arXiv:physics/0406026v1 [physics.gen-ph] 7 Jun 2004J.B.Almeida
UniversidadedoMinho,DepartamentodeF´isica,
4710-057Braga,Portugal.E-mail:bda@fisica.uminho.pt
Avariationalprincipleisappliedto4DEuclideanspaceprovidedwithatensor
refractiveindex,definingwhatcanbeseenas4-dimensionaloptics(4DO).Thege-ometryofsuchspaceisanalysed,makingnophysicalassumptionsofanykind.How-ever,byassigninggeometricentitiestophysicalquantitiesthepaperallowsphysicalpredictionstobemade.Amechanismisproposedfortranslationbetween4DOandGR,whichinvolvesthenullsubspaceof5Dspacewithsignature(−++++).
Atensorequationrelatingtherefractiveindextosourcesisestablishedgeometri-callyandthesourcestensorisshowntohavecloserelationshiptothestresstensorofGR.ThisequationissolvedforthespecialcaseofzerosourcesbutthesolutionthatisfoundisonlyapplicabletoNewtonmechanicsandisinadequateforsuchpredictionsaslightbendingandperiheliumadvance.Itisthenarguedthattestinggravityinthephysicalworldinvolvestheuseofatestchargewhichisitselfasource.Solvingthenewequation,withconsiderationofthetestparticle’sinertialmass,producesanexponentialrefractiveindexwheretheNewtonianpotentialappearsinexponentandprovidesaccuratepredictions.ResortingtohypersphericalcoordinatesitbecomespossibletoshowthattheUniverse’sexpansionhasapurelygeometricexplanationwithoutappealtodarkmatter.
1Introduction
Accordingtogeneralconsensusanyphysicstheoryisbasedonasetofprinciplesuponwhichpredictionsaremadeusingestablishedmathematicalderivations;thevalidityofsuchtheorydependsonagreementbetweenpredictionsandobservedphysicalreality.Inthatsensethispaperdoesnotformulateaphysicaltheorybecauseitdoesnotpresumeanyphysicalprinciples;forinstanceitdoesnotassumespeedoflightconstancyorequivalencebetweenframeaccelerationandgravity.Thisisapaperaboutgeometry.Allalongthepaper,inseveraloccasions,aparallelismadewiththephysicalworldbyassigningaphysicalmeaningtogeometricentitiesandthisallowspredictionstobemade.Howeverthevalidityofderivationsandoverallconsistencyoftheexpositionisindependentofpredictioncorrectness.
Theonlypostulatesinthispaperareofageometricalnatureandcanbecondensedinthedefinitionofthespacewearegoingtoworkwith:4-dimensionalspacewithEuclideansignature(++++).Forthesolepurposeofmakingtransitionstospacetimewewillalsoconsiderthenullsubspaceofthe5-dimensionalspacewithsignature(−++++).Thischoiceofspacedoesnotimplyanyassumptionaboutitsphysicalmeaninguptothepointwheregeometric
1
EuclideanformulationofgeneralrelativityJ.B.Almeida
Table1:Normalisingfactorsfornon-dimensionalunitsusedinthetext;→Planckconstant
dividedby2π,G→gravitationalconstant,c→speedoflightande→protoncharge.
LengthG
Massc
c5
e
entitieslikecoordinatesandgeodesicsstartbeingassignedtophysicalquantitieslikedistancesandtrajectories.Someofthoseassignmentswillbemadeveryearlyintheexpositionandwillbekeptconsistentlyuntiltheendinordertoallowthereadersomeassessmentoftheproposedgeometricmodelasatoolforthepredictionofphysicalphenomena.Mappingbetweengeometryandphysicsisfacilitatedifonechoosestoworkalwayswithnon-dimensionalquantities;thisiseasilydonewithasuitablechoiceforstandardsofthefundamentalunits.Inthisworkallproblemsofdimensionalhomogeneityareavoidedthroughtheuseofnormalisingfactorsforallunits,listedinTable1,definedwithrecoursetothefundamentalconstants:Planckconstant,gravitationalconstant,speedoflightandprotoncharge.Thisnormalisationdefinesasystemofnon-dimensionalunitswithimportantconsequences,namely:1)allthefundamentalconstants,,G,c,e,becomeunity;2)aparticle’sComptonfrequency,definedbyν=mc2/,becomesequaltotheparticle’smass;3)thefrequenttermGM/(c2r)issimplifiedtoM/r.
