An Introduction to Special Relativity for Radiation and Plasma Physics

331 Pages·2022·2.95 MB·English

by  Greg Tallents

AN INTRODUCTION TO SPECIAL RELATIVITY
FOR RADIATION AND PLASMA PHYSICS
This textbook introduces the topic of special relativity, with a particular emphasis upon
light–matter interaction and the production of light in plasma. The physics of special
relativityisintuitivelydevelopedandrelatedtotheradiativeprocessesoflight.Thebook
reviews the underlying theory of special relativity, before extending the discussion to
applicationsfrequentlyencounteredbypostgraduatesandresearchersinastrophysics,high-
powerlaserinteractions,andtheusersofspecializedlightsources,suchassynchrotronsand
freeelectronlasers.Ahighlypedagogicalapproachisadoptedthroughout,andnumerous
exercisesareincludedwithineachchaptertoreinforcethepresentationofkeyconceptsand
applicationsofthematerial.
GREG TALLENTS isProfessorinPhysics attheYorkPlasmaInstituteattheUniversity
ofYork.HecompletedaPhDinEngineeringPhysicsattheAustralianNationalUniversity
in the late 1970s and has since researched the physics and applications of laser-produced
plasmas and short wavelength plasma lasers. He has written a textbook An Introduction
to the Atomic and Radiation Physics of Plasmas published in 2018 by Cambridge
UniversityPress.
Published online by Cambridge University Press
Published online by Cambridge University Press
AN INTRODUCTION TO SPECIAL
RELATIVITY FOR RADIATION
AND PLASMA PHYSICS
GREG TALLENTS
UniversityofYork
Published online by Cambridge University Press
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Informationonthistitle:www.cambridge.org/9781009236065
DOI:10.1017/9781009236072
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Contents
Preface pageix
1 ABriefIntroductiontoSpecialRelativityandLight 1
1.1 ThePropagationofLight 6
1.2 TimeDilationandLengthContraction 11
1.3 LightEmissionfromChargeAcceleration 15
Exercises 19
2 AnIntroductiontoRelativisticKinematics,KineticsandEnergy 22
2.1 GalileanTransformations 23
2.2 LorentzTransformations 25
2.3 TheTransformationofVelocities 30
2.4 TheRelativisticRocketEquation 32
2.5 TheTransformationofthePropagationDirectionofLight 36
2.6 TheDopplerEffect 37
2.7 RelativisticMomentum 40
2.8 RelativisticEnergy 43
2.9 TheQuantumNatureofMatterandRadiation 47
2.10 Four-Vectors 51
Exercises 55
3 RelativityandElectromagneticFields 59
3.1 RevisitingthePropagationofLight 60
3.2 TheLorentzTransformationoftheElectricandMagneticFields 61
3.3 AccelerationandForceinaMovingFrameofReference 68
3.4 EmissionfromRelativisticParticles 71
3.5 “Beaming”ofEmissionfromRelativisticBodies 73
v
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vi Contents
3.6 EmissionwithAccelerationParallelorPerpendiculartov 77
Exercises 80
4 ElectromagneticWaveInteractionswithPlasmaandOtherMatter 83
4.1 NonrelativisticSingleElectronMotion:LinearPolarization 84
4.2 NonrelativisticSingleElectronMotion:EllipticalPolarization 89
4.3 RelativisticSingleElectronMotion 92
4.4 ThePropagationofLightinaPlasma 96
4.5 CherenkovRadiation 101
4.6 ThomsonScatter 105
4.7 CoherentScatter 111
4.8 ComptonScatter 117
Exercises 123
5 ParticleMotionandRadiationinElectricandMagneticFields 127
5.1 TheMotionofaPointChargeinanElectricorMagneticField 128
5.2 AccelerationofParticles 133
5.3 EmissionfromParticlesinaMagneticField 138
5.4 SynchrotronRadiation 142
5.5 TheSpectrumofSynchrotronRadiation 146
5.6 BendingMagnetsandUndulators 150
5.7 FreeElectronLasers 154
Exercises 159
6 PhotonandParticleDistributionFunctions 162
6.1 ParticleDistributionFunctions 164
6.2 TheSaha-BoltzmannEquationandBoltzmannRatio 170
6.3 TheRelativisticFormoftheSaha-BoltzmannEquation 175
6.4 Black-BodyRadiation 178
