Theory of Gravitational Interactions

339 Pages·2013·2.35 MB·English

by  Maurizio Gasperini

Undergraduate Lecture Notes in Physics
Forfurthervolumes:
www.springer.com/series/8917
UndergraduateLectureNotesinPhysics(ULNP)publishesauthoritativetextscoveringtop-
ics throughout pure and applied physics. Each title in the series is suitable as a basis for
undergraduateinstruction,typicallycontainingpracticeproblems,workedexamples,chapter
summaries,andsuggestionsforfurtherreading.
ULNPtitlesmustprovideatleastoneofthefollowing:
• Anexceptionallyclearandconcisetreatmentofastandardundergraduatesubject.
• Asolidundergraduate-levelintroductiontoagraduate,advanced,ornon-standardsubject.
• Anovelperspectiveoranunusualapproachtoteachingasubject.
ULNPespeciallyencouragesnew,original,andidiosyncraticapproachestophysicsteaching
attheundergraduatelevel.
ThepurposeofULNPistoprovideintriguing,absorbingbooksthatwillcontinuetobethe
reader’spreferredreferencethroughouttheiracademiccareer.
Maurizio Gasperini
Theory of
Gravitational
Interactions
MaurizioGasperini
DepartmentofPhysics
UniversityofBari
Bari,Italy
ISSN2192-4791 ISSN2192-4805(electronic)
UndergraduateLectureNotesinPhysics
ISBN978-88-470-2690-2 ISBN978-88-470-2691-9(eBook)
DOI10.1007/978-88-470-2691-9
SpringerMilanHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013930607
©Springer-VerlagItalia2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To myparents
This page intentionally left blank
Preface
This book grew out of lectures given by the author at the University of Turin and
attheUniversityofBari.Itisprimarilyintendedforundergraduatestudentstaking
classesingravitationaltheory,asprescribedbymodernacademicplanstograduate
in Physics with a theoretical/high-energy physics or astrophysics curriculum. The
challengeistoprovidestudentswithatextbookwhich,ononehand,canrepresent
a self-contained reference for a semester cycle of lectures and, on the other hand,
maybeaccessibleandof profitableuse also for studentshavingdifferentinterests
andfollowingdifferentacademictracks.
To this aim the book includes a first, conventional part introducing general rel-
ativity as a geometric theory of the macroscopic gravitational field, and a second,
more advanced part, connecting general relativity to the gauge theories of funda-
mentalinteractions.Adiscussionofthedeepanalogies(andofthephysicaldiffer-
ences)existingbetweengravityandtheotherstandard-modelinteractionsfillsagap
whichispresentwithinthetraditionalgeometricapproachtogeneralrelativity,and
whichusuallypuzzlesstudentsabouttheroleofgravityinthecontextofaunified
modelofallinteractions.
Inthisspirit,theformalismofdifferentialgeometryhasbeenreducedtothenec-
essaryminimum,leavingmoreroomtocurrentinterestingaspectsofgravitational
physicsofbothapplicative/observationaltype(suchasthephenomenologyofgrav-
itationalwaves)andtheoretical/fundamentaltype(suchasthegravitationalinterac-
tions of spinors, supergravity and higher-dimensional gravity). We have included,
however, a final appendix introducing the so-called “Cartan calculus” of exterior
(or differential) forms, in view of the importantapplications of this formalism not
onlytothegravitationaltheorybutalsotomanyotherfieldsoftheoreticalphysics.
A second appendix introduces various possible approaches to the problem of em-
beddingafour-dimensionaltheoryofgravityinthecontextofahigher-dimensional
space–timemanifold.
Formostprofitableuseofthisbookthereaderisexpectedtohaveabasicknowl-
edgeofspecialrelativity,electromagnetictheoryandclassicaltheoryoffields.Ex-
cept for the above input, however, the book aims at being self-contained as much
