Hyperbolic triangle in the special theory of relat

arXiv:nucl-th/0204037v115Apr2002HyperbolictriangleinthespecialtheoryofrelativityYongkyuKo∗DepartmentofPhysics,YonseiUniversity,Seoul120-749,Korea(Dated:February8,2008)ThevectorformofaLorentztransformationwhichisseparatedwithtimeandspacepartsisstudied.Itisnecessarytointroduceanewdefinitionoftherelativevelocityinthistransformation,whichplaysanimportantroleforthecalculationsofvariousinvariantphysicalquantities.TheLorentztransformationexpressedwiththisvectorformisgeometricallywellinterpretedinahyper-bolicspace.ItisshownthattheLorentztransformationcanbeinterpretedasthelawofcosinesandsinesforahyperbolictriangleinhyperbolictrigonometry.Sothetrianglemadebythetwooriginsofinertialframesandamovingparticlehastheangleswhosesumislessthan180o.PACSnumbers:02.40Yy,03.30+p,11.30.Cp,45.20.-d,98.80HwI.INTRODUCTIONThespecialtheoryofrelativitystartswiththefundamentaltwopostulateswhichareforrelativityandtheconstancyofthespeedoflightinallinertialframes[1,2,3].ThetransformationbetweeninertialframesisknownastheLorentztransformationwhichsatisfiesthetwopostulates.Thereforephysicalquantitiesandformulasarethesameformsinallinertialframes.Thisisachievedbyusingthefourvectornotationwhichiscombinedwiththespaceandtimecomponents,suchasvμ=dxμ/dτ,Fμ=dpμ/dτandsoon[4].Howevertheseparatenotationoffourvectorsdoesnotshowthecovariancemanifestlyinthetransformationequations.Thetransformedcoordinatesystemusedinthestandardtexts[1,2]hasalittlecomplicatedtransformationformulas,suchas,t′=γ(t−v·r),r′=r+γ−1v2(v·r)v−γvt,(1)thus,itisoftendoubtfulthatthetransformedcoordinatesystemhasthesamephysicalformulasastheoriginalcoordinatesystemhas.Thesetransformationequationsarenottruevectorforms.Theyaremerelycomponentsofthevectorequation,becausethebasisvectorsareusedincommoninthetwoframes.Physicalquantitiesaremeasuredinaninertialframewithitsownunitsoftimeandlength.Physicalformulascomposedofthesequantitieswhicharescalars,vectorsortensorsareformedinaframewithitsownunitvectors.Thereforeitisdifficulttosayrelativityinthetwoframeswiththeaboveequations,becausethetransformedequationsoftensufferfromunwantedfactors,suchas,theγfactor.Duetothistroublemaker,theBiot-Savartlaw,foranexampleinelectromagnetism,isexplainedwithsophisticatedterms,suchas,thedurationtimeofmeasurementforthemagneticfieldcomefromthetransformedelectricfieldofamovingchargedparticle[2].Arotatingcoordinatesystemisagoodexamplefortheabovearguments,ofwhichtransformationisx′i=[eiJkθk]ijxj,(2)whereJkisageneratoroftherotationgroupO(3).Itstimederivativeisthevelocity,whichcanbecalculatedwiththetransformationequationasfollowsv′i=dx′idt=[eiJkθk]ij(dxjdt+i[Jk]jldθkdtxl)=[eiJkθk]ij(vj+ǫjklωkxl)(3)where[Jk]jl=iǫjklintheregularrepresentationoftherotationgroup[5].Thenthevectorformofthevelocityisv′=v′iˆe′i=[eiJkθk]ij(vj+ǫjklωkxl)ˆe′i=(vj+ǫjklωkxl)ˆej=v+ω×r,(4)∗[e-mail];~yongkyu2wheretheunprimedbasisvectorsaretransformedasˆej=ˆe′i[eiJkθk]ij.Theaccelerationisa′i=dv′idt=[eiJkθk]ij(aj+ǫjklωkǫlmnωmxn+2ǫjklωkvl),(5)whereweconsideraconstantangularvelocity.Thevectorformoftheaboveequationisa′=a+ω×(ω×r)+2ω×v,(6)and,ifamassismultipliedtotheequation,theNewton’slawisF′=ma′=ma+mω×(ω×r)+2mω×v,(7)wherethesecondtermisthecentrifugalforceandthelasttermistheCoriolisforceasshowninthestandardtexts[1,3].Thereforethepositionvectorsinthetwoframesarewrittenbyr′=x′iˆe′i=[eiJkθk]ijxjˆe′i=xjˆej=r,(8)irrespectiveoftherotationanglewhichisconstantorvaryingwithtime.Sincethemasstimesaccelerationisexpressedasthesameformsinthetwoframes,theNewton’slawcanbesaidtobeexpressedascovariantmannerinthevectorrepresentationratherthaninthecomponentrepresentationofthevector.Thereisnorotationmatrix,namely,anunwantedfactor,inEq.(7)comparedtoEq.(5).ThereforethecorrectvectorformoftheLorentztransformationshouldbeexpressedwithitsowncomponentsandunitvectors.ItismorenaturalthatrelativitycanbeexpectedafterobtainingthecorrectvectorformoftheLorentztransformation.Aslearnedfromnon-relativistickinematics,ifaphysicalphenomenonisdescribedwithageometricalpicture,itismucheasiertocomprehendit.ThecorrectvectorformsoftheLorentztransformationarewellinterpretedwithgeometricalpictures,whicharethelawofcosinesandsinesinhyperbolictrigonometry.ItisofcoursealreadyknownthatthevelocityspaceofthespecialtheoryofrelativityisaLobatchewskyspace,thatis,ahyperbolicspace[6],butourinterpretationgivesmoreclearextendedexplanationforthehyperbolicspace.SotheThomasprecessioncanbeinterpretedasthetimerateoftheangledefect.TheoutlineofthispaperisthatarotationaltransformationisadaptedtoaLorentztransformationinsectionIIinordertoobtainthecorrectvectorformoftheLorentztransformation,themethodofcalculationforinvariantquantitiesundertheLorentztransformationandthegeometricalinterpretationsoftheLorentztransformation.InsectionIII,mostofbasicphysicalquantitiesareinvestigatedundertheLorentztransformationaccordingtotheguidelinesofsectionII,anddevelopedfurther.InsectionIV,theLorentztransformationforanenergyandamomentumisshowntobethelawofcosinesandsinesinhyperbolictrigonometry.Finallysomeconclusionsaregiven.Si