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ENJOYOURUNIVERSE AlbertEinsteinholdingthevisibleUniverse,towhosecurrentunderstandinghecontributedsosignificantly.ThevisibleUniverse, ©WMAP/NASA.
ENJOYOUR UNIVERSE YouHaveNoOtherChoice
AlvaroDeRújula
OriginalillustrationsbyAlvaroDeRújula
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©AlvaroDeRújula2018
Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2018
Impression:1
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Preface Mostpeople—andamongthem,mostscientists—arefamiliarwiththe rulesofsomesportsandtheresultsoftheirrecentgames.Thismaybe becausetheypracticethesesportsthemselves,orsimplybecausesports arefuntofollow.Mostscientistspracticetheirtradeand,evenifthey donot,theyconsiderthatprogressinscienceisfuntofollow.And hereendsthesimilaritybetweensportsandscience,orbetween“most scientists”and“mostpeople.”
Alargefractionofnon-scientistswouldnotenthusiasticallyadhere totheopinionthatunderstandingtheUniverseinwhichwehappento be,orsimplytoattempttodoit,isalsogreat“fun.” Iwouldn’tunderstand anything isnotanuncommonreactiontoanytextonscience.Inmy opinion,themainreasonisnotthatscienceisboringandunfathomable butthat,moreoftenthannot,itisnotproperlytaught.
Butevenina(good)kindergarten,kidsmaybetaughtthe“scientificmethod”;doingexperimentsanddrawingconclusionsfromtheir results.Tobespecific,fairlyyoungchildrenmaybegivenabalance,a kettle,akitchenutensilthatmeasuresvolumesofliquidandvarious typesofballs:atennis-ball,agolf-ball,atable-tennisball,abilliardball Theaimofthegameistofigureoutwhatitisthatmakesballs floatorsink.
Theresultsaresurprising:Childrenfindoutfastthatitisnotthe sizethatmatters,butsomethingalsoinvolvingtheweight.Iftheyare nottoddlers,givenasmallhint,orballsofthesamesizeanddifferent weight,theymayevendiscoverthattheanswerisintherelation ofweighttovolume(theirratioisthedensity).Thegameismore efficient—andteachesthebenefitsofcollaboration—ifthechildrenare organizedinteams.Italsoprovidesthemwithenjoyment,auseful wayofthinking,andanapproachtoproblemsbasedoncuriosityand constructivedoubt.
Thepreviousparagraphisnotbasedona thoughtexperiment,atype ofreasoningofwhichscientistsaresoveryfond.Itistheresultof many“experiments”performedwith—andby—realkids.Thereisa successfulthoughinsufficientlyfundedteachingprogram,pioneered byLeonLederman(anAmericanrecipientofthePhysicsNobelPrize) called“HandsOn”(1).Itemployspreciselythistrialanderror—andtry
onceagain—teachingandlearningmethodology.Ithasbeentested,in particular,withkidsoryoungstersfrom“problematicneighborhoods” inChicago.TheprogramhasspreadfromtheUSAtoothercountries suchasFrance,whereitiscalled Lamainàlapâte (handsinthedough)(2). ItwaspioneeredtherebyGeorgesCharpak,yetanotherNobelPhysics Laureate.
“Activelearning”projectssuchasHandsOnalsoexistforeducation uptosecondaryschoolorevenuniversitylevel.Atthehigherlevels theproblemmaybethatteachershavenotbeenproperlytaught,or sosaysCarlWieman—alsoaNobelPhysicslaureateforachange— whohasbeenveryactivelyinvolvedwiththistypeofeducation(3). Thepointistosubstitutemedievalteachingtechniques,essentially theendlessrepetitionoftextstobememorized,withsomethingmore constructive.
Thisbookisnotintendedfor(very)youngkidsnorforphysicists.Itis intendedforanyone—independentlyoftheeducation(s)hesuffered— whoisinterestedinourbasiccurrentscientificunderstandingofthe Universe.By“Universe”Imeaneverythingobservablefromthelargest object,theUniverseitself,tothesmallestones,the elementaryparticles that“function”asiftheyhadnosmallerparts.Thisisonemore ofmanybooksonthesubject.Whywriteyetanotherone?Becausethe attemptstounderstandourUniverseareindeed fun andIcannotresist thetemptationofputtinginwriting—andattemptingtopartakeofmy ownshareofthisfun.
