Derivation of the Missing Equations of Special Relativity from de-broglie s Matter Wave Concept and the Correspondence between Them

Transcription

1 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene beween The M.O.G. Taukder, Mushfiq Ahad Varendra Uniersiy, Rajshahi-64, BANGLADESH Dearen of Physis, Rajshahi Uniersiy, Rajshahi-65, BANGLADESH Absra In his aer, we hae onsidered ha he arie eoiy in he aer wae one of de Brogie is reaie. I reeas ha aer wae assoiaed wih a oing arie is a hoon wae whih has ongiudina and ranserse aues. The oneniona de-brogie s waeenghoenu reaion orresonds o he ongiudina wae. As a onsequene, we hae found he ongiudina and ranserse reaie aues of he ass, ie, engh and eoiy inuding he issing equaions of Einsein s heory of seia reaiiy. I aso reeas ha oenu and energy are reaie quaniies and he orresonding exressions for he ongiudina and ranserse aues for boh of he hae been resened. Moreoer, he addiion rues, wih a genera exression, for a he reaie quaniies based on Lorenz-Einsein ransforaion aw for he addiion of eoiies hae been gien. Finay, he axiu and iniu aues of he reaie quaniies hae been exressed in ers of Pank s onsan and he seed of igh. Their nueria aues hae aso been auaed and resened in his aer. Keywords: Maer wae, Phoon, Waeengh, Moenu, Reaie ass, Tie, Lengh and eoiy, Transerse, Longiudina and Pank s onsan. INTRODUCTION Wae arie duaiy is a fundaena roery of boh aer and wae. In 94, de- Brogie roosed a ode, whih an be regarded as he wae ode of aer, o exain he waeenghs of aer waes. In his ode, i was assued ha waeengh for a wae assoiaed wih a oing arie is rooriona o is oenu wih Pank s onsan as he onsan of roorionaiy,. Syboiay, i an be wrien as h h d () where 3, () In he aboe equaions, λd = de Brogie waeengh Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 57

2 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) h = Pank s onsan = eoiy of he arie = = oenu of he arie = res ass of he arie = reaiisi ass of he arie Moreoer, he eoiy (w) of he aer wae is gien by w ; w (3) is he eoiy of he arie and is he seed of igh. On he oher hand, he arie naure of wae was firs inrodued by Einsein. He roosed a arie or hoon ode, whih is aso known as he quanu ode, of igh. In ha ode, he iewed igh as onsising of sreas of aries, aed hoons, raher han of wae. The energy onen of eah hoon is equa o he rodu of Pank s onsan and he frequeny of igh 4. Tha is E h (4) E = energy of hoon h = Pank s onsan ν = frequeny of igh I shoud be oined here ha he res ass of a hoon is zero. Howeer, i has oenu whih an be obained fro he reaion h (5) is he oenu and is he seed of igh. In his aer, we deonsrae ha he aer wae assoiaed wih a oing arie is hoon wae whih has ongiudina and ranserse aues. The oneniona de-brogie s aer wae orresonds o he ranserse wae. Moreoer, we derie he exising and issing equaions of Einsein s heory of seia reaiiy and he exressions for he iniu and axiu aues for he reaie quaniies. Finay, aording o he Lorenz Einsein Transforaion aw 5.6, he reaiisi addiion of eoiies u and an be obained as foows: u u u (6) sign indiaes L-E addiion and is he seed of igh. In his aer, we resen he addiion rues for a oher reaie quaniies ike ass, ie and engh and a genera exression for ha. Page 58 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE

3 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) RESULTS AND DISCUSSIONS RATIVE VOCITY Equaion (3) does no iose any resriion on he aue of. Howeer, Seia Reaiiy (SR) requires ha does no exeed he seed of igh. This an be aouned for by exressing Eq. (3) as foows: w (7) Now, exressing as a fraion of (i.e. w Therefore, using Eq. (3), we an wrie and ), in he aboe equaion, we ge (8) (9) () () () Equaion () indiaes ha he wae assoiaed wih a oing body is a hoon of seed wih is ongiudina aue reaed o he arie eoiy and he ranserse aue reaed o he oneniona de-brogie s aer wae. RATIVE TIME Now, e us onsider ha he hoon has a waeengh λ and ie eriod, so ha = λ/. Puing his aue of in Eq. (9), we ge or (3) (4) Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 59

