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6 0 0 2 n a J 5 2 Can physics laws be derived from ] h monogenic functions? p - n e g José B. Almeida . s c Universidade do Minho, Physics Department i s y Braga, Portugal, email: [email protected] h p [ 1 v This is a paper about geometry and how one can derive several 4 fundamentallawsofphysicsfrom asimplepostulateofgeometrical 9 nature. The method uses monogenicfunctions analysed in thealge- 1 1 bra of 5-dimensionalspacetime, exploring the4-dimensionalwaves 0 thattheygenerate. Withthismethodoneisabletoarriveatequations 6 0 ofrelativisticdynamics,quantummechanicsandelectromagnetism. / s Fields as disparate as cosmology and particle physics will be influ- c encedbythisapproachinawaythatthepaperonlysuggests. Thepa- i s y per provides an introduction to a formalism which shows prospects h of one day leading to a theory of everything and suggests several p areas offuturedevelopment. : v i X r 1 Introduction a The editor’s invitation to write a chapter for this book about ether and the Uni- verseledmetothinkhowmyrecentworkhadanythingtodowithether,because the word was never used previously in my writings. It will become clear in the followingsections that the concept of a privilegedframe or absolutemotion un- derlies all the argument. When one accepts the existence of a preferred frame, 1 thequestionofattachingthatframetosomeobservablefeatureoftheUniverseis immediate. ThisquestionisaddressedinSec.8butwecananticipatethatgalaxy clusters are fixed and can be seen as the anchors for the preferred frame. This statementseemsinconsistentwiththeobservationthatclustersofgalaxiesmove relative to each other but it is resolved invoking an hyperspherical symmetry in theUniversethatisrevealed bythechoiceofappropriatecoordinates. The relationship between geometry and physics is probably stronger in the General Theory of Relativity (GTR) than in any other physics field. It is the author’s belief that a perfect theory will eventually be formulated, where geometry and physics become indistinguishable, so that the complete understanding of space properties, together with proper assignments between geometric and physical entities, will provide all necessary predictions, not only in relativistic dynamicsbut inphysicsas awhole. Wedon’thavesuchperfecttheoryyet,howevertheauthorintendstoshowthat GTRandQuantumMechanics(QM)canbeseenasoriginatingfrommonogenic functions in the algebra of the 5-dimensional spacetime G . These functions 4,1 can generate a null displacementcondition, thus reducing the dimensionalityby one to the number of dimensions we are all used to. Besides generating GTR and QM, the same space generates also 4-dimensional Euclidean space where dynamicscanbeformulatedandisquiteoftenequivalenttotherelativisticcoun- terpart; Euclidean relativistic dynamics resembles Fermat’s principle extended to4 dimensionsand is thusdesignatedas4-DimensionalOptics(4DO). Ourgoalis toshowhowtheimportantequationsofphysics,suchas relativity equations and equations of quantum mechanics, can be put under the umbrella of a common mathematical approach[1, 2]. We use geometric algebra as the frameworkbutintroducemonogenicfunctionswiththeirnullderivativesinorder to advance the concept. Furthermore, we clarify some previous work in this directionandidentifythestepstotakeinordertocompletethisambitiousproject. SinceA.Einsteinformulateddynamicsin4-dimensionalspacetime,thisspace isrecognizedbythevastmajorityofphysicistsas beingthebestforformulating thelaws ofphysics. However,mathematicalconsiderationslead to severalalter- native 4D spaces. For example, the Euclidean 4-dimensional space of 4DO is equivalent to the 4D spacetime of GTR when the metric is static, and therefore the geodesics of one space can be mapped one-to-one with those of the other. Then one can choose to work in the space that is more suitable. We build upon previousworkbyourselvesandbyotherauthorsaboutnullgeodesics,regarding theconditionthat allmaterial particlesmustfollownullgeodesics of5D space: Theimplicationofthisforparticlesisclear: theyshouldtravelon null 5D geodesics. This idea has recently been taken up in the literature, andhas a considerablefuture. It means thatwhat weperceive as massiveparticles in4D areakin tophotonsin5D.