Somepotentialreadersmaybeallergictomath.Fearnot.Ishalluse verylittle algebra (symbolstorepresentconcepts,asin E = mc2 ,ortheir relations,suchas =, ≈ , ∼ , ∝, > ,fornotequal,approximatelyequal, roughlyequal,proportionalto,greaterthan)and arithmetic: thefour rules,powerssuchas103 (thecubicpoweroften,akatentothethird, akaonethousand).Imaygoasfarasusinganoccasional squareroot ora vector,butnotwithoutwarning.
Threeextraadvancenotices.Therewillbeawarningstarortwoin thetitleoftheslightlymorealgebraicorunavoidablyintricatechaptersandsub-sections.Theycanbeskippedaccordingtothereader’s preference.Somefootnotescontaintechnicalclarificationsintended topacifythemostknowledgeablereaders.Attheendofthebook thereisaGlossaryofTerms,sinceitwouldbedifficulttomemorizein passingalloftheinnumerableconceptsandnamesintroducedatone pointandlaterreused,suchas fermion, boson,andthenamesofthe powersof10.
References [1]See,forinstance,LookingBack:InnovativeProgramsoftheFermilabEducationOffice,Availableonlineathttp://ed.fnal.gov/office/marge/retro.html (accessedOctober2017).
[2]FondationdeCoopérationScientifiquepourL’ÉducationàlaScience.Website.Availableat:www.fondation-lamap.org/
[3]CarlWieman, ScientificAmerican,August2014,page60.
Acknowledgments IamindebtedtoAliciaRivera,FabioTruc,andfive(!)anonymousOUP refereesfortheirhavingcarefullyreadtheoriginalmanuscriptandfor manyusefulsuggestions.IamparticularlythankfultoElisaMuxella, whoconvincedmetowritethisbook.
1 PhysicsasanArtForm ShouldtheAlmightyhaveconsultedmeaboutCreation,Iwould haverecommendedsomethingsimpler.
AlphonsoXthofCastille(1221–1284) KingoftheThreeReligions.
Theroyalquotewasacommentaryonthethenaccurateandpervading butverycomplexPtolemaicviewoftheSolarSystem,inwhichplanets revolvedincomplexorbitsaroundtheEarth,asopposedtosimpleones aroundtheSun.Thisexceptionalking,AlphonsetheWise,hadgood intuition:Thingsareindeedsimpler!
Forascientist,“simplicity”isalsosynonymouswith“beauty.”The beauty,forinstance,oftheequationsthatdescribetheUniverse.But theveryfirstruleinwritingabookonphysicsforreaderswhoare notnecessarilyfamiliarwithmathis not towriteanyequations,with thepossibleexceptionof E = mc2 ,whichissooftenmisinterpreted.1 Whocanresistthetemptationofbreakingtherules?Tocamouflagemy sinfulmanners,Ibreaktherulebywritingyetanotherequation,butin afigure,Figure1.
ThemystifyingscribblingsonUncleAlbert’slowerblackboardin Figure1arethe Einstein’sequationsof GeneralRelativity. Thisrather simpleformulaencompassesagoodfractionof“everythingphysicists know”—atabasiclevel.2Itsaysthatgravityisdescribedbythe propertiesofagravitational field,embeddedintheleft-handside oftheequation;thatis,beforethe“=”sign.The“source”ofgravity (theright-handside)isanythingthathasmassand/orenergyand
1 ApointtowhichweshalldedicatethesecondpartofChapter4.
2 Basic,asopposedto emergent.Thepropertiesofasolid,forinstance,“emerge”from theonesoftheouterelectronsofitsconstituentatoms.Weknowandunderstandan awfullotaboutemergentphenomena.
EnjoyOurUniverse:YouHaveNoOtherChoice.AlvaroDeRújula. ©AlvaroDeRújula2018.Publishedin2018byOxfordUniversityPress. DOI:10.1093/oso/9780198817802.001.0001
Figure1 AlbertEinsteinandhisblackboards.Onthelowerone,theleft-hand sidedescribesgravity,theright-handsideits“source,”discussedindetailin thesecondpartofChapter7.Thesymbols c and G arethe speedoflight andNewton’sconstant,whichdeterminesthemagnitudeofthegravitational forces. μ and ν are“indices”that“run”from0to3fortimeandthethree dimensionsofspace.The8and π areotherratherwell-knownnumbers. gμν describesthe“geometry”ofspace-time,whichmoreoftenthannotis “curved,”likethesurfaceofasphere.Thesymbols R areexplicitfunctionsof gμν anditsspace-timevariations.Formoreabout ,seethetext. momentum.3Theequationdescribesorpredicts—amongotherthings andwithincredibleaccuracy—Newton’sproverbialfallingapple,the behaviorofclocksinGPSsatellites,theorbitsoftheplanets(including Mercury’sandthepeculiar“advance”ofitsperihelion4),thedeflection ofstarlightbytheSun,themotionsofstarsandgalaxies,theexistence ofblackholes,theemissionof gravitationalwaves bybinarypulsars
