Onthehistoryofgeometrizationofspace-time:FromMinkowskitoFinslergeometry. (100yearsafterMinkowskisCologneadress.)HubertGoennerInstituteforTheoretical PhysicsUniversity ofGottingenGermany1 Introduction: on the geometrization of physicsThistributetoHermannMinkowski will consistofthreeparts: abriefhis-torical introductionconcerninggeometrizationof physics, amiddle cater-ingtomathematical themes, andanal chapter dealingwitha(specula-tive)endeavouratapplyingFinslergeometrytophysicslooslyconnectedtoMinkowski.From the historyof physicsweknowthat, at rst,physicalsystemsweredescribedinagivenspaceandbyagiventimewhichbothwereregardedasindependent of matter or any physical inuence- not just by the philosopherKant. Athistime, theideaofgeometrizingspacewouldhavebeenabsurd.JohannesKeplerwhoforsomeperiod in hislife had relatedphysicalbodies,the planets, to geometric objects, i.e., to the ve regular polyhedra, certainlywas far from what we now understand by geometrization of physics,i. e. theembeddingof physicalobjects(matter, elds) intoa geometrical framework.A weakening of the rigid understanding of space seems to have occured whenthenotionof non-euclideangeometrycameup, inthe19thcentury(C. F.Gauss, N. I. Loba cevski, J. Bolyai). Theanswer tothequestionof whatkind of geometrythe space welivein exhibits,now could be delegatedto anempirical test [1]. Whether the anecdote about Gauss withhis geodesicmeasurement of the angles of atriangle formedbythree hills is true ornot, inanycasetheastronomerK. Schwarzschildinvestigatedthequestionscienticallywithbodies far awayintheheavens (1900) [2]. Alsointhe19th century, the mechanics of rigid bodies became reformulated within non-euclidean geometry (F. Klein, W. A. Cliord, R. S. Heath) [3]). Yet, with theexceptionofCliord, therestill wasnoquestionaboutspaceortimebeinginuencedbymaterialsystems.Asiswell known, thejoinderof spaceandtimetospace-timebyHer-mann Minkowskiwhosefamous speechabout theunion ofspaceand time[4] wascommemoratedinSeptember2008, becamearststepinthisstillongoingprocess of geometrization. As holds for manyinnovations insci-ence, the ideaof space-time didnot appear like ashootingstar. Afewmathematicians, ctionwriters, andphilosopherspresenteditquiteclearlybeforeMinkowski, butnotasamathematical theory. Inthe18thcenturyDAlembert mentioned time as a fourth dimension in 1754 in his encyclopedia(writtenwithDiderot); thenagainthiswasdonebyLagrangein17971[5].For the 19th century, let me rst mention Charles Howard Hintons article of1880: What is the fourth dimension?[6] whichconsidereda fourth spatialdimension. Inhisreplytoitinthejournal Natureof 1885, ananonymousletter writer signingS., introduced time as the fourth dimensionand dealtwith a 4-dimensionaltime-space. S. mastered(verbally)what wenow callthespace-timepicture, andevenmanagedtocorrectlydescribethehyper-cube by looking at the motion of a cube in time-space [7]. In another article,Hintontriedtogeometrizeelectrical chargeandcurrents[8]. Betterknownis H. G. Wells novel Time Machine of 1894 in which again a 4-dimensionaljunction of time and space called Space is considered [9]. There, it is madeclear that time is not considered as a fourth spacelike dimension. As the Ger-man translation of Wells book came out in 1904 [10], Minkowskicould havereadit, inprinciple. Finally, aphilosopherof Hungarianorigin, Menyhert(Melchior)Palagyi,whohad becomeaprofessorin Darmstadt publishedhisNewtheoryof space andtimein1901[11]. He joinedspace andtimerathervaguelytoa4-dimensional entitynamedowingspace(ieenderRaum),drewaMinkowski-diagramandintroducedatimeanglebetweentheworldline