Talukder and Ahmad: Relativistic Rule of Multiplication of Velocities Consistent with Lorentz Einstein Law of Addition and Derivation (26-41)

Transcription

1 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion of he Missing Eqaions of Speia Reaiiy M.O.G. Takder, Mshfiq Ahad Varendra Uniersiy, Rajshahi-64, BAGLADESH Deparen of Physis, Rajshahi Uniersiy, Rajshahi-65, BAGLADESH Absra In his paper, we presen he re for reaiisi ipiaion of a eoiy by a nber. We hae reasoned on he basis of a hogh experien and we hae aken ino onsideraion he L-E aw of addiion of eoiies. The forais gies he res of repeaed L-E addiion js as ordinary ipiaion gies he effe of repeaed Gaiean addiion. In he assia ii, i opies wih he Gaiean aw of ipiaion. The forais presened here an exend he horizon of reaiiy. The hogh experien aso reeas he aes of boh he reaie engh and ie in he ongidina direion. Frher, i has been deonsraed, as ipiaions, ha eah reaie qaniy has wo aes - one in he ongidina and he oher in he ranserse direions. As a onseqene, we hae fond o he issing eqaions whih are neessary o ake Einsein s heory of speia reaiiy sef-onsisen and opee. Moreoer, we se he geoeri ean o ge he ean ae of he reaie qaniies. The jsifiaion of doing so is aso deonsraed in his paper. Finay, in he appendix, we presen he reaiisi ipiaion res for he reaie qaniies ike eoiy, ass, ie and engh by a nber. We aso presen he genera res for he prod of wo reaie qaniies of he sae eniy. Keywords: Reaiisi addiion and ipiaion, eoiy, ass, ie, engh, speia reaiiy, issing eqaions, geoeri ean. ITRODUCTIO The reaiisi addiion of eoiies sing Lorenz-Einsein ransforaion, is gien by he we known fora V r () where, Vr is he reaie eoiy, obsered fro an ineria referene frae S, of a body oing wih a speed in anoher frae S'; when S' oes nifory wih speed reaie o S and is he speed of igh. We hae inroded he sybo o ean L-E addiion. This eans ha eoiy shod be added o he eoiy by L- Page 6 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE

2 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) E addiion () and no by Gaiean addiion (+). ow, if here is nber of eqa eoiy o be added reaiisiay, hen V r where, for he L-E s of ers we hae sed he expression words, he sybo has been sed for L-E ipiaion. (). In oher The reaiisi addiion of eoiies hae aso been deried wiho sing he L-E ransforaions b sing hogh experiens and he foras ha aon for ie diaion and engh onraion 3,4, fro he inariane 5 of and sing he ie diaion fora 6. We hae sed he forais, presened in his paper, o find he reaiisi expression of oen onseraion aw 7. Frher, in a reen paper 8, we hae shown he wae represenaion of parie kineais and he eqiaene beween oninos and disree ie sing he sae. On he oher hand, Mr. Ahad 9, has sdied he disree and oninos represenaions of he sae oion epoying his forais. The ain objeies of his work are (a) o widen he sope of speia reaiiy (o aoodae qan ehanis) by inding soe issing eqaions ino Einsein s heory of SR and (b) o ndersand he roe of he speed of igh. DERIVATIO OF THE FORMALISM Le s onsider ha he frae of referene S' oes aong he -axis wih a nifor eoiy reaie o a saionary frae of referene S as shown in Fig.. There is a igh bea ok, wih is wo parae irrors paed horizonay aong -axis, in he frae of referene S'. The ok raps a igh pse beween wo parae irrors ha bones off he irrors a perfey regar ineras of ie. The igh pse akes ie / o rae fro one irror o anoher. Sppose, iniiay he origins of S and S' are oiniden. Y Y / / / / / S / + / S M M M M, Z Z Fig. : S (, Y, Z) is a Saionary and S' (', Y', Z') is oing frae of referene. M', M' are he horizonay paed irror posiions, of a igh bea ok, afer ie /, wih Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 7

3 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) Page 8 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE M and M being heir posiions afer ie. The separaion beween he irrors is. The soid ine wih arrows represens he pah of igh bea as seen fro S. Then an obserer in S wi see ha in ie, he frae S' wi oe a disane (d) gien in ers of he pah of igh by d (3) Dring he sae ie inera, he igh pse raerses a pah (d') gien by d (4) d d (5) Mipying boh sides by and rearranging he ers, (6) Hene, we an onde ha a eoiy an be represened as a fraion of by he aboe reaion. Ths, he L E aw (Eq. ) an be wrien as V r (7) Hene, foowing Eq. (6), we an wrie - V r (8) Therefore, for =

4 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) / / V r (9) / / where, he sybo indiaes L-E ipiaion. Siiary, we an show ha () Where, is a nber. The aboe eqaion represens he L-E ipiaion of he eoiy by he nber. I is eqiaen o he L-E s of nber of eqa eoiy. I has soe adanages oer he oneniona for as foows. Sppose here is nber of eqa eoiies o be added reaiisiay. If he oneniona L-E aw is sed for his prpose, nber of seps is needed o ge he fina res. B he operaion beoes bersoe afer 3 or 4 seps. Whereas, sing he presen for, he fina res an be obained in js a singe sep een for arge aes of. The oneniona for an be fond siabe for erain ases, whereas, he presen for an be sed o noer differen aspes of physia phenoena. As a res, he doain