A Nonstandard Derivation for the Special Theory of

arXiv:physics/0005031v5[physics.gen-ph]30Nov2007ANonstandardDerivationfortheSpecialTheoryofRelativity*RobertA.HerrmannMathematicsDepartmentU.S.NavalAcademy572CHollowayRdAnnapolis,MD21402-5002USA21SEP1993Abstract:Usingpropertiesofthenonstandardphysicalworld,anewfundamentalderiva-tionforalloftheeffectsoftheSpecialTheoryofRelativityisgiven.ThisfundamentalderivationremovesallthecontradictionsandlogicalerrorsintheoriginalderivationandleadstothefundamentalexpressionsfortheSpecialTheoryLorentztransformations.Nec-essary,theseareobtainedbymeansofhyperbolicgeometry.ItisshownthattheSpecialTheoryeffectsaremanifestationsoftheinteractionbetweenournaturalworldandanon-standard,theNSEM-field.Thisderivationeliminatesthecontroversyassociatedwithanyphysicallyunexplainedabsolutetimedilationandlengthcontraction.Itisshownthatthereisnosuchthingasaabsolutetimedilationandlengthcontractionbut,rather,al-terationsinpurenumericalquantitiesassociatedwithanelectromagneticinteractionwithannonstandardphysicalworld(NSP-world)substratum.1.TheFundamentalPostulates.TherearevariousPrinciplesofRelativity.ThemostgeneralandleastjustifiedistheoneasstatedbyDingle“Thereisnomeaninginabsolutemotion.Bysayingthatsuchmotionhasnomeaning,weassertthatthereisnoobservableeffectbywhichwecandeterminewhetheranobjectisabsolutelyatrestorinmotion,orwhetheritismovingwithonevelocityoranother.”[1:1]ThenwehaveEinstein’sstatementsthat“I.Thelawsofmotionareequallyvalidforallinertialframesofreference.II.Thevelocityoflightisinvariantforallinertialsystems,beingindependentofthevelocityofitssource;moreexactly,themeasureofthisvelocity(oflight)isconstant,c,forallobservers.”[7:6–7]IpointoutthatEinstein’soriginalderivationinhis1905paper(Ann.derPhys.17:891)usescertainwell-knownprocessesrelatedtopartialdifferentialcalculus.In1981[5]and1991[2],itwasdiscoveredthattheintuitiveconceptsassociatedwiththeNew-tonianlawsofmotionwereinconsistentwithrespecttothemathematicaltheoryofinfinitesimalswhenappliedtoatheoryforlightpropagation.Theapparentnonballisticnatureforlightpropaga-tionwhentransferredtoinfinitesimalworldwouldalsoyieldanonballisticbehavior.Consequently,thereisanabsolutecontradictionbetweenEinstein’spostulateIIandthederivationemployed.ThiscontradictionwouldnothaveoccurredifithadnotbeenassumedthattheetherfollowedtheprinciplesofNewtonianphysicswithrespecttoelectromagneticpropagation.[Note:OnNov.14,1992,whentheinformationinthisarticlewasformallypresented,Ilistedvariouspred-icatesthatEinsteinusedandshowedthespecificplaceswithinthederivationswherethepredicate’sdomainwasalteredwithoutanyadditionalargument.Thus,Igavespecificexamplesofthemodeltheoreticerrorofgeneralization.Seepage49.]ImentionthatLorentzspeculatedthatethertheoryneednotcorresponddirectlytothemath-ematicalstructurebutcouldnotshowwhatthecorrectcorrespondencewouldbe.Indeed,ifoneassumesthatthe(mathematical)NS-substratumsatisfiesthemostbasicconceptassociatedwithaninertialsystemthatabodycanbeconsideredinastateofrestoruniformmotionunlessacteduponbyaforce,thentheexpressionF=ma,amongothers,maybealteredforinfinitesimalNS-substratumbehavior.Further,theNS-substratum,whenlightpropagationisdiscussed,doesnot*Thisisanexpandedversionofthepaperpresentedatthe14NOV1992MeetingoftheMathe-maticalAssociationofAmerica,CoppinStateCollege,BaltimoreMDandasitappearsinHerrmann,R.A.,Specialrelativityandanonstandardsubstratum,Speculat.Sci.Technol.,17(1994),2-10.1followtheGalileanrulesforvelocitycomposition.TheadditiverulesarefollowedbutnonegativerealNS-substratumvelocitiesexteriortotheEuclideanmonadsareusedsinceweareonlyinterestedinthepropagationpropertiesforelectromagneticradiation.Thederivationinsection3removesallcontradictionsbyapplyingthemostsimplisticGalileanpropertiesofmotion,includingtheballisticproperty,butonlytobehaviorwithinaEuclideanmonad.Asdiscussedinsection3,theuseofanNSP-world(i.e.nonstandardphysicalworld)NS-substratumallowsfortheeliminationofthewell-knownSpecialTheory“interpretation”contradic-tionthatthemathematicalmodelusestheconceptsofNewtonianabsolutetimeandspace,and,yet,oneofthemajorinterpretationsisthatthereisnosuchthingasabsolutetimeorabsolutespace.CertaingeneralprinciplesforNS-substratumlightpropagationwillbespecificallystatedinsection3.Theseprinciplescanbegatheredtogetherasfollows:(1)Thereisanonstandardelectro-magneticfield-theNSEM-field-thatsustainsN-worldelectromagneticpropagation.Suchpropa-gationfollowstheinfinitesimallypresentedlawsofGalileandynamics,whenrestrictedtomonadicclusters,andthemonadicclustersfollowanadditiveandanactualmetricpropertyforlinearrel-ativemotionwhenconsideredcollectively.[Theterm“nonstandardelectromagneticfield”shouldonlybeconstruedasanentity,aNS-substratum,wherethepropagationofelectromagneticradiationfollowsslightlydifferentprinciplesthanwithinthenaturalworld.](2)Themotionoflight-clockswithintheN-world(naturalworld)isassociatedwithonesingleeffect.Thiseffectisanalterationinanappropriatelight-clockmechanism.[Thelight-clockconceptwillbeexplicitlydefinedattheendofsection3.]Itwillbeshownlaterthatanactualphysicalcausemaybeassociatedwith