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 naturalsetting for discussions of special relativity. 11 Chapter 1Special relativity and spacetime

1.1Basic conceptsofspecialrelativity 1.1.1Events, frames of referenceand observers

When dealing with special relativity it is important to uselanguage very precisely in order to avoidconfusionand error.Fundamental to the precisedescriptionof physical phenomenaisthe conceptofanevent,the spacetimeanalogue of apoint in space or an instant in time.

Events An event is an instantaneous occurrenceataspecificpoint in space.

An exploding firecracker or asmall light that flashesonceare good approximationstoevents, sinceeach happens at adefinite timeand at adefinite position. To knowwhenand whereanevent happened, we need to assign some coordinates to it: atimecoordinate t andanorderedset of spatial coordinates such as the Cartesian coordinates (x, y,z),though we might equally well use spherical coordinates (r, θ, φ) or anyother suitable set. Theimportant point is that we shouldbeabletoassign aunique setofclearly defined coordinates to anyevent. This leadsustoour second important concept, a frameofreference.

Framesofreference A frame of reference is asystem for assigning coordinates to events.It consists of asystem of synchronized clocksthatallows aunique valueofthe timetobeassignedtoany event, andasystem of spatial coordinates that allows aunique positiontobeassignedtoany event.

In much of what follows we shall make useofaCartesian coordinate system with axes labelled x, y and z.The precisespecification of such asystem involves selectinganoriginand specifying theorientation of thethree orthogonalaxesthat meet at the origin. As farasthe system of clocksisconcerned, you can imagine that space is filled with identical synchronized clocksall tickingtogether (we shall need to saymore about howthismight be achievedlater). When usingaparticular frame of reference, the timeassignedtoanevent is thetimeshown on theclock at thesite of theevent when the eventhappens.Itisparticularly important to note that thetimeofanevent is not thetimeatwhich the eventisseen at some faroff point —itisthe timeatthe eventitself that matters. Reference framesare often represented by theletter S. Figure 1.2 provideswhat we hope is amemorableillustrationofthe basicidea, in this casewith just two spatial dimensions.Thismight be called the frame Sgnome. Among all the framesofreference that might be imagined,there is aclassof framesthatisparticularly important in special relativity.Thisisthe classof inertialframes.Aninertial frame of referenceisone in which abody that is not subject to anynet forcemaintainsaconstantvelocity.ThisisNewton’sfirstlaw of motion,sowecan saythe following. 12 1.1 Basic concepts of special relativity

Figure1.2 Ajocular representation of aframe of referenceintwo spatial dimensions.Gnomes pervadeall of space andtime. Each gnome hasaperfectly reliable clock. When an eventoccurs,the gnomenearesttothe event communicates the time t and location (x, y) of theevent to theobserver.

Inertial frames of reference An inertialframe of reference is aframe of referenceinwhich Newton’s first lawofmotionholds true.

Anyframe that moveswith constantvelocity relative to an inertial frame will also be an inertial frame.So, if you can identifyorestablishone inertial frame,then you can findaninfinite numberofsuchframes, each having aconstantvelocity relative to anyofthe others.Any frame that accelerates relative to an inertial frame cannot be an inertial frame.Since rotation involves changing velocity,any frame that rotates relative to an inertial frame is alsodisqualified from being inertial. Oneother conceptisneeded to complete thebasic vocabularyofspecial relativity. This is theideaofanobserver.

Observers An observer is an individualdedicated to usingaparticular frame of referencefor recording events.

13 Chapter 1Special relativity and spacetime

We might speak of an observerOusingframe S, or adifferent observer O% (read as ‘O-prime’) using frame S% (read as ‘S-prime’). Though you maythink of an observerasaperson, just likeyou or me,atrestin their chosen frame of reference, it is important to realize that an observer’s locationisofnoimportancefor reportingthe coordinates of events in special relativity.The positionthatanobserverassigns to an eventisthe place whereit happened. Thetimethatanobserverassigns is thetimethatwouldbeshown on a clock at the site of theevent when the eventactually happened, andwhere the clock concernedispartofthe network of synchronized clocksalwaysusedinthat observer’sframe of reference. An observermight seethe explosionofadistant star tonight, butwouldreport thetimeofthe explosionasthe timelong agowhen theexplosionactually occurred, notthe timeatwhich the lightfrom theexplosion reached the observer’slocation.Tothisextent, ‘seeing’and ‘observing’ arevery differentprocesses. It is best to avoidphrases such as ‘anobserversees ...’ unlessthatiswhatyou really mean.Anobservermeasures andobserves. Anyobserverwho uses an inertial frame of referenceissaid to be an inertial observer.Einstein’sspecial theory of relativity is mainly concernedwith observations made by inertial observers.That’swhy it’scalled special relativity —the term‘special’ is used in the senseof‘restricted’ or ‘limited’. We shall not really getawayfrom this limitationuntil we turn to general relativity in Chapter 4.

Exercise 1.1 Formanypurposes,aframe of referencefixedinalaboratory on theEarth providesagood approximation to an inertial frame.However,sucha frame is not really an inertial frame.How might its true,non-inertial, nature be revealed experimentally,atleastinprinciple? ■

1.1.2The postulates of special relativity

Physicists generally treat thelawsofphysics as though they holdtrue everywhere andatall times.There is some evidence to support such an assumption, though it is recognized as ahypothesisthatmight fail underextremeconditions.Tothe extentthatthe assumptionistrue,itdoesnot matter whereorwhenobservations aremadetotestthe laws of physics sincethe timeand place of atestof fundamental laws shouldnot have anyinfluenceonits outcome. Whereand when laws aretested might not influence the outcome,but what about motion? We knowthatinertial andnon-inertial observers will not agreeabout Newton’sfirstlaw.But what about differentinertial observers in uniformrelative motionwhere one observermovesatconstantvelocity with respect to the other? Apair of inertial observers would agree about Newton’sfirstlaw;might they also agreeabout otherlawsofphysics? It haslong been thought that they wouldatleastagree about thelawsof mechanics.Evenbefore Newton’slawswereformulated, the greatItalian physicistGalileo Galilei (1564–1642) pointed out that atraveller on asmoothly moving boathad exactly the same experiences as someone standing on theshore. Aball game couldbeplayed on auniformlymovingshipjustaswell as it could be played on shore. To theearly investigators,uniformmotionalone appeared to have no detectable consequences as farasthe laws of mechanics were concerned. An observershut up in asealed box that prevented anyobservationofthe outside 14 1.1 Basic concepts of special relativity worldwouldbeunabletoperform anymechanics experimentthatwouldreveal the uniformvelocity of thebox, even though anyacceleration couldbeeasily detected.(We areall familiar with thefeeling of beingpressedback in our seats when atrain or car accelerates forward.) Thesenotions providedthe basisfor the first theory of relativity,which is nowknown as Galileanrelativity in honour of Galileo’s originalinsight.Thistheory of relativity assumesthatall inertial observers will agreeabout thelawsofNewtonian mechanics. Einstein believedthatinertial observers wouldagree about thelawsofphysics quite generally,not just in mechanics.But he wasnot convinced that Galilean relativity wascorrect, which brought Newtonian mechanics into question. The onlystatementthathewanted to presumeasalawofphysics wasthatall inertial observers agreed about thespeed of light in avacuum.Startingfrom this minimal assumption, Einstein wasled to anew theory of relativity that wasmarkedly differentfrom Galilean relativity.The newtheory,the special theory of relativity, supportedMaxwell’slawsofelectromagnetismbut causedthe laws of mechanics to be substantially rewritten.Italsoprovidedextraordinary newinsightsinto space andtimethatwill occupy us for therestofthischapter. Einstein basedthe special theory of relativity on two postulates,thatis, two statements that he believedtobetrue on thebasis of thephysics that he knew. The first postulate is oftenreferredtoasthe principle of relativity.

The first postulate of special relativity Thelawsofphysics can be written in thesameform in all inertial frames.

This is aboldextension of theearlier belief that observers wouldagree about the laws of mechanics,but it is not at first sight exceptionally outrageous.Itwill, however, have profound consequences. Thesecond postulate is theone that givesprimacy to thebehaviour of light, asubject that wasalreadyknown as asource of difficulty.Thispostulate is sometimes referredtoasthe principle of theconstancy of thespeed of light.

The second postulate of special relativity Thespeed of light in avacuum hasthe same constantvalue, c =3×108 ms−1,inall inertial frames.

This postulate certainly accountsfor Michelson andMorley’sfailuretodetect anyvariationsinthe speed of light,but at first sight it still seemscrazy.Our experiencewith everydayobjects moving at speedsthatare small compared with thespeed of light tells us that if someone in acar that is travelling forwardat speed v throwssomethingforwardatspeed w relative to thecar,then, according to an observerstanding on theroadside, thethrown object will move forwardwith speed v + w.But thesecond postulate tells us that if thetraveller in thecar turns on atorch, effectively throwing forwardsomelight moving at speed c relative to thecar,thenthe roadsideobserverwill findthatthe light travels at speed c,not the v + c that might have been expected. Einstein realized that for this to be true, space andtimemustbehaveinpreviouslyunexpected ways. 15 Chapter 1Special relativity and spacetime

Thesecond postulate hasanotherimportant consequence. Sinceall observers agreeabout thespeed of light,itispossibletouse light signals (or anyother electromagnetic signalthattravels at thespeed of light)toensure that thenetwork of clocksweimagine each observertobeusing is properlysynchronized.Weshall not go into thedetails of howthisisdone,but it is worthpointingout that if an observersentaradar signal(which travels at the speed of light)sothatitarrivedat an eventjustasthe eventwas happening andwas immediately reflected back,then thetimeofthe eventwouldbemidwaybetween the times of transmission and reception of theradar signal. Similarly,the distance to the eventwouldbegiven by half theround trip travel timeofthe signal, multiplied by thespeed of light.

