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·IfalineisdrawlaheSuheplastheSun，thatlinewillsweepoutanequalareaofspaytwointervalsofequaltime.
2.Anillustrationofaplaaryorbit.Theshadedregions，markedAandB，areofequalarea，iftheyaresweptoutbytheplaime.
·Thesquareofthetimetakenforaplaopleteoisproportionaltothecubeofthedistawofurthestpointsonthatorbit（themajoraxisoftheellipse）.
&iesdesthesethreelawsareallillustratedihtheexoftheorbitalperiod）.ThegoodthingaboutKepler'slawswasthattheyseemedtoapplytoallks，eventhoughatthetimetheyhadnoknownbasisinanyphysicaltheory.Theyweresimplyseentobeagoodfittothedata.Kepler'slaws，andinhisPrincipiaMathematiexplanationofhowtheycouldbederivedfromhislawsofmotionandfromhisuheravitatioiohegreatsuccessstoriesofphysics.Empiricallaws，derivedfromg-edgeobservatios，wereexplaihefirsttimeintermsofsimplemathematicalequatioonhadshownthatthesamelawsthatdescribethemotionofaballfiredonthesurfaceoftheEarthcouldalsobeusedtodescribethemotiosthemselves.Thiswasthebeginningofmuchofphysicsaswekno>
&ein’stheravity
&haeritspubliewton'stheravitylantedbyAlbertEinstein's.Ifon'stheoryleaheein'swasbeautifulandtrulyuniversal.Eijustgetheequatioon'stheory，hepulleduptheveryfoundationsuponwhichitrested.Einsteihing.
Aswithmuchressthathappensinphysistein'stheoryrimarilybytheincyofexistioheoryofhowgravityandmotioon'sideaswerenotpatiblewiththethehtthatJamesClerkMaxedinthemiddleofthe19thtury.Maxwell'sedthateveryoheUniverseshouldmeasurethespeedoflighttohavethesamevalue:justunder300milliorespersed.Thismightnotsoundterriblyprofound，untilyoutakeafewmomentstosiderwhatitmeans.
&isthat，agtoon'smeics，ifIfireabulletforwardsat1，000milesperhourwhileseatedonatrainmovingat100milesperhour，thehesideofthetrackwillseethatbulletmovingat1，100milesperhour.Iermswesaythatthevelocityofthebulletayofthetrainaddlinearly.NowsiderthatIturnonatorch，whilestillseatedfagforwardsorain.FrommyseatedpositionIwillseethelightfratefhthetraihespeedoflight（i.e.atabout300milliorespersesideragaigmefromthesideofthetraintrack.Ifyoulisteon，youmightexpectthispersohelighttravelatthe300milliorespersed，plus100milesperhour（thespeedofthetrain）.ButagtoMaxwellthisisnotehatthepersonatthesideofthetrackseesthelightpropagateatthesamespeedasthepersohatis，Maxwell’sequationsimplythatvelocitiesdonotaddlinearly.
&radijustdescribedisaprofoundone.Ifweareuoagreeonhowtoaddvelocities，theousephysicstocalculatethemotioall.Itossibleforbothonaobecorrect.Atleastobewrong.Alesserstistmighthavetriedtore-writeeitheron’sorMaxwell’stheories，butthiswasnot>
&eireatedbothon’sandMaxiththeutmostrespect.Hereizedtheirgreatstrengths，aosolvethetradiatrulyingeniousway.Eihesizedthatifthespeedoflightwasthesameforeveryoimeabeuniversalstead，hereasoned，eausthavetheirotoftime，andtheirotofspace.AgtoEinsteiheory，aclockcarriedbyapersorainisseeahetrack，totickslowerthanaclocktheycarrythemselves.Likewise，thepersoraiheclockofthepersonstahetracktotickslowerthanhisorherown.
