The Meaning of Einstein’s Equation

1 TheMeaningof einstein ' andEmoryF. BunnyJanuary4, 2006 AbstractThisis a briefintroductionto generalrelativity, designedforbothstudentsandteachersof excellent expositionsofgeneralrelativity, fewadequatelyexplainthegeometricalmeanin gof thebasicequationof thetheory: einstein ' give a simpleformulationof thisequationin termsof themotionof alsosketch someof theconsequencesof thisformulationandexplainhow it is equivalent to theusualonein termsof , weincludeanannotatedbibliography of books,articlesandwebsitessuitableforthes tudent of explainsgravity as thecurvatureof 'sallaboutgeometry. Thebasicequationof generalrelativity is calledEinstein' unitswherec= 8 G= 1, it saysG =T :(1)It lookssimple,butwhatdoes it mean?Unfortunately, thebeautifulgeometri-calmeaningof thisequationis a bithardto ndin mosttreatments of nicepopularizationsthatexplainthephiloso phy behindrela-tivity andtheideaof curvedspacetime,butmostof themdon'tgetaroundtoexplainingEinstein's equationandshowinghow to explainEinstein'sequationin detail| butherethegeometryis oftenhiddenunderpilesof a pity, becausein factthereis aneasyway toexpressthewholecontent of einstein 'sequationin fact,aftera suitableprelude,onecansummarizeit in a singlesentence!

2 Oneneedsa lotof mathematicstoderive alltheconsequencesof thissentence,butit is stillworthseeing|andwe canworkoutsomeof itsconsequencesquiteeasily. Department of Mathematics,University of University of Richmond,Richmond,VA whatfollows,we startby outliningsomedi erencesbetweenspecialandgeneralrelativit y. Nextwe give a verbalformulationof einstein ' derive a fewof itsconsequencesconcerningtidalforces,gra vitationalwaves,gravitationalcollapse,an dthebigbangcosmology. In thelastsectionwe explainwhy ourverbalformulationis equivalent to theusualonein mainlyaimedat thosewhoteach relativity , butexceptforthelastsection,we have triedto make it accessibleto students,as a sketchof how thesubjectmight be concludewitha bibliography ofsourcesto helpteach 'sequation,we needa assumethereaderis somewhatfamiliarwithspecialrelativity | otherwisegeneralrelativitywillbe too erencesbetweenspecialandgeneralrelativit y, which cancauseimmenseconfusionif specialrelativity, we cannottalkaboutabsolutevelocities, ,we cannotsensiblyaskif a particleis at rest,onlywhetherit is at restrelative to thatin thistheory.

3 Velocitiesaredescribedas vectorsin adi erent inertialcoordinatesystemcanchangewhich way thesevectorspointrelative to ourcoordinateaxes,butnotwhethertwo of thempoint generalrelativity, we cannoteven talkaboutrelativevelocities,exceptfortwo particlesat thesamepoint of spacetime| thatis, at thesameplaceat thesameinstant. Thereasonis thatin generalrelativity, we take veryseriouslythenotionthata vectoris a littlearrow sittingat a particularpoint in comparevectorsat di erent points of spacetime,we mustcarryoneover carryinga vectoralonga pathwithoutturningorstretchingit is called`paralleltransport'.Whenspacetimei s curved,theresultof paralleltransportfromonepoint to anotherdependsonthepathtaken!Infact,this is theveryde nitionof whatit meansforspacetimeto be it is ambiguousto askwhethertwo particleshave thesamevelocity vectorunlesstheyareat thesamepoint of is hardto imaginethecurvatureof 4-dimensionalspacetime,butit is easytoseeit in a 2-dimensionalsurface,like a tsnicelyin3-dimensional atEuclideanspace,so we canvisualizevectorsonthesphereas`tangent vectors'.

