Table Of ContentRelativity 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. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub-
lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany
errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect
tothematerialcontainedherein.
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
