General Relativity and Gravitational Waves - Essentials of Theory and Practice PDF
Preview General Relativity and Gravitational Waves - Essentials of Theory and Practice
UNITEXT for Physics Sanjeev Dhurandhar Sanjit Mitra General Relativity and Gravitational Waves Essentials of Theory and Practice UNITEXT for Physics SeriesEditors MicheleCini,UniversityofRomeTorVergata,Roma,Italy AttilioFerrari,UniversityofTurin,Turin,Italy StefanoForte,UniversityofMilan,Milan,Italy GuidoMontagna,UniversityofPavia,Pavia,Italy OresteNicrosini,UniversityofPavia,Pavia,Italy LucaPeliti,UniversityofNapoli,Naples,Italy AlbertoRotondi,Pavia,Italy PaoloBiscari,PolitecnicodiMilano,Milan,Italy NicolaManini,UniversityofMilan,Milan,Italy MortenHjorth-Jensen,UniversityofOslo,Oslo,Norway UNITEXT for Physics series publishes textbooks in physics and astronomy, char- acterized by a didactic style and comprehensiveness. The books are addressed to upper-undergraduateandgraduatestudents,butalsotoscientistsandresearchersas importantresourcesfortheireducation,knowledge,andteaching. Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/13351 · Sanjeev Dhurandhar Sanjit Mitra General Relativity and Gravitational Waves Essentials of Theory and Practice SanjeevDhurandhar SanjitMitra Inter-UniversityCentreforAstronomy Inter-UniversityCentreforAstronomy andAstrophysics(IUCAA) andAstrophysics(IUCAA) Pune,Maharashtra,India Pune,Maharashtra,India ISSN2198-7882 ISSN2198-7890 (electronic) UNITEXTforPhysics ISBN978-3-030-92334-1 ISBN978-3-030-92335-8 (eBook) https://doi.org/10.1007/978-3-030-92335-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Iliketheauthors’styleandthechoiceof topics.Theyhavetakenanunusualapproach sothatthereadergetstoseehowrelativity worksinpracticalsituations. Normallyatext inrelativitypresentsthesubjectasan abstractmathematicalexercise.Theauthors haveworkedoutexampleswhereinthereader appreciateshowthetheoryworks. JayantNarlikar Tomyparents(especiallymymother)and BhalchandraDhurandhar(myfather’suncle) whoindulgedmychildhoodscientificantics. —SanjeevDhurandhar Tomyparentsandmyuncle,whodidtheir bestformyeducation. —Sanjit Mitra Preface Several high-quality textbooks at various levels have been published on General Relativity(GR).Thenwhy,onemayask,writeanotherbook?Therearetworeasons whywechosetowritethisbook.Thefirstreasonisthat,iftherehadbeenasingle book available that could have served our purpose, we might not have been moti- vated enough to write this book! A number of universities and institutions offer a half-semestercourseonGeneralRelativitytofinalyearMasters’studentsinPhysics oraspartofPh.D.coursework.Wehavebeenteachingsuchcoursesforyears(oreven decades),butwecouldnotrecommendforsuchacourseasingleself-containedbook. Wehadtorefertomultiplebooksfordifferenttopicssuchasspecialrelativity,tensor calculus, differential geometry, black holes, etc. inevitably with different conven- tions,notations,etc.Often,inordertodojustice,thebooksbecametoovoluminous with interlinked concepts which were difficult to disentangle. It is not uncommon to encounter students who have completed a course on GR without having any insights into the rationale/prescription for parallel transport in Riemannian geom- etry. Referring students to advanced texts has the consequence of bogging them down and making them lose sight of the main objective of learning general rela- tivity. We have, therefore, adopted a middle-ground, in which the contents can be followed through smoothly without omissions. The instructor may teach a short coursewithoutcompromisingontheessentialmathematicalconceptswhichprovide areasonablysolidfoundation.Thisisourmainmotivationinwritingthisbook.We expect that students will find it easy to follow the introductory topics covered and gainvaluableinsightsfromthisbook.Moreover,aclearunderstandingofthebasics willalsoempowerthestudentstoaskadvancedquestionsandfurtherseekanswers fromadvancedtexts. Theothermajormotivationistoprovideanintroductiontogravitationalwaves (GW). With the detection of GW and given the ever-increasing sensitivity of GW detectors,thefieldissizzling.Alargenumberofresearchershavebeenmotivatedto enterintothisfieldtocarryoutresearchintheobservationalaspects.Againthereis noshortintroductionreadilyavailablethatcanbeunderstoodinashorttimewithout sacrificingcontent.Infact,fundamentalsonGWdataanalysiscanrarelybefound in introductory books. Moreover, many of these researchers may not be familiar ix x Preface with GR either, but we believe that they would appreciate the field more if they areprovidedwithabasicintroductionnotrequiringtoomucheffort.Thisstrongly motivated us to put significantly more emphasis on GW and make it an essential one-stopintroductionpackageforsomeonebeginningaresearchcareerinGW. The authors would like to thank Jayant Narlikar and Tarun Souradeep for their encouragement.Wearegratefultoourstudentsandyoungercolleagueswithwhom wehaveheldanimatedandspiriteddiscussionsfromwhichthebookhassignificantly benefited. Finally,SVDwouldliketothankNASI,India,forthePlatinumJubileeFellowship andSMwouldliketoacknowledgeDST,India,fortheSwarnaJayantiFellowship. Apartofthisbookwaswrittenduringbothfellowships. Pune,India SanjeevDhurandhar SanjitMitra Contents 1 OverviewofSpecialRelativity ................................... 1 1.1 Introduction .............................................. 1 1.2 PostulatesofSpecialRelativity(SR) ......................... 2 1.3 Space-TimeDiagrams:APictureIsWorthaThousandWords ... 4 1.3.1 SettingUptheAxesforaMovingObserver ............ 6 1.3.2 LorentzContractionandTimeDilation ................ 9 1.4 LorentzTransformations ................................... 11 1.4.1 TheLorentzTransformationEquations ................ 11 1.4.2 VelocityAddition .................................. 13 1.5 Four-VectorNotationandCovariantFormalism ................ 14 1.6 RelativisticMechanics ..................................... 16 1.7 CovariantFormulationofElectrodynamics .................... 19 2 TheEquivalencePrinciple ...................................... 27 2.1 EquivalencePrinciple:WeakForm ........................... 28 2.2 EquivalencePrinciple:StrongForm(SEP) .................... 30 2.3 GravityandCurvatureofSpace-Time ........................ 30 2.3.1 GravityBendsLightRays ........................... 31 2.3.2 GravityAffectsClocks:GravitationalRedshift ......... 32 3 TensorAlgebra ................................................ 37 3.1 Introduction .............................................. 37 3.2 BasisVectors ............................................. 39 3.3 ContravariantandCovariantComponentsofaVector ........... 41 3.3.1 TheJacobianmatrices .............................. 43 3.4 TheMetric ............................................... 44 3.4.1 ScalarProduct ..................................... 44 3.4.2 RaisingandLoweringofIndices ..................... 45 3.5 TensorsofHigherRank .................................... 45 3.5.1 TransformationLaws ............................... 45 3.5.2 SymmetricandAnti-symmetricTensors ............... 47 xi
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