Table Of Content

Relativity for Everyone Kurt Fischer Relativity for Everyone How Space-Time Bends KurtFischer DepartmentofMechanicalandElectrical Engineering TokyuamaCollegeofTechnology Shunan-Shi,Japan ISBN978-3-319-00586-7 ISBN978-3-319-00587-4(eBook) DOI10.1007/978-3-319-00587-4 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013939093 ©SpringerInternationalPublishingSwitzerland2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To Yukiko: You editedthetexttobecomereadable. Preface DearReader, Iencounteredthetheoryofrelativityatfirstwhileinmyteens,inthepubliclibrary of my hometown. There were two kinds of books: The easy ones did not really explain,butdisplayedlarge,coloredpicturesofthescience-fictionkind.Theserious books seem to explain something, but that was hidden in a mass of mathematical symbols,andthosetheydidnotexplain,soIwasbacktosquareone.Nevertheless, theyarousedmyinterestinphysics.It alsoincitedmeto fillthegap.Theresult is thisbook. Thisbookisaboutlight,energy,mass,space,time,andgravity:Iexplaintoyou howthetheoryofspecialrelativityandthetheoryofgeneralrelativityworkout. Wewillusemanythoughtexperiments,andshowyouhowphysicistscreateand solvemodels.ThismethodistheoneusedbyEinsteinhimself:Heunderstoodthe theoryinphysicalandgeometricalpictures.WewillfollowEinstein’soriginaltrain ofthoughtascloselyaspossible,usingevensomeofhisownthoughtexperiments. I will present some involved arguments, so you need your imagination, but no complicatedmathematicstounderstandtheessenceofitall.However,thebeautyof physicsis,thatwecancalculatehowthenumbersworkout.ThereforeIpreparedthe equationsofthetheoryofrelativitytogetherwiththemostimportantexactsolutions, usingonlyelementarymathematics.EventheEinsteinequationofgravity,showing thebendingofspaceandtime,wepresentindetail,incommonlanguage.Wewill see why it is the simplest theory of gravity. We will not be content with analogies like“spacebendslikethesurfaceofasphere!” Inthefirstfourchapters,Iexplainwhatiscalledthetheoryofspecialrelativity: Idescribetherelationbetweenlight,matter,space,andtime. 1. InChap.1,Idescribeafterintroducingthebasics,thatmassandenergyarethe oppositesiteofthesamecoin. 2. InChap.2,wewillseewhytimeandlengthare“relative”. 3. InChap.3,wewillseewhyanyelectricwireshowsrelativityineverydaylife. 4. InChap.4,welearnthatwhileridingamerry-go-round,school-geometryceases tobetrue. vii viii Preface IntheChaps.5to9Idescribegravity,thatisthetheoryofgeneralrelativity. 5. InChap.5,Ishowyouthatearth’sgravitydoesnotpullatyouatall,andthatit bendsspaceandtime. 6. In Chap. 6, we will see in detail which effects the bending space and time are causing. 7. InChap.7,weexplainthoroughlythemeaningofthefamousEinsteinequation ofgravity,andwhyitisthesimplestpossiblewayofdescribinggravity. 8. InChap.8,weintroducethemostfamousexactsolutionoftheEinsteinequation, thatistheSchwarzschildsolution,insimpleterms. 9. InChap.9,weusethesolutionstotheEinsteinequation,andexplorethefamous predictionsofthetheoryofgeneralrelativity,asforexamplehowmuchalight beambendswhilepassingatthesun,howlargeandheavyblackholesare,why theorbitsoftheplanetsaroundthesunturnslowlyaroundthesun,andwhythe universehadabigbang,butwhyitsfutureisuncleartous. Onewordabouthighlightedtext:Indexedwordsappearinboldface.Suchyoucan easily find them on their page, searching from the index. Before going on with describing the strange properties of light, we better tell in what units we measure andcomparethesizeofthings. UnitsandSymbols In physics, we use certain units to measure things. Lengths we measure only in meters,timeonlyinseconds,notinminutes,hours,ordays.Masswemeasureonly