Digression into Geometry - University of Illinois at ... · PDF fileDigression into Geometry Status of geometry and logic Euclidean vs Non-Euclidean Geometry Topology Is space real?

DigressionintoGeometryStatusofgeometryandlogicEuclideanvsNon-EuclideanGeometryTopologyIsspacereal?Sklar“Space,TimeandSpace@me”pgs.13-46;“PhilosophyofPhysics”pgs.53-91

•  Egyp@ansdevelopedgeometryasaprac@calscience.

•  GreekgeometryculminatedinEuclid’swork(300BC)

•  Star@ngfromdefini@ons,axioms,postulates,logiconederivestheorems:a2+b2=c2

•  Defini@onsrelatewordstothereader’sintui@on“alineisanelementoflengthwithnobreath.”Nophysicalobjectisaperfectline.Howtodefine?

•  Axiomsarecommonno@ons(logic).•  Postulatesrelatetogeometricalideas.

Euclid

Axioms:1.  Thingsequaltothesamethingareequaltoeachother.2.  Equalsaddedtoequalsyieldequals.3.  Equalsremovedfromequalsyieldequals.4.  Coincidentfiguresareequaltooneanotherinallrespects.5.  Awholeisgreaterthananyofitsparts.

Postulates1.  Twopointsdetermineastraightline.2.  [email protected].  Aboutanypointacircleofaspecifiedradiusexists.4.  Allrightanglesareequal.5.  Ifastraightlinefallingacrosstwostraightlinesmakesthesumoftheinterior

anglesonthesamesidelessthantworightangles,thenthetwostraightlinesintersect,ifsufficientlyextended,onthatside.

OR5.’Throughapointoutsideagivenlineoneandonlyonelinecanbedrawnwhichdoesnotintersectagivenline,nomaaerhowfaritisextended.

•  Therearefurtherimplicitassump@ons:No@onofalimi@ngopera@on,e.g.theareaandcircumferenceofacircle.

•  Whyarethepostulatestrueinourworld?Especiallythe5thpostulate.

•  Whyislogicortheprocedurecorrect?•  [email protected]/shouldwemakephysicslikegeometry?

•  Straightlinescanbeconstructedoncewehaveastraightedge.Empricalfact:dolinesmeet?– Defini@on:whatisaline?

Furtherdevelopments

•  Descartes(1596-1650)relatedgeometrytoalgebra–  P=(x,y)=r–  straightline:a.r=b–  Circle:(r-c)2=d2

•  Wedon’thavetodrawpicturesbutonlymanipulatesymbols.Butwhatdothesymbolsmeanintherealworld?

•  CalculusdevelopedbyLeibnitzandNewtonallowedmoregeneralobjectsthanlinesandcircles.

•  Newton:geometryisthesimplestscience.Itisempirical!•  Leibnitz(Plato)theaxiomsareself-evident•  Kant:Geometryformulatesthestructureofoursenses.

Non-EuclidianGeometry•  Isthe5thpostulatederivablefromotherassump@ons?

Thereweremanyaaemptsduring200yearsbutnosuccess.

•  Inthe18thand19thcentury,non-Euclidiangeometrieswereconstructed.Replacethe5thpostulateby:–  Noparallels(Reimann):Throughagivenpointoutsidealinealllines

intersectthefirstlineandtheyareoffinitelength–  Manyparallels(LobachevskiandGauss)Aninfinityoflinescanbe

drawnfromagivenpointwithoutintersec@on.

"triangle"

Sphere:Saddle:

"triangle"

Howtoprovealternategeometriesareconsistent

Riemann–  Point–  Line–  Triangle–  distance

Euclid–  Pointonasphere–  Greatcircle–  SphericalTriangle–  Arclength

Relatethemtoaknownconsistenttheory.•  Mapthemintoalgebralikeanaly@cgeometry•  Riemanngeometryin2Dislikesphericalgeometryin3D.

•  EverystatementinRiemanngeometryistranslatedintoastatementinsphericalgeometry.QED

Hence:alterna@vegeometriesexist!!