3 WhattheseapparentlyobviousconceptsactuallymeanisdiscussedinthebeginningofChapter4.
4 ThepointofitsorbitclosesttotheSun.
andblack-holemergers ... allthewaytotheexpandingUniverse.5Who wouldnotadmitthatthereisamannerofbeautyinthissimplicity?
TheupperblackboardinFigure1isanadd-onbyEinstein,wholater thoughtthatitmighthavebeenhisbiggesterror:Yet,itmayturnout tobeoneofhismajorcontributions;notaneasycontesttowin.What symbolizesisthe CosmologicalConstant.Itistobeaddedtothe otherequation,anditmaybeinterpretedasthe“energydensityofthe vacuum.”If isnotzero,the vacuum isnotvoid,asweshalldiscussin welldeserveddetailinthefirstsectioninChapter22.Atthemoment, theCosmologicalConstantisthesimplestexplanationoftheobserved factthattheexpansionoftheUniverseiscurrentlyaccelerating.Fora positive ,a“chunk”ofvacuumintheUniversewouldgravitationally repel anyotherone,thusprovidingtheacceleration.Isthisnotalso “beautiful”or,tosaytheleast,mesmerizing?
Incidentally,adeepquestiontowhichwehavenoseriousanswer is:whyarethefundamentallawsofNatureelegantandconcise?Yet anotherPhysicsNobelPrizewinner,whounliketheonesinthePreface Ishallnotname,hasatentativeanswer:Itissufficienttolookatour planettoconcludethat,ifSomeonecreatedtheUniverse,Shediditat random.Howcouldyouotherwiseexplain,forinstance,thepolitical situationinmanycountries,includingmineandperhapsyoursaswell? Youmayhavenoticedthatmycolleagueisafeminist,tothepoint ofassumingthatthehypotheticalcreatorisaGoddess.ThisGoddess happenstohavegoodtaste,inthesensethatonweekends,whenSheis supposedtorest,Shereadsthepreviousweek’sphysicsliterature.When Shecomesacrosssomethingirresistiblybeautiful—andIhavegiven anexample—ShedecidesthatitisTheTruth;thatis,aninescapable LawofNature.SincetheGoddessisalmighty,thenewLawofNature becomesforevertrue ... itevenappliestothepastwhenitwasalso unbreakable.
Thestatementthatoftentherightanswer—notonlyinscience— isthesimplestoneisoftenreferredtoas Ockham’sRazor,afterthe EnglishFranciscanfriarandphilosopher(ca.1287–1347).Inmany
5 Aknowledgablereadermayquipthatanextrahypothesisisrequired:the“CosmologicalPrinciple,”statingthat,atalllocations,theUniversehasthesamepropertiesif averagedoversufficientlylargevolumes.Evenmorepedantically,Iwouldanswerthat thisisnotanassumption,providedtheUniverseunderwentaperiod“inflation”shortly afteritsbirth,asdiscussedinChapter29.
Figure2 WilliamofOckhamwithhisrazorandspoon.
cases—particularlyinscience—thetooltodistinguishthesimpler fromthemorecomplexhypothesesdoesnothavetobeassharpas arazor,evenaspoonwouldsuffice,seeFigure2.Aclearexampleis theheliocentric(Suncentered)viewoftheorbitsoftheplanetsversus themorecomplexgeocentricone,whereinwearethenavelofthe Universe.
ScienceasaSport ThequestforthelawsofNatureisalsoacompetitive“sport,”very muchakintoarace.Onemaythinkthatgettingitrightisallthat matters.ForNaturebeingaplayerandalsoaninfallible“referee,” whatelsemaymatter?Oneproblemisthat,whenthehistoricalcontextisripe,aspecificdiscovery—beittheoreticalorexperimental—is oftenmadealmostsimultaneouslybymorethanonepersonorteam. Aclassicalcaseinfruitfulmathematicsistheinventionofdifferential calculus(theuseofminusculestepstobuildacompleteobject,such asaplanet’strajectory).IsaacNewton,GottfriedWilhelmLeibniz,and theirfollowersbitterlyfoughtoverwhohaditfirstandwhetheror notplagiarismwasinvolved.Scientistscompetingforthesamewinand reachingitalmostsimultaneouslylacka“photo-finish”—theinfallible refereeofrunningraces—andtheproblemofpriorityispervasive.