[3] 2 Accordingly, particles movingon nullpaths in 5D (dS2 =0) will appearasmassiveparticlesmovingontimelikepathsin4D(ds2>0) ...[4] Weactuallyimproveonthesenulldisplacementideasbyintroducingthemore fundamentalmonogeniccondition,derivingtheformerfromthelatterandestab- lishingacommonfirst principle. The only postulates in this paper are of a geometrical nature and can be sum- marized in the definition of the space we are going to work with; this is the 4- dimensional null subspace of the 5-dimensional space with signature ( +++ − +). Thechoiceofthisgeometricspacedoesnotimplyanyassumptionforphysi- calspaceuptothepointwheregeometricentitieslikecoordinatesandgeodesics start being assigned to physical quantities like distances and trajectories. Some of those assignments will be made very soon in the exposition and will be kept consistently until the end in order to allow the reader some assessment of the proposed geometric model as a tool for the prediction of physical phenomena. Mapping between geometry and physics is facilitated if one chooses to work always with non-dimensional quantities; this is done with a suitable choice for standards of the fundamental units. From this point onwards all problems of dimensional homogeneity are avoided through the use of normalizing factors listed below for all units, defined with recourse to the fundamental constants: h¯ Planck constant divided by 2p , G gravitational constant, c speed of → → → lightand e protoncharge. → Length Time Mass Charge Gh¯ Gh¯ h¯c e c3 c5 G r r r This normalization defines a system of non-dimensional units (Planck units) with important consequences, namely: 1) All the fundamental constants, h¯, G, c, e, become unity; 2) a particle’s Compton frequency, defined by n = mc2/h¯, becomesequaltotheparticle’smass;3)thefrequenttermGM/(c2r)issimplified toM/r. 5-dimensionalspacecanhaveamazingstructure,providingcountlessparallels tothephysicalworld;thispaperisjustalimitedintroductorylookatsuch struc- ture and parallels. The exposition makes full use of an extraordinary and little knownmathematicaltoolcalledgeometricalgebra(GA),a.k.a.Cliffordalgebra, which received an important thrust with the works of David Hestenes [5]. A good introduction to GA can be found in Gull et al. [6] and the following para- graphsusebasicallythenotationandconventionstherein. Acompletecourseon physical applications of GA can be downloaded from the internet [7]; the same authorspublishedamorecomprehensiveversioninbookform[8]. Anaccessible 3 presentationofmechanics inGA formalismis providedby Hestenes[9]. Thisis the subject of first section, where some essential GA concepts and notation are introduced. Sectiontwodealswithmonogenicfunctioninflat5Dspacetime,derivingspe- cialrelativityandthefreeparticleDiracequationfromthissimpleconcept. 4DO appears here as a perfect equivalent to special relativity, where trajectories can beunderstoodasnormalsto4-dimensionalplane-likewaves. Thefollowingsec- tion improves on this by allowing for curved space, introducing the notion of refractive index tensor. Section five examines the variational principle applied in both 4DO and GTR spaces to justify the equivalence of geodesics between the two spaces for static metrics. Refractive index is then related to its sources andthesourcestensorisdefined. Thecaseofacentralmassisexaminedandthe linkstoSchwarzschild’smetricarethoroughlydiscussed. Electromagnetismand electrodynamics are formulated as particular cases of refractiveindex in section seven and the sources tensor is here related to a current vector. The next section introduces the hypothesis of an hyperspherical symmetry in the Universe, which would call for the use of hyperspherical coordinates; the consequences for cosmology would include a complete dismissal of dark matter for a flat rate Hubble expansion. Before the conclusion, section nine shows how the monogenic condition is effective in generating an SU(4) symmetry group and makes someadvancestowardsarelation withthestandard modelofparticlephysics. 