Table Of Content
Einstein’s Space-Time
An Introduction to Special
and General Relativity
Einstein’s Space-Time
An Introduction to Special
and General Relativity
Rafael Ferraro
UniversidaddeBuenosAires
ConsejoNacionaldeInvestigacionesCientíficasyTécnicas
(RepúblicaArgentina)
RafaelFerraro
UniversidaddeBuenosAires
ConsejoNacionaldeInvestigacionesCientíficasyTécnicas(RepúblicaArgentina)
Email:[email protected]
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ISBN978-0-387-69946-2 eISBN978-0-387-69947-9
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Preface
Between 1994 and 1999, I had the pleasure of lecturing Special and General
Relativity in the Facultad de Ciencias Exactas y Naturales of the Universidad
de Buenos Aires. These lectures were targeted to undergraduate and graduate
studentsofPhysics.However,itisincreasinglyapparentthatinterestinRelativity
extendsbeyondtheseacademiccircles.Becauseofthisreason,thisbookintends
tobecomeusefultostudentsofrelateddisciplinesandtootherreadersinterested
in Einstein’s work, who will be able to incorporate entirely the fundamental
ideasofRelativitystartingfromtheverybasicconceptsofPhysics.
To understand the Theory of Relativity it is necessary to give up our
intuitivenotionsofspaceandtime,i.e.,thenotionsusedinourdailyrelationwith
theworld.Theseclassicalnotionsofspaceandtimearealsothefoundationsof
Newtonianmechanics,whichdominatedPhysicsforovertwocenturiesuntilthey
clashedwithMaxwell’selectromagnetism.Classicalphysicsassumedthatspace
is immutable and its geometry obeys the Euclidean postulates. Furthermore,
distancesandtimeintervalsarebelievedinvariant,i.e.,independentofthestate
of motion. Both preconceptions about the nature of space and time rely firmly
on our daily experience, in such a way that the classical notions are imprinted
inourthoughtwiththestatusof“true.”Therefore,wetendtoresistabandoning
the classical notions of space and time, and to replace them with other notions
that are not evident in the phenomena observed in our daily life. This state
of affairs can take the student of Relativity to a mode of thinking vitiated by
the coexistence of old and new conceptions of space and time, which would
lead to several perplexities and paradoxes. The historical approach of this text
attempts to guide the reader along the same road followed by Physics between
theseventeenthandtwentiethcenturies,thusreproducingtheintellectualprocess
that led to the relativistic way of conceiving the space-time. After this journey,
the reader will become more receptive to the consequences of the change of
paradigm.
Chapters 1–7 contain a complete course on Special Relativity that can
be adapted to an introductory level by means of a suitable choice of issues.
In this sense, the text is conceived to allow its reading in several levels, since
some subjects that could hinder the reading by beginners have been located
as Complements inside boxes. The historical road toward Special Relativity is
v
vi Preface
covered in the first two chapters. They begin with the dispute between Leibniz
and Newton about the relational or absolute character of space—an issue that
pavesthewayforintroducingGeneralRelativityinChapter8—andtheanalysis
oftheroleplayedbyabsolutedistancesandtimesinthefoundationsofNewton’s
Dynamics. The two mechanical theories of light—corpuscular model and wave
theoryofluminiferousether—aredevelopedindetailinChapter2,thusarriving
at the tensions arisen between Maxwell’s electromagnetic ether theory and the
Principleofrelativityinthesecondhalfofthenineteenthcentury.Thechallenge
posedbyexperimentalresults(Arago,Hoek,Airy,Michelson-Morley,etc.)and
the theoretical efforts to solve it (Fresnel, Lorentz, Poincaré, etc.) prepare the
reader for the arrival of Einstein’s theory. Chapter 3 introduces the postulates
of Special Relativity and the notions of space and time derived from them.
Here mathematics is entirely basic, because no differential calculus is needed.
