Gravity, Black Holes, and the Very Early Universe: An Introduction to General Relativity and Cosmology

284 Pages·2008·4.91 MB·English

Gravity, Black Holes, and the Very Early Universe
Tai L. Chow
Author
Gravity, Black Holes,
and the Very Early Universe
An Introduction to General Relativity
and Cosmology
TaiL.Chow
CaliforniaStateUniversityStanislaus
Turlock,CA
USA
ISBN-13:978-0-387-73629-7 e-ISBN-13:978-0-387-73631-0
LibraryofCongressControlNumber:2007936678
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
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DedicatedtoMyWife
SylviaHsiuYungChen
About the Author
Dr. Tai Chow was born and raised in China. He received a Bachelor of Science
degree in physics from National Taiwan University, a Master’s degree in physics
from Case Western Reserve University in Cleveland, and a Ph.D. in physics from
the University of Rochester in New York. Since 1970, Dr. Chow has been in the
Department of Physics at California State University, Stanislaus, and served as
department chairman for 17 years. He has also served as a Visiting Professor at
the University of California at Davis and Berkeley and has worked as a Summer
Faculty Fellow at Stanford University and at NASA/Ames Center. Dr. Chow has
publishedover40articlesinphysicsandastrophysicsjournalsandistheauthorof
three textbooks: Classical Mechanics, published in 1995 by John Wiley & Sons,
MathematicalMethodsforPhysicists,in2000byCambridgeUniversityPress,and
IntroductiontoElectromagneticTheory,in2005byJones&BartlettPublishers.
vii
Preface
Intheearly1900s,threeeventstookplacethatdramaticallychangedthecourseof
modern physics. In 1905 Albert Einstein formulated the Special Theory of Rela-
tivity. Then, in 1915, he developed the General Theory of Relativity, and around
1925quantummechanicstookitspresentform.Sincethen,physicshasprogressed
rapidly. Beginning in 1930, quantum mechanics and special relativity were united
into what is known as the relativistic quantum field theory. This merger was very
rewardinginthatitprovides,attheleast,partialexplanationofthelawsandinter-
actionsgoverningelementaryparticlephysics.
Amongthefourtypesofforces(strong,electromagnetic,weak,andgravitational)
knowntoday,gravityisperhapsthestrangest.Weakthoughitis,gravitydominates
theotherthreeforcesovercosmicdistances.Anycosmologymustbefoundedona
logicallysecuretheoryofgravitation.
Thefirstthreeforcescouldbeexplainedthroughparticleinteractionstakingplace
intheflatspace-timeofspecialrelativity.However,gravitydefiessuchanexplana-
tion. In order to describe the mysterious force known as gravity, Einstein in 1915
wascompelledtogeneralizetheideasofhisspecialrelativity,andheeventuallycon-
nectedgravitywiththegeometryofspace-time.Inotherwords,Einstein’sGeneral
TheoryofRelativityisarelativistictheoryofgravitation.
For a long time, Einstein’s Theory of General Relativity occupied an isolated
position within the domain of general physics. This was attributable in part to the
mathematicalframeworkofthetheory,whichisbasedonRiemanniangeometry,a
kindofgeometrynotneededinmostotherphysicalapplications.Theextremedif-
ficultyindevisingsuitableexperimentsthatmightverifythetheoryandthegrowth
ofmorefertilefieldsofinvestigation,suchasatomicandnuclearphysicsaswellas
thestudyofelementaryparticles,alsocontributedtotheisolationofthetheory.
However, Einstein’s Theory of General Relativity is now enjoying renewed
interest. This is due partly to the development of new technological capabilities
thatopeneduppreviouslyinaccessibleavenuesfortheexperimentalverificationof
general relativity and partly to the conjecture of some theoretical physicists that
the fundamental difficulties confronting quantum field theory may find their reso-
lutioninasuitablecombinationofthetwodisciplines.Thediscoveryofextremely
ix
x Preface
compactcelestialobjects—neutronstarsandblackholes,forinstance—providedthe
finalturningpoint.ThestudyoftheseobjectsdemandedtheapplicationofEinstein’s
Theory of General Relativity. Today, physics and astronomy have joined forces to
formthedisciplinecalledrelativisticastrophysics.Einstein’sTheoryofGeneralRel-
ativityisalsoessentialtomoderncosmology,sincetheoverallspace-timestructure
isintimatelyrelatedtothegravitationalfield.Inthepastdecadeinterestincosmol-
ogyandgeneralrelativityhasgrownconsiderably.