1.2Coordinate transformations

A theoryofrelativity concerns therelationship between observations made by observers in relative motion. In thecaseofspecial relativity,the observers will be inertial observers in uniformrelative motion, andtheir most fundamental observations will be thetimeand space coordinates of events. Forthe sake of definitenessand simplicity,weshall consider twoinertial observers Oand O% whoserespective framesofreference, Sand S%,are arranged in the following standard configuration (see Figure 1.3): 1. Theoriginofframe S% movesalong the x-axisofframe S, in thedirectionof increasingvalues of x,with constantvelocity V as measured in S. 2. The x-, y-and z-axes of frameSarealwaysparallel to the corresponding x%-, y%-and z%-axes of frameS%. 3. Theevent at which theorigins of Sand S% coincideoccurs at time t =0in frame Sand at time t% =0in frame S%. We shall make extensive useof‘standard configuration’ in what follows. The arrangementdoesnot entail anyreal lossofgenerality sinceany pair of inertial framesinuniformrelative motioncan be placed in standard configurationby choosingtoreorientate the coordinate axes in an appropriate way, shiftingthe origin, andresetting the clocksappropriately. In general, theobservers using theframesSandS%will not agreeabout the coordinates of an event, butsince each observerisusing awell-defined frame of reference, theremustexist aset of equations relating the coordinates (t, x, y,z) assignedtoaparticular eventbyobserverO,tothe coordinates (t%,x%,y%,z%) assignedtothe same eventbyobserverO%.The setofequations that performsthe taskofrelating the twosets of coordinates is called a coordinate transformation. This section considersfirstthe Galileantransformations that provide thebasis of Galilean relativity,and then the Lorentztransformations on which Einstein’s special relativity is based.

1.2.1The Galileantransformations

Beforethe introductionofspecial relativity,mostphysicists wouldhavesaid that the coordinate transformation between Sand S% was‘obvious’, andtheywould

16 1.2 Coordinate transformations have written down thefollowing Galileantransformations: t% = t, (1.1) x% = x − Vt, (1.2) y% = y, (1.3) z% = z, (1.4) where V = |V | is therelative speed of S% with respect to S.

in standard configuration z frameorigins coincide at t = t" =0

t z"

t"

y frameS

" frameS" y

x

x"

Figure1.3 Twoframesofreference in standardconfiguration.Notethatthe speed V is measured in frame S. To justifythisresult, it might have been arguedthatsince theobservers agree about thetimeofthe eventatwhich the origins coincide(seepoint 3inthe definitionofstandard configuration), they must alsoagree about thetimes of all otherevents. Further, sinceattime t theoriginofS%will have travelled adistance Vtalongthe x-axisofframe S, it must be thecasethatany eventthatoccurs at time t with positioncoordinate x in frame Smustoccuratx%=x−Vtin frame S%,while thevalues of y and z will be unaffected by themotion. However, as Einstein realized, such an argumentcontains many assumptions about the behaviour of timeand space, andthoseassumptions might not be correct. For example, Equation1.1 implies that timeisinsomesense absolute,bywhich we mean that the timeintervalbetween anytwo events is the same for all observers. Newton certainly believedthistobethe case, butwithout supporting evidence it wasreally nothing more than aplausibleassumption. It wasintuitively appealing, butitwas fundamentally untested.

17 Chapter 1Special relativity and spacetime

1.2.2The Lorentztransformations

Rather than rely on intuitionand run theriskofmaking unjustified assumptions, Einstein chosetoset out histwo postulates anduse them to deducethe appropriate coordinate transformation between Sand S%.Aderivationwill be givenlater,but beforethatlet’sexamine theresultthatEinstein found.The equations that he derivedhad alreadybeen obtained by theDutch physicist Hendrik Lorentz(Figure 1.4) in thecourseofhis owninvestigationsintolight andelectromagnetism. Forthatreason, they areknown as the Lorentz transformations even though Lorentzdid not interpret or utilize them in thesame waythatEinstein did. Here arethe equations: t − Vx/c2 t% = - , 1 − V 2/c2 x − Vt x% = - , 1−V2/c2 y% = y, z% = z. Figure1.4 Hendrik Lorentz (1853–1928) wrotedownthe It is clear that theLorentztransformationsare very differentfrom theGalilean Lorentztransformationsin transformations. They indicate athorough mixing together of space andtime, 1904. He wonthe 1902 Nobel sincethe t%-coordinate of an eventnow depends on both t and x,justasthe Prize for Physics for work on x%-coordinate does. According to theLorentztransformations, thetwo observers electromagnetism, andwas do notgenerally agreeabout thetimeofevents, even though they still agree greatly respected by Einstein. about thetimeatwhich the origins of their respective framescoincided.So, timeisnolongeranabsolutequantity that all observers agreeabout.Tobe meaningful,statements about thetimeofanevent must nowbeassociated with a particular observer. Also,the extenttowhich the observers di-sagree about the x-coordinate of an eventhas been modifiedbyafactor of 1/ 1 − V 2/c2.Infact, this multiplicative factor is so commoninspecial relativity that it is usually referredtoasthe Lorentz factor or gamma factor andisrepresented by the symbol γ(V ),emphasizing that its valuedepends on therelative speed V of the twoframes. Using this factor,the Lorentztransformationscan be written in the following compact form.

The Lorentz transformations

t% = γ(V )(t − Vx/c2), (1.5) x% = γ(V )(x − Vt), (1.6) y% = y, (1.7) z% = z, (1.8) where 1 γ(V )=- . (1.9) 1−V2/c2

Figure 1.5 showshow theLorentzfactor grows as the relative speed V of the twoframesincreases. Forspeedsthatare small compared with thespeed of 18 1.2 Coordinate transformations

γ(V ) ≈ 1 light, ,and theLorentztransformationsapproximate the Galilean 5 transformationsprovidedthat x is not toolarge.Asthe relative speed of thetwo 4 framesapproachesthe speed of light,however,the Lorentzfactor grows rapidly 3 andsodothe discrepancies between theGalilean andLorentztransformations. γ 2 Exercise 1.2 Computethe Lorentzfactor γ(V ) when the relative speed V is (a) 10%ofthe speed of light,and (b) 90%ofthe speed of light. ■ 1 TheLorentztransformationsare so important in special relativity that you will see 0 c/4 c/2 3c/4 c them written in many differentways. They areoften presented in matrix form,as V     ct% γ(V ) −γ(V )V/c 00 ct Figure1.5 Plot of x%  −γ(V)V/cγ(V)00x   =   . y%   0010y  (1.10) theLorentzf-actor, γ(V )=1/ 1−V2/c2.The z% 0001 z factor is closeto1for speeds Youshouldconvinceyourself that this matrix multiplicationgives equations much smaller than thespeed of equivalenttothe Lorentztransformations. (The equationfor transformingthe light,but increases rapidlyasV timecoordinate is multiplied by c.) We can alsorepresent this relationshipbythe approaches c.Notethat γ>1 equation for all values of V . %µ µ ν [x ]=[Λ ν][x ], (1.11) where we use thesymbol [xµ] to represent thecolumnvector with components 0 1 2 3 µ (x ,x ,x ,x )=(ct,x,y,z),and thesymbol [Λ ν ] to represent the Lorentz transformationmatrix  0 0 0 0 Λ 0 Λ 1 Λ 2 Λ 3  1 1 1 1  µ Λ 0 Λ 1 Λ 2 Λ 3 [Λ ν ] ≡ 2 2 2 2 Λ 0 Λ 1 Λ 2 Λ 3 Λ3 Λ3 Λ3 Λ3  0 1 2 3  γ(V ) −γ(V )V/c 00 −γ(V)V/cγ(V)00 =  .  0010 (1.12) 0001 µ At this stage, when dealing with an individualmatrix element Λ ν ,you can simply regard thefirstindexasindicating the rowtowhich it belongs andthe second indexasindicating the column. It then makessense that each of the elements xµ in thecolumnvector [xµ] shouldhavearaised index.However,as you will seelater,inthe contextofrelativity thepositioning of theseindices actually hasamuch greater significance. Thequantity [xµ] is sometimes called the four-position sinceits fourcomponents (ct,x,y,z)describe thepositionofthe eventintimeand space. Note that by using ct to convey thetimeinformation, ratherthanjust t,all fourcomponentsof thefour-positionare measured in units of distance. Also notethatthe Greek indices µ and ν takethe values 0 to 3.Itisconventionalinspecial andgeneral relativity to startthe indexing of thevectorsand matrices from zero, where x0 = ct.Thisisbecause thetimecoordinate hasspecial properties. Using theindividualcomponentsofthe four-position, anotherway of writingthe Lorentztransformation is in terms of summations: A3 %µ µ ν x = Λ ν x (µ =0,1,2,3). (1.13) ν=0 19 Chapter 1Special relativity and spacetime

This one linereally representsfour differentequations,one for each valueofµ. When an index is used in this way, it is said to be a free index,since we arefree to give it anyvalue between 0 and 3,and whateverchoice we make indicates a differentequation. Theindex ν that appearsinthe summation is not free, since whatevervalue we choosefor µ,weare required to sumoverall possiblevalues of ν to obtain the final equation. This meansthatwecouldreplace all appearances of ν by some otherindex, α say, without actually changing anything. An index that is summed overinthisway is said to be a dummy index. Familiarity with thesummationform of theLorentztransformationsisparticularly useful when beginning thediscussion of general relativity;you will meet many such sums. Beforemovingon, you shouldconvinceyourself that you can easily switch between theuse of separateequations,matrices (including theuse of four-positions)and summationswhenrepresentingLorentztransformations. Giventhe coordinates of an eventinframe S, theLorentztransformationstell us thecoordinates of that same eventasobservedinframe S%.Itisequally important that thereissomeway to transform coordinates of an eventinframe S% back into the coordinates in frame S. Thetransformationsthatperform this taskare known as the inverseLorentz transformations.

The inverse Lorentz transformations

t = γ(V )(t% + Vx%/c2), (1.14) x = γ(V )(x% + Vt%), (1.15) y = y%, (1.16) z = z%. (1.17)

Notethatthe onlydifferencebetween the Lorentztransformationsand their inverses is that all the primed andunprimed quantities have been interchanged, andthe relative speed of thetwo frames, V ,has been replaced by thequantity −V .(This changesthe transformationsbut not thevalue of the Lorentzfactor,which depends onlyonV2,socan still be written as γ(V ).) This relationshipbetween the transformationsisexpected, sinceframe S% is moving with speed V in the positive x-direction as measured in frame S, while frame Sis moving with speed V in the negative x%-direction as measured in frame S%. Youshouldconfirm that performingaLorentztransformation andits inverse transformation in succession really doeslead back to the originalcoordinates,i.e. (ct,x,y,z)→(ct%,x%,y%,z%)→(ct,x,y,z). ● An eventoccurs at coordinates (ct =3m,x=4m,y=0,z=0)in frame Saccording to an observerO.Whatare thecoordinates of thesame eventinframe S% according to an observerO%,movingwith speed V =3c/4 in thepositive x-direction,asmeasured in S? ❍ First,the Lorentzfactor γ(V ) shouldbecomputed: - √ γ(3c/4) =1/ 1−32/42=4/ 7.