of amovingparticleandthetimeaxis. Heabstainedfromgivingamathematicalschemeexceptforpointingout thatthecoordinatesofapointinowingspacecouldberepresentedbyx + it, y + it, z + it([11], p. 32).2Afterhehadbecomeawareof special relativityand, then,of Minkowskis famousspeech, Palagyi claimedpriorityforthespace-timepicturebutrejectedMinkowskisspace-timemanifold. Fromhiswritingsit1Ainsi on peut regarder la mecanique comme une geometrie `a quatre dimensions [..].2This isacuriouscombinationof timewithoneabsoluteandtworelativespacelikecoordinates. Let x= x+i t , y= y+i t , z= z+i t ; then yx= yx ; zx= zxaretherelativecoordinates. x x=itcouldatbestdescribepartofthelightcone, aconceptPalagyididnothave. ForthephilosophyofPalagyi cf. [15].isobviousthathisthoughtsremainwithinpsychology, andthathewasin-competent both in mathematics and physics. [12]. In sharp contrast, around1905, and before Minkowski,Poincare also had a 4-dimensional (space-time)formalismforthewaveequationandelectrodynamics [13].3Possibly, duetohisepistemological positionasaconventionalist, hemightnothavebeeninterestedatallintheissueofgeometrization.2 MinkowskianspacesDid Minkowski geometrize electrodynamics by formulating it on a space-timemanifold? Notinthesenseof havingfoundageometryinwhichtheelec-tromagnetic eld corresponded to a geometric object. This would come onlylater-afterthegeometrizationofthegravitationaleld-intheframeworkof unied eld theory. Right after Minkowski, M. Planck (1906) [14], G. Her-glotz(1910),F.Kottler(1912),G.N.Lewis&R.C.Tolman(1909)amongothersputmechanicsandelectrodynamicsintoaspace-timepicture: theyrelativized suchtopics.TherstgeometrizationinthenarrowermeaningwasachievedbyEin-steinandGromann(1913-15)[16]. Atrst, Einsteinhadwantedtokeepthe constancy of the velocity of light only for areas of almost constant grav-itationalpotential([17], p. 713)Thus, inhisattempttowardarelativistictheory of gravitation, he assumed the velocity of light to be a function of the(Newtonian) gravitational potential . He replaced the space-time metric ofMinkowskibyds2= c()2dt2dxdx(, = 1, 2, 3) , (1)Inanimportantnextstep, withGromannshelp, heintroduceda(semi-)Riemannianmetricandidentieditscomponentswiththenowmorenu-merousgravitationalpotentials.43InGermany, atthetime, Poincarespapersseemtohavebeenneglected. Manywellknownscientiststhen(e.g., M. Planck, inhispaperonrelativisticmechanics)andevenlater historians of science do not refer to Poincares short paper of 1905 - before Einsteins-,butonlytoPoincares extendedpresentationof1906.4Forthemost detailedandexperthistory oftheformation ofgeneral relativity cf. the4volumesof[18],[19].3 Minkowski space-time and Minkowski SpaceInthis part, we rst distinguishbetweenthe physicists andthe mathe-maticians use of the expressionMinkowski space andpresent some ofMinkowskisresultsconcerningthegeometryofnormedspaces.3.1 Minkowski space-timeTheintroductionof animaginarytime-coordinateT =i ct byMinkowskiintothelineelementofspace-timeds2= dT2+ dxdx(, = 1, 2, 3) (2)turnedouttobealittlemisleadingforphysicsandmoresoforthegeneralpublic. On the surface, (2) looked as if physics now played in a 4-dimensionaleuclidean space - with four spatial dimensions corresponding exactly to whatRiemann had had in mind. The 19th century had been full of talk and papersabout4-dimensional spacewithitsstrikingpossibilitytoenteralocked(3-dimensional)roomwithoutbreakingaseal [27]5. Itwasquicklyrealized,though, that a real representation of the metric suited physicsbetter, i.e., byaLorentz-metricwithsignature 