Thisresultinitiallysoundsodd，butthat’sonlybecausewe’vebeenprogrammedfromayouhinkoftimeasuniversal.WhatEinsteihatourchildhoftimeismistakeauniversalfoldieforeveryoimeisapersonalthing，anddependsoivemotiooothers.Likewise，spaotthefixedbackdropthatwethinkitis.Whatwethinkofasdistahelengthsofobjects，areactuallydepeonh.
&artlifirsttheyseemuling，asifthecrutcheswe'veusedtouheworldhavesuddenlybeenkickedaway.Butwedespair.Thereisavolviimethatsurvivesiheory，andthatmaintainsanobserver-iy.Thisiswhatisknoace-time.Insteadofosofuimeanduniversalspace，whatweareleftwithisalargerstructurethatehemboth.Aperso，likeyouorme，followsalihroughthisstruownasourworld-liimeismeasuredalongourworld-line，andwhilemyworld-libediferenttoyours，theybothexistiime（seeFigure3）.
So，itisthepromotioimetospace-timethatallowsustomakeon’smeisistentwithMaxwell’s.ThisrealizatioeiributionstosditisthebaeofwhatisnoecialTheoryofRelativity.Ithasawidevarietyofprofoundyofwhichhavebeenedexperimentally.Themostfamousoftheseisprobablytheequation:
3.Awoworld-liwiime.Theobserversmeetwherethelinescross.
whichtellsusthatmassandeelyected（afactthatbecamedevastatinglyapparentwiththeadventofnuclearons）.Othercesarethelouiclesappeartohavewhentheymovequidthefaothingevermht.
Itisthislastresult，togetherwiththeofspace-time，whisteintohistheaiusforthisdevelopmentarenttradidagain，itwasheorythatroblems.Thistime，however，itseemedthattherewasatradiwithEinstein’sownwork.Thiswasbeewton’sgravityactedbetweeahatis，iftheSuosuddenlyexpltoonweshouldfeelthegravitatiohisattheverysamemomentithappeeihisossible.First，hehadfoundoutthatnothinght.Sed，hehadshownthattherewasnosugasuime，sotheideaoftwothingshappeningsimultaneouslyiplaoseall（iftheyhappeaneouslyforoheywouldaneousforanyotherwhowasieofmotion）.So，on，somethingwaswrongandneededfixing.
&ein’ssolutiontothisproblemwasevenHehypothesizedthatgravity，insteadofbeisimplypulledthingsthroughspace，wastheresultofthecurvatureofspace-time.Thefactthatmassiveobjectsardseachotherwasthen，agtoEiaresultofthoseobjectsfollowipathstheythecurvedspawhichtheyexisted.Theideawasthatmassandenergycausedspace-timetodthatthiscurvaturecausedthepathsoftheobjectsthatmhspacetoappeartobendtowardseachother.ThebeautyofthisideaisthatwenowoiyasaraforcethatexistsintheUhisure，theonlythingrespoheattraassivebodiesisspace-timeitself（whichhastobethereanyway）.ThisisthefualideabehiheoryofRelativity.
EvenmoreimpressiveisthatEinstein’sideaexplainedGalileo’sresultthatallobjectsfallatthesamerate.Recallthatiheorythisresultlaiall.Itlytakenasafadalawofgravitywasdevisedthatatiblewithit.Eioer.eihereisernalforcecalledgravity;themotioisjustaresultofthecurvatureofspace-time.Butalliime，soallobjectsmustfolloaths.Inotherwords，allobjectsmustfallatthesamerate，justasGalileohadobserved.
Theseideasesbeg，solet’sthinkaboutanexample.ImagihsoftwoobjectsthathaveingontheInaflatspacethepathsofthesehtlines，asshowninFigure4.
Ifthespaceiscurved，however，thenthisisrue.siderthesimplestcurvedspace:thesurfaceofasphere.Theshortestpathbetointsonthesurfaceofasphereiscalledagreatcircle（theequatorisanexampleofagreattheglobe）.Iftwoobjectsfollowtwreatesphere，thentheywillinitiallymoveawayfromeachother，beforefinallyiheragain，asshownihisishowEinsteiyw.Heimagiwascurvaturethatohepathsofobjeg，andnotahatpullsthemlefthtastheymove.Thecurvatureofspace-timeisusuallymularthanthesurfaceofasphere，butthebasicideaisthesame.Asfarasthecesfestefestein'sheorywerefoundtolookverysimilartothelawthatonhadprescribed200yearsearlier.Thediferethislaithaandingofspadtime.Italsoeswiththeprediofavarietyofsmaller，moresubtle，efects（tobedisthefolloters）.