Theparticularspacewechosetoworkwithcanhaveamazingstructure,providingcountlessparallelstothephysicalworld;thispaperisjustalimitedintroductorylookatsuchstructureandparallels.Theexpositionmakesfulluseofanextraordinaryandlittleknownmathematicaltoolcalledgeometricalgebra(GA),a.k.a.Cliffordalgebra,whichreceivedanimportantthrustwiththeintroductionofgeometriccalculusbyDavidHestenes[1].AgoodintroductiontoGAcanbefoundinGulletal.[2]andthefollowingparagraphsusebasicallythenotationandconventionstherein.AcompletecourseonphysicalapplicationsofGAcanbedownloadedfromtheinternet[3]withamorecomprehensiveversionpublishedrecentlyinbookform[4]whileanaccessiblepresentationofmechanicsinGAformalismisprovidedbyHestenes[5].
2Introductiontogeometricalgebra
WewilluseGreekcharactersfortheindicesthatspan1to4andLatincharactersforthosethatexcludethe4value;inrarecaseswewillhavetouseindicesspanning0to3andthesewillbedenotedwithGreekcharacterswithanoverbar.Thegeometricalgebraofthehyperbolic5-dimensionalspacewewanttoconsiderG4,1isgeneratedbytheframeoforthonormalvectors{i,σµ},µ=1...4,verifyingtherelations
i2=−1,
iσµ+σµi=0,
σµσν+σνσµ=2δµν.
(1)
Wewillsimplifythenotationforbasisvectorproductsusingmultipleindices,i.e.σµσν≡σµν.Thealgebrais32-dimensionalandisspannedbythebasis
1,{i,σµ},{iσµ,σµν},{iσµν,σµνλ},{iI,σµI},I;
1scalar5vectors10bivectors10trivectors5tetravectors1pentavector
(2)
whereI≡iσ1σ2σ3σ4isalsocalledthepseudoscalarunit.Severalelementsofthisbasissquaretounity:
(σµ)2=1,(iσµ)2=1,(iσµν)2=1,(iI)2=1;(3)
2
Euclideanformulationofgeneralrelativityandtheremainingsquareto−1:
i2=−1,
(σµν)2=−1,
(σµνλ)2=−1,
(σµI)2,
I2=−1.
J.B.Almeida
(4)
Notethatthesymboliisusedheretorepresentavectorwithnorm−1andmustnotbeconfusedwiththescalarimaginary,whichwedon’tusuallyneed.
Thegeometricproductofanytwovectorsa=a0i+aµσµandb=b0i+bνσνcanbedecomposedintoasymmetricpart,ascalarcalledtheinnerproduct,andananti-symmetricpart,abivectorcalledtheexteriorproduct.
ab=a·b+a∧b,
ba=a·b−a∧b.
(5)
Reversingthedefinitiononecanwriteinternalandexteriorproductsas
a·b=
1
2
(ab−ba).
(6)
Whenavectorisoperatedwithamultivectortheinnerproductreducesthegradeofeachelementbyoneunitandtheouterproductincreasesthegradebyone.Therearetwoexceptions;whenoperatedwithascalartheinnerproductdoesnotproducegrade−1butgrade1instead,andtheouterproductwithapseudoscalarisdisallowed.
3Displacementandvelocity
Anydisplacementinthe5-dimensionalhyperbolicspacecanbedefinedbythedisplacementvector
ds=idx0+σµdxµ;(7)andthenullspaceconditionimpliesthatdshaszerolength
ds2=ds·ds=0;
whichiseasilyseenequivalenttoeitheroftherelations
02µ24202(dx)=(dx);(dx)=(dx)−(dxj)2.
(8)
(9)
Theseequationsdefinethemetricsoftwoalternativespaces,oneEuclideantheotherone
Minkowskian,bothequivalenttothenull5-dimensionalsubspace.
ApathonnullspacedoesnothaveanyaffineparameterbutwecanuseEqs.(9)toexpress4coordinatesintermsofthefifthone.Wewillfrequentlyusetheletterttorefertocoordinatex0andtheletterτforcoordinatex4;totalderivativeswithrespecttotwillbedenotedbyan
ˇ.Dividingoverdotwhiletotalderivativeswithrespecttoτwillbedenotedbya”check”,asinF
bothmembersofEq.(7)bydtweget
s˙=i+σµx˙µ=i+v.