4 If we paralleltransporta tangent vectorfromthenorthpoleto theequatorby goingstraight downa meridian,we geta di erent resultthanif we godownanothermeridianandthenalongtheequa tor:2 Becauseof thisanalogy, in generalrelativity vectorsareusuallycalled`tangentvectors'. However,it is important notto take thisanalogytoo seriously. Ourcurved spacetimeneednotbe embeddedin somehigher-dimensional atspace-timeforus to understanditscurvature,or theconceptof tangent tensorcalculusis designedto letus handletheseconcepts`in-trinsically'| ,workingsolelywithinthe4-dimensionalspac etimein whichwe onereasontensorcalculusis so important in ,in specialrelativity we canthinkof aninertialcoordinatesystem,or`inertialfr ame',as beingde nedby a eldof clocks,allat restrelative to generalrelativity thismakes no sense,sincewe canonlyunambiguouslyde netherelative velocity of two clocks if theyareat inertialframe,so important in specialrelativity, isbannedfromgeneralrelativity!

5 If we arewillingto putupwithlimitedaccuracy, we canstilltalkabouttherelative velocity of two particlesin thelimitwheretheyareveryclose,sincecurva turee ectswillthenbe thisapproximatesense,we cantalkabouta `local' ,we mustremember thatthisnotionmakes perfectsenseonlyin thelimitwheretheregionof spacetimecoveredby thecoordinatesystemgoes to zeroin 'sequationcanbe expressedas a statement abouttherelative ac-celerationof veryclosetestparticlesin `testparticle'is anidealizedpoint particlewithenergyandmomentumso smallthatitse particleis saidto be in `freefall'whenitsmotionis a ectedby noforcesexceptgravity. Ingeneralrelativity, a testparticlein freefallwilltraceouta `geodesic'.Thismeansthatitsvelocity vectoris paralleltransportedalongthecurve it tracesoutin geodesicis theclosestthingthereis to a straight lineincurved ,allthisis easierto visualizein a sphere`followingtheirnose'willtraceouta geodesic| thatis, a peoplestandside-by-sideon theequatorandstartwalkingnorth, each other,thedistancebetweenthemwillgraduall ystartto shrink,until nallytheybumpinto each otherat theydidn'tunderstandthecurved geometryof thesphere,theymight thinka `force'was , in generalrelativity gravity is notreallya `force',butjusta man-ifestationof thecurvatureof :notthecurvatureof space,butofspacetime.

6 Thedistinctionis youtossa ball,it followsa farfrombeinga geodesicinspace: spaceis curved by theEarth'sgravitational eld,butit is certainlynotso curved as allthat!Thepoint is thatwhiletheballmoves a shortdistancein space,it moves anenormousdistanceintime, sinceonesecondequalsabout300,000kilomete rsin unitswherec= slight amount of spacetimecurvatureto have a noticeablee 'sEquationTo stateEinstein'sequationin simpleEnglish,we needto considera roundballof testparticlesthatareallinitiallyat restrelative to each haveseen,thisis a sensiblenotiononlyin thelimitwheretheballis startwithsuch a ballof particles,it will,to secondorderin time, too surprising,becauseanylineartransformatio nappliedto a ballgives anellipsoid,andas thesayinggoes,\everythingis linearto rstorder".

7 Herewe geta bitmore:therelative velocityof theparticlesstartsoutbeingzero,so to rstorderin timetheballdoes notchangeshape at all:thechangeis a second-ordere (t) be thevolumeof theballaftera proper timethaselapsed,asmeasuredby theparticleat thecenterof 'sequationsays: VV t=0= 120BB@ ow oft momentumintdirection+ ow ofx momentuminxdirection+ ow ofy momentuminydirection+ ow ofz momentuminzdirection1 CCAwherethese owsaremeasuredat thecenterof theballat timezero, owsarethediagonalcomponents of a 4 4 matrixTcalledthe`stress-energytensor'.Th ecomponentsT of thismatrixsay howmuch momentumin the directionis owingin the directionthrougha givenpoint of spacetime,where ; =t; x; y; z. The ow oft-momentumin thet-directionis justtheenergydensity, oftendenoted . The ow ofx-momentumin thex-directionis the`pressurein thexdirection'denotedPx, andsimilarlyforyandz.