Today,thereisincreaseddemandforundergraduatecoursesinrelativityandcos-
mology. There are many advanced books on the Theory of General Relativity and
cosmology for the specialist, and many elementary expositions for the lay reader.
Butthereisagapattheundergraduatelevel.Thisbookisanattempttofillthegap.
Wewilltrytomakeavailabletothestudentaworkingacquaintancewiththecon-
cepts and fundamental ideas in general relativity and modern cosmology. For the
modes of calculation we choose the old-fashioned tensor calculus for pedagogical
reasons.Mostundergraduateshavenotbeenexposedtothemanynewformalisms
developed in general relativity. Hopefully after reading this book, the student can
continue delving more deeply intoparticular aspects or topics ingeneral relativity
andcosmologythatinteresthimorher.
ThisbookevolvedfromasetoflecturenotesforacoursethatIhavetaughtover
thepast10years.Iammakingtheassumptionthatthestudenthasbeenexposedto
a calculus-based course in general physics and a course in calculus (including the
handling of differentiations of field equations). Some exposure to tensor analysis
wouldbehelpfulbutisnotnecessary;thissubjectiscoveredinthetext.
Thestudentwillfindthatinthederivationsofequations,agenerousamountof
detailhasbeengiven.However,toensurethatthestudentdoesnotlosesightofthe
developmentunderway,someofthemorelengthyandtediousalgebraicmanipula-
tionshavebeenomitted.
Turlock,California TaiL.Chow,Ph.D.
Contents
AbouttheAuthor .................................................. vii
Preface............................................................ ix
1 BasicIdeasofGeneralRelativity................................. 1
1.1 InadequacyofSpecialRelativity.............................. 1
1.2 Einstein’sPrincipleofEquivalence............................ 3
1.3 ImmediateConsequencesofthePrincipleofEquivalence ......... 7
1.3.1 TheBendingofaLightBeam.......................... 7
1.3.2 GravitationalShiftofSpectralLines
(GravitationalRedshift)............................... 8
1.4 TheCurvedSpace-TimeConcept ............................. 8
1.5 ThePrincipleofGeneralCovariance .......................... 12
1.6 DistanceandTimeIntervals.................................. 13
1.7 Problems ................................................. 15
References..................................................... 17
2 CurvilinearCoordinatesandGeneralTensors..................... 19
2.1 CurvilinearCoordinates ..................................... 19
2.2 ParallelDisplacementandCovariantDifferentiation ............. 23
2.3 SymmetryPropertiesoftheChristoffelSymbols................. 27
2.4 ChristoffelSymbolsandtheMetricTensor ..................... 28
2.5 Geodesics................................................. 29
2.6 TheStationaryPropertyofGeodesics.......................... 30
2.7 TheCurvatureTensor ....................................... 32
2.8 TheConditionforFlatSpace................................. 36
2.9 GeodesicDeviation......................................... 37
2.10 LawsofPhysicsinCurvedSpaces ............................ 38
2.11 TheMetricTensorandtheClassicalGravitationalPotential ....... 39
2.12 SomeUsefulCalculationTools ............................... 40
2.13 Problems ................................................. 43
References..................................................... 44
xi
xii Contents
3 Einstein’sLawofGravitation.................................... 45
3.1 Introduction(SummaryofGeneralPrinciples) .................. 45
3.2 AHeuristicDerivationofEinstein’sEquations .................. 46
3.2.1 VacuumFieldEquations .............................. 46
3.2.2 FieldEquationsWhereMatterisPresentinSpace......... 48
3.3 Energy-MomentumTensor................................... 51
3.4 GravitationalRadiation...................................... 52
3.5 Problems ................................................. 54
References..................................................... 54
4 TheSchwarzschildSolution ..................................... 55
4.1 TheSchwarzschildMetric ................................... 55
4.2 TheSchwarzschildSolutionoftheVacuumFieldEquations....... 56
4.3 SchwarzschildGeodesics .................................... 60
4.4 QuasiuniformGravitationalField ............................. 62
4.5 Problems ................................................. 63
References..................................................... 63