Hilbert(1898)axiomizedgeometriesmuchmoreexplicitlythanEucliddid

– Everythingwasdefinedintermsofprimi@ves:points,lines,planes….

– Byremovingaxiomsone-by-one,manytypesofgeometriescanbeconstructed

– Defini@onsisaprocesswherebynewtermsacquiremeaningintermsofprimi@ves.

– HilbertcandisentangletheEuclideansystem.– Notonlyisthe5thpostulatearbitrary.

Gauss’s(1777-1855)CurvedGeometry–  Al@tudeshouldnotenterin.(E.g.Thesurfaceoftheearth.)–  Define“coordinatesasacon@nuousfamilyofcurves,agrid–  Geometryisspeciifedbyadistancemetricg12(r)–  Definea“geodesic”astheshortestpathbetween2points.

[email protected]+b2=c2???–  Therecanbemul@plelinesbetween2points(considercirclesfrom

thenorthtosouthpole)–  GeometryislocallyEuclidean:smalltrianglesobeyEuclid’slaws.–  Thereisanintrinsiccurvaturedeterminedbyanglesanddistances

withinthesurface.Inahomogenousspacegisconstanteverywhere.–  ButspecifyinggisNOTenough.Wemustalsospecifythetopology.

•  AplaneandthesurfaceofacylinderarebothEuclideanbuttheyaredifferent.

•  Theycanonlybedis@nguishedbyglobalmeasurements.

ChoicesofCoordinates

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Eitherofthesegrids(blackorblue)isOK!Howcanyoukeepthesamelawsofphysics?

Distancesareafunc@onnotjustofthedifferenceinthecoordinatelabelsbutalsoofa“metrictensor”fielddefinedateachcoordinatepoint.

Topology•  Curvedgeometryallowsdifferentconnectedness,

i.e.“topologies”,e.g.howmanyholes.•  Consideracylindricaluniverse:

t

x

The local geometry of a cylinder is flat, but one coordinate is cyclic. If our universe had this topology, there would be a preferred reference frame, the one with the time axis along the cylinder, as shown.

•  WouldthismeanthatGalileanrela@vityiswrong?•  Isthegeometryoftheuniversea“law”orjusta“fact”?

•  Consider a donut universe : •  The local geometry is curved. •  The time coordinate is cyclic. •  How can one distinguish past from future here? •  What about causality?

There may be valid solutions to GR with this problem.

t x

IsEuclideangeometrycorrect?

•  Plato:Thesensedworldisanephemeralapproxima@ontothe‘true’worldofidealessences.Reasontellsusmoreaboutthattrueworldthanmeresensa@oncanprovide.

•  Aristotle:thefundamentalproposi@onsareself-evident.•  Hume(1711-1776):Whateverweclaimaboutreality,onlysensesare

available.Thereare2typesofstatements:–  Apriori:theonlyunpackmeaningsofwords“Allbachelorsare

unmarried”–  Aposterioricanberejectedinlightofexperience.

•  Kant(1724-1804):Therearenon-trivialaprioriproposi@ons;meaningfulstatementsthatareirrefutable.–  Thereexistsarealworld;sensa@onscomefromoutsidethereforethereexistsan

outside.–  Ourpercep@onsarestructuredbyourmind–  Spaceand@meareapriori.–  Wecan’timagineaworldwithout@meorspace.–  Wecanimaginebeingblindordeafbut@meless?–  Statementsareirrefutablebecausethat’showwethinkaboutit.

Empiricistposi@on

•  Anythingmeaningfulcanberefuted.•  Geometryisanexperimentalscience.•  Whatisaline?Weneedanopera@onaldefini@on.

–  Astraightedge(rulerscanbend)–  Abeamoflight(lightdiffracts)

•  Whatisapoint?Numbersdescribingwheretoputaninfinitelysmallpar@cle.–  Butpar@cleshaveadefinitesizeandwecan’tmeasuretheirposi@on

toarbitraryaccuracy.Quantumeffectslimitus.–  Maybetheunderlingrealityisdiscrete.Coordinatesshouldnotbe

con@nuousbutintegerlikestreetaddresses.Istherealwaysapointbetweentwootherpoints?