The understanding of length contraction, time dilatation, Lorentz transforma-
tions, relativistic addition of velocities, etc. only entails elementary concepts
of kinematics. This Chapter deals with replacing classical notions of space
and time with those notions that emerge from postulating that Maxwell’s laws
are valid in any inertial frame. Therefore, the central goal here is to enable
a full understanding of the interrelation between length contraction and time
dilatation and its consequences. Chapter 3 ends with the transformation of a
planewave,whereitsconnectionwiththefoundationsofQuantumMechanicsis
emphasized. Chapter 4 develops the geometric properties of Minkowski space-
time (invariance of the interval, causal structure, light cone, etc.) together with
some advanced topics like Wigner rotation. Chapter 5 teaches the transfor-
mations of the electromagnetic field, charge and current densities, potentials,
etc. in the context of the ordinary vector language, starting from the preser-
vation of the polar and axial characters of electric and magnetic fields, and the
essential features of a plane electromagnetic wave. It also contains the fields
of moving charges, dipoles, and continuous media. Chapter 6 is devoted to
relativisticDynamics,andexplainsthechangessufferedbyNewton’sDynamics
inordertobringtheirlawsintoagreementwiththePrincipleofrelativityunder
Lorentztransformations.ThereformulationoftheDynamicsleadstothe“mass-
energy equivalence,” so the rich phenomenology in atomic and nuclear physics
concernedwiththistopiciscovered.Chapter7introducesthefour-tensorformu-
lation of Special Relativity, and applies it to several subjects that enter into
play also in General Relativity: volume and hypersurfaces, energy-momentum
tensor of a fluid, electromagnetism, Fermi–Walker transport, etc. Chapter 8
displays General Relativity—the relativistic theory for the gravitational-inertial
field—starting from Mach’s criticism of Mechanics as a trigger of Einstein’s
thought.Onceagain,theexplanationtransitsthehistoricalwaybyfocusingonthe
beginningofthenewconcepts.Thischapterintroducesthemathematicaltoolsof
the geometric language—covariant derivative, Riemann tensor, etc.—avoiding
excesses of mathematical complexity. The Einstein equations for the geometry
ofspace-timeareformulated,andtheirconsequencesintheweakfieldapproxi-
mationarestudiedtoconnectwithNewtoniangravity.Theissueofthenumber
Preface vii
of degrees of freedom and constraint equations is tackled, and exemplified in
the context of gravitational waves. Chapter 9 explains the main features of the
Schwarzschild black hole, and applies General Relativity to Cosmology. The
isotropicandhomogeneouscosmologicalmodelsaredisplayedwithinthecontext
oftherecentprogressoftheobservationalcosmology.Thechapterendswithan
updatedaccountofthesetofexperimentalresultsconfirmingGeneralRelativity.
Somespecialtopics,notessentialforafirstreadingofthebookbutmeaningful
foradvancedstudents,arecontainedintheAppendix.Inshort,thefinalchapters
areadequateforanintroductorycourseonGeneralRelativityandCosmology.
IwishtothanktheInstitutodeAstronomíayFísicadelEspacio(CONICET
-UBA), where most of the lectures took place, for its hospitality. IAFE is the
institutewhereasignificantnumberofrelativisticphysicistsfromBuenosAires
got our start under the wise guidance of Mario Castagnino. I am indebted to
GerardoMilesi,DanielSforza,ClaudioSimeone,andMarcThibeault,whohave
contributed in several ways to the realization of this book: by adding biblio-
graphic cites, correcting proofs, or enriching the contents with their comments.
A special gratitude for my wife Mónica Landau—who has also collaborated
in preparing this work—and my son Damián and my daughter Sofía, for their
encouragementandpatienceduringthislongperiodofwriting.
RafaelFerraro
BuenosAires,January2005
Preface to the English Edition
This English edition is basically the translation of the first Spanish edition
published last year. A few changes and additions were made, and errors were
corrected.
I wish to express my gratitude to Luis Landau and Claudio Simeone
for their cooperation in the realization of this edition. I am especially grateful
to Alicia Semino for her thorough supervision of the English version of the
manuscript.