aspossible,followingtheinformalstyleofclasslectureswherealltherequiredno-
vii
viii Preface
tions and techniquesare explicitlyrecalledand/or introducedwhenevernecessary.
Also,forabetterpedagogicefficiency,allcomputationsareexplicitlycarriedoutin
themaintext(leavingno“voids”tobefilledbythereaders),orpresentedassolved
exercisesattheendofeachchapter.
Thepresentbookiscertainlynotintendedtorepresentacompletereferencefor
a rigorous and comprehensive study of all theoretical aspects of the gravitational
interaction.Itsmainpurposeistoprovidestudentswiththebasicstartingnotions,
enabling them to do further independent work and subsequent deeper studies on
moreprofessionaltextbooksandpapers.Thereadersinterestedinadvanceddiscus-
sions of some specific topic are strongly advised to refer to the list of specialistic
bookspresentedinthebibliography.
Finally,itshouldbenotedthatthisbookdeliberatelyavoidsanygravitationalap-
plicationtocosmologyandlarge-scaleastrophysics,because—accordingtomodern
academicplansofstudies—theyareamatterofspecificcoursesandlectures,well
separatedfromacourseonthetheorygravity.Thefieldofrelativisticcosmologyis
todaysoextended,withsomanybranchesandapplications,astodeservebyitself
a dedicated book. We refer, for this purpose, to the excellent books quoted in the
bibliography, as well as to an introduction to theoretical cosmology which repre-
sentsthenaturalcontinuationofthisbook,andwhichcurrentlyexistsasaSpringer
Italianedition[20].
Acknowledgements
Itisapleasure,aswellasaduty,tothankallmycolleaguesandstudentsfortheir
comments,suggestions,andcriticismthatcontributedovertheyearstocorrectand
improvetheselecturenotes.Listingallofthemwouldbeanimpossibletask,solet
methankthemjointlyfortheirimportanthelp.
AwarmacknowledgementisalsoduetoVenzoDeSabbata,whowasoneofmy
Professorswhen(manyyearsago!)IwasastudentofPhysicsattheUniversityof
Bologna.Professor De Sabbataintroducedmeto thestudy of gravitationandcos-
mology,andtheinteresthewasabletostimulatetowardsthosebranchesofphysics
was so intense as to be still alive, and fully effective, even today in my present
scientificactivity.
Finally, I wish to thank Marina Forlizzi, Executive Editor for Springer-Verlag,
forherkindencouragement,advice,andmanyusefulsuggestions.
Cesena,Italy MaurizioGasperini
Notations, Units and Conventions
Throughoutthisbookwewillusetheindex0forthetime-likecomponentsofvector
andtensorobjects,whiletheindices1,2,3willrefertothespace-likecomponents.
Forthespace–timemetricg wewilladoptthesignaturewithapositivetime-like
μν
eigenvalue,namely:
g =diag(+,−,−,−).
μν
Ourconventionsfor thecurvatureandthecovariantderivativeswillbeas follows.
Riemanntensor:
R β =∂ Γ β+Γ βΓ ρ−{μ↔ν},
μνα μ να μρ να
wherethesymbol{μ↔ν}meansthatwemustinsertalltheprecedingtermswith
theindicesμandν interchangedbetweenthemselves.Riccitensor:
R =R μ.
να μνα
Covariantderivative:
∇ Vα=∂ Vα+Γ αVβ; ∇ V =∂ V −Γ βV ;
μ μ μβ μ α μ α μα β
Lorentzcovariantderivative:
D Va=∂ Va+ω a Vb; D V =∂ V −ω b V .
μ μ μ b μ a μ a μ a b
Also,thesymbol(cid:2)willdenotetheusuald’AlembertoperatoroftheflatMinkowski
space–time,i.e.:
1 ∂2
(cid:2)=ημν∂ ∂ = −∇2,
μ ν c2∂t2
where η is the Minkowski metric and ∇2 =δij∂ ∂ the Laplacian operator of the
i j
Euclideanthree-dimensionalspace.
Unlessotherwisestated,wewillusesmallLatinlettersi,j,k,... forthespatial
indices1,2,3;smallGreeklettersμ,ν,α,... forthespace–timeindices0,1,2,3.
Inhigher-dimensionalspace–times,withanumberd>3ofspatialdimensions,the
space–time indices will be denoted instead by capital Latin letters: A,B,C,...=
0,1,2,3,...,d.
ix