20 1.2 Coordinate transformations

Thenew coordinates arethengiven by th√eLorentztransformations: ct% = cγ(3c/4)(t − 3x/4c)=(4/ 7)(3 m − 3c × 4 m/4c)=0m, √ √ x%=γ(3c/4)(x − 3tc/4) =(4/ 7)(4 m − 3 × 3 m/4) = 7 m, y% = y =0m, ct z%=z=0m. S Exercise 1.3 ? @ ?Thematrix equation @? @ ct% γ(V ) −γ(V )V/c ct = x% −γ(V)V/cγ(V) x no acceleration canbeinverted to determine thecoordinates (ct,x)in terms of (ct%,x%).Show that invertingthe 2 × 2 matrix leadstothe inverseLorentztransformationsin Equations 1.14 and1.15. ■ O x ct" 1.2.3Aderivation of theLorentz transformations S" This subsection presents aderivation of theLorentztransformationsthatrelates the coordinates of an eventintwo inertial frames, Sand S%,thatare in standard configuration. It mainly ignores the y-and z-coordinates andjustconsidersthe particle observed transformation of the t-and x-coordinates of an event. Ageneral transformation to accelerateif relatingthe coordinates (t%,x%)of an eventinframe S% to thecoordinates (t, x) of higher-order terms thesameevent in frame Smay be written as are left in % 2 2 t = a0 + a1t + a2x + a3t + a4x + ··· , (1.18) O" x" % 2 2 x = b0 + b1x + b2t + b3x + b4t + ··· , (1.19) Figure1.6 Leaving x t where thedotsrepresent additionalterms involvinghigherpowersof or . higher-order terms in the Now, we knowfrom thedefinitionofstandard configurationthatthe event coordinate transformations marking thecoincidence of theorigins of framesSandS%hasthe coordinates wouldcause uniformmotionin (t, x)=(0, 0) in Sand (t%,x%)=(0,0) in S%.Itfollows from Equations 1.18 one inertial frame Stobe and1.19 that theconstants a0 and b0 arezero. observedasaccelerated motion % ThetransformationsinEquations 1.18 and1.19 can be further simplified by the in theother inertial frame S . requirementthatthe observers areusing inertial framesofreference. Since Thesediagrams, in which the Newton’sfirstlaw must holdinall inertial framesofreference, it is necessary that vertical axis representstime an object not accelerating in one setofcoordinates is alsonot accelerating in the multiplied by thespeed of light, t2 otherset of coordinates.Ifthe higher-order terms in x and t were not zero, then an show that if the terms were object observedtohavenoacceleration in S(such as aspaceshipwith its thrusters leftinthe transformations, then offmovingonthe line x = vt,shown in theupperpartofFigure 1.6) wouldbe motionwith no acceleration in observedtoaccelerate in terms of x% and t% (i.e. x% =4 v%t%,asindicated in the lower frame Swouldbetransformed into motionwith non-zero part of Figure 1.6).ObserverOwouldreport no force on thespaceship, while % observerO%wouldreport some unknown forceacting on it. In this way, thetwo acceleration in frame S .This wouldimply change in velocity observers wouldregister differentlawsofphysics,violatingthe first postulate of % special relativity.The higher-order terms arethereforeinconsistentwith the without force in S ,inconflict required physics andmustberemoved, leaving onlyalinear transformation. with Newton’sfirstlaw. So we expect the special relativistic coordinate transformation between two framesinstandard configurationtoberepresented by linear equations of theform % t = a1t + a2x, (1.20) % x = b1x + b2t. (1.21)

21 Chapter 1Special relativity and spacetime

Theremainingtaskistodetermine thecoefficients a1, a2, b1 and b2. To do this,use is made of known relations between coordinates in bothframesof reference. Thefirststep is to usethe fact that at anytime t,the originofS%(which is always at x% =0in S%)will be at x = Vtin S. It follows from Equation1.21 that

0=b1Vt+b2t, from which we seethat

b2 = −b1V. (1.22)

Dividing Equation1.21 by Equation1.20,and usingEquation1.22 to replace b2 by −b1V ,leadsto x% b x−b Vt = 1 1 . % (1.23) t a1t+a2x Now, as asecond step we can usethe fact that at anytime t%,the originofframe S (which is always at x =0in S) will be at x% = −Vt% in S%.Substitutingthese values for x and x% into Equation1.23 gives −Vt% −b Vt = 1 , % (1.24) t a1t from which it follows that

b1 = a1.

If we nowsubstitute a1 = b1 into Equation1.23 anddividethe numerator and denominatoronthe right-hand side by t,then x% b (x/t) − Vb = 1 1. % (1.25) t b1+a2(x/t)

As athird step,the coefficient a2 can be found usingthe principle of theconstancy of thespeed of light.Apulse of light emitted in the positive x-direction from (ct =0,x=0)hasspeed c = x%/t% andalso c = x/t.Substitutingthese values into Equation1.25 gives b c − Vb c = 1 1, b1+a2c which can be rearrangedtogive 2 2 a2 = −Vb1/c = −Va1/c . (1.26)

Nowthat a2, b1 and b2 areknown in terms of a1,the coordinate transformations between thetwo framescan be written as % 2 t = a1(t − Vx/c ), (1.27) % x = a1(x − Vt). (1.28)

Allthatremains for thefourth step is to findanexpression for a1.Todothis, we first write down theinverse transformationstoEquations 1.27 and1.28,which arefound by exchanging primes andreplacing V by −V .(We areimplicitly assuming that a1 depends onlyonsomeevenpower of V .) This gives % % 2 t = a1(t + Vx/c ), (1.29) % % x = a1(x + Vt). (1.30)

22 1.2 Coordinate transformations

SubstitutingEquations 1.29 and1.30 into Equation1.28 gives ? ? @@ V x% = a a (x% + Vt%)−Va t%+ x% . 1 1 1 c2 % Thesecond andthird terms involving a1Vt cancel in this expression, leaving an expression in which the x% cancels on bothsides: ? @ V 2 x% = a2 1 − x%. 1 c2

By rearrangingthisequationand takingthe positive square root, thecoefficient a1 is determinedtobe 1 a1= - . (1.31) 1−V2/c2

Thus a1 is seen to be theLorentzfactor γ(V ),which completes thederivation. Some furtherarguments allowthe Lorentztransformationstobeextendedtoone timeand threespace dimensions.There can be no y and z contributions to the transformationsfor t% and x% sincethe y-and z-axes couldbeoriented in anyof theperpendicular directions without affecting the events on the x-axis. Similarly, therecan be no contributions to thetransformationsfor y% and z% from anyother coordinates,asspace wouldbecome distortedinanon-symmetric manner.

1.2.4Intervals andtheir transformationrules

Knowinghow thecoordinates of an eventtransform from oneframe to another, it is relatively simple to determine howthe coordinate intervals that separatepairsof events transform.Asyou will seeinthe next section,the rulesfor transforming intervals areoften very useful.

Intervals An interval between twoevents, measured alongaspecified axis in agiven frame of reference, is the difference in the corresponding coordinates of the twoevents.

To developtransformation rules for intervals,consider theLorentz transformationsfor thecoordinates of twoeventslabelled 1 and 2: % 2 % t1 = γ(V )(t1 − Vx1/c ),x1=γ(V)(x1 − Vt1), % % y1 =y1,z1=z1 %2 % t2=γ(V)(t2 − Vx2/c ),x2=γ(V)(x2 − Vt2), % % y2 =y2,z2=z2.

% % Subtractingthe transformation equationfor t1 from that for t2,and subtractingthe % % transformation equationfor x1 from that for x2,and so on, givesthe following transformation rules forintervals:

23 Chapter 1Special relativity and spacetime

Δt% = γ(V )(Δt − V Δx/c2), (1.32) Δx% = γ(V )(Δx − V Δt), (1.33) Δy% =Δy, (1.34) Δz% =Δz, (1.35)

where Δt = t2 − t1, Δx = x2 − x1, Δy = y2 − y1 and Δz = z2 − z1 denotethe varioustimeand space intervals between theevents. Theinverse transformations for intervals have thesameform,with V replaced by −V : Δt = γ(V )(Δt% + V Δx%/c2), (1.36) Δx = γ(V )(Δx% + V Δt%), (1.37) Δy =Δy%, (1.38) Δz =Δz%. (1.39)

Thetransformation rules for intervals areuseful because they depend onlyon coordinate differences andnot on thespecificlocationsofeventsintimeorspace.

1.3Consequencesofthe Lorentz transformations

In this section,someofthe extraordinary consequences of theLorentz transformationswill be examined.Inparticular,weshall consider thefindings of differentobservers regarding therate at which aclock ticks, thelengthofarod andthe simultaneity of apair of events.Ineach case, the trick for determining howthe relevantproperty transformsbetween framesofreference is to carefully specifyhow intuitive concepts such as length or durationshouldbedefined consistently in differentframesofreference. This is most easily done by identifying each conceptwith an appropriate interval between twoevents: 1 and 2.Oncethishas been achieved, we can determine which intervals areknown and then usethe intervaltransformation rules (Equations1.32–1.35 and1.36–1.39)to findrelationshipsbetween them.The restofthissection will give examples of this process.

1.3.1Time dilation

Oneofthe most celebrated consequences of special relativity is thefinding that ‘movingclocksrun slow’. More precisely,any inertial observermustobserve that theclocksusedbyanotherinertial observer, in uniformrelative motion, will run slow.Since clocksare merely indicatorsofthe passage of time, this is really the assertionthatany inertial observerwill findthattimepassesmore slowly for any otherinertial observerwho is in relative motion. Thus,according to special relativity,ifyou andIare inertial observers,and we areinuniformrelative motion, then Ican perform measurements that will show that timeispassing more slowly for youand, simultaneously,you can perform measurements that will show that timeispassing more slowly for me.Bothofuswill be right because timeisa relative quantity,not an absolute one.Toshowhow this effect follows from the Lorentztransformations, it is essential to introduceclear,unambiguous definitions of thetimeintervals that aretoberelated. 24 1.3 Consequences of the Lorentz transformations