RafaelFerraro
BuenosAires,October2006
ix
Contents
ListofComplements................................................. xv
ListofBiographies...................................................xvii
1. SpaceandTimeBeforeEinstein.................................. 1
1.1. AbsoluteSpaceandTime............................................................. 1
1.2. GeometricPropertiesoftheSpace............................................... 2
1.3. GalileoandtheLawsofMotion................................................... 4
1.4. ChangeofCoordinatesBetweenFramesinRelativeMotion...... 6
1.5. PrincipleofInertia........................................................................ 8
1.6. PrincipleofRelativity................................................................... 9
1.7. PhysicalPhenomenahavingaPrivilegedReferenceSystem...... 11
1.8. Maxwell’sElectromagnetism........................................................ 13
2. InSearchoftheEther........................................... 17
2.1. TwoModelsfortheLight............................................................. 17
2.2. FirstDeterminationoftheSpeedofLight................................... 21
2.3. TheAberrationofLight................................................................ 23
2.4. FirstTerrestrialMethodtoMeasurec.......................................... 26
2.5. TheLuminiferousEther ............................................................... 27
2.6. SearchingfortheAbsoluteTerrestrialMotion:
DraggingofEther.......................................................................... 29
2.7. Fizeau’sExperiment...................................................................... 32
2.8. Hoek’sExperiment........................................................................ 34
2.9. Airy’sExperiment......................................................................... 36
2.10. Michelson–MorleyExperiment..................................................... 36
2.11. FitzGerald–LorentzLengthContraction....................................... 43
2.12. TheTwilightoftheEther............................................................. 44
xi
xii Contents
3. SpaceandTimeinSpecialRelativity............................. 47
3.1. PostulatesofSpecialRelativity.................................................... 47
3.2. LengthContractionsandTimeDilatations................................... 49
3.3. TheMuonJourney........................................................................ 52
3.4. LengthsTransversaltoMotion..................................................... 53
3.5. CompositionofMotions............................................................... 54
3.6. InterpretationofFizeau’sExperiment.......................................... 57
3.7. TransversalComponentsoftheVelocity..................................... 58
3.8. TheNotionofSimultaneity.......................................................... 58
3.9. EventsandWorldLines................................................................ 62
3.10. CoordinateLinesofS(cid:2) intheSpace-Time
DiagramofS.................................................................................. 65
3.11. LorentzTransformations............................................................... 69
3.12. ComparingClocksinDifferentFrames....................................... 74
3.13. VelocityandAccelerationTransformations................................. 78
3.14. “Paradoxes”:RemnantsofClassicalThought.............................. 81
3.15. DopplerEffect............................................................................... 86
3.16. TransformationofLightRays...................................................... 90
3.17. TransformationofaPlaneWave.................................................. 92
3.18. PropagationofLightinMaterialMedia....................................... 95
4. GeometricStructureofSpace-Time.............................. 97
4.1. Interval........................................................................................... 97
4.2. CalibrationHyperbola................................................................... 98
4.3. LightCone..................................................................................... 100
4.4. Timelike-SeparatedEvents........................................................... 101
4.5. TwinParadox................................................................................ 103
4.6. Spacelike-SeparatedEvents.......................................................... 107
4.7. VelocityParameter:Rapidity........................................................ 109
4.8. WignerRotation............................................................................ 111
5. TransformationoftheElectromagneticField..................... 117
5.1. TheElectromagneticPlaneWave................................................. 117
5.2. TransformationofEandB........................................................... 119
5.3. ChargeandCurrentTransformations........................................... 122
5.4. FieldofaUniformlyMovingCharge.......................................... 126
5.5. TransformationofPotentials ....................................................... 127
5.6. FieldsinMaterialMedia............................................................... 128
5.7. MovingDipolesFields.................................................................. 130
5.8. LorentzForceTransformation ..................................................... 132
5.9. ElectromagneticFieldInvariants.................................................. 133