Poincaréparable•  Considera2dworldofadisk.•  Measurings@ckesaresensi@vetotemperaturebutif

getscolderthefurtheroutyouare.Length=R2-r2.•  Theythinkthediskisinfinitesincetheirs@ckscan

neverreachtheedgeinafinitenumberofsteps.SpaceisLobachevskiwithnega@vecurvature.

•  Supposetheyhavelightrayswithindexofrefrac@on1/(R2-r2).Sameconclusion.

HowdoweconvincethemthattheworldisreallyEuclidean?•  Poincaré(1854-1912):Anygeometryispossible.

Decideonthebasisofconven@onorconvenience.SinceEuclideangeometryissimpler,useit.

•  Einstein:Beamsoflightdefinegeodesics.Equivalenceprincipleimpliesspace-@meiscurved.

EmpiricistreplytoPoincaré(Eddington,Reichenbach)

Alterna@vegeometriescanalwaysexplainanynewdata.WecansaveEuclidbyconven@on.•  Thisisatrivialreworkingofthemeaningsoflengthand@me.•  Ifpredic@onsarethesame,thetheoriesarethesame.•  Inphysicswedefinewhatwemeanbylength.Thenthereisareal

dis@nc@onbetweenEuclideanandnon-Euclideangeometries.•  Ifwesayrodsshrinkandexpandthatisanewdefini@on.•  Poincarémustintroduceuniversalforces,somethingthatshrinksall

lengths.•  Thereisaninterconnec@onbetweengeometricallawsandphysicallaws.

•  Howaboutthetopologyofspace?Isthatconven@on?

TAXONOMYOFBELIEFINGEOMETRY

•  Disbelievers:everythingistheory,evendefini@onsoflength•  Believersinobserva@onalbasis

–  Reduc@onists:theory=observa@on,alterna@vesarerewordings.–  An@reduc@onists:theoryismorethanobserva@ons

•  Skep@cs:nora@onalchoiceispossiblebetweenalterna@vetheories.Wemayneverknowwhatliesbeyondourmeasurements

•  Conven@onalists:simplychooseatheory.Thereisnotruerealityonlyconven@on.

•  Apriorists:chooseatheoryonthebasisofsimplicity,elegance,ini@alplausibility

Whatwillourstraightlinesbe?•  Ifagooddefini@onofastraightlineis"thepathfollowedbyanobjectwhich

experiencesnoforces"wemustrecognize:•  Gravityisnotexperienced:inprincipleagravita@onalfieldcannotbefeltlocally.•  Astraightlinewouldthenbeapathfollowedbysomesmallobjectinthepresence

ofnoinfluencesotherthangravity,i.e.infreefall.–  Thesearecalledspace@megeodesics–  Thisisthenon-Euclideangeneraliza@on

ofNewton’s1stlaw.•  Ageodesicisdefinedtobetheshortestlinebetween

twopointsonasurface.Inspace@me,thisisgeneralizedtobetheworldlinewhichhasthelargestintervalbetweentwoevents(4-dpoints).

–  (“Largest”isanar@factofthe-sign.)

x

ctStay-at-home sister's interval is greater than her travelling sister's.

x

ct

Onthespheretherearenoparallellines.Allgeodesicsintersect.Similarly,theinterpreta@onofthetrajectoryofadroppedballneartheEarthisthatitsworldlineappearstostartoutparalleltotheEarth’s,butinevitablyintersectsit.

Doweneedcurvedspace@me?•  The2-dexamplesofcurvedsurfacesallowustolookatthingsfrom

“outside.”However,a2-dpersonconfinedtothesurfaceandwithnoknowledgeofthe3rddimensioncoulds@llinferthecurvaturefromhisowngeometricalmeasurements(e.g.,theanglesoftriangles).That’sthesitua@onwefindourselvesin.Thecurvaturecanbecompletelydefinedandallexperimentalpredic@ons,etc.madewithnoreferencetosomeotherdimensions.–  Thisclaimdoesnotmeanthatnootherdimensionsexist,justthattheyare

irrelevanttotheques@onofwhetherourfamiliardimensionsformaflatspace.