Rather than deal with tickingclocks, our discussion here will refertoshort-lived ct" sub-nuclear particles of thesort routinely studied at CERN andother particle S" " physics laboratories. Forthe purposeofthe discussion, ashort-lived particle is ctct2 event2 considered to be apoint-like object that is created at some event, labelled 1,and subsequently decaysatsomeother event, labelled 2.The timeintervalbetween c Δτ thesetwo events,asmeasured in anyparticular inertial frame,isthe lifetime " of theparticle in that frame.Thisintervalisanalogous to thetimebetween ctct1 event1 successive ticksofaclock. We shall consider thelifetimeofaparticular particle as observedbytwo different inertial observers Oand O%.ObserverOuses aframe Sthatisfixedinthe x" x" laboratory,inwhich the particle travels with constantspeed V in the positive ct x-direction.Weshall call this the laboratoryframe.ObserverO%uses aframe S% S Δx that moveswith theparticle. Such aframe is called the rest frame of theparticle ct2 event2 sincethe particle is always at rest in that frame.(Youcan thinkofthe observerO% as riding on theparticle if you wish.) c ΔT According to observerO%,the birth anddecay of the(stationary) particle happen % % % % ct1 event1 at the same place, so if event 1 occurs at (t1,x),thenevent 2 occurs at (t2,x), % % % andthe lifetimeofthe particle will be Δt = t2 − t1.Inspecial relativity,the time between twoeventsmeasured in aframe in which theeventshappenatthe same x x x positioniscalled the proper time between the events andisusually denoted by 1 2 thesymbol Δτ.So, in this case, we can saythatinframe S% theintervals of time % % % % Figure1.7 andspace that separatethe twoeventsare Δt =Δτ=t2−t1and Δx =0. Events and intervals for establishing the 1 (t ,x ) According to observerOin thelaboratory frameS,event occurs at 1 1 and relationbetween the lifetimeof 2 (t ,x ) Δt = t − t event at 2 2 ,and thelifetimeofthe particle is 2 1,which we shall aparticle in its rest frame (S%) ΔT call .Thus in frame Sthe intervals of timeand space that separatethe two andinalaboratory frame(S). Δt =ΔT =t −t Δx = x − x events are 2 1and 2 1. Notethatweshowthe Theseeventsand intervals arerepresented in Figure 1.7, andeverything we know coordinate on thevertical axis as about them is listed in Table1.1. Suchatable is helpful in establishing which of ‘ct’rather than ‘t’toensure that theintervaltransformationswill be useful. bothaxeshavethe dimensionof length. To converttimeintervals Ta ble1.1 Atabular approachtotimedilation. Thecoordinates of theevents such as Δτ and ΔT to this arelisted andthe intervals between them worked out,taking account of any coordinate, simply multiply known values.The lastrow is used to show which of theintervals relates to a them by theconstant c. namedquantity (suchasthe lifetimes ΔT and Δτ)orhas aknown value(such as Δx% =0). Anyintervalthatisneither known nor related to anamed quantity is shownasaquestionmark.

EventS(laboratory) S% (restframe) %% 2(t2,x2)(t2,x) %% 1(t1,x1)(t1,x) % % Intervals (t2 − t1,x2 −x1)(t2−t1,0) ≡ (Δt, Δx) ≡ (Δt%, Δx%) Relation to knownintervals (ΔT,?) (Δτ,0)

Four of theintervaltransformation rules that were introduced in the previous section involvethree intervals.But onlyEquation1.36 involves thethree known intervals.Substitutingthe known intervals into that equationgives

25 Chapter 1Special relativity and spacetime

ΔT = γ(V )(Δτ +0).Thereforethe particle lifetimes measured in Sand S% are related by ΔT = γ(V )Δτ. (1.40)

Since γ(V ) > 1,thisresulttells us that theparticle is observedtolivelongerin thelaboratory framethanitdoesinits ownrestframe.Thisisanexample of the effect known as time dilation.Aprocess that occupies a(proper) time Δτ in its ownrestframe hasalongerduration ΔT when observedfrom some otherframe that movesrelative to therestframe.Ifthe process is thetickingofaclock,thena consequenceisthatmovingclockswill be observedtorun slow. Thetimedilationeffect hasbeen demonstrated experimentally many times.It providesone of themostcommonpieces of evidence supporting Einstein’s theory of special relativity.Ifitdid not exist, many experiments involvingshort-lived Figure1.8 Henri Poincare´ particles,suchasmuons,wouldbeimpossible, whereas theyare actually quite (1854–1912). routine. It is interestingtonotethatthe Frenchmathematician Henri Poincare(´ Figure 1.8) proposed an effect similar to timedilationshortlybefore Einstein formulated special relativity.

Exercise 1.4 Aparticular muon livesfor Δτ =2.2µsinits ownrestframe.If that muon is travelling with speed V =3c/5 relative to an observeronEarth, what is its lifetimeasmeasured by that observer? ■ ct" S" 1.3.2Lengthcontraction " ctct2 event2 Thereisanothercurious relativistic effect that relates to thelengthofanobject c Δt" event1 observedfrom differentframesofreference. Forthe sake of simplicity,the object ctct" that we shall consider is arod, andweshall startour discussion with adefinition 1 L P of therod’slengththatapplies whether or not therod is moving. In anyinertial frame of reference, the length of arod is thedistance between its x" x" x" 1 2 end-pointsatasingletimeasmeasured in that frame. ct Thus,inaninertial frame Sinwhich the rod is oriented alongthe x-axisand S movesalong that axis with constantspeed V ,the length L of therod can be related to twoevents, 1 and 2,thathappenatthe ends of therod at the same time t.Ifevent 1 is at (t, x1) andevent 2 is at (t, x2),thenthe lengthofthe rod, as measured in Sattime t,isgiven by L =Δx=x2−x1. event1 event2 % ct Nowconsider thesesametwo events as observedinaninertial frame S in which L % therod is oriented alongthe x -axisbut is always at rest.Inthiscasewestill know that event 1 andevent 2 occuratthe end-pointsofthe rod, butwehavenoreason x1 x2 x to supposethattheywill occuratthe same time, so we shall describe them by the % % % % coordinates (t1,x1)and (t2,x2).Although theseeventsmay not be simultaneous, % x% Figure1.9 Events and we knowthatinframe S therod is not moving, so its end-pointsare always at 1 x% intervals for establishing the and 2.Consequently,wecan saythatthe lengthofthe rod in its ownrest relationbetween the length of a frame —aquantity sometimes referredtoasthe proper length of therod and % L L =Δx%=x% −x% rod in its rest frame (S )and in a denoted P —isgiven by P 2 1. laboratory frame(S). Theseeventsand intervals arerepresented in Figure 1.9, andeverything we know about them is listed in Table1.2. 26 1.3 Consequences of the Lorentz transformations

Ta ble1.2 Events andintervals for lengthcontraction.

EventS(laboratory) S% (restframe) %% 2(t, x2)(t2,x2) %% 1(t, x1)(t1,x1) %%% % Intervals (0,x2 −x1)(t2−t1,x2 −x1) ≡(Δt, Δx) ≡ (Δt%, Δx%) Relation to knownintervals (0,L)(?,LP)

On this occasion,the one unknown intervalisΔt%,sothe intervaltransformation rulethatrelates the three known intervals is Equation1.33.Substitutingthe known intervals into that equationgives LP = γ(V )(L − 0).Sothe lengths measured in Sand S% arerelated by L = L /γ(V ). P (1.41) Figure1.10 George Fitzgerald (1851–1901) wasan Since γ(V ) > 1,thisresulttells us that therod is observedtobeshorterinthe Irish physicistinterested in laboratory framethaninits ownrestframe.Inshort, moving rods contract. This is electromagnetism. He was an exampleofthe effect known as length contraction.The effect is not limited to influential in understanding that rods. Anymovingbody will be observedtocontractalong its directionofmotion, lengthcontracts. though it is particularly important in this casetorememberthatthisdoesnot mean that it will necessarily be seen to contract. Thereisasubstantial body of literature relatingtothe visual appearance of rapidlymovingbodies,which generally involves factorsapart from theobservedlengthofthe body. Lengthcontractionissometimes known as Lorentz–Fitzgerald contraction after the physicists (Figure 1.4 andFigure 1.10) whofirstsuggested such a ct phenomenon, though their interpretation wasrather differentfrom that of Einstein. S V ct2 Exercise 1.5 Thereisanalternative wayofdefining lengthinframe Sbased event2 on twoevents, 1 and 2,thathappenatdifferent times in that frame.Supposethat event 1 occurs at x =0as the frontend of therod passesthatpoint,and event 2 alsooccurs at x =0butatthe later timewhenthe rearend passes. Thus event 1 is V at (t1, 0) andevent 2 is at (t2, 0).Since therod moveswith uniformspeed V in ct1 frame S, we can definethe lengthofthe rod, as measured in S, by therelation event1 L = V (t2 − t1).Use this alternative definitionoflengthinframe Stoestablish that the length of amovingrod is lessthanits proper length. (The events are 0 x represented in Figure 1.11.) ■ Figure1.11 An alternative setofeventsthatcan be used to 1.3.3The relativityofsimultaneity determine thelengthofa uniformlymovingrod. It wasnoted in the discussion of lengthcontractionthattwo events that occurat thesametimeinone framedonot necessarily occuratthe same timeinanother frame. Indeed,looking again at Figure 1.9 andTable 1.2 butnow calling on the intervaltransformation ruleofEquation1.32,itisclear that if theevents 1 and 2 areobservedtooccuratthe same timeinframe S(so Δt =0)but areseparated by adistance L along the x-axis, then in frame S% they will be separated by thetime Δt% = −γ(V )VL/c2.

27 Chapter 1Special relativity and spacetime

Twoeventsthatoccuratthe same timeinsomeframe aresaid to be simultaneous in that frame.The above result showsthatthe conditionofbeing simultaneous is a relative one not an absolute one;two events that aresimultaneous in one frameare notnecessarily simultaneous in everyother frame.Thisconsequenceofthe Lorentztransformationsisreferredtoasthe relativity of simultaneity.

