Introduction On thecosmicscale, gravitationdominates theuniverse. Nuclear and electromagnetic forces account for thedetailed processesthatallowstarstoshine andastronomerstosee them.But it is gravitationthatshapesthe universe, determining thegeometryofspace andtimeand thus thelarge-scale distribution of galaxies.Providing insight into gravitation–its effects,its nature andits causes –isthereforerightly seen as one of themostimportant goals of physics and astronomy. Through more than athousandyearsofhuman history thecommonexplanation of gravitationwas basedonthe Aristotelian belief that objects hadanaturalplace in an Earth-centred universethattheywouldseek out if freetodoso. Forabout two andahalf centuries the Newtonian idea of gravity as aforce held sway.Then, in thetwentieth century,cameEinstein’sconception of gravity as amanifestation of spacetimecurvature. It is this latter view that is themain concernofthisbook. Thestory of Einsteinian gravitationbeginswith afailure. Einstein’s theory of special relativity,publishedin1905 while he wasworking as aclerkinthe Swiss Figure1 Albert Einstein PatentOffice in Bern, marked an enormous step forwardintheoretical physics (1879–1955) depicted during the andsoon brought himacademic recognitionand personalfame. However, it also timethatheworked at the Patent showed that the Newtonian idea of agravitationalforce wasinconsistentwith the OfficeinBern. While there, he relativistic approachand that anew theory of gravitationwas required.Ten years publishedaseries of papers later,Einstein’sgeneral theory of relativity metthatneed,highlightingthe relating to special relativity, important role of geometryinaccountingfor gravitationalphenomenaand leading quantum physics andstatistical on to concepts such as black holes andgravitationalwaves.Within ayear anda mechanics.Hewas awardedthe half of its completion, thenew theory wasproviding thebasis for anovelapproach NobelPrize for Physics in 1921, to cosmology –the science of theuniverse–that wouldsoon have to takeaccount mainly for hiswork on the of theastronomyofgalaxies andthe physics of cosmicexpansion. Thechange in photoelectric effect. thinking demandedbyrelativity wasradical andprofound. Itsmasteryisone of thegreat challenges andgreatestdelights of anyseriousstudy of physical science. This book begins with twochaptersdevoted to special relativity.These are followedbyamainly mathematical chapter that providesthe background in geometrythatisneeded to appreciate Einstein’s subsequent developmentofthe theory.Chapter 4examinesthe basicprinciples andassumptions of general relativity –Einstein’stheory of gravity –while Chapters5and6applythe theory to an isolated spherical body andthenextendthatanalysistonon-rotating and rotating black holes.Chapter 7concerns thetesting of general relativity,including theuse of astronomical observations andgravitationalwaves.Finally,Chapter 8 examines modern relativistic cosmology,setting the scenefor further andongoing studies of observationalcosmology. Thetextbefore you is theresultofacollaborative effort involvingateam of authors andeditors working as part of thebroadereffort to produce theOpen University courseS383 TheRelativistic Universe.Details of theteam’s membership andresponsibilities arelisted elsewhere butitisappropriate to acknowledge here theparticular contributions of JimHague regarding Chapters 1 and2,Derek Capperconcerning Chapters3,4and7,and AidenDrooganin relationtoChapters5,6and 8. Robert Lambourne wasresponsible for planning andproducingthe final unifiedtextwhich benefited greatly from theinput of the S383 CourseTeam Chair,AndrewNorton, andthe attentionofproductioneditor 9 Introduction Peter Twomey.The wholeteam drewheavily on thework andwisdom of an earlier Open University CourseTeam that wasresponsible for theproductionof thecourseS357 Space, Time and Cosmology. Amajor aim for this book is to allowupper-levelundergraduate studentsto developthe skills andconfidenceneeded to pursue theindependent study of the many more comprehensive textsthatare nowavailable to studentsofrelativity, gravitationand cosmology.Tofacilitate this the current text haslargely adopted the notation used in the outstanding book by Hobson et al. GeneralRelativity : An Introductionfor Physicists,M.P.Hobson, G. Efstathiou andA.N.Lasenby,Cambridge University Press,2006. Otherbooks that provide valuable furtherreading are(roughly in order of increasingmathematical demand): An IntroductiontoModern Cosmology,A.Liddle, Wiley, 1999. Relativity,Gravitationand Cosmology : ABasic Introduction,T-P.Cheng, Oxford University Press:2005. IntroducingEinstein’sRelativity,R.d’Inverno,OxfordUniversity Press,1992. Relativity : Special, Generaland Cosmological,W.Rindler,OxfordUniversity Press,2001. Cosmology,S.Weinberg, CambridgeUniversity