ThereisincreasinglyconvincingevidencethattheVikingssetfoot inAmericaasearlyasthetenthcentury.Thereisnoquestionthat theAmerindiansweretheremuchbeforethat.Andyet,theglory of“discovering”AmericagoestoColumbus.Thus,thepointisnot beingthefirsttodiscoversomething,butthelast.Ilearnedthisfrom acolleaguewiseenoughtoadmitthathismostquotedworkwas onlyanimprovementoverpreviouslyknownstuff.Themoralisthat mostscientistsareextremelysensitivetorecognition,deservedornot; mammy!,daddy!,look!
Thebesteverseriesofscientificgamesmusthavebeen“Einsteinvs. Nature.”Thefirstthreegameshewonwithapparenteaseallinone year:1905.Inthat AnnusMirabilis hediscoveredRelativity,understood Brownianmotion andexplainedthe photoelectriceffect.Whethermuchof relativitywascontainedinpreviousworkbyHendrikLorentz,Henri Poincaré,HermannMinkowski,andothersactuallydidleadtoacontroversy,thoughnonebutEinsteinwrotethebestknownresult, E = mc2 . Brownianmotion—theobservedjitterofpollengrainsinwater— isduetothewatermoleculescollidingwiththem.Thisconclusion
EnjoyOurUniverse:YouHaveNoOtherChoice.AlvaroDeRújula. ©AlvaroDeRújula2018.Publishedin2018byOxfordUniversityPress. DOI:10.1093/oso/9780198817802.001.0001
definitelyestablishedthenon-continuous“atomic”natureof matter. Toexplainindetailthephotoelectriceffect—howlightimpingingon metalsextracts electrons fromit—Einsteinhadtoassumethatlight alsooccursin“packets,”theelementaryparticlesthatwenowcall photons.ThisearnedEinsteinaNobelPrize,arguablynotthemost deservedoneofthefourhemighthavegotten.
ThebestofallEinstein-vs-Naturegameswasnodoubttheonewon byEinsteinwhenheunveiledtheenigmaofgravitybyfiguringout theequationsinFigure1.Thegamestartedin1907withadisarmingly simplethoughtexperiment:Einsteinrealizedthatsomeoneinsidea closedfallingelevatorwouldfloatinitandnotknowthat(s)heisin apotentiallyfatalacceleratingmotion.Fromthisheconcludedthat accelerationandgravityare“locallyequivalent.”Theroadfromthere totheequationsofGeneralRelativityisarduousevenforsomeone studyingthemratherthanattemptingtoworkthemoutfromscratch. IttookEinsteinupto1915tojourneyoverit.Wedonotknowhowhe feltuponreachingtheroad’sendpoint,perhapsthewayillustratedin Figure3.Fortherestofhislifehestruggledtofinda“UnifiedTheory” ofgravitationandelectromagnetism,ataskatwhichhefailed.
TheBothersomeQuestionofUnits Theverymanyunitsusedtomeasurethesamereality,forinstance distance,isestheticallyinteresting,butarealpaininpractice.Notthat onehastoremembertheexactrelativemagnitudesofmiles,kilometers, nauticalmiles,verstas,royalfeet,andwhatnot,but,whentraveling bycar,itmayproveusefultoconvertmilespergallontolitersfora hundredkilometers,orviceversa ... anditisnotsoeasy.Nottomention gettingusedto,say,whattheupperfiftiesinFahrenheitorsixdegrees Celsius feellike whentheyarenottheunitsyougrewupwith.Perhaps, notsosurprisingly,scientistschoose1—exactlyone—asthevalueofas manyfundamentalquantitiesastheycan.
Insomerespects,therehasbeenprogressconcerningunits.Nineteen countriesoftheEuropeanUnion6andfiveothergeographicallysmall Europeancountries7shareacommoncurrency:TheEuro.Thismayor maynotmakeeconomicsense,butitimpliesagiganticsimplificationof banking,accounting,traveling,andsoon.Onemay,forinstance,cross mostof(continental)Europewithawallet,asopposedtoabriefcasefull ofdiversebanknotes.