inkilograms.Anyotherunitweuseinthisbookisacombinationoftheseunits.For example,speedwemeasureinmeterspersecond.Otherunitslikepoundsorinches orsuchlikeweneveruse.Themeritofthisisthatwecanleavetheunitsawayinall calculations, because we know anyway what units to add afterwards, just because wefixedthematthebeginning. Wewillencounteroftenreallylargeorverysmallnumbers.Forexample,while numberslike“onethousand”wecanwritedownas1000,thenumberofonebillion two hundred fifty-two million seven hundred eighty-four 1,250,000,784 is much harder to read. Mostly we are only interested in the first three or so digits, for a roughestimateofhowlargethingsare.Herephysicistscountthenumberofdigits afterthefirstone,thatisnineinthiscase,andwrite 1.25×109 Inthesameway,averysmallnumberlike0.000145wewriteas1.45×10−4 by countingthenumberofleadingzeros.Thenwecaneasilymultiplysuchnumbers: Wemultiply1.25×109×1.45×10−4bymultiplyingatfirst1.25and1.45whichis roughly1.81,andaddingtheexponents9−4=5,sothattheresultis≈1.81×105. Herethesymbol≈means“isroughlyequalto”. Forexample,weuseforthespeedoflightmostlytheroughvalue speedoflight≈3.00×108meterspersecond Preface ix Wewillencounterotherimportantnumbersofnatureinthetext.Wecollectedthem inTableA.1forreference. SpecialExpressions Atlast,onewordabouthowweusephraseslike“farenoughawayfromsomething”, “fastenough”,andthelike.Forexample,wesay “iftheastronautisfarenoughawayfromearth,theastronautcanneglectearth’sgravity.” We are aware of the fact, that gravity is never exactly zero, even very far away fromearth.Thepointishere,thattheastronautwantstomeasuresomeeffect,with gravityspoilingthiseffect,to,say,onepercentoftheresultthattheastronautwould get in really empty space. So if the astronaut finds that gravity is still spoiling his experimenttoomuch,heisfreetomovetoaplacewhichissofarawayfromearth thatthere,gravityreallydoesspoilhisexperimentonlyuptothedesiredonepercent atthemost.Ofcourse,ifhewantstomeasureevenmoreaccurately,hemustmove even farther away from earth. That is why we say, in short, that if he moves “far enoughaway”fromearth,healwayscanneglectearth’sgravitytotheextentthathe wantstoneglectit. Tokuyama,Japan KurtFischer Contents 1 Light,Matter,andEnergy . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 LightBeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 FirstLawofRelativity:Straight,SteadySpeedIsRelative . . . . . 1 1.3 MeasuringtheSpeedofLight . . . . . . . . . . . . . . . . . . . . 2 1.4 SecondLawofRelativity:SpeedofLightIsAbsolute . . . . . . . 3 1.5 FasterthanLight? . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 TheoryandPractice . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 MassandInertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.8 InertiaandWeight:AFirstGlance . . . . . . . . . . . . . . . . . 6 1.9 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.10 MassandMotionEnergy . . . . . . . . . . . . . . . . . . . . . . 7 1.11 Resting-MassandMotionEnergy . . . . . . . . . . . . . . . . . . 8 1.11.1 InternalMotion-Energy . . . . . . . . . . . . . . . . . . . 8 1.11.2 PureEnergy . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.12 InertiaofPureEnergy . . . . . . . . . . . . . . . . . . . . . . . . 9 1.13 MassIsEnergyIsMass . . . . . . . . . . . . . . . . . . . . . . . 11 1.14 InformationNeedsEnergy . . . . . . . . . . . . . . . . . . . . . . 11 2 Light,Time,Mass,andLength . . . . . . . . . . . . . . . . . . . . . 13 2.1 LightandTime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 TheGammaFactor . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 WhoseClockIsRunningSlower? . . . . . . . . . . . . . . . . . . 17 2.4 Light,Time,andLength . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 LengthinDirectionoftheSpeed . . . . . . . . . . . . . . 18 2.4.2 LengthatRightAnglestoSpeed . . . . . . . . . . . . . . 19 2.5 AttheSameTime?. . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 TimeandMass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 SpeedAddition . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Light,Electricity,andMagnetism . . . . . . . . . . . . . . . . . . . . 25 3.1 ElectricChargeandSpeed . . . . . . . . . . . . . . . . . . . . . . 25 xi xii Contents 3.2 ElectricChargesandMagnets . . . . . . . . . . . . . . . . . . . . 26 3.3 ElectricandMagneticFields . . . . . . . . . . . . . . . . . . . . 28 3.4 MagneticFieldfromElectricCurrent . . . . . . . . . . . . . . . . 29 3.4.1 TheFaradayParadox . . . . . . . . . . . . . . . . . . . . 30 3.4.2 NoAttractionWithoutRelativity . . . . . . . . . . . . . . 31 3.4.3 AttractionwithRelativity . . . . . . . . . . . . . . . . . . 31 4 AccelerationandInertia . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 RotatingMotion:TwinParadox1 . . . . . . . . . . . . . . . . . . 35 4.2 RotatingMotion:NottheSchoolGeometry . . . . . . . . . . . . . 37 4.3 StraightMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 ProperTimeandInertia:TwinParadox2 . . . . . . . . . . . . . . 40 5 InertiaandGravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.1 GravityIsNoForce . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 GravityBendsSpace-Time . . . . . . . . . . . . . . . . . . . . . 46 5.2.1 BendedSurface . . . . . . . . . . . . . . . . . . . . . . . 47 5.2.2 BendedSpace-Time . . . . . . . . . . . . . . . . . . . . . 48 5.3 MeasuringtheBendingofSpace-Time . . . . . . . . . . . . . . . 50 6 EquivalencePrincipleinAction . . . . . . . . . . . . . . . . . . . . . 53 6.1 TimeandGravity . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 ProperTimeinBendedSpace-Time:TwinParadox3 . . . . . . . . 55 6.3 MovingStraightlyinBendedSpace-Time. . . . . . . . . . . . . . 57 6.4 LengthUnderGravityofaPerfectBall . . . . . . . . . . . . . . . 57 6.5 GravityAroundaPerfectBall . . . . . . . . . . . . . . . . . . . . 60 6.6 MassUnderGravity . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.7 LightUnderGravity . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.8 BlackHoles:AFirstLook . . . . . . . . . . . . . . . . . . . . . . 64 6.9 EquivalencePrinciple:Summary . . . . . . . . . . . . . . . . . . 65 7 HowMassCreatesGravity . . . . . . . . . . . . . . . . . . . . . . . 67 7.1 GravityinaLonelyCloud . . . . . . . . . . . . . . . . . . . . . . 67 7.2 EinsteinEquationofGravity . . . . . . . . . . . . . . . . . . . . 69 7.3 EnterPressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.4 EnterSpeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.5 EnterOutsideMasses . . . . . . . . . . . . . . . . . . . . . . . . 72 7.6 LocalandGlobalSpace-Time . . . . . . . . . . . . . . . . . . . . 73 7.7 HowtoSolvetheEinsteinEquationofGravity . . . . . . . . . . . 74 8 SolvingtheEinsteinEquationofGravity . . . . . . . . . . . . . . . . 77 8.1 GravityCausesLawofMotion . . . . . . . . . . . . . . . . . . . 77 8.2 GravityInsideaPerfectBallofMass . . . . . . . . . . . . . . . . 78 8.3 FlatSpace-TimeInsideaBall-ShapedHollow . . . . . . . . . . . 81 8.4 GravityOutsideaPerfectBallofMass . . . . . . . . . . . . . . . 81 8.5 SchwarzschildExactSolution . . . . . . . . . . . . . . . . . . . . 83 8.6 NewtonLawofGravity . . . . . . . . . . . . . . . . . . . . . . . 86