(10)
Thisisthedefinitionforthevelocityvectorv;itisimportanttostressagainthatthevelocity
vectordefinedhereisageometricalentitywhichbearsforthemomentnorelationtophysicalvelocity,beitrelativisticornot.Thevelocityhasunitnormbecauses˙2=0;evaluationofv·vyieldstherelation
v·v=(x˙µ)2=1.(11)
3
EuclideanformulationofgeneralrelativityJ.B.Almeida
Thevelocityvectorcanbeobtainedbyasuitablerotationofanyoftheσµframevectors,inparticularitcanalwaysbeexpressedasarotationoftheσ4vector.
Atthispointwearegoingtomakeasmalldetourforthefirstparallelwithphysics.Inthepreviousequationwereplacex0bythegreekletterτandrewritewithτ˙2inthefirstmember
2τ˙=1−(x˙j)2.(12)Therelationaboveiswellknowninspecialrelativity,seeforinstanceMartin[6];seealsoAlmeida
[7],Montanus[8]forparallelsbetweenspecialrelativityanditsEuclideanspacecounterpart.1WenotethattheoperationperformedbetweenEqs.(11)and(12)isaperfectlylegitimatealgebraicoperationsincealltheelementsinvolvedarepurenumbers.ObviouslywecouldalsodividebothmembersofEq.(7)bydτ,whichisthenassociatedwithrelativisticpropertime;
sˇ=iˇx0+σjxˇj+σ4.
(13)
j2
Squaringthesecondmemberandnotingthatitmustbenullweobtain(ˇx0)2−(ˇx)=1.This
0j
meansthatwecanrelatethevectoriˇx+σjxˇtorelativistic4-velocity,althoughthenormofthisvectorissymmetrictowhatisusualinSR.Therelativistic4-velocityismoreconvenientlyassignedtothe5Dbivectoriσ4xˇ0+σj4xˇj,whichhasthenecessaryproperties.Themethodwehaveusedtomakethetransitionbetween4DEuclideanspaceandhyperbolicspacetimeinvolvedthetransformationofa5Dvectorintoscalarplusbivectorthroughproductwithσ4;thismethodwilllaterbeextendedtocurvedspaces.
Equation(10)appliestoflatspacebutcanbegeneralisedforcurvedspace;wedothisintwosteps.Firstofallwecanincludeascalefactor(v=nσµx˙µ),whichcanchangefrompointtopoint
s˙=i+nσµx˙µ.(14)Inthiswayweareintroducingthe4-dimensionalanalogueofarefractiveindex,thatcanbeseen
asageneralisationofthe3-dimensionaldefinitionofrefractiveindexforanopticalmedium:thequotientbetweenthespeedoflightinvacuumandthespeedoflightinthatmedium.Thescalefactornusedhererelatesthenormofvectorσµx˙µtounityandsoitdeservesthedesignationof4-dimensionalrefractiveindex;wewilldropthe”4-dimensional”qualificationbecausetheconfusionwiththe3-dimensionalcasecanalwaysberesolvedeasily.Thematerialpresentedinthispaperis,inmanyrespects,alogicalgeneralisationofopticsto4-dimensionalspace;so,evenifthepaperisonlyaboutgeometry,itbecomesnaturaltodesignatethisstudyas4-dimensionaloptics(4DO).
FullgeneralisationofEq.(10)impliestheconsiderationofatensorrefractiveindex,similartothenon-isotropicrefractiveindexofopticalmedia
s˙=i+nµνx˙νσµ;
(15)
thevelocityisthengenerallydefinedbyv=nµνx˙νσµ.Thesameexpressioncanbeusedwith
anyorthonormalframe,includingforinstancesphericalcoordinates,butforthemomentwewillrestrictourattentiontothosecaseswheretheframedoesnotrotateinadisplacement;thisposesnorestrictionontheproblemstobeaddressedbutisobviouslyinconvenientwhensymmetriesareinvolved.Equation(15)canbewrittenwiththevelocityintheformv=gνx˙νifwedefinetherefractiveindexvectors
gν=nµνσµ.(16)
EuclideanformulationofgeneralrelativityJ.B.Almeida
Thesetoffourgµvectorswillbedesignatedtherefractiveindexframe.ObviouslythevelocityisstillaunitaryvectorandwecanexpressthisfactevaluatingtheinternalproductwithitselfandnotingthatthesecondmemberinEq.(15)haszeronorm.
v·v=nαµx˙µnβνx˙νδαβ=1.
