Composition of Lorentz Transformations in Terms of Their Generators

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1、COMPOSITIONOFLORENTZTRANSFORMATIONSINTERMSOFTHEIRGENERATORSBartolom´eCOLL∗andFernandoSANJOSEMART´´INEZ†AbstractTwo-formsinMinkowskispace-timemaybeconsideredasgeneratorsofLorentztransformations.Here,thecovariantandgeneralexpressionforthecompositionlaw(Baker-Campbell

2、-Hausdorffformula)oftwoLorentztransformationsintermsoftheirgeneratorsisobtained.EverysubalgebraoftheLorentzalgebraofsuchgenerators,uptoone,maybegeneratedbyasolepairofgenerators.Whenthesubalgebraisknown,theaboveBCHformulaforthetwotwo-formssimplifies.Itssimplifiedexpres

3、sionsforallsuchsubalgebrasarealsogiven.Keywords:Lorentztransformations,Liealgebrasoftwo–forms,Baker–Campbell–Hausdorffformula.IIntroductionInMinkowskispace-time,globalLorentztransformationsareusedtorelateiner-tialobservers.Inageneralspace-time,localLorentztransforma

4、tions[1],alongtime-likecongruencesofcurvesorspace-likefamiliesofhypersurfaces,areusedarXiv:gr-qc/0106052v115Jun2001torelatearbitraryframestocomovingobservers[2]ortosynchronizations[3].Inmostoftheproblems,thetransformationsinvolvedbelongtotheproperorthocronousLorent

5、zgroup(itsconnectedcomponentoftheidentity),sothattheyareunivocallygivenbytheexponentialoftheelementsoftheLorentzalgebra[4].LocalLorentztransformationsofthespace-timemaythusbegivenbyexponentialsofthetwo-forms[5]ofthespace-time.Thisrepresentationbytwo-formsoflocalLor

6、entztransformationshastheimportantadvantageofinvolvingexclusivelyitsintrinsicelements[6];thesearetheelementsinwhichthecorrespondingtwo-formsdecompose[7].Nevertheless,thisimportantadvantageisobscuredbythepracticalandformaldifficultiesthatariseinthecompositionoftransfo

7、rmations,wherethecorrespondingtwo-formsarerelatedbytheBaker-Campbell-Hausdorff(BCH)formula.ItistheabsenceofasimpleandcompactexpressionfortheBCHformulathatoriginatesthesedifficulties.Themainpurposeofthispaperistoobtainsuchanexpression.∗Syst`emesder´ef´erencespatio–temp

8、orels;ObservatoiredeParis–CNRSUMR8630;61,Avenuedel’Observatoire;75014Paris;France†DeptodeMatem´aticaAplicada;E.T.S.I.Agr´onomos;UniversidadPolit´