2 Introduction to geometric algebra Geometricalgebra is not usually taught in universitycourses and its presence in the literature is scarce; good reference works are [5, 7, 8]. We will concentrate onthealgebraof5-dimensionalspacetimebecausethiswillbeourmainworking space; this algebra incorporates as subalgebras thoseof the usual 3-dimensional Euclidean space, Euclidean 4-space and Minkowski spacetime. We begin with thesimpler5D flat spaceand progressto a5Dspacetimeofgeneral curvature. ThegeometricalgebraG ofthehyperbolic5-dimensionalspaceweconsider 4,1 isgenerated by thecoordinateframeoforthonormalbasisvectorss a suchthat (s )2 = 1, 0 − (s )2 =1, (2.1) i s a s b =0, a =b . · 6 Note that the English characters i, j, k range from 1 to 4 while the Greek characters a ,b ,g range from 0 to 4. See the Appendix A for the complete notation conventionused. Any two basis vectors can be multiplied, producing the new entity called a bivector. This bivector is the geometric product or, quite simply, the product; 4 this product is distributive. Similarly to the product of two basis vectors, the productofthreedifferentbasisvectorsproducesatrivectorandsoforthuptothe fivevector,becausefiveis thedimensionofspace. Wewillsimplifythenotationforbasisvectorproductsusingmultipleindices, i.e.s a s b s ab .Thealgebrais32-dimensionaland isspanned bythebasis ≡ • 1 scalar, 1, • 5 vectors, s a , • 10 bivectors(area), s ab , • 10 trivectors(volume), s abg , • 5 tetravectors(4-volume), is a , • 1 pseudoscalar(5-volume), i s . 01234 ≡ Severalelementsofthisbasissquareto unity: (s )2 =(s )2 =(s )2 =(is )2 =1. (2.2) i 0i 0ij 0 Itiseasytoverifytheequationsabove;supposewewanttocheck that(s )2 = 0ij 1. Start by expanding the square and remove the compact notation (s )2 = 0ij s s s s s s , then swap the last s twice to bring it next to its homonymous; 0 i j 0 i j j each swap changes the sign, so an even number of swaps preserves the sign: (s )2=s s (s )2s s . Fromthethirdequation(2.1)weknowthatthesquared 0ij 0 i j 0 i vector is unity and we get successively (s )2 = s s s s = (s )2(s )2 = 0ij 0 i 0 i 0 i (s )2; usingthefirst equation(2.1)weget finally(s )2 =1 a−s desired. 0 0ij − The remaining basis elements square to 1 as can be verified in a similar − manner: (s )2 =(s )2 =(s )2 =(is )2 =i2 = 1. (2.3) 0 ij ijk i − Note that the pseudoscalar i commutes with all the other basis elements while being a square root of 1; this makes it a very special element which can play − theroleofthescalarimaginaryincomplexalgebra. We can now address the geometric product of any two vectors a=aa s a and b=bb s b makinguseofthedistributiveproperty ab= a0b0+(cid:229) aibi + (cid:229) aa bb s ab ; (2.4) − i ! a =b 6 and we notice it can be decomposed into a symmetric part, a scalar called the inner or interior product, and an anti-symmetricpart, a bivector called the outer orexteriorproduct. ab=a b+a b, ba=a b a b. (2.5) · ∧ · − ∧ 5 Reversingthedefinitiononecan writeinnerand outerproductsas 1 1 a b= (ab+ba), a b= (ab ba). (2.6) · 2 ∧ 2 − The inner product is the same as the usual ”dot product,” the only difference being in the negative sign of the a b term; this is to be expected and is similar 0 0 to what one finds in special relativity. The outer product represents an oriented area; in Euclidean 3-space it can be linked to the "cross product" by the relation cross(a,b)= s a b; here weintroduced boldcharacters for3-dimensional 123 − ∧ vectorsandavoideddefining asymbolforthecrossproductbecausewewillnot useitagain. Wealsousedtheconventionthatinteriorandexteriorproductstake precedenceovergeometricproductinan expression. When a vector is operated with a multivector the inner product reduces the grade of each element by one unit and the outer product increases