(17)
UsingEq.(16)wecanrewritetheequationaboveasgµ·gνx˙µx˙ν=1anddenotingbygµνthe
scalargµ·gνtheequationbecomes
gµνx˙µx˙ν=1.(18)ThegeneralisedformofthedisplacementvectorarisesfrommultiplyingEq.(15)bydt,using
thedefinition(16)
ds=idt+gµdxµ.(19)Thiscanbeputintheformofaspacemetricbydottingwithitselfandnotingthatthefirst
membervanishes
(dt)2=gµνdxµdxν.(20)Noticethatthecoordinatesarestillreferredtothefixedframevectorsσµandnottotherefractive
indexvectorsgµ.InGRthereisnosuchdistinctionbetweentwoframesbutMontanus[8]clearlyseparatestheframefromtensorgµν.
Wearegoingtoneedthereciprocalframe[4]{−i,gµ}suchthat
gµ·gν=δµν.
(21)
Fromthedefinitionitbecomesobviousthatgµgν=gµ·gν+gµ∧gνisapurebivectorandsogµgν=−gνgµ.WenowmultiplyEq.(19)ontherightandontheleftbyg4,simultaneouslyreplacingx4byτtoobtain
dsg4=ig4dt+gjg4dxj+dτ;
g4ds=g4idt+g4gjdxj+dτ.
(22)
Whentheinternalproductisperformedbetweenthetwoequationsmembertomemberthefirstmembervanishesandthesecondmemberproducestheresult
(dτ)2=g44(dt)2−gjkdxjdxk.(23)Ifthevariousgµarefunctionsonlyofxjtheequationisequivalenttoametricdefinitionin
generalrelativity.Wewillexaminethespecialcasewhengµ=nµσµ;replacinginEq.(23)
(dτ)2=
1
n4
dxj
2
.
(24)
Thisequationcoversalargenumberofsituationsingeneralrelativity,includingtheveryimpor-tantSchwarzschild’smetric,aswasshowninAlmeida[9]andwillbediscussedbelow.NoticethatEq.(20)hasmoreinformationthanEq.(23)becausethestructureofg4iskeptintheformer,throughthecoefficientsgµ4,butismostlylostintheg44coefficientofthelatter.
4Thesourcesofspacecurvature
Equations(20)and(23)definetwoalternative4-dimensionalspaces;intheformer,4DO,tisanaffineparameterwhileinthelatter,GR,itisτthattakessuchrole.Thegeodesicsofonespacecanbemappedonetoonewiththoseoftheotherandwecanchoosetoworkonthespacethatbestsuitsus.
5
Euclideanformulationofgeneralrelativity
Thegeodesicsof4DOspacecanbefoundbyconsiderationoftheLagrangian
L=
gµνx˙µx˙ν
2.
J.B.Almeida
(25)
ThejustificationforthischoiceofLagrangiancanbefoundinseveralreferencebooksbutsee
forinstanceMartin[6].FromtheLagrangianonedefinesimmediatelytheconjugatemomenta
vµ=
∂L
Euclideanformulationofgeneralrelativity
VectorTiscalledthesourcesvectorandcanbeexpandedintosixteentermsas
T=Tµνσµx˙ν=(2nµν)σµx˙ν.
J.B.Almeida
(32)
ThetensorTµνcontainsthecoefficientsofthesourcesvectorandwecallitthesourcestensor;
itisverysimilartothestresstensorofGR,althoughitsrelationtogeometryisdifferent.Thesourcestensorinfluencestheshapeofgeodesicsbutweshallnotexamineherehowsuchinfluencearises,exceptforveryspecialcases.
BeforewebeginsearchingsolutionsforEq.(31)wewillshowthatthisequationcanbede-composedintoasetofequationssimilartoMaxwell’s.Considerfirstthevelocityderivativev=·v+∧v;theresultisamultivectorwithscalarandbivectorpartG=v.NowderiveagainG=·G+∧G;weknowthattheexteriorderivativeofGvanishesandthedivergenceequalsthesourcesvector.Maxwell’sequationscanbewritteninasimilarform,aswasshowninAlmeida[10],withvelocityreplacedbythevectorpotentialandmultivectorGreplacedbytheFaradaybivectorF;DoranandLasenby[4]offersimilarformulationforspacetime.