1.3.4The Doppler effect

Aphysical phenomenon that waswell known long beforethe advent of special relativity is the Doppler effect.Thisaccountsfor thedifferencebetween the emitted andreceivedfrequencies (or wavelengths)ofradiation arising from the relative motionofthe emitter andthe receiver. Youwill have heardanexample of theDoppler effect if you have listened to thesiren of apassing ambulance: the frequency of thesiren is higherwhenthe ambulance is approaching (i.e.travelling towards thereceiver) than when it is receding (i.e.travellingawayfrom the receiver). Astronomersroutinely usethe Doppler effect to determine thespeed of approach or recession of distantstars. They do this by measuring thereceivedwavelengths of narrow lines in thestar’s spectrum,and comparing their results with theproper wavelengths of thoselines that arewell known from laboratory measurements and represent thewavelengths that wouldhavebeen emitted in the star’s rest frame. Despite thelong history of theDoppler effect, one of theconsequences of special y y" relativity wasthe recognitionthatthe formula that hadtraditionally been used to describe it waswrong. We shall nowobtain the correct formula. Consider alampatrestatthe originofaninertial frame Semitting electromagnetic wavesofproper frequency fem as measured in S. Nowsuppose that thelampisobservedfrom anotherinertial frame S% that is in standard V configurationwith S, moving away at constantspeed V (see Figure 1.12).A detectorfixedatthe originofS%will show that theradiation from thereceding % lampisreceivedwith frequency frec as measured in S .Our aim is to findthe relationship between frec and fem. lamp detector x x" Theemitted waveshaveregularly positioned nodes (pointsofzerodisturbance) that areseparated by aproper wavelength λem = fem/c as measured in S. Figure1.12 TheDoppler In that frame thetimeintervalbetween the emission of one node andthe Δt T effect arises from therelative next, ,representsthe proper period of thewave, em,sowecan write Δt = T =1/f motionofthe emitter and em em. receiverofradiation. Duetothe phenomenon of timedilation, an observerinframe S% will findthatthe timeseparating the emission of successive nodesisΔt%=γ(V)Δt.However, this is not thetimethatseparates the arrivalofthosenodesatthe detectorbecause thedetector is moving away from theemitter at aconstantrate. In fact, during the interval Δt% thedetector will increaseits distance from theemitter by V Δt% as measured in S%,and this will cause thereception of thetwo nodestobeseparated by atotal time Δt% + V Δt%/c as measured in S%.Thisrepresentsthe received period of thewaveand is therefore thereciprocal of thereceivedfrequency,sowe can write ? @ 1 VΔt% V =Δt%+ =γ(V)Δt 1+ . frec c c 28 1.3 Consequences of the Lorentz transformations

We can nowidentify-Δt with thereciprocal of theemitted frequency anduse the identity γ(V )=1/ (1 − V/c)(1 + V/c) to write ? @ 1 1 1 V = - 1+ , frec fem (1 − V/c)(1 + V/c) c which can be rearrangedtogive # c − V f = f . unshifted rec em c + V (1.42)

This showsthatthe radiation receivedfrom areceding source will have a redshifted frequency that is lessthanthe proper frequencywith which theradiation was emitted. It follows that thereceivedwavelength λrec = c/frec will be greater than theproper wavelength λem.Consequently,the spectral lines seen in the blueshifted lightofreceding starswill be shifted towards thered endofthe spectrum;a phenomenon known as redshift (seeFigure 1.13).Inasimilar way, thespectra Figure1.13 Spectral lines are of approaching starswill be subject to a blueshift describedbyanequation redshifted (thatis, reduced in similar to Equation1.42 butwith V replaced by −V throughout. Thecorrect frequency)whenthe source is interpretation of these Doppler shifts is of greatimportance. receding,and blueshifted (increased in frequency)when Exercise 1.6 Some astronomersare studying an unusualphenomenon, close thesource is approaching. to the centreofour galaxy, involvingajet of material containingsodium.The jet is moving almostdirectly along thelinebetween the Earthand thegalactic centre. In alaboratory,astationarysampleofsodium vapour absorbs light of wavelength λ =5850 × 10−10 m. Spectroscopicstudies show that thewavelength of the sodium absorptionlineinthe jet’sspectrum is λ% =4483 × 10−10 m. Is thejet approaching or receding?Whatisthe speed of thejet relative to Earth? (Notethat themain challengeinthisquestionisthe mathematical one of usingEquation1.42 to obtain an expression for V in terms of λ/λ%.) ■

1.3.5The velocitytransformation

Supposethatanobject is observedtobemovingwith velocity v =(vx,vy,vz)in an inertial frame S. What will its velocity be in aframe S% that is in standard configurationwith S, travelling with uniformspeed V in the positive x-direction? % TheGalilean transformation wouldlead us to expect v =(vx−V,vy,vz),but we knowthatisnot consistentwith theobservedbehaviour of light.Onceagain we shall usethe intervaltransformation rules that followdirectly from theLorentz transformationstofind thevelocity transformation ruleaccording to special relativity. We knowfrom Equations 1.32 and1.33 that thetimeand space intervals between twoevents 1 and 2 that occuronthe x-axisinframe S, transform according to Δt% = γ(V )(Δt − V Δx/c2), Δx% = γ(V )(Δx − V Δt).

Dividing thesecond of theseequations by thefirstgives Δx% γ(V )(Δx − V Δt) = . Δt% γ(V )(Δt − V Δx/c2) 29 Chapter 1Special relativity and spacetime

Dividing theupperand lowerexpressions on theright-hand side of this equation by Δt,and cancelling the Lorentzfactors, gives Δx% (Δx/Δt − V ) = . Δt% (1 − (Δx/Δt)V/c2) Now, if we supposethatthe twoeventsthatweare consideringare very close together —indeed,ifweconsider thelimit as Δt and Δx go to zero—then thequantities Δx/Δt and Δx%/Δt% will become theinstantaneous velocity % components vx and vx of amovingobject that passesthrough theevents 1 and 2. Extending thesearguments to three dimensions by consideringeventsthatare not confinedtothe x-axisleadstothe following velocitytransformation rules: v − V v% = x , x 2 (1.43) 1 − vxV/c v v% = y , y 2 (1.44) γ(V )(1 − vxV/c ) v v% = z . z 2 (1.45) γ(V )(1 − vxV/c )

Theseequations maylook ratherodd at first sight buttheymakegood senseinthe contextofspecial relativity.When vx and V aresmall compared to the speed of 2 light c,the term vxV/c is very small andthe denominatorisapproximately 1.In % such cases,the Galilean velocity transformation rule, vx = vx − V ,isrecovered as alow-speed approximation to the special relativistic result. At high speedsthe situationisevenmore interesting, as the following questionwill show. ● An observerhas establishedthattwo objects arereceding in opposite directions.Object 1 hasspeed c,and object 2 hasspeed V .Using thevelocity transformation,computethe velocity with which object 1 recedes as measured by an observertravellingonobject 2. ❍ Letthe linealong which theobjects aretravellingbethe x-axisofthe original observer’sframe,S.Wecan then supposethataframe of referenceS%that has its originonobject 2 is in standard configurationwith frame S, andapplythe velocity transformation to the velocity componentsofobject 1 with v =(−c, 0, 0) (see Figure 1.14).The velocity transformation predicts that as % % % observedinS,the velocity of object 2 is v =(vx,0,0),where v − V −c−V v% = x = = −c. x 2 2 1 − vxV/c 1−(−c)V/c So,asobservedfrom object 2,object 1 is travelling in the −x%-direction at the speed of light, c.Thiswas inevitable, sincethe second postulate of special relativity (which wasusedinthe derivationofthe Lorentztransformations) tells us that all observers agreeabout thespeed of light.Itisnonetheless pleasing to seehow thevelocity transformation delivers therequired resultin this case. It is worthnotingthatthisresultdoesnot depend on thevalue of V .

Exercise 1.7 According to an observeronaspacestation,two spacecraftare moving away,travellinginthe same directionatdifferent speeds. Thenearer spacecraftismovingatspeed c/2,the further at speed 3c/4.Whatisthe speed of one of thespacecraftasobservedfrom theother? ■

30 1.4 Minkowski spacetime

y y" S S"

V

v =(−c, 0, 0) object 2 object 1 x x"

Figure1.14 Twoobjects move in opposite directions alongthe x-axisof frame S. Object 1 travels with speed c;object 2 travels with speed V andisthe originofasecond frameofreference S%.

1.4Minkowskispacetime

In 1908 Einstein’s former mathematics teacher,Hermann Minkowski (Figure 1.15), gave alectureinwhich he introduced the idea of spacetime.He said in thelecture: ‘Henceforth space by itself,and timebyitself aredoomed to fade away into mere shadows, andonlyakind of union of thetwo will preserve an independent reality’. This section concerns that four-dimensionalunion of space andtime, theset of all possibleevents, which is nowcalled Minkowski spacetime. Figure1.15 Hermann Minkowski(1864–1909) was 1.4.1Spacetime diagrams,lightcones andcausality one of Einstein’s mathematics teachersatthe Swiss Federal We have alreadyseen howthe Lorentztransformationslead to some very PolytechnicinZurich. In 1907 counter-intuitive consequences.Thissubsection introduces agraphical tool he movedtothe University of known as a spacetime diagram or a Minkowski diagram that will help you to Got¨ tingen, andwhile there visualize events in Minkowskispacetimeand therebydevelop abetter intuitive he introduced the idea of understanding of relativistic effects.The spacetimediagramfor aframe of spacetime. Einstein wasinitially ct x referenceSis usually presented as aplotof against ,and each point on unimpressedbut later y thediagramrepresentsapossibleevent as observedinframe S. The -and acknowledgedhis indebtedness z -coordinates areusually ignored. to Minkowskifor easing the Giventwo inertial frames, Sand S%,instandard configuration, it is instructive to transitionfrom special to plot the ct%-and x%-axes of frameS%on thespacetimediagramfor frameS.The general relativity. x%-axisofframe S% is defined by theset of events for which ct% =0,and the ct%-axisisdefinedbythe setofeventsfor which x% =0.The coordinates of theseeventsinSarerelated to their coordinates in S% by thefollowing Lorentz transformations. (Notethatthe timetransformation of Equation1.5 hasbeen multiplied by c so that each coordinate can be measured in units of length.) ct% = γ(V )(ct − Vx/c), x% = γ(V )(x − Vt). Setting ct% =0in thefirstofthese equations gives 0=γ(V)(ct − Vx/c).This showsthatinthe spacetimediagramfor frameS,the ct%-axisofframe S% is 31 Chapter 1Special relativity and spacetime

represented by theline ct =(V/c)x,astraight linethrough theoriginwith gradient V/c.Similarly,setting x% =0in thesecond transformation equation gives 0=γ(V)(x − Vt),showing that the x%-axisofframe S% is represented by theline ct =(c/V )x,astraight linethrough theoriginwith gradient c/V in the spacetimediagramofS.These lines areshown in Figure 1.16.

ct

increasing V

x ) V s c/ xi -a =( " ct x ct " = x ct " = ct increasing V

Figure1.16 Thespacetime )x V/c diagramofframe S, showing the =( ct xis events that make up the ct%-and x"-a x%-axes of frameS%,and thepath of alight ray that passesthrough 0 x theorigin.