5 ExperimentalTestsofEinstein’sTheory .......................... 65
5.1 PrecessionofthePerihelionofMercury........................ 65
5.2 DeflectionofLightRaysinaGravitationalField ................ 71
5.3 LightRetardation(TheShapiroExperiment).................... 75
5.4 TestofGravitationalRadiation(Hulse-Taylor’sMeasurement
oftheOrbitalDecayoftheBinaryPulsarPSR-1913+16) ......... 77
5.5 Problems ................................................. 79
References..................................................... 79
6 ThePhysicsofBlackHoles ...................................... 81
6.1 TheSchwarzschildBlackHole ............................... 81
6.2 InsideaBlackHole......................................... 84
6.3 HowaBlackHoleMayForm ................................ 86
6.4 TheKerr-NewmanBlackHole ............................... 89
6.4.1 Energy Extraction from a Rotating Black Hole:
ThePenroseProcess.................................. 92
6.4.2 TheAreaTheorem ................................... 93
6.4.3 EnergyExtractionfromTwoCoalescingBlackHoles...... 94
6.5 ThermodynamicsofBlackHoles ............................. 95
6.6 QuantumMechanicsofBlackHoles:HawkingRadiation ......... 97
6.7 TheDetectionofBlackHoles ................................101
6.7.1 DetectionofStellar-MassBlackHoles ..................101
6.7.2 SupermassiveBlackHolesintheCentersofGalaxies......104
6.7.3 Intermediate-MassBlackHoles ........................106
6.8 HowDoElectricalandGravitationalFields
GetOutofBlackHoles?.....................................106
6.9 BlackHolesandParticlePhysics..............................107
6.10 Problems .................................................108
References.....................................................109
Contents xiii
7 IntroductiontoCosmology ......................................111
7.1 Introduction ...............................................111
7.2 TheDevelopmentofWesternCosmologicalConcepts ............112
7.2.1 AncientGreece......................................112
7.2.2 TheRenaissanceofCosmology ........................113
7.2.3 NewtonandtheInfiniteUniverse .......................114
7.2.4 Newton’sLawofGravityandaNonstationaryUniverse....115
7.2.5 Olbers’Paradox .....................................118
7.3 TheDiscoveryoftheExpansionoftheUniverse.................119
7.4 TheBigBang..............................................123
7.5 TheMicrowaveBackgroundRadiation.........................124
7.6 AdditionalEvidencefortheBigBang .........................128
7.7 Problems .................................................130
References.....................................................131
8 BigBangModels...............................................133
8.1 TheCosmicFluidandFundamentalObservers ..................133
8.2 PropertiesoftheRobertson-WalkerMetric .....................135
8.3 CosmicDynamicsandFriedmann’sEquations ..................139
8.4 TheSolutionsofFriedmann’sEquations .......................142
8.4.1 FlatModel(k =0)...................................143
8.4.2 ClosedModel(k =1) ................................144
8.4.3 OpenModel(k =−1)................................146
8.5 DarkMatterandtheFateoftheUniverse.......................148
8.6 TheBeginning,theEnd,andTime’sArrow.....................152
8.7 AnAcceleratingUniverse?...................................156
8.8 TheCosmologicalConstant..................................158
8.9 Problems .................................................161
References.....................................................161
9 Particles,Forces,andUnificationofForces........................163
9.1 Particles ..................................................163
9.1.1 Spin ...............................................163
9.1.2 Fermions ...........................................164
9.1.3 Bosons.............................................165
9.1.4 HadronsandLeptons .................................165
9.1.5 Quarks .............................................167
9.1.6 QuarkColors........................................168
9.1.7 QuarkConfinement ..................................169
9.2 FundamentalInteractionsandConservationLaws ...............171
9.3 SpontaneousSymmetryBreaking .............................177
9.4 UnificationofForces(Interactions)............................180
9.5 TheNegativeVacuumPressure ...............................184
References.....................................................186