... 471 E.3 Acronyms ...... 473 E.4 Miscellaneous units and symbols ...... 474