Thereisanotherfeatureofinterestinthe diagram. Thestraight linethrough theoriginofgradient 1 linksall theeventswhere x = ct andthus showsthe path of alight ray that passesthrough x =0at time t =0.Using theinverse Lorentztransformationsshows that this linealsopassesthrough all the events where γ(V )(x% + Vt%)=γ(V)(ct% + Vx%/c),thatis(after some cancelling and rearranging), where x% = ct%.Sothe lineofgradient 1 passing through theorigin alsorepresentsthe path of alight ray that passesthrough theoriginofframe S% at ct t% =0.Infact, anylinewith gradient 1 on aspacetimediagrammustalways represent thepossiblepath of alight ray,and thanks to thesecond postulate of special relativity,wecan be sure that all observers will agreeabout that. % y As therelative speed V of theframesSand S increases,the lines representingthe x%-and ct%-axes of S% closeinonthe lineofgradient 1 from either side,rather like the closing of aclapperboard. This behaviour reflects thefact that Lorentz x transformationswill generally alter thecoordinates of events butwill not change thebehaviour of light on which all observers must agree. In thesomewhatunusualcasewhenweinclude asecond spatial axis (the y-axis, say) in thespacetimediagram, theoriginallineofgradient 1 is seen to be part of a cone,asindicated in Figure 1.17. This cone,which connects the eventatthe origintoall thoseevents, past andfuture, that might be linkedtoitbyasignal Figure1.17 In three travelling at the speed of light,isanexample of a lightcone.Ahorizontal slice (at ct = dimensions (one timeand two constant) through the(pseudo) three-dimensionaldiagramatany particular space) it becomesclear that a timeshows acircle, butinafully four-dimensionaldiagramwith all threespatial lineofgradient 1 in aspacetime axes included, such afixed-time slice wouldbeasphere, andwouldrepresent a t =0 diagramispartofalightcone. spherical shell of light surroundingthe origin. At times earlier than ,the shell wouldrepresent incoming light signals closing in on theorigin. At times later than t =0,the shell wouldrepresent outgoing light signals travelling away 32 1.4 Minkowski spacetime from theorigin. Although observers Oand O%,using frames Sand S%,wouldnot generally agreeabout thecoordinates of events,theywouldagree about which events were on thelightcone,which were inside thelightcone andwhich were outside.Thisagreement between observers makeslightconesveryuseful in discussions about which events might cause,orbecausedby, otherevents. Goingback to an ordinary two-dimensionalspacetimediagramofthe kind shown in Figure 1.18, it is straightforward to read offthe coordinates of an eventin frame Sorinframe S%.The event 1 in the diagramclearly hascoordinates % (ct1,x1)in frame S. In frameS,ithas adifferent setofcoordinates.These can be determinedbydrawing constructionlines parallel to thelines representingthe primed axes.Where aconstructionlineparallel to one primed axis intersects the otherprimedaxis, thecoordinate can be found. By doing this on bothaxes, both coordinates arefound. In thecaseofFigure 1.18, thedashedconstructionlines show that, as observedinframe S%,event 1 occurs at the same timeasevent 2,and at the same positionasevent 3.

ct

increasing V

s xi " -a t0 ct en ev of event1 event2 ne increasing V ct1 co ht lig event3 s Figure1.18 " -axi Aspacetime x diagramfor frameSwith four events, 0, 1, 2 and 3.Event % event0 x1 x coordinates in S can be found at (0, 0) by drawing constructionlines parallel to theappropriate axes.

Anotherlessonthatcan be drawn from Figure 1.18 concerns theorderofevents. Starting from thebottomofthe ct-axisand working upwards, it is clear that in frame S, thefour events occurinthe order 0, 2, 3 and 1.But it is equally clear from thedashedconstructionlines that in frame S%,event 3 happens at the same timeasevent 0 (theyare simultaneous in S%), andbothhappenatanearlier time than event 2 andevent 1,which arealsosimultaneous in S%.Thisillustrates the relativity of simultaneity,but more importantly it alsoshows that theorderof events 2 and 3 will be differentfor observers Oand O%. At first sight it is quite shocking to learnthatthe relative motionoftwo observers can reversethe order in which they observe events to happen. This hasthe potential to overthrowour normal notionofcausality,the principle that all observers must agreethatany effect is preceded by its cause.Itiseasytoimagine observing thepressing of aplungerand then observing theexplosionthatit causes. It wouldbeveryshocking if some otherobserver, simply by moving 33 Chapter 1Special relativity and spacetime

sufficiently fast in theright direction, wasabletoobserve theexplosionfirstand then thepressing of theplungerthatcausedit. (It is important to remember that we arediscussing observing, not seeing.) Fortunately,suchanoverthrowofcausality is not permitted by special relativity, provided that we do not allowsignalstotravel at speedsgreater than c.Although observers will disagree about theorderofsomeevents, they will not disagree about theorderofany twoeventsthatmight be linkedbyalight signalorany signalthattravels at lessthanthe speed of light.Sucheventsare said to be causally related. To seehow theorderofcausally related events is preserved, look again at Figure 1.18, notingthatall theeventsthatare causally related to event 0 are contained within its lightcone,and that includesevent 2.Eventsthatare not causally related to event 0,suchasevent 1 andevent 3,are outsidethe lightcone of event 0 andcouldonlybelinkedtothatevent by signals that travel faster than light. Now, remember that as the relative speed V of theobservers Oand O% increases,the linerepresentingthe ct%-axisclosesinonthe lightcone.Asaresult, therewill not be anyvalue of V that allows thecausally related events 0 and 2 to change their order.Event 2 will always be at ahighervalue of ct% than event 0. However, when you examine the corresponding behaviour of events 0 and 3, which arenot causally related,the conclusion is quite different. Figure 1.18 showsthe conditioninwhich event 0 andevent 3 occuratthe same time t% =0, according to O%.WhenOandO%have alower relative speed,event 3 occurs after event 0,but as V increases andthe linerepresentingthe x%-axis(whereall events occuratct% =0)closesinonthe lightcone,wesee that therewill be avalue of V above which theorderofevent 0 andevent 3 is reversed. So,ifevent 0 representsthe pressing of aplungerand event 2 andevent 3 represent explosions,all observers will agreethatevent 0 might have caused event 2,which happenedlater.However,thosesameobservers will not agree about theorderofevent 0 andevent 3,though they will agreethatevent 0 could not have caused event 3 unlessbodies or signals can travel faster than light. It is thedesiretopreservecausalrelationshipsthatisthe basisfor therequirementthat no material body or signalofany kind shouldbeabletotravelfaster than light. ● Is event 1 in Figure 1.18 causally related to event 0?Isevent 1 causally related to event 3?Justifyyour answers. ❍ Event 1 is outside thelightcone of event 0,sothe twocannot be causally related.The diagramdoesnot show thelightcone of event 3,but if you imaginealineofgradient 1,parallel to the shownlightcone,passing through event 3,itisclear that event 1 is inside the lightconeofevent 3,sothosetwo events arecausally related.The earlier eventmay have causedthe later one, andall observers will agreeabout that. An important lessontolearnfrom this questionisthe significance of drawing lightconesfor events otherthanthoseatthe origin. Everyevent hasalightcone, andthatlightcone is of greatvalue in determining causalrelationships.

34 1.4 Minkowski spacetime

1.4.2Spacetime separation andthe Minkowskimetric

In three-dimensionalspace, the separationbetween twopoints (x1,y1,z1)and (x2,y2,z2)can be conveniently describedbythe square of thedistance Δl between them: (Δl)2 =(Δx)2+(Δy)2+(Δz)2, (1.46) where Δx = x2 − x1, Δy = y2 − y1 and Δz = z2 − z1.Thisquantity hasthe useful propertyofbeing unchangedbyrotationsofthe coordinate system.So, if we choosetodescribe the pointsusing anew coordinate system with axes x%, y% and z%,obtained by rotatingthe oldsystem about one or more of its axes,thenthe spatial separationofthe twopointswouldstill be describedbyanexpression of theform (Δl%)2 =(Δx%)2+(Δy%)2+(Δz%)2, (1.47) andwewouldfind in additionthat (Δl)2 =(Δl%)2. (1.48) We describe this situationbysaying that the spatial separationoftwo pointsis invariant underrotationsofthe coordinate system used to describe thepositions of thetwo points. Theseideas can be extendedtofour-dimensionalMinkowskispacetime, where the most useful expression for the spacetime separation of twoeventsisthe following.

Spacetime separation

(Δs)2 =(cΔt)2−(Δx)2 − (Δy)2 − (Δz)2. (1.49)

Thereason whythisparticular form is chosen is that it turns out to be invariant underLorenz transformations. So,ifOandO%areinertial observers using frames Sand S%,theywill generally not agreeabout thecoordinates that describe two events 1 and 2,orabout thedistance or thetimethatseparates them,but they will agreethatthe twoeventshaveaninvariant spacetimeseparation (Δs)2 =(cΔt)2−(Δl)2 =(cΔt%)2−(Δl%)2 =(Δs%)2. (1.50)

Exercise 1.8 Twoeventsoccurat(ct1,x1,y1,z1)=(3, 7, 0, 0) mand (ct2,x2,y2,z2)=(5, 5, 0, 0) m. What is their spacetimeseparation? Exercise 1.9 In thecasethat Δy =0and Δz =0,use theinterval transformation rules to show that thespacetimeseparation givenbyEquation1.49 really is invariantunderLorentztransformations. ■ Aconvenientway of writingthe spacetimeseparation is as asummation: A3 2 µ ν (Δs) = ηµν Δx Δx , (1.51) µ,ν=0

35 Chapter 1Special relativity and spacetime

wherethe four quantities Δx0, Δx1, Δx2 and Δx3 arethe componentsof µ [Δx ]=(cΔt, Δx, Δy,Δz),and thenew quantities ηµν that have been introduced arethe sixteen componentsofanentity called the Minkowski metric, which can be represented as     η00 η01 η02 η03 1000     η10 η11 η12 η13 0−100 [ηµν] ≡ = . (1.52) η20 η21 η22 η23 00−10 η30 η31 η32 η33 000−1 It’s worthnotingthatthe Minkowskimetric hasbeen shownasamatrix onlyfor convenience; Equation1.51 is not amatrix equation, though it is awell-defined sum. Theimportant point is that thequantity [ηµν] hassixteen components, and from Equation1.52 youcan uniquely identifyeach of them.The metric provides avaluable reminderofhow thespacetimeseparation is related to thecoordinate intervals.Metrics will have acrucial roletoplay in therestofthisbook. The Minkowskimetric is justthe first of many that you will meet. Thespacetimeseparation of twoeventsisanimportant quantity for several reasons.Its sign alonetells us about thepossiblecausalrelationship between the events.Infact, we can identifythree classesofrelationship,corresponding to the cases (Δs)2 > 0, (Δs)2 =0and (Δs)2 < 0.