•  Theimmediateques@onisdoweneedcurvedspace@me,orcanwegetawaywith“flat”geometry?(Sklar,55-67)Wehavealreadyseenthatifwewanttoputacceleratedobserversonaparwithiner@alones,weneedcurvature.Ifwedon’tdothis,thenweneedtosaythattheforcesacceleratedobserversexperiencearepseudoforcesakintothecentrifugalforce.Arethesetwoviewsequivalent?

Isspacereal?•  IfGRforcesusintonon-Euclideangeometry,doesthisrequirethatspaceis“real”?

–  Arela@onistwouldsaythatthisonlyimpliesthatthegeometricalrela@onsbetweenobjectsisnotwhatwethought.Nosubstan@vespace@meisrequired.

•  However,therearetwonewsubstan@valistarguments.–  InGRthegravita@onalinterac@onbetweenobjectsismediatedbythe

geometry.Thatis,geometryplaysthesameroleas,e.g.,theelectricfield.Anobjectdistortsthegeometryinitsvicinity.Thisdistor@onaffectsthemo@onofotherobjects,becausethegeodesicsaremodified.Thus,space7meplaysamoredirectroleinthedynamicsthaninNewton’sphysics.

–  Finitepropaga@onspeedsgivefieldsmoreofarealityinSR(andGR)thantheyhadbefore.Wealreadysawquitedrama@callyinelectromagne@sm,inwhichwavemo@onofthefieldswaspredictedandthenobserved.GRmakesasimilarpredic@onforthegravita@onalfield.IftheSunweretomovewewouldfeelthechangedgravityattheEarth500secondslater.AstheSunshakesbackandforth,sayastheplanetsorbit,GRsaysitwouldemitgravita@onalradia@on(waves)withmanyofthesameproper@esofEMwaves.

–  Gravita@onalwavesareasrealasanyotherobject.Theycarryenergyandmomentum.Theyhaverecentlybeenobserved.

[email protected]@valist•  Space-@meseemstohaveobservableproper@esinitself,likeelectromagne@c

fields,etc.•  Theseproper@esarefarfromresemblingthoseexpectedforNewton'sspace.•  Einstein'soriginalmo@va@onwastodevelopphysicsthatfollowedMach's

principle,butGRdoesnotfollowthatprinciple.(AndMachwasunabletofollowSR,muchlessGR.)ItmaybepossibletoaddMach'sprincipleasaseparaterequirement,i.e.toruleoutthoseGRspace@meswhichdonotobeyMach'sprinciple,butnothingaboutthestructureofGRitselftellsyoutodoso.

•  Newtonsaid(ineffect)thattwomasses@edtogetherandspunwouldstretchastringtaut,becausetheywouldneedaforcetokeepthembothaccelera@ngtowardthemiddle,[email protected]"absoluteaccelera@on".Machsaidthestringcouldnotgotautbecause"absoluteaccelera@on"ismeaningless-youneedtheotherstuffintheuniversetocreatetheforces.Einsteinabandonsthephrase"absoluteaccelera@on",[email protected]@oncaneitherbedescribedastwomassesrota@nginnearlyflatspace-@meorasastrangetwistyspace-@meexer@[email protected]@onally,NewtonandEinsteinagreeonwhatthepossibili@esare,andtheyincludethepossibili@esexcludedbyMach.

GRoutgrewitsphilosophicalancestry.

Space@meisbeginningtobecomemoresubstan@althanitwas(contrarytoEinstein’soriginalmo@va@on).Thegeometryofspace@mevariesfromplacetoplaceinawaythatisobservable.Geometryisempirical.KantwaswrongthatitwasonlypossibletoconceiveoftheworldinEuclideanterms.TheworldviolatesEuclid'saxioms.