8 It takes a whileto gureoutwhy pressureis reallythe ow ofmomentum,butit is any event, we may summarizeEinstein'sequationas follows: VV t=0= 12( +Px+Py+Pz):(2)4 Thisequationsaysthatpositive energydensity andpositive pressurecurvespacetimein a way thatmakes a freelyfallingballof point areworkingin unitswherec= 1, ordinarymassdensity counts as a formof energydensity. Thusa massive objectwillmakea swarmof freelyfallingparticlesat restaroundit startto promisedto stateEinstein'sequationin plainEnglish,buthave notdoneso is:Givena smallballof freelyfallingtestparticlesinitiallyatres twithrespectto each other,therateat which it beginsto shrinkisproportionalto itsvolumetimes:theenergydensity at thecenterof theball,plusthepressurein thexdirectionat thatpoint, plusthepressurein theydirection,plusthepressurein to prove thisis by usingtheRaychaudhuriequation,discussions of which canbe foundin thetextbooksby Waldandby CiufoliniandWheelercitedin thebibliography.

9 Butanelementaryproof canalsobe givenstartingfrom rstprinciples,as we willshow in the nalsectionof may be somewhatskepticalthatallof einstein 'sequationis , einstein 'sequationin itsusualtensorialformis reallya bunch ofequations:theleftandright sidesof Equation (1)are4 4 is hardto believe thatthesingleequation(2) does,though,as longas we includeonebitof neprint: in orderto getthefullcontentof theEinsteinequationfromequation(2),we mustconsidersmallballswithall possibleinitialvelocities| ,ballsthatbeginat restin begin,it is worthnotinganeven simplerformulationof einstein 'sequationthatapplieswhenthepres sureis thesamein everydirection:Givena smallballof freelyfallingtestparticlesinitiallyatres twithrespectto each other,therateat which it beginsto shrinkisproportionalto itsvolumetimes:theenergydensity at thecenterof theballplusthreetimesthepressureat onlysu cient for`isotropic'situations:thatis,thosein whichalldirectionslookthesamein ,sincethesimplestmodelsof cosmologytreattheuniverseasisotropic|atl eastapproximately, onlargeenoughdistancescales|thisis allwe shallneedtoderive anequationdescribingthebigbang!

10 4 SomeConsequencesTheformulationof einstein 'sequationwe have givenis certainlynotthebestformostapplicationsof generalrelativity. For example,in 1915 Einsteinused5generalrelativity to correctlycomputetheanomalousprecessionof theorbitof Mercuryandalsothede ectionof starlight by theSun'sgravitational veryhardstartingfromequation(2);theyreal lycallforthefullapparatusof ,we caneasilyuseourformulationof einstein 'sequationto getaqualitative | andsometimeseven quantitative | understandingofsomeconse-quencesof generalrelativity. We have alreadyseenthatit explainshow sketch a 'sinverse-squareforcelaw,which holdsin thelimitof weakgravitational eldsandsmallvelocities, ,GravitationalWavesWe beginwithsomequalitative consequencesof einstein ' (t)be thevolumeof a smallballof testparticlesin freefallthatareinitiallyat restrelative to each thevacuumthereis noenergydensity or pressure,so Vjt=0= 0, butthecurvatureof ,supposeyoudropa smallballof instant co eewhenmakingco eein co eecloserto theearthacceleratetowardsit abitmore,causingtheballto startstretchingin ,as thegrainsallacceleratetowardsthecenterof theearth,theballalsostartsbeingsquashedi n thetwo 'sequationsays thatif we treattheco eegrainsas testparticles,thesetwo e ectscanceleach otherwhenwe calculatethesecondderivative of theball'svolume,leavinguswith Vjt=0= 0.