Time-like, light-likeand space-likeseparations Events with apositive spacetimeseparation, (Δs)2 > 0,are said to be time-like separated.Sucheventsare causally related,and therewill exista frame in which thetwo events happenatthe same place butatdifferent times. Events with azero(or null) spacetimeseparation, (Δs)2 =0,are said to be light-like separated.Sucheventsare causally related,and all observers will agreethattheycouldbelinkedbyalight signal. Events with anegative spacetimeseparation, (Δs)2 < 0,are said to be space-like separated.Sucheventsare notcausally related,and therewill existaframe in which thetwo events happenatthe same timebut at differentplaces.

Thesedifferent kindsofspacetimeseparation correspond to differentregions of spacetimedefinedbythe lightcone of an event. Figure 1.19 showsthe lightcone of event 0.All theeventsthathaveatime-likeseparation from event 0 arewithin the future or past lightcone of event 0;all theeventsthatare light-like separated from event 0 areonits lightcone;and all the events that arespace-likeseparated from event 0 areoutside its lightcone.Thisemphasizes the rolethatlightconesplay in revealing the causalstructure of Minkowskispacetime. Anotherreason whyspacetimeseparation is important relates to propertime.You will recall that in the earlier discussion of timedilation, it wassaid that the proper timebetween twoeventswas the timeseparating those events as measured in aframe wherethe events happenatthe same position. In such aframe,the spacetimeseparation of theeventsis(Δs)2 = c2(Δt)2 = c2(Δτ)2.However,

36 1.4 Minkowski spacetime

ct

time-like

lig e ik ht -l -l space-like ik ht y space-like e lig

Figure1.19 event0 x Events that lig aretime-likeseparated from e ht event 0 arefound inside its ik -l space-like -l ik space-like ht e lightcone.Eventsthatare lig light-like separated arefound on time-like thelightcone,and events that are space-likeseparated from event 0 areoutside thelightcone. sincethe spacetimeseparation of events is an invariantquantity,wecan useitto determine theproper timebetween twotime-likeseparated events,irrespective of theframe in which theeventsare described. Fortwo time-likeseparated events with positive spacetimeseparation (Δs)2,the proper time Δτ between thosetwo events is givenbythe following.

Proper time related to spacetime separation

(Δτ)2 =(Δs)2/c2. (1.53)

Therelation between proper timeand theinvariant spacetimeseparation is extremely useful in special relativity.The reasonfor this relates to thelengthofa particle’spathway through four-dimensionalMinkowskispacetime. Such a pathway, with all its twists andturns,records thewholehistory of theparticle andissometimes called its world-line.(Onewell-known relativist called his autobiography My worldline.) By adding together the spacetimeseparations between successive events alongaparticle’sworld-line, anddividingthe sum by c2,wecan determine thetotal timethathas passedaccording to aclock carried by theparticle. This simple principle will be used to help to explain atroublesome relativistic effect in the next subsection. In this book, apositive sign will always be associated with thesquare of thetime intervalinthe spacetimeseparation,and anegative sign with thespatial intervals. This choice of sign is just aconvention, andthe opposite setofsigns couldhave been used.The convention used here ensuresthatthe spacetimeseparation of events on theworld-lineofanobject moving slower than lightispositive. Nonetheless, you will findthatmanyauthors adopt theopposite convention,so when consultingother works, always payattentiontothe sign convention that theyare using.

Exercise 1.10 Giventwo time-likeseparated events,showthatthe proper time between thoseeventsisthe leastamount of timethatany inertial observerwill measure between them. ■ 37 Chapter 1Special relativity and spacetime

1.4.3The twin effect

We endthischapter with adiscussion of awell-known relativistic effect, the twin effect.Thiscausedagreatdeal of controversyearly in the theory’shistory.Itis usually presented as athought experimentconcerning thephenomenon of time dilation. Thethought experimentinvolvestwo twins, Astra andTerra.The twins areidentical in everyway,exceptthatAstra likes to travel around very fast in her spaceship, while Terra prefers to stay at homeonEarth. As wasdemonstrated earlier in this chapter,fast-moving objects aresubject to observabletimedilationeffects.Thisindicates that if Astra jets offinsomefixed directionatclose to thespeed of light,then, as measured by Terra,she will age more slowly because ‘movingclocksrun slow’. This is fine—itisjustwhat relativity theory predicts,and agrees with theobservedbehaviour of high-speed particles.But nowsupposethatAstra somehowmanages to turnaround and return to Earthatequally high speed.Itseemsclear that Terra will again observe that Astra’s clock will run slow andwill therefore not be surprisedtofind that on herreturn, Astra hasagedlessthanher stay-at-hometwinTerra. Thesupposed problemariseswhenthisprocessisexaminedfrom Astra’spoint of view.Woulditnot be thecase, some argued, that Astra wouldobserve thesame events apartfrom areversalofvelocities,sothatTerra wouldbethe travelling twin anditwouldbeTerra’s clock that wouldberunning slow during bothparts of thejourney? Consequently,shouldn’tAstra expect Terra to be theyoungerwhen they were reunited?Clearly,it’snot possiblefor each twin to be youngerthanthe otherwhentheymeet at the same place, so if thearguments areequally sound, it wassaid, theremustbesomethingwrongwith special relativity. In fact, the arguments arenot equally sound. Thebasic problemisthatthe presumed symmetrybetween Terra’s view andAstra’s view is illusory.ItisAstra who wouldbethe youngeratthe reunion, as will nowbeexplained with theaid of aspacetimediagramand aproper useofspacetimeseparationsinMinkowski space. Thefirstpoint to make clear is that although velocity is apurelyrelative quantity, accelerationisnot.According to thefirstpostulate of special relativity,the laws of physics do not distinguish one inertial frame from another, so atraveller in aclosedbox cannot determine hisorher speed by performingaphysics experiment. However, such atraveller wouldcertainly be able to feel the effect of anyacceleration,asweall knowfrom everydayexperience. In order to leave the Solar System,jet around thegalaxy andreturn, Astra must have undergone a change in velocity,and that wouldinvolve adetectable acceleration.Toafirst approximation,Terra does not accelerate (hervelocity changesdue to therotation andrevolutionofthe Earthare very small compared with Astra’s accelerations). Asingleinertial frame of referenceissufficienttorepresent Terra’s view of events,but no single inertial frame can adequately represent Astra’s view.There is no symmetrybetween thesetwo observers;onlyTerra is an (approximately) inertial observer. In order to be clear about what’sgoing on andtoavoid theuse of non-inertial frames, it is convenient to usethree inertial frameswhendiscussing thetwin effect. ThefirstisTerra’s frame,which we can treat as fixed on anon-rotating, non-revolving Earth. Thesecond, which we shall call Astra’s frame,movesata

38 1.4 Minkowski spacetime high butconstantspeed V relative to Terra’s frame.You can thinkofthisasthe frameofAstra’s spaceship, andyou can thinkofAstra as simply jumpingaboard herpassing ship at thedeparture, event 0,whenshe leavesTerra to beginthe outward legofher journey. Thethird inertial frame,called Stella’sframe,belongs to anotherspace traveller whohappens to be approaching Earthatspeed V along the same linethatAstra leavesalong. At some point,Stella’sshipwill pass Astra’s,and at that point we can imagine that Astra jumpsfrom hershipto Stella’sshiptomakethe return legofher journey. Of course, this is unrealistic sincethe ‘jump’ wouldkill Astra,soyou mayprefertoimagine that Astra is actually aconsciousrobot or even that shecan somehow‘teleport’ from one ship to another. In anycase, the important point is that thetransferisabrupt andhas no effect on Astra’s age. Theevent at which Astra makesthe transfer to Stella’sshipweshall call event 1, andthe eventatwhich Astra andTerra areeventually reunited we shall call event 2.Astra’s quick transfer from oneshiptothe otherallows us to discussthe essential features of thetwineffect without gettingboggeddownindetails about thenatureofthe acceleration that Astra experiences.Itisvital that Astra is accelerated,but exactly howthathappens is unimportant.Notethatwemay treat each of theseframesasbeing in standard configurationwith either of theothers. We can setupthe frames in such away that the origins of Terra’s frame and Astra’s frame coincideatevent 0,the origins of Astra’s frame andStella’sframe coincideatevent 1,and theorigins of Stella’sframe andTerra’s frame coincideat event 2. Figure 1.20 is aspacetimediagramfor Terra’s frame,showing all theseeventsand making clear the coordinates that Terra assigns to them.

ct event2(cT, 0) Terra’sframe

As tr aa nd St el la

rra ? @

Te cT VT event1 , 2 2

Figure1.20 a Aspacetime tr diagramfor Terra’s frame, As showingthe departure,transfer andreunion events together event0 x with their coordinates.The t at (0, 0) -coordinate hasbeen multiplied by c,asusual.

It is clear from thefigure that theproper timebetween departure andreunion (both of which happenatTerra’s location) is T .Alittle calculationusing the

39 Chapter 1Special relativity and spacetime

relation (Δτ)2 =(Δs)2/c2 makesitequally clear that theproper timebetween event 0 andevent 1 is givenby '? @ ? @ < (Δs )2 1 cT 2 VT 2 (Δτ )2 = 0,1 = − 0,1 c2 c2 2 2 ? @ ? @ T2 V2 T 2 = 1− = . 4 c2 2γ (1.54) So T Δτ = . 0,1 2γ (1.55) Although we have arrivedatthisresultusing thecoordinates assignedbyTerra,it is important to notethatproper timeisaninvariant, so all inertial observers will agreeonthe proper timebetween twoeventsnomatter howitiscalculated. Asimilar calculationfor theproper timeseparating event 1 andevent 2 showsthat T Δτ = . 1,2 2γ (1.56) So thetotal proper timethatelapsesalong theworld-linefollowedbyAstra is Δτ0,1 +Δτ1,2 =T/γ.Asexpected, this showsthatAstra will be theyounger twin at thetimeofthe reunion. Howisitpossiblefor Terra andAstra to disagree about thetimebetween events 0 and 2?The answertothisquestionisthatwhenthe wholetripisconsidered,Astra is not an inertial observer; sheundergoesanacceleration that Terra does not. Giventwo time-likeseparated events,the timethatelapsesbetween those events, as measured by an observerpresent at bothevents, will depend on theobserver’s world-line. Thetotal timebetween the events,measured alongthe world-lineofa non-inertial observer, is generally less than theproper timebetween theseevents as measured alongthe world-lineofaninertial observer. Theanalysisthatwehavejustcompleted is really sufficient to settle anyquestions about thetwineffect. However, it is still instructive to examine the same events from Astra’sframe (which sheleavesatevent 1). Thespacetimediagramfor Astra’s frame is showninFigure 1.21. Thecoordinates of theeventshavebeen worked out from thosegiven in Terra’s frame usingthe Lorentztransformations. ● Confirm thecoordinate assignments showninFigure 1.21. ❍ In Terra’s frame,event 0 is at (ct,x)=(0, 0),event 1 at (cT/2,VT/2),and event 2 at (cT, 0).Treating Terra’s frame as frame Sand Astra’s frame as S%, andusing theLorentztransformations t% = γ(t − Vx/c2) and x% = γ(x − Vt),itfollows immediately that in Astra’s frame,event 0 is at % % % % (ct ,x)=(-0,0),event 1 is at (ct ,x)=(cT/2γ,0) (rememberthat γ(V )=1/ 1−V2/c2),and event 2 is at (ct%,x%)=(cγT,−γV T). Note that again thereisakink in Astra’s world-linedue to theacceleration that sheundergoes. ThereisnosuchkinkinTerra’s world-linesince sheisaninertial observer. Once again we can work outthe proper timethatAstra experiences while passing between thethree events:thisrepresentsthe timethatwouldhave elapsed according to aclock that Astra carries between each of theevents. The proper timebetween event 0 andevent 1 is simply Δτ0,1 = T/2γ,since those 40 1.4 Minkowski spacetime events happenatthe same place in Astra’s frame.The proper timebetween event 1 andevent 2 is givenby

Astra’sframe ct"

event2(cγT,−γV T)

As tr aa nd St el la Te rra

? @ cT event1 , 0 2γ Figure1.21

a Aspacetime

tr diagramfor Astra’s frame, As showingthe departure,transfer event0 x" andreunion events with their at (0, 0) coordinates.NotethatAstra leavesthisframe at event 1.