Likecurrencies,thesystemofbasicscientificunits,suchasthoseof lengthandtime,hasalsoevolvedintelligently.Themeterwasoriginally definedasafixedfraction(atenthofamillionth)ofthequadrantof theterrestrialmeridian(thedistancefromthePoletotheEquator). Cross-countryexpeditionswereonceuponatimesentto“measurethe meter,”adelightfulabsurdity.
From1889to1960,themeterwasthelengthofarodofplatinum andiridiummaintainedsomewhereinParisatice-meltingtemperature. Thesecondusedtobethefraction1/(86,400)ofthemeansolarday, whateverthatwas.Thespeedoflight,generallydenotedwith c,was averylargenumberofkilometerspersecond,withverymanydigits
6 Austria,Belgium,Cyprus,Estonia,Finland,France,Germany,Greece,Ireland, Italy,Latvia,Lithuania,Luxembourg,Malta,theNetherlands,Portugal,Slovakia,Slovenia,andSpain.
7 Andorra,Cyprus,Monaco,SanMarino,andtheVatican.
EnjoyOurUniverse:YouHaveNoOtherChoice.AlvaroDeRújula. ©AlvaroDeRújula2018.Publishedin2018byOxfordUniversityPress. DOI:10.1093/oso/9780198817802.001.0001
andaninevitableuncertainty.Absurd,inparticular,becausethespeed oflightinvacuumisanunperturbableconstantofNature.
The second isnowdefinedasa fixed number(9,192,631,770)ofthe periodsofthelightemittedbyanatomofCesium133inthe“hyperfine” transitionofitsgroundstates,8whichisyetanotherconstantofNature. Insteadoflookingforaridiculousdefinitionofthemeter,thespeed oflight(invacuum)iscurrentlyfixedtobe precisely 299,792,458meters persecond.Combinedwiththecurrentmeaningofthesecond,this defines the meter,withconsiderablesavingsinplatinum,iridium,craftsmanship,andtravel.
Noticethat,inthevacuum,inananosecond9lighttravelsadistance ofaboutonefoot(29.9792458cm,tobeexact).IfIweretodefine“my royalfoot”aspreciselythatdistance,thespeedoflightwouldbe,in unitsof(myroyal)feetpernanosecond, preciselyc = 1.Afairlyeasynumbertoremember.Moreover,with c = 1,theunitsoftime(nanoseconds) anddistance(nanosecondsoflighttravel)areessentiallythesame,see Figure4.Setting c = 1isevenbetterthanmeasuringmaritimedistances
Figure4 Lighttravels29.9792458cm(thelengthof myRoyalFoot)inone nanosecond(ns)or,moresimply,thedistanceof1light-nsinans.The horizontalandverticalscalesaretimesanddistances,respectively.Footfigure: courtesyofPearsonScottForesman.
8 Thetwostatescorrespondtothetworelativeorientationsofthespinofthenucleus andthatoftheoutermostelectron.SpinisdiscussedinChapter11.
9 Onenanosecond(ns)is10 9 = 1/109 seconds.109 isa(North)Americanbillion;a Britishoneis1000timesmore,asouveniroftheolderEmpire.WeshallusetheAmerican conventionhere.A1followedby n zeroesis10n ,itsinverseis10 n
Figure5 Themeasurementofdistancesingoodoldunits(fathomsandnauticalmiles)orinmeters.Thehorizontalandverticalscalesareallfordistances,in thevariousunits.Ship:https://pixabay.com/en/ship-boat-sailing-travel-ocean29244/Submarine:https://pixabay.com/en/amarillo-blue-el-mar-mar-sea-sub1300164/. anddepthsinkilometersasopposedtonauticalmilesandfathoms; trytofigureout,usingthelatterpairofunits,thedistancefromyour submarinetoashipatwhichyouareaiminga(dummy!)torpedo.By thetimeyouaredone,theshipisgone.Theunitstobenaturallyused fortorpedoesandshipsareillustratedinFigure5.
Physicistsuse naturalunits,fixingsomefundamentalquantitiesto thevalue“1.”“Naturally,”thespeedoflightis c = 1.Forphysicists, consequently,massandenergyarealsomeasuredinthesameunits: E = mc2 simplybecomes E = m.Thisdoesnotmean,asweshallseeanon, thatEinsteindiscoveredthatthemassofanobjectchangeswithits energy,thoughhe did saysointroducingamisconceptionthat,more thanacenturylater,stilllingers,evenintextbooks.