4.Twhtlines，indigthepathsthattwoparticlesmightfollowiftheytravelledthroughaflatspace，withoutaeringuponthe
5.Twogreatcirdigthepathsthatparticlesmighttravelinasphericallycurvedspaoexterupohelinesarenolongerparallelforever，butmeetatapoint.
Thinknowaboutwhatallofthismeansinapracticalsituation.sideraskydiverjumpingoutofaheskydiverfallsfreely，uptotheefectsofairresistance.AgtoEiheskydiver’spathisthestraightestlinepossiblethroughthecurvedspace-timearouh.Fromtheskydiver’sperspectivethisseemsquitefpasther，theskydiverfeelsurbiall.Infact，ifitweren’tfortheairresistance，shewouldexperielesshesamewaythatanastronautdoesinorbit.TheohinktheskydiverisagisbecauseweareusedtousingthesurfaceoftheEarthasourframeofreference.Ifwefreeourselvesfromthis，thenwehavehinktheskydiverisagatall.
Nowsideryourselfontheground，lookingupatthefallingdaredevil.Normally，youriionofyourownmotioyouarestatiainthisisonlybecauseofourslavishregardtotheEarthasthearbiterofwhatisatrestandwhatismoving.Freeyourselffromthisprison，ahatyouare，infact，ag.Youfeelaforthesolesofyourfeetthatpushesyouupwards，iyouwouldifyouwereinaliftthatacceleratedupwardsveryquiEiurethereisweenyourexperieahandyourexperie.Inbothsituationsyouareagupwards.Iersituationitistheliftthatisresponsibleforyourac.Intheformer，itisthefactthattheEarthissolidthatpushesyhspace-time，knogyouofyourfree-falltrajectory.ThatthesurfaceoftheEarthaccelerateupointonitssurfadremainasolidobject，isbecauseitexistsiimeandnotinaflatspace.
Withthisperspectivethetruenatureofgravitybeesapparent.ThefreelyfallingskydiverisbroughttoEarthbecausethespace-timethroughwhichshefallsiscurved.Itiserugsherdownwards，butherownnaturalmhacurvedspatheotherhand，asapersonstandingontheground，thepressureyoufeelonthesolesofyourfeetisduetidityoftheEarthpushingyain，thereisernalforgyoutoEarth.Itisoaticfortherocksbelowyourfeetthatkeepthegrid，andthatpreventsywhatwouldotherwisebeyournaturalmotion（whichwouldalsobefreefall）.
So，ifwefreeourselvesfromdefiningourmotioothesurfaceoftheEarthwerealizethattheskydiverisihepersonwhostandsonthesurfaceoftheEarthisag.Justtheoppositeofwhatweusuallythink.GoingbacktoGalileo’sexperimentoowerofPisa，hyheobservedallofhisballstofallatthesamerate.ItwasheballsthatwereagawayfromGalileoatall，itwasGalileothatwasagawayfromtheballs!IfIleaveanumberofobjectsatrestatsomepositioninspadaccelerateawayfromthem，Ishouldallsurprisedthatthedistaweeheseobjecreasesatthesamerate.SoitiswithGalileoandhisballs.
Forsome，thebeautyiionisobvious.ForothersitisthefactthatitispossibletoprobeEiheoryexperimentallythatismostpelling.TheseexperimentsrangefromlookingforsmallperturbatiosoftheplahebendingoflightaroundtheSun，andmany，manymorebesides.Wewillexploretheseexgphehechaptersthatfollow.Butbearinmindthroughout:itisthecurvatureofspace-timethatisrespoall.
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