Anisotropicspacemustbecharacterisedbyorthogonalrefractiveindexvectorsgµwhosenormcanchangewithcoordinatesbutisthesameforallvectors.Weusuallyrelaxthisconditionbyacceptingthatthethreegjmusthaveequalnormbutg4canbedifferent.Thereasonforthisrelaxedisotropyisfoundintheparallelusuallymadewithphysicsbyassigningdimensions1to3tophysicalspace.Isotropyinaphysicalsenseneedonlybeconcernedwiththesedimensionsandignoreswhathappenswithdimension4.Wewillthereforecharacteriseanisotropicspacebytherefractiveindexframegj=nrσj,g4=n4σ4.Indeedwecouldalsoacceptanon-orthogonalg4withintherelaxedisotropyconceptbutwewillnotdosointhiswork.
Wewillonlyinvestigatesphericallysymmetricsolutionsindependentofx4;thismeansthattherefractiveindexcanbeexpressedasfunctionsofrinsphericalcoordinates.Thevectorderivativeinsphericalcoordinatesisofcourse
=
1
rσθ∂θ+
1
n4
σ4∂4.
(33)
TheLaplacianistheinnerproductofwithitselfbuttheframederivativesmustbeconsidered
∂rσr=0,∂rσθ=0,∂rσϕ=0,
∂θσr=σθ,∂θσθ=−σr,∂θσϕ=0,
∂ϕσr=sinθσϕ,
∂ϕσθ=cosθσϕ,
∂ϕσϕ=−sinθσr−cosθσθ.cotθ
r2
1
(34)
AfterevaluationtheLaplacianbecomes
2=
1
r∂r−
n′r
r2
∂θθ+
∂ϕϕ
+
r
−
2
(n′r)
r
d+
a
EuclideanformulationofgeneralrelativityJ.B.Almeida
Whenappliedton4andequatedtozeroweobtainsolutionswhichimposen4=nrandsothespacemustbetrulyisotropicandnotrelaxedisotropicaswehadallowed.ThesolutionwehavefoundfortherefractiveindexcomponentsinisotropicspacecancorrectlymodelNewtondynamics,whichledtheauthortoadheretoitforsometime[11].HoweverifinsertedintoEq.(24)thissolutionproducesaGRmetricwhichisverifiablyindisagreementwithobservations;consequentlyithaspurelygeometricsignificance.
Theinadequacyoftheisotropicsolutionfoundaboveforrelativisticpredictionsdeservessomethought,sothatwecansearchforsolutionsguidedbytheresultsthatareexpectedtohavephysicalsignificance.Inthephysicalworldweareneverinasituationofzerosourcesbecausetheshapeofspaceortheexistenceofarefractiveindexmustalwaysbetestedwithatestparticle.Atestparticleisanabstractioncorrespondingtoapointmassconsideredsosmallastohavenoinfluenceontheshapeofspace.Butinrealityatestparticleisalwaysasourceofrefractiveindexanditsinfluenceontheshapeofspacemaynotbenegligibleinanycircumstances.Ifthisisthecasethesolutionsforvanishingsourcesvectormayhaveonlygeometricmeaning,withnoconnectiontophysicalreality.
ThequestionisthenhowdoweincludethetestparticleinEq.(31)inordertofindphysicallymeaningfulsolutions.Herewewillmakeoneaddhocproposalwithoutfurtherjustificationbe-causetheauthorhasnotyetcompletedtheworkthatwillprovidesuchjustificationingeometricterms.ThesecondmemberofEq.(31)willnotbezeroandwewillimposethesourcesvector
J=−∇2n4σ4.
Equation(31)becomes
2v=−∇2n4σ4;
(38)(39)
asaresulttheequationfornrremainsunchangedbuttheequationforn4becomes
n4+
′′
2n′4
nr
=−n4+
′′
2n′4
nr.Thiscannowbeen-teredintoEq.(24)andthecoefficientscanbeexpandedinseriesandcomparedtoSchwarzschild’s
forthedeterminationofparametera.Thefinalsolution,forastationarymassMis
nr=e2M/r,
n4=eM/r.