TheScientificMethod Thetruthis,theScienceofNaturehasbeenalreadytoolongmade onlyaworkoftheBrainandtheFancy:Itisnowhightimethatit shouldreturntotheplainnessandsoundnessofObservationson materialandobviousthings.
RobertHooke[1635–1703]
Itissometimesstatedthatascienceisfundamentaliftheanswersto thequestionsitposes liebeyond itsestablishedlimits.Thatmaybea fairdescriptionand—ifonesubstitutestheword“fundamental”with “fascinating”—itmaybethereasonwhysomeofthemostsuccessfulpopularscientificbooksaretheonesthatmainlydiscusssubjects that“liebeyond.”Infundamentalphysicsthesesubjectsincludethe meaningoftime,affordabletraveltothepast,dimensionsbeyondthree (ofspace)andone(oftime),universesotherthanours,theveryfirst instantsoftheUniverse,theday“before,”andsoon.10
Exceptonrareoccasions,Ishallonlydealwithitemsthatlaboratory experimentsorcosmologicalobservationshaveshowntobecorrect. “Correct,”asopposedto“true,”inthesensethatscientificprogressis obtainedasasuccessionofincreasinglygoodapproximations.When Ideviatefromthisself-imposedlimitationofstickingtothescientific methodnodoubtthereaderwillrealizethatIdoitwithtonguein cheek.11
10 Thereisnothingwrongindiscussingthesesubjects,except,inmyopinion,doingit withoutaveryclearcutdistinctionbetweenfacts,reasonableconjectures,andoutright fantasies.
11 Anexample:myinterviewofEinstein,whomIvisitinhisofficein1905. https://www.youtube.com/watch?v=iTWI-jV8HJY.
EnjoyOurUniverse:YouHaveNoOtherChoice.AlvaroDeRújula. ©AlvaroDeRújula2018.Publishedin2018byOxfordUniversityPress. DOI:10.1093/oso/9780198817802.001.0001
Mass,energy,andmomentum Thediscoveryofthe Higgsboson at CERN12announcedonJuly4th, 2012,hadatotallyunprecedentedresponsefromthemedia.Amongst themanyreactionsinthecommentstotheelectronicversionsof newspapers,therewereendlessdiscussionsontheconceptof“mass,” triggeredbythefact,whichweshalldiscuss,thattheHiggs boson hastodowiththeoriginofthemassesof(theother)elementary particles.Thistaughtmethattherathersimpleconceptof mass isoften extremelywellmisunderstood,amisunderstandingtowhichEinstein himselfsignificantlycontributed.Consequently,Ishallusethisconcept toillustratethe scientificmethod.
Themethodworkslikeasimple,sofarendlesscomputerprogram:
1.Accept—perhapstemporarily—someconceptsasfundamental.
2.Definethemempirically(i.e.,inpracticalterms).
3.Demandthattheconceptuallinksbetweentheconceptsbeconsistentwithobservation.
4.Buildupfromthereand,ifthisdoesnotwork,gobackto(1)to startallover.
Letus,likemathematicians,befastidiouslyprecise,seeFigure6.And letusaccepttheconceptsoftimeandspaceinthesimpleandpractical sensethatwecanallagreeaboutthemeasurementoftimelapsesand spacedistanceswithagivenclockandaparticularyard-stick.Weshall saythatanobjectcoveringthesamestraight-linedistanceinsuccessive identicaltimeintervalsisin uniformmotion. Nowtakeabilliardtable andsomebilliardballs.Cleanandpolishthemtothepointthatthe ballsmoveuniformlyonthetabletotheprecisionofourclockand yard-stick.
Wearegoingtohavetouseanunavoidableingredientofthescientific method:Math.Hopefullysolittlethatonemaychoosenottopay particularattentiontoit.Foraballinuniformmotion,oratrest,we can define a velocity astheratioofagivendistanceandthetimeittakes theballtocoverit.Onthetablethevelocityhastwocomponents,along itslength(callit vx )andwidth(vy ),whichwecangatherasapairina
12 CERNistheEuropeanLaboratoryforHigh-EnergyPhysics,locatedcloseto Geneva(Switzerland).AnAmericancompetitor,Fermilab,isclosetoGeneva(Illinois). Formoreontheselabs,see CERN and HighEnergyLaboratories,intheGlossaryattheend ofthebook.
Figure6 Thediverseperceptionsofdifferentprofessionals.