'? @ < (Δs )2 1 cT 2 (Δτ )2 = 1,2 = cγT − − (−γV T)2 1,2 c2 c2 2γ '? @ ? @ < 1 2 V 2 = T2 γ − − γ 2γ c 7 F 1 γ2V2 = T2 γ2 − 1+ − 4γ2 c2 7 ? @ F V2 1 =T2 γ2 1− −1+ . c2 4γ2

Since γ2(1 − V 2/c2)=1,the above expression simplifies to give T Δτ = . 1,2 2γ So onceagain thetheory predicts that the timefor theround trip recordedby Astra is Δτ0,1 +Δτ1,2 =T/γ. Thereisone otherpoint to notice usingAstra’s frame.Timedilationtells us that, as measured in Astra’s frame,Terra’s clock will be running slow.From Astra’s frame,a1-second tick of Terra’s clock will be observedtolast γ seconds.But in Astra’s frame,itisalsothe casethatthe timeofthe reunion is γT,which is greater than thetimeofthe reunion as observedinTerra’s frame.According to an observerwho uses Astra’s frame,thislongerjourneytimecompensates for the slower tickingofTerra’s clock,with theresultthatsuchanobserverwill fully expect Terra to have aged by T while Astra herself hasagedbyonly T/γ.Using 41 Chapter 1Special relativity and spacetime

thecoordinates of event 0 andevent 2 in Astra’s frame,itiseasytoconfirm that theproper timebetween them is T ,which is anotherway of stating the same result.

Exercise 1.11 Using thevelocity transformation,showthatAstra observes the speed of approachofStella’sspaceshiptobe2V/(1 + V 2/c2). Exercise 1.12 SupposethatTerra sends regular timesignals towards Astraand Stella at one-second intervals.Write down expressions for thefrequency at which Astra receivesthe signals on theoutward andreturnlegsofher journey. ■

Summary of Chapter 1

1. Basic termsinthe vocabularyofrelativity include:event,frame of reference, inertial frame andobserver. 2. Atheory of relativity concerns therelationshipsbetween observations made by observers in aspecified state of relative motion. Special relativity is essentially restricted to inertial observers in uniformrelative motion. 3. Einstein basedspecial relativity on twopostulates:the principle of relativity (thatthe laws of physics can be written in thesameform in all inertial frames) andthe principle of theconstancy of thespeed of light (thatall inertial observers agreethatlight travels through emptyspace with thesame fixed speed, c,inall directions). 4. Giventwo inertial framesSandS%in standard configuration, thecoordinates of an eventobservedinframe Sare related to thecoordinates of thesame eventobservedinframe S% by theLorentztransformations t% = γ(V )(t − Vx/c2), (Eqn 1.5) x% = γ(V )(x − Vt), (Eqn 1.6) y% = y, (Eqn 1.7) z% = z, (Eqn 1.8) where theLorentzfactor is 1 γ(V )=- . (Eqn 1.9) 1−V2/c2 Thesetransformationsmay alsoberepresented by matrices,     ct% γ(V ) −γ(V )V/c 00 ct x%  −γ(V)V/cγ(V)00x   =   ,  y%   0010y  (Eqn 1.10) z% 0001 z

or as aset of summations A3 %µ µ ν x = Λ ν x (µ =0,1,2,3). (Eqn 1.13) ν=0

42 SummaryofChapter 1

5. Theinverse Lorentztransformationsmay be written as t = γ(V )(t% + Vx%/c2), (Eqn 1.14) x = γ(V )(x% + Vt%), (Eqn 1.15) y = y%, (Eqn 1.16) z = z%. (Eqn 1.17)

6. Similar equations describe thetransformation of intervals, Δt, Δx,etc., between thetwo frames. 7. Theconsequences of special relativity,deduced by consideringthe transformation of events andintervals,include thefollowing.

(a) Time dilation: ΔT = γ(V )Δτ. (Eqn 1.40)

(b) Lengthcontraction:

L = LP/γ(V ). (Eqn 1.41)

(c) Therelativity of simultaneity. (d) Therelativistic Doppler effect (Eqn 1.42): - frec = fem (c + V )/(c − V ) (for an approaching source), - frec = fem (c − V )/(c + V ) (for areceding source).

(e) Thevelocity transformation: v − V v% = x , x 2 (Eqn 1.43) 1 − vxV/c v v% = y , y 2 (Eqn 1.44) γ(V )(1 − vxV/c ) v v% = z . z 2 (Eqn 1.45) γ(V )(1 − vxV/c ) 8. Four-dimensionalMinkowskispacetimecontains all possibleevents. 9. Spacetimediagrams showingeventsasobservedbyaparticular observerare avaluable toolthatcan provide pictorial insights into relativistic effects and the structure of Minkowskispacetime. 10. Lightconesare particularly useful for understanding causalrelationships between events in Minkowskispacetime. 11. Theinvariant spacetimeseparation between twoeventshas the form (Δs)2 =(cΔt)2−(Δx)2 − (Δy)2 − (Δz)2, (Eqn 1.49) andmay be positive (time-like),zero(light-like)ornegative (space-like). 12. Thespacetimeseparation maybeconveniently written as A3 2 µ ν (Δs) = ηµν Δx Δx , (Eqn 1.51) µ,ν=0

43 Chapter 1Special relativity and spacetime

wherethe ηµν arethe componentsofthe Minkowskimetric     η00 η01 η02 η03 1000     η10 η11 η12 η13 0−100 [ηµν ] ≡ = .(Eqn 1.52) η20 η21 η22 η23 00−10 η30 η31 η32 η33 000−1

13.The proper time Δτ between twotime-likeseparated events is givenby (Δτ)2 =(Δs)2/c2. (Eqn 1.53) This is thetimethatwouldberecordedonaclock that movesuniformly between thetwo events. 14. Theproper timebetween twoeventsisaninvariant underLorentz transformations. 15. Thetimebetween twotime-likeseparated events,asmeasured by an observerpresent at bothevents, depends on theworld-lineofthe observer. Thetimebetween the events,measured by anon-inertial observer, is generally less than theproper timebetween theseeventsasmeasured by an inertial observer. This is thebasis of thetwineffect.

44 Acknowledgements

Grateful acknowledgement is made to thefollowing sources: Figures Coverimage courtesy of NASA/CXC/Wisconsin/D.Pooley andCfA/A.Zezas and NASA/CXC/M.Weiss; Figure 1: Science PhotoLibrary; Figure 1.1:Emilio SegreV` isualArchives/American Institute of Physics/Science PhotoLibrary; Figure 1.4:Science PhotoLibrary; Figures2.2 and2.5: Copyright )c CERN; Figure 2.6: US DepartmentofEnergy/Science PhotoLibrary; Figure 2.8: )c Corbis;Figure 4.1: Sheila Terry/Science PhotoLibrary; Figure 5.1:Emilio Segre` Visual Archives/American Institute of Physics/Science PhotoLibrary; Figure 6.1: Physics TodayCollection/American Institute of Physics/Science PhotoLibrary; Figure 6.2:Los Alamos NationalLaboratory/Science PhotoLibrary; Figure 6.3: Emilio SegreV` isualArchives/American Institute of Physics/Science Photo Library; Figure 6.18: Time andLifePictures/Getty Images;Figure 6.21: Corbin O’GradyStudio/Science PhotoLibrary; Figure 7.10: NASA Marshall Space Flight Center (NASA-MSFC); Figure 7.12: ESA, NASA andFelix Mirabel (FrenchAtomicEnergyCommissionand institute for Astronomyand Space Physics/Conicet of Argentina);Figure 7.13: http://www.star.le.ac.uk; Figure 7.17: NASA Johnson Space Center Collection; Figure 7.18: ESA, NASA, RichardEllis (Caltech, USA) andJean-PaulKneib (ObservatoireMidi-Pyrenees); Figure 7.19: Weisberg,J.M.and Taylor,J.H.(2004) ‘Relativistic binary pulsarB1913+16: thirty yearsofobservations andanalysis’, BinaryRadioPulsars,Vol TBD, 2004. ASP ConferenceSeries;Figure 7.20: Volker Steger/Science Photo Library; Figure 8.1: M. Collness, Mount Stromlo Observatory; Figure 8.2: Robert Smith,The 2dF GalaxyRedshiftSurvey;Figure 8.3: Bennett, C. L. et al. ‘FirstYear WilkinsonMicrowaveAnistropyProbe(WMAP) Observations: PreliminaryMapsand BasicResults’, AstrophysicalJournal (submitted); Figure 8.6:NASA Goddard Space Center;Figure 8.11: RiaNovosti/Science PhotoLibrary; Figure 8.13: Adapted from Landsberg,P.T.and Evans, D. A. (1977) MathematicalCosmology,OxfordUniversity Press;Figure 8.15: Sky Publishing Corporation; Figures 8.17 and8.18:MarkWhittle, University of Virginia from Carr, B. and Ellis,G.(2008) ‘Universeormultiverse?’, Astronomy and Geophysics,Vol.49, April2008. Everyeffort hasbeen made to contact copyrightholders.Ifany have been inadvertently overlookedthe publishers will be pleased to make thenecessary arrangements at the first opportunity.

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