the grade by one. We will generalize the definition of inner and outer products below; underthisgeneralized definitiontheinnerproduct between avectorand ascalar produces a vector. Given a multivector a we refer to its grade-r part by writing <a> ;thescalarorgradezero part issimplydesignatedas <a>. By operating r avectorwithitselfweobtainascalar equalto thesquareofthevector’slength a2 =aa=a a+a a=a a. (2.7) · ∧ · Thedefinitionsofinnerand outerproductscanbeextendedtogeneral multivec- tors (cid:229) a b = <a>a <b>b a b , (2.8) · a ,b | − | (cid:229) (cid:10) (cid:11) a∧b = <a>a <b>b a +b . (2.9) a ,b (cid:10) (cid:11) Twootherusefulproductsarethescalarproduct,denotedas<ab>andcommu- tatorproduct,defined by a b=ab ba. (2.10) × − In mixed product expressions we will always use the convention that inner and outerproductstakeprecedenceovergeometricproducts,thusreducingthenum- berofparenthesis. We will encounter exponentials with multivector exponents; two particular cases of exponentiationare specially important. If u is such that u2 = 1 and q − 6 isascalar q 2 q 3 q 4 euq = 1+uq u + +... − 2! − 3! 4! q 2 q 4 = 1 + ... =cosq + − 2! 4! − { } q 3 +uq u +... =usinq (2.11) − 3! { } = cosq +usinq . Converselyifhis suchthath2 =1 q 2 q 3 q 4 ehq = 1+hq + +h + +... 2! 3! 4! q 2 q 4 = 1+ + +... =coshq + 2! 4! { } q 3 +hq +h +... =hsinhq (2.12) 3! { } = coshq +hsinhq . Theexponential of bivectorsis useful for defining rotations; a rotation of vector aby angleq onthes planeis performed by 12 a =es 21q /2aes 12q /2 =RãR; (2.13) ′ thetildedenotes reversionand reverses theorder of all products. As a check we makea=s 1 q q e s 12q /2s es 12q /2 = cos s sin s − 1 12 1 2 − 2 ∗ (cid:18) (cid:19) q q cos +s sin (2.14) 12 ∗ 2 2 (cid:18) (cid:19) = cosqs +sinqs . 1 2 Similarly,ifwehadmadea=s ,theresultwouldhavebeen sinqs +cosqs . 2 1 2 If we use B to represent a bivectorwhose plane is normal−to s and define its 0 normby B =(BB˜)1/2, ageneral rotationin4-spaceis represented bytherotor | | B B B R e B/2 =cos | | sin | | . (2.15) − ≡ 2 − B 2 (cid:18) (cid:19) | | (cid:18) (cid:19) Therotationangleis B andtherotationplaneisdefinedbyB.Arotorisdefined | | as a unitary even multivector (a multivector with even grade components only) which squares to unity; we are particularly interested in rotors with bivector 7 components. It ismoregeneral to define a rotationby aplane (bivector)then by anaxis(vector)becausethelatteronlyworksin3Dwhiletheformerisapplicable inanydimension. When theplaneofbivectorB containss , asimilaroperation 0 does not produce a simple rotation but produces a boost, eventually combined with a rotation. Take for instance B = s q /2 and define the transformation 01 operatorT =exp(B);atransformationofthebasis vectors produces 0 a = T˜s T =e s 01q /2s es 01q /2 ′ 0 − 0 q q = cosh s sinh s 01 0 2 − 2 ∗ (cid:18) (cid:19) q q cosh +s sinh (2.16) 01 ∗ 2 2 (cid:18) (cid:19) = coshqs +sinhqs . 0 1 In 5-dimensional spacetime of general curvature, we introduce 5 coordinate frame vectors ga , the indices follow the conventions set forth in Appendix A. We will also assume this spacetime to be a metric space whose metric tensor is givenby gab =ga gb ; (2.17) · thedoubleindexis used with g to denotetheinnerproduct offrame vectors and nottheirgeometricproduct. Thespacesignatureis( ++++),whichamounts − tosayingthatg <0andg >0. Areciprocalframeisdefinedbythecondition 00 ii ga gb =d a b . (2.18) · ab Defining g as the inverse of g , the matrix product of the two must be the ab identitymatrix;usingEinstein’ssummationconventionthisis gag ggb =d a b . (2.19) Usingthedefinition(2.17) wehave gag gg gb =d a b ; (2.20) · comparingwithEq.(2.18)we(cid:0)determ(cid:1)inega with a ag g =g gg . (2.21) Ifthecoordinateframevectorscanbeexpressedasalinearcombinationofthe orthonormedones, wehave ga =nb a s b , (2.22) b where n a is called the refractive index tensor or simply the refractive index; its 25 elements can vary from point to point as a function of the coordinates.