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1、RECIPROCITYINDEPENDENTLORENTZTRANSFORMATIONMushfiqAhmadDepartmentofPhysics,RajshahiUniversity,Rajshahi,Bangladeshe-mail:mushfiqahmad@ru.ac.bdAbstractWehavedefinedslowness(orreciprocalvelocity,correspondingtovelocityv)ascc/v,wherecisthespeedoflight.Itisobservedthattherelativeve
2、locityremainsinvariantifthevelocitiesarereplacedbycorrespondingslownessesi.e.relativemotioninonedimensionisreciprocalsymmetric.Reciprocityoperation,whichconvertsavelocitytothecorrespondingslowness,isfound.LorentztransformationisgeneralizedsothatLorentzinvarianceismaintainedifv
3、elocitiesarereplacedbycorrespondingslownesses.1.IntroductionConsideravectorina2dimensionalCartesianspace,V=xi+yj.Interchangingbetweenxandy,wegetthevectorV’=yi+xj.V’isrotatedwithrespecttoV,butthelengthremainsunchanged,
4、V’
5、=
6、V
7、.Wenowconsiderrelativisticmotionin1spacedimension.It
8、isaneventina2dimensionalSpace-Timespace.Interchangingbetweenspaceandtimecomponentsshouldonlyrotateitinthe2dimensionalspacekeepingtheLorentzlengthinvariant(Lorentzinvariant).Butinterchangingbetweenspaceandtimecomponentsmeanschangingvelocitytoitsreciprocal.Therefore,reciprocatio
9、nshouldmaintainLorentzinvariance.Thisweshallstudybelow.2.MotioninOneSpaceDimensionWeshallmeasurevelocity,v,inunitsofc.Therefore,ourvelocitywillbev/c.Weshalldefinethecorrespondingslownessorreciprocalvelocity,v*,bytherelation2v*.v=c(2.1)Ifuisthevelocityofamovingbodyandvisthevelo
10、cityoftheobserver,therelativevelocityisu±vu⊕(±v)=(2.2)u.v1±2cWeobservethattherelativevelocityremainsinvariantifthevelocitiesarereplaceby1correspondingslownesses.Thisweshallcallreciprocalsymmetry.Using(2.1)u⊕(±v)=(u*)⊕(±v*)(2.3)xisadistancecoveredintimet,asobservedbyaanobserver
11、atrest.Thedistancex’and2timet’asobservedbyanobservermovingwithvelocityvarex−vtx'=(2.4)()21−vc2t−vx/ct'=(2.5)()21−vcLorentzinvariancerequirementis2222(ct')−(x')=(ct)−x(2.6)3.RotationinReciprocitySpace~~Letvandxbevandxrotatedinreciprocityspacethroughangleϕ~v+ictan(φ/2)v+icrv==(3
12、.1)1+i(v/c)tan(φ/2)1+i(v/c)rAnd~x+icttan(φ/2)v+ictrx==(3.2)1+