6.5 TheMaxwellianDistribution 185
6.6 Maxwell-JuttnerDistributionsforRelativisticParticles 186
6.7 RelativisticEquationsofState 191
Exercises 199
7 RadiationEmission,Absorption,andScatterinPlasmas 204
7.1 RadiativeTransfer 205
7.2 TheEinsteinAandBCoefficients 210
7.3 SpectralLineEmissionCoefficients 214
7.4 SpectralLineAbsorptionCoefficients 215
7.5 DopplerBroadening 217
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Contents vii
7.6 DopplerBroadeninginHighlyRelativisticPlasmas 219
7.7 ContinuumRadiation 224
7.8 ContinuumEmissionwithRelativisticElectrons 232
7.9 InverseComptonScattering 239
7.10 InverseComptonScatteringinaPlasma 243
7.11 ComptonScatterofaLightBeam 245
7.12 TheRadiationReactionForce 248
Exercises 252
8 CollisionalProcesses 255
8.1 RelativisticCollisionKinematics 256
8.2 CollisionsinPlasmas 257
8.3 NonrelativisticCollisionalRates 260
8.4 RelativisticCollisionalRates 265
Exercises 270
9 RelativisticOptics 272
9.1 TheFizeauEffect 273
9.2 ThePhaseVelocityofLightinaMovingMedium 275
9.3 AnglesofReflectionandDopplerShiftsfromaMovingMirror 281
9.4 TheLorentzTransformationofthePolarizationofLight 286
Exercises 290
10 SolutionstoSelectedExercises 292
10.1 TheTwinParadox 292
10.2 TheBiot-SavartEquation 298
AppendixA LorentzInvariantQuantities 302
AppendixB ModifiedBesselFunctionsoftheSecondKind 304
AppendixC TheChemicalPotentialVariationwithDensity 307
References 309
Index 315
Published online by Cambridge University Press
Published online by Cambridge University Press
Preface
This book introduces special relativity with an emphasis on the ways relativity
impacts the interaction of light with matter, particularly the interaction of light
with plasma material. Often called the fourth state of matter after solids, liquids,
and gases, plasma is created when a gas is heated, so that free electrons and ions
are produced. Free electrons, in particular, interact strongly with light and other
electromagneticradiation.
Rather than a treatment with a collection of formulas, the book emphasizes the
physicsofrelativityasrelatedtotheradiativeprocessesoflight.Priorexposureto
specialrelativityisnotassumed.Manyresearchersandstudentsundertakeprelimi-
narycourseswheretheresultsof,forexample,timedilationandlengthcontraction
for objects moving at speeds approaching the speed of light are introduced. How-
ever,whenrelativisticphysicsisneededforresearchormoreadvancedcourses,the
jumprequiredinunderstandingcanbeahurdle,partlyduetotheuseinmanytexts
oftensorfieldmathematics(asawayofpavingthewayforstudiesofgeneralrel-
ativity).Thetreatmentinthisbooksetsouttheideasofspecialrelativityassuming
no prior relativistic study and uses US junior and European year 2 undergraduate
mathematics. Initial chapters end with the idea of the Lorentz transformation of
time and space between observers with different velocities. Electric and magnetic
fields,particleacceleration,andthe“beaming”effectofhighvelocityontheemis-
sion of light are then treated. Later chapters deal with the interaction of light with
plasmaandtheuseofmagneticfieldsforlightproduction(withspecifictreatment
of synchrotrons and free electron lasers). The final chapters present detailed treat-
mentsofplasmaradiativeprocesses,collisions,andrelativisticoptics.Theaimisto
presentspecialrelativityandthewayspecialrelativityimpactsradiativeprocesses
in astrophysics and laboratory plasmas as a continuous and interconnected story
usingthemathematicscommonlyusedbyexperimentalphysicists.1
1Themathematicalknowledgeassumedincludesanunderstandingofvectorandmatrixoperations,for
example,theoperations∇2,∇×,and∇·onvectors.Standardcalculusisalsousedextensively,forexample,
ix
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