2(nullcone,Cauchyproblemetc.):ds2= ijdxidxj= c2dt2dxdx(i, j= 0, 1, 2, 3) . (3)Space-time, asMinkowski hadintroducedit, becametheframeworkforallphysical theories in which velocities comparable to the velocity of light couldoccur: itisanaturalrepresentationspaceoftheLorentz(Poincare-)group.Alsoincurvedspace-timeitplaysarole: asthetangentspaceatanypointof the manifold ofevents. Thisis all too wellknownsuchthat nothingmoreneedstobesaid.3.2 Minkowski SpaceThesecondmeaningofthetermMinkowskiSpaceinthewaymathemati-ciansuseit, isbarelyknowntophysicists. Onthisoccasionof commemo-ratingMinkowski inatributetohimitismandatorytoincludesomeofhismathematicalachievements.5Forahistoryof4-dimensionalspaceanditsmodernusescf. [28]Minkowski Space is a real, nite-dimensional (d 2) normed(vector)spaceV (= Rd)([29],p. 138).If completeness is added, then it is just a special case of a Banach space. ThenormX ofanelementX V satisesthefollowingconditions:i)a)X 0; b)X = 0ifandonlyifX= 0;ii)X = ||X; R, X V ;iii)X + Y X+Y .If 1) b) does not hold, a semi(pseudo)-norm of the kind needed in space-timeobtains.Theunitball B MdwithB:= {X V | X 1}isa(compact)convexandsymmetric set6Withthehelpofthenorm,a(canonical)metric(semi-metric)(X, Y ) := X Y(4)canalwaysbeintroduced. Theunitball maybeverydierentof whatweimagineinaneuclideansituation. Infact, ifandonlyiftheunitball isanellipsoid then MinkowskiSpace turns out to be euclideanspace ([20], p. 38).Thetopologyof anyd-dimensional Minkowski Spaceiseuclideantopology([21], section1.2). WhenMinkowski Spaceisseenasametrical spacewespeakofMinkowskigeometry.LetmegiveanexampleforMinkowski geometry: LetKbeacompact,convex set in euclidean space, x = ytwo points of Kand = points on theboundary Kof Kmetby straightlinesjoining x and ywith the zero-pointO = x; O = y. ThenadistancefunctionFonKisdenedbyF(x y) :=x y , F(0) = 0. (5)Iftheeuclideannormisused,thenF(x y) =(x y)2( )2. (6)Thedistancefunctionisaconvexfunction7withF(x) 1.6Minkowski calleditEichkorper. AsetWiscalledsymmetric(withregardtothezeropointO)if W= W. ThismeansthatallstraightlinesegmentspassingthroughOofthesetarehalvedbyO.7A convex function satises the same conditions like a norm, i.e., the triangle inequalityF(x + y) F(x) + F(y),etc.Thestudyof normsother thantheeuclidean(andnotderivedfromametric)isprimarilyduetoMinkowski. Becauseofa1-1correspondencebe-tweennormsonthelinearspaceV andsymmetric, closedconvexsetsinVwithnon-emptyinterior, convexitybecomesanessential ingredientforthestudyof Minkowski Space8[21]. Inthiscontext, Minkowski introducedavector sumof twoconvexsets (bodies) nowcalledMinkowski sum. Itscombinationwiththeconceptof volumeledhimtoimportantresults. Werealizethat, withonlyanormavailable, volumeandorthogonalityarenot immediatelyat hand. As to volume,we assumethat Vis equippedwithanauxiliaryeuclideanstructureandthatthevolumeistheLebesquemea-sureinducedbythisstructure.9Inconnectionwithhisresearchinnumbertheory,Minkowski usedtheconceptsvolumeandareaofconvexbodies(cf. hisgeometryofnumbers, [22]and[23]). Tomakeprogress, heintro-duced fundamental quantities as, e.g., the supportfunction of convex bodies.Another one is mixed volume generalizing the euclidean volume: it comprisesandconnectstheconceptsofvolume,area,andtotalmeancurvature.10In this context, one of the well-knownresults of Minkowskiamong math-ematicians perhaps is the Brunn-Minkowski inequalityconcerningthe (n-dimensional)volumesn(K0)andn(K1)of2compactconvexsetsK0, K1inn-dimensionaleuclideanspaceEn. Let0 t 1,then:[n((1 t)K0 + tK1)]1n (1 t)[n(K0)]1n+ t[n(K1)]1n. (7)Equalityforsome0