Mocanu Paradox of Different Types of Lorentz Transformations

Transcription

1 Page Moanu Parado of Differen Types of Lorenz Transformaions A R aizid and M S Alam * Deparmen of usiness Adminisraion Leading niersiy Sylhe 300 angladesh Deparmen of Physis Shahjalal niersiy of Siene and Tehnology Sylhe 34 angladesh Absra: *Corresponding auhor: susaik@gmailom The Lorenz ransformaion is well known In his paper we hae represened he differen ypes of Lorenz ransformaion depending upon he naure of moemen of one inerial frame relaie o he oher inerial frame hen one frame moes along he - ais wih respe o he res frame hen he relaion beween he spae and ime o-ordinaes of he wo frames gies he speial Lorenz ransformaion SLT hen he moion of he moing frame is no along he -ais relaie o he res frame bu he moion is in any arbirary direion hen he relaion beween spae and ime o-ordinaes beween he wo frames gies he mos general Lorenz ransformaion MGLT e an generae differen ypes of mos general Lorenz ransformaion using mied number quaernion and geomeri produ hose are alled Mied number Lorenz ransformaion MNLT quaernion Lorenz ransformaion QLT and geomeri produ Lorenz ransformaion GPLT respeiely e hae disussed abou he relaiisi eloiy addiion formula of hese differen ypes of Lorenz ransformaions e hae also sudied he Moanu parado for differen ypes of Lorenz ransformaions Key ords: Speial relaiiy Lorenz ransformaion relaiisi eloiy addiion and Moanu parado PACS: p P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

2 Page Inroduion The Moanu parado [ ] is an ineresing parado of Relaiisi Mehanis The angle beween wo suessie Lorenz ransformaions is known as Thomas roaion parado whih was firs inened by Moanu 99 [7] This parado is also known as Moanu parado In his paper we hae sudied his Moanu parado for differen ypes of Lorenz Transformaions Consider hree inerial frames of referene S S and S where he frame S is a res and he frame S is moing wih eloiy wih respe o S frame and he frame S moing wih eloiy wih respe o S frame Then he eloiy of S wih respe o S an be wrien as ; where represens he Lorenz sum Similarly he eloiy of S wih respe o S an be wrien as Aording o he eor riangular law mus be equal o u ; his is Moanu parado is Figure : Moanu parado Speial Lorenz Transformaion Consider wo inerial frames of referene S and S where he frame S is a res and he frame S is moing along -ais wih eloiy wih respe o S frame The spae and ime oordinaes of S and S are y z and y z respeiely Then he relaion beween he oordinaes of S and S is alled SLT whih an be wrien as [8] Figure : Speial Lorenz ransformaion P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

3 z z y y z z y y Mos General Lorenz Transformaion hen he eloiy of S wih respe o S is no along -ais ie he eloiy has hree omponens y and z Then he relaion beween he oordinaes of S and S is alled MGLT whih an be wrien as [9] 3 And 4 here z k yj i k z j y i and 3 Quaernion Lorenz Transformaion In his ase he eloiy of S' wih respe o S has also hree omponens y and z as he MGLT Le in his ase and be he spae pars in S and S' frames respeielythen using he he quaernion produ A A A he QLT [0-] an be wrien as 5 Figure 3: Mos General Lorenz Transformaion Page 3 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

4 And 6 4 Mied Number Lorenz Transformaion In he ase of he MGLT he eloiy of S' wih respe o S is no along he -ais; ie he eloiy has hree omponens y and z Le in his ase and ' be he spae pars in S and S' frames respeiely Then using he mied produ ia A A he MNLT [3-7] an be wrien as i 7 And i 8 5 Geomeri Produ Lorenz Transformaion In his ase he eloiy of S' wih respe o S also has hree omponens y and z as he MGLT Le in his ase and be he spae pars in S and S' frames respeiely Then using he he geomeri produ of wo eors A A A he GPLT [8 9] an be wrien as 9 And 0 eloiy Addiion Formula of Differen Types of Lorenz Transformaions eloiy Addiion Formula for Speial Lorenz Transformaion Consider hree inerial frames of referene S S and S where he frame S is a res and he frame S is moing along -ais wih eloiy wih respe o S frame and he frame S is moing along -ais wih eloiy wih respe o S frame Figure 4: eloiy addiion for SLT Page 4 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

5 Now from equaion we hae Or Or whih is he eloiy addiion formula for SLT eloiy Addiion Formula for Mos general Lorenz Transformaion From he equaion 4 we hae = Diiding numeraor and denominaor of equaion by we ge where Puing we an wrie 3 whih is he eloiy addiion formula for MGLT 3 eloiy Addiion Formula for Quaernion Lorenz Transformaion sing Equaion 5 and 6 or Page 5 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