Press,2008. Twouseful sourcesofreprintsoforiginalpapersofhistorical significance are: ThePrinciple of Relativity,A.Einstein et al.,Dover, NewYork, 1952. CosmologicalConstants,edited by J. Bernstein and G. Feinberg,Columbia University Press,1986. Thosewishing to undertake background reading in astronomy, physics and mathematics to support their study of this book or of anyofthe others listed above might findthe followingparticularly helpful: An IntroductiontoGalaxies and Cosmology,edited by M. H. Jonesand R. J. A. Lambourne, CambridgeUniversity Press,2003. Theseven volumesinthe series ThePhysicalWorld,edited by R. J. A. Lambourne, A. J. Norton et al.,Institute of Physics Publishing, 2000. (Gotowww.physicalworld.org for furtherdetails.) Thepaired volumes BasicMathematics forthe Physical Sciences,edited by R. J. A. Lambourne and M. H. Tinker,Wiley, 2000. FurtherMathematics forthe Physical Sciences,edited by M. H. Tinker and R. J. A. Lambourne, Wiley, 2000. Numbered exercises appear throughout this book. Complete solutions to these exercises can be found at the back of thebook. Thereare anumberoflengthy worked examples;these arehighlighted by abluebackground. Thereare also severalshorterin-text questions that areimmediately followedbytheir answers. Thesemay be treated as questions or examples.The questions areindented and indicated by afilled circle; their answers arealsoindented andshown by an open circle. 10 Chapter 1Special relativity and spacetime Introduction In twoseminal papers in 1861 and1864, andinhis treatiseof1873, JamesClerk Maxwell (Figure 1.1), Scottishphysicistand genius,wrotedownhis revolutionary unifiedtheory of electricity andmagnetism, atheory that is nowsummarized in the equations that bear hisname. Oneofthe deep results of thetheory introduced by Maxwell wasthe prediction that wave-likeexcitations of combined electric andmagnetic fieldswouldtravelthrough avacuum with thesamespeed as light. It wassoon widely accepted that lightitself wasanelectromagnetic disturbance propagating through space, thusunifying electricity andmagnetismwith optics. Thefundamental work of Maxwell openedthe wayfor an understanding of the universeatamuch deeper level. Maxwell himself,incommonwith many scientists of thenineteenth century,believedinanall-pervading medium called the ether,through which electromagnetic disturbances travelled, justasocean wavestravelled through water.Maxwell’stheory predicted that lighttravels with thesamespeed in all directions,soitwas generally assumedthatthe theory Figure1.1 JamesClerk predicted the results of measurements made usingequipmentthatwas at rest with Maxwell (1831–1879) respect to the ether.Since theEarth wasexpected to move through theether as it developedatheory of orbited the Sun, measurements made in terrestrial laboratories were expected to electromagnetismthatwas show that light actually travelled with differentspeedsindifferent directions, alreadycompatible with special allowing thespeed of theEarth’smovement through theether to be determined. relativity theory severaldecades However, experiments,mostnotably by A. A. Michelson andE.W.Morleyin beforeEinstein andothers 1887, failed to detect anyvariationsinthe measured speed of light.Thisled some developedthe theory.Heisalso to suspect that measurements of thespeed of light in avacuum wouldalways famous for majorcontributions yield the same result irrespective of themotionofthe measuring equipment. to statistical physics andthe Explaininghow this couldbethe casewas amajor challengethatprompted inventionofcolour photography. ingeniousproposals from mathematiciansand physicists such as Henri Poincare,´ George Fitzgerald andHendrik Lorentz. However, it wasthe young Albert Einstein whofirstput forward acoherent andcomprehensivesolutioninhis 1905 paper‘On theelectrodynamics of moving bodies’, which introduced the special theoryofrelativity.With thebenefitofhindsight,wenow realize that Maxwell hadunintentionally formulated the first majortheory that wasconsistentwith special relativity,arevolutionary newway of thinking about space andtime. This chapter reviewsthe implications of special relativity theory for the understanding of space andtime. Thenarrative covers thefundamentals of the theory,concentrating on some of themajor differences between our intuition about space andtimeand thepredictionsofspecial relativity.Bythe endofthis chapter,you shouldhaveabroad conceptual understanding of special relativity, andbeabletoderiveits basicequations, theLorentz transformations,from the postulates of special relativity.You will understandhow to use events and intervals to describe properties of space andtimefar from gravitatingbodies.You will alsohavebeen introduced to Minkowskispacetime, afour-dimensional fusionofspace andtimethatprovidesthe