4 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) (5) (6) and (7) I shoud be oined ou here ha he ie eriod reresens an inera of ie. Hene, Eq. (6) reresens he reaie ie in he ongiudina direion. On he oher hand, Eq. (7) reresens he reaie ie in he ranserse direion. Eq. (5) indiaes ha is an inarian quaniy and i is equa o he geoeri ean of he ongiudina and ranserse ies. RATIVE LENGTH The waeengh λ reresens an inera beween wo sae oins. Hene, uing λ = in Eq. (3), we ge (8) (9) () () is he reaie engh in he ongiudina direion and () is he reaie engh in he ranserse direion. Equaion () indiaes ha is an inarian quaniy and i is equa o he geoeri ean of he ongiudina and ranserse enghs. RATIVE MASS Aording o Einsein s ass-energy equiaene reaion, he res-ass energy (E) for a arie of ass () is E (3) Page 6 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE

5 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) Hene, obining Eqs. (4) and (3), E (4) h Using Eq. (4) in Eq. (7), we ge h w Therefore, using Eq. (3) and in he aboe equaion, we an wrie and (5) h (6) h (7) (8) (9) (3) Exressing as a fraion of, Eq. (7) an be wrien as C h h is he Coon waeengh and CT (3) (3) C CT (33) is he Coon waeengh in he ranserse direion. Furher, Eq. (7) an aso be wrien as Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 6

6 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) Hene, using Eq. (4), we ge w w h As before, exressing w as a fraion and using, we ge (34) (35) h (36) h (37) (38) is he reaie ass in he ranserse direion and (39) is he waeengh in he ongiudina direion. Equaion (37) reresens he waeengh of he de-brogie s aer wae gien by Eq. (). Howeer, fro Eqs. (8) and (38), we ge (4) i.e. he res ass is an inarian quaniy and i is equa o he geoeri ean of he ongiudina and ranserse asses. RATIVE MOMENTUM AND ENERGY Now, using Eqs. (3) and (4) in Eq. (7), we an wrie E (4) (4) E (43) is he Coon oenu in he ongiudina direion and E E (44) Page 6 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE

7 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) is he Coon wae energy in he ongiudina direion. Furher, using Eqs. (3) and (4) in Eq. (34), we ge E (45) (46) is he Coon oenu in he ranserse direion and E E (47) is he Coon wae energy in he ranserse direion. Hene, fro Eqs. (43) and (46), we ge (48) (49) is he oenu when or he Coon oenu. Equaion (48) indiaes ha is an inarian quaniy and is aue is equa o he geoeri ean of he ongiudina and ranserse oenus. Furher, fro Eqs. (44) and (47), we an wrie E E E (5) The aboe equaion indiaes ha E is an inarian quaniy and i is equa o he geoeri ean of he ongiudina and ranserse energies. Fro he aboe disussions i is ear ha boh he oenu and energy are reaiisi quaniies. ADDITION RULES FOR THE RATIVE QUANTITIES The L-E aw gien by Eq. (6) an aso be exressed as foows: u u u Hene, for reaie eoiies, and, we an wrie (5) (5) Therefore, using Eq. (), we obain Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 63

8 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) i.e. he reaiisi addiion of / and / is equa o he arihei aerage of he wo. Now, uing = λ/ ( λ is he waeengh and is he ie eriod) in Eqs. () and (), reseiey, we ge (53) and (54) or and (55) RATIVE LENGTH Using Eq. (54) in Eq. (5), we ge Now, uing λ =, we ge Tha is he reaiisi addiion of and is equa o he arihei aerage of he wo. RATIVE TIME Furher, using Eq. (55) in (5), we an wrie Tha is he reaiisi addiion of and is equa o he arihei aerage of he wo. RATIVE MOMENTUM Fro Eqs. (43) and (49), we ge / = / and fro Eqs. (46) and (49), we ge / = /. Puing hese aues in Eq. (5), we an wrie (56) (57) (58) (59) (6) Page 64 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE

9 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) Tha is he reaiisi addiion of E and E is equa o he arihei aerage of he wo. RATIVE MASS Furher, fro Eqs. (8) and (43), we ge / = / and fro Eqs. (38) and (46), we ge / = /. Puing hese aues in Eq. (5), we ge Tha is he reaiisi addiion of and is equa o he arihei aerage of he wo. RATIVE ENERGY Foowing Eqs. (4), (45) and (49), we ge / = E/E and / = E/E. Puing hese aues in Eq. (5), we obain E E E E Tha is he reaiisi addiion of E and E is equa o he arihei aerage of he wo. (6) (6) (63) (64) (65) GENERALIZED EXPRESSIONS I is ear fro he aboe disussions ha he reaiisi addiion of he ongiudina and ranserse aues of any reaie quaniy an be generaized as X X X X X X X X X X is a reaie quaniy and X, X and X are i s ongiudina, ranserse and res aues, reseiey. Anoher for of he aboe exression is a b a b ab a = X/X and b = X/X. a b (66) (67) Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 65