(41)
Equation(39)canbeinterpretedinphysicaltermsascontainingtheessenceofgravitation.Whensolvedforsphericallysymmetricsolutions,aswehavedone,thefirstmemberprovidesthedefinitionofastationarygravitationalmassasthefactorMappearingintheexponentandthesecondmemberdefinesinertialmassas∇2n4.Gravitationalmassisdefinedwithrecoursetosomeparticlewhichundergoesitsinfluenceandisanimatedwithvelocityvandinertialmasscannotbedefinedwithoutsomefieldn4actinguponit.Completeinvestigationofthesourcestensorelementsandtheirrelationtophysicalquantitiesisnotyetdone.Itisbelievedthatthe16termsofthistensorhavestronglinkswithhomologouselementsofstresstensorinGRbutthiswillhavetobeverified.
Finallyweturnourattentiontohypersphericalcoordinates.Thepositionvectorisquitesimplyx=τστ,wherethecoordinateisthedistancetothehyperspherecentre.Differentiatingthepositionvectorweobtainthedisplacementvector,whichisanaturalgeneralisationof3Dsphericalcoordinatescase
dx=στdτ+τσρdρ+τsinρσθdθ+τsinρsinθσϕdϕ;
(42)
8
EuclideanformulationofgeneralrelativityJ.B.Almeida
ρ,θandϕareangles.Thevelocityinanisotropicmediumshouldnowbewrittenas
˙+sinρsinθσϕϕ˙).v=n4σττ˙+nrτ(σρρ˙+sinρσθθ
(43)
Inordertoreplacetheangularcoordinateρwithadistancecoordinaterwecanmaker=τρ
andderivewithrespecttotime
r˙=ρτ˙+τρ˙=
r
τ
˙+rsinθσϕϕ˙).˙+nr(σrr˙+rσθθσrτ
(45)
wehavealsoreplacedσρbyσrforconsistencywiththenewcoordinates.
Wehavejustdefinedaparticularlyimportantsetofcoordinates,whichappearstobeespeciallywelladaptedtodescribethephysicalUniverse,withτbeinginterpretedastheUniverse’sageoritsradius;notethattimeanddistancecannotbedistinguishedinnon-dimensionalunits.Whenrτ/τ˙issmallinEq.(45),therefractiveindexvectorsbecomeorthogonalandweusen4andnrinconjunctionwithEq.(24)toobtainaGRmetricwhosecoefficientsareequivalentsoSchwarzschild’sonthefirsttermsoftheirseriesexpansions.Whenrτ˙/τcannotbeneglected,however,theequationcanexplaintheUniverse’sexpansionandflatrotationcurvesingalaxieswithoutdarkmatterintervention.AmorecompletediscussionofthissubjectcanbefoundinRef.[9].
5Conclusions
EuclideanandMinkowskian4-spacescanbeformallylinkedthroughthenullsubspaceof5-dimensionalspacewithsignature(−++++).Theextensionofsuchformalismtonon-flatspacesallowsthetransitionbetweenspaceswithbothsignaturesandthepaperdiscussessomeconditionsformetricandgeodesictranslation.Foritssimilaritieswithoptics,thegeometryof4-spaceswithEuclideansignatureiscalled4-dimensionaloptics(4DO).Usingonlygeometricargumentsitispossibletodefinesuchconceptsasvelocityandtrajectoryin4DOwhichbecomephysicalconceptswhenproperandnaturalassignmentsaremade.
Oneimportantpointwhichisaddressedforthefirsttimeintheauthor’sworkisthelinkbetweentheshapeofspaceandthesourcesofcurvature.Thisisdoneongeometricalgroundsbutitisalsoplacedinthecontextofphysics.TheequationpertainingtothetestofgravitybyatestparticleisproposedandsolvedforthesphericallysymmetriccaseprovidingasolutionequivalenttoSchwarzschild’sasfirstapproximation.SomementionismadeofhypersphericalcoordinatesandthereaderisreferredtopreviousworklinkingthisgeometrytotheUniverse’sexpansionintheabsenceofdarkmatter.
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Thegeometricalgebraofspacetime,Found.Phys.23,1175,1993,URLhttp://www.mrao.cam.ac.uk/~clifford/publications/abstracts/imag_numbs.html.
9
EuclideanformulationofgeneralrelativityJ.B.Almeida
hand-URL
[3]A.LasenbyandC.Doran,Physicalapplicationsofgeometricalgebra,
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