[2, 8 10] When the refractive index is the identity, we have ga =s a for the main or directframeandg0 = s ,gi =s forthereciprocal frame, sothatEq.(2.18)is 0 i − verified. In this work we will not consider spaces of general curvature but only thosesatisfyingcondition(2.22). The first use we will make of the reciprocal frame is for the definition of two derivativeoperators. In flat space wedefinethevectorderivative (cid:209) =s a ¶ a . (2.23) It will be convenient, sometimes, to use vector derivatives in subspaces of 5D space; these will be denoted by an upper index before the (cid:209) and the particular indexuseddeterminesthesubspacetowhichthederivativeapplies;Forinstance m(cid:209) =s m¶ =s 1¶ +s 2¶ +s 3¶ . In 5-dimensional space it will be useful to m 1 2 3 splitthevectorderivativeintoitstimeand 4-dimensionalparts (cid:209) = s ¶ +s i¶ = s ¶ +i(cid:209) . (2.24) 0 t i 0 t − − The second derivative operator is the covariant derivative, sometimes called a theDiracoperator,and itisdefined inthereciprocal frameg D=ga ¶ a . (2.25) Taking into account the definition of the reciprocal frame (2.18), we see that the covariant derivative is also a vector. In cases such as those we consider in this work, where there is a refractive index, it will be possible to define both derivativesinthesamespace. We define also second order differential operators, designated Laplacian and covariantLaplacian respectively,resultingfrom theinnerproduct of onederiva- tive operator by itself. The square of a vector is always a scalar and the vector derivative is no exception, so the Laplacian is a scalar operator, which conse- quently acts separately in each component of a multivector. For 4+1 flat space itis ¶ 2 (cid:209) 2 = +i(cid:209) 2. (2.26) −¶ t2 Oneseesimmediatelythata4-dimensionalwaveequationisobtainedbyzeroing theLaplacian ofsomefunction ¶ 2 (cid:209) 2y = +i(cid:209) 2 y =0. (2.27) −¶ t2 (cid:18) (cid:19) Thisprocedurewillbeused in thenextsectionforthederivationofspecial rela- tivityand willbeextendedlatertogeneral curved spaces. 9 3 Monogenic functions and waves in flat space It turns out that there is a class of functions of great importance, called monogenic functions,[8] characterized by having null vector derivative; a function y ismonogenicifand onlyif (cid:209) y =0. (3.1) AmonogenicfunctionhasbynecessitynullLaplacian,ascanbeseenbydotting Eq.(3.1)with(cid:209) ontheleft. Wearethen allowedtowrite (cid:229) ¶ y =¶ y . (3.2) ii 00 i This can be recognized as a wave equation in the 4-dimensional space spanned bys whichwillaccept planewavetypesolutionsofthegeneral form i y =y ei(pa xa +d ), (3.3) 0 where y is an amplitude whose characteristics we shall not discuss for now, d 0 isaphaseangleand pa areconstantssuch that (cid:229) (p )2 (p )2 =0. (3.4) i 0 − i By setting the argument of y constant in Eq. (3.3) and differentiating we can getthedifferentialequation a pa dx =0. (3.5) The first member can equivalently be written as the inner product of the two vectors p dx= 0, where p =s a pa . In 5D hyperbolic space the inner product · of two vectors can be null when the vectors are perpendicular but also when the two vectors are null; since we have established that p is a null vector, Eq. (3.5) can be satisfied either by dx normal to p or by (dx)2 = 0. In the former case the condition describes a 3-volume called wavefront and in the latter case it describes the wave motion. Notice that the wavefronts are not surfaces but volumes,becauseweare workingwith4-dimensionalwaves. Theconditiondescribingwavemotioncan beexpandedas (dx0)2+(cid:229) (dxi)2 =0. (3.6) − This is a purely scalar equation and can be manipulated as such, which means we are allowed to rewrite it with any chosen terms in the second member; some ofthosemanipulationsare particularlysignificant. Supposewedecide toisolate (dx4)2 in the first member: (dx4)2 = (dx0)2 (cid:229) (dxm)2. We can then rename coordinate x4 as t , to get the interval squared−of special relativity for space-like displacements dt 2 =(dx0)2 (cid:229) (dxm)2. (3.7) − 10