10 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) MAXIMUM AND MINIMUM VALUES OF RATIVE QUANTITIES RATIVE MASS When a arie of res ass is onered ino energy hen, foowing Eqs. (4) and (3), we an wrie h (68) ν is he frequeny of he energy wae, h is Pank s onsan and is he seed of igh. Sine energy is reaiisi, is aue in he ongiudina direion an be wrien as h h (69) h (7) and ν are reaiisi ass and frequeny, reseiey, in he ongiudina direion and is he reaiisi ie in he ranserse direion. On he LHS of he aboe equaion we hae he rodu of ie and energy. In a subsequen aer 7, we sha show ha in he rodu for boh ie and energy anno be reaie. Hene, onsidering energy as reaiisi, we ge =. h (7) Hene, he iniu aue of E (in) an be obained as foows. h in (7) h in (73) Now, we an infer fro Eq. (39) ha for = in, wi be equa o ax, so ha (74) in ax h (75) ax in RATIVE TIME Considering ie as reaie, Eq. (7) akes he for h (76) Hene, he iniu aue of (in) an be obained as foows: h in (77) Therefore, foowing Eq. (5), we ge Page 66 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE

11 Asian Journa of Aied Siene and Engineering, Voue, No /3 ISSN 35-95X(); (e) ax (78) in h RATIVE VOCITY Rearranging he ers, Eq. (7) an aso be wrien as h E (79) h in Furher, foowing Eq. (), we ge (8) ax (8) in h RATIVE LENGTH Sine =, rearranging he ers, Eq. (76) an aso be wrien as h (8) h in Furher, foowing Eq. (), we ge (83) ax (84) in h I shoud be oined ou here ha Pau Daies and his o-workers suggesed ha he seed of igh was uh higher in he as and had droed o he resen aue oer he ife ie of he unierse. On he oher hand, A. Abreh and J. Magueijo ai ha if igh-seed was signifiany higher (abou 6 ies higher) hen a nuber of asronoia robes an be readiy resoed 8,9. Our resus show (uing = se and = kg in Eq. (8)) ha igh-seed an be u o.36 x 5 whih is ery ose o he required aue for suh resouion. CONCLUSIONS We hae found ha aer wae assoiaed wih a oing body an be reresened by a hoon wae whih has ongiudina and ranserse aues. de-brogie s waeengh-oenu reaion he orreaes he waeengh of he ongiudina wae and he ranserse oenu of he arie. The waeengh of he ranserse wae is reaed o he ongiudina oenu. Thus we hae found: (a) Boh he ongiudina and ranserse aues for a he reaie quaniies ike ass, ie, engh and eoiy. Ou of whih, he ongiudina aues for ass Coyrigh CC-BY-NC, Asian Business Consoriu AJASE Page 67

12 Taukder and Ahad: Deriaion of he Missing Equaions of Seia Reaiiy fro de-brogie s Maer Wae Cone and he Corresondene (57-68) and ie and he ongiudina and ranserse aues for eoiy are onsidered o be he issing equaions of seia reaiiy. The exising equaions aong wih hese issing equaions u forward a oee se of sef onsisen equaions. I is onuded ha his oeed ersion an exend he doain of reaiiy uh beyond is resen horizon. (b) Soe roeries of he reaie quaniies suh as: (i) The rodu of he ongiudina and ranserse aues of eah reaie quaniy is equa o he square of is aue a res. I eans ha a he reaie quaniies reain inarian a heir aues a res. (ii) The reaiisi addiion of he ongiudina and ranserse aues for a he reaie quaniies foow Lorenz Einsein ransforaion aw for he addiion of eoiies and beoe equa o he arihei aerage of he wo. Whereas, he geoeri ean of he wo for eah reaie quaniy is equa o is aue a res. For <<, he arihei aerage and he geoeri ean beoe equa. (iii) Reaie quaniies anno be zero and, hene, eah of he wi hae iniu and axiu aues. This iies ha hey are disree in naure. Neerheess, we hae found he exressions for he iniu and axiu aues for reaie ass, ie, engh and eoiy. REFERENCES. L. de Brogie, Naure,, 54 (93).. L. de Brogie, Ann. Phys. (Paris) 3, (95). Rerined in Ann. Found. Louis de Brogie, 7 (99).. 3. A. Einsein, Annaen der Physik,, 67 (96). 4. A. Einsein, Annaen der Physik, 7, 3 (95). 5. H.A Lorenz, KNAW, Proeedings, Aserda, 6, 89 (94). 6. A. Einsein, Annaen der Physik, 7, 89 (95). 7. M.O.G. Taukder and Mushfiq Ahad, Proeedings, ISSR-, Rajshahi Uni. J. of Si. 38, 9 (). 8. A. Abreh and J. Magueijo, arxi.org/abs/asro-h/ J. Magueijo, arxi.org/abs/asro-h/ Page 68 Coyrigh CC-BY-NC, Asian Business Consoriu AJASE