6 So he eloiy addiion formula for QLT is 4 4 eloiy Addiion Formula for Mied Number Lorenz Transformaion sing Equaion 7 and 8 i i i or So he eloiy addiion formula for MNLT i 5 5 eloiy Addiion Formula for Geomeri Produ Lorenz Transformaion sing Equaion 9 and 0 or So he eloiy addiion formula for GPLT is 6 3 Moanu Parado of Differen Types of Lorenz ransformaion 3 Moanu Parado of Speial Lorenz Transformaion The eloiy addiion formula for he SLT an be wrien as see figure 4 Aording o figure 4 S moes wih eloiy wih respe o S and S moes wih eloiy wih respe o S hen aording o he eloiy addiion formula for he SLT he resulan eloiy of and an be wrien as Page 6 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

7 7 So Speial Lorenz ransformaion has no Moano parado 3 Moanu Parado of for Mos General Lorenz Transformaion From he eloiy addiion formula for he MGLT we hae see figure Aording o figure S moes wih eloiy wih respe o S and S moes wih eloiy wih respe o S hen aording o he eloiy addiion formula for he MGLT he resulan eloiy of and an be wrien as 8 So MGLT has Moano parado 33 Moanu Parado of for Quaernion Lorenz Transformaion From he eloiy addiion formula for he QLT we hae See figure Aording o figure S moes wih eloiy wih respe o S and S moes wih eloiy wih respe o S hen aording o he eloiy addiion formula for he QLT he resulan eloiy of and an be wrien as 9 Page 7 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

8 So QLT has no Moano parado 34 Moanu Parado of Mied Number Lorenz Transformaion From he eloiy addiion formula for he MNLT we hae See figure i Aording o figure S moes wih eloiy wih respe o S and S moes wih eloiy wih respe o S hen aording o he eloiy addiion formula for he MNLT he resulan eloiy of and an be wrien as i i i i 0 So MNLT has no Moano parado 35 Moanu Parado of Geomeri Produ Lorenz Transformaion From he eloiy addiion formula for he GPLT we hae See figure Aording o figure S moes wih eloiy wih respe o S and S moes wih eloiy wih respe o S hen aording o he eloiy addiion formula for he GPLT he resulan eloiy of and an be wrien as So GPLT has no Moano parado Page 8 P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

9 Page 9 4 Comparaie sudy of Moano parado for differen ypes of Lorenz ransformaions Names of Lorenz eloiy addiion formula Moano parado ransformaions Speial Lorenz ransformaion Mos general Lorenz ransformaion Quaernion Lorenz ransformaion Mied number Lorenz ransformaion Geomeri produ Lorenz ransformaion i No Yes No No No 5 Conlusion Moanu Parado of differen ypes of Lorenz ransformaions has been disussed learly e hae found ha he Moanu Parado is presen only for MGLT bu here is no eisene of his for any oher Lorenz ransformaions Referenes Moanu C I: Some diffiulies wihin he framework of relaiisi elerody namis Arh Elekroeh Moanu C I: On he relaiisi eloiy omposiion parado and he Thomas roaion Found Phys Le ngar A A: The relaiisi eloiy omposiion parado and he Thomas roaion Found Phys ngar A A: The Relaiisi Composie-eloiy Reiproiy Priniple Found Phys Good I J: Lorenz maries: A reiew In J Theor Phys Maolsi T Goher A: Spaeime wihou Referene Frames: An Appliaion o he eloiy Addiion Parado Sud His Philos Mod Phys Muanu CIIs Thomas roaion a Parado?Elerial Engineering deparmen Poly ehnial Insiue of uhares seor I Srada Rozelor nr uhares Romania 8 Resnik R 994 Inroduion o speial relaiiy iley Easern limied P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET

10 Page 0 9 Moller C 97 The Theory of Relaiiy Oford niersiy press London 0 Kyrala A 967 Theoreial Physis Saunders Company Philadelphia & London Toppan Company Limied Tokyo Japan hp://mahworldwolframom/quaernionhml hp://wwwsappsaeedu/~sjg/lass/30/mahfesalg000/quaernionshml 3 Alam MS 000 Sudy of Mied Number Pro Pakisan Aad of Si 37: 9-4 Alam MS 00 Mied produ of eors Journal of Theoreis 34 hp://wwwjournalofheoreisom/ 5 Alam MS 003 Comparaie sudy of mied produ and quaernion produ Indian J Physis A 77: Alam MS and Khurshida egum Differen Types of Lorenz Transformaions Jahangirnagar Physis Sudies Alam MS 003 Differen ypes of produ of eors NewsullCalMahSo 6 & Daa K De Sabbaa and Ronhei L 998 Quanizaion of graiy in real spae ime Il Nuoo Cimeno 3 9 Daa K Daa R 998 Einsein field equaions in spinor formalism Foundaions of Physis leers P-ID 0 Inernaional Conferene on Physis Susainable Deelopmen & Tehnology ICPSDT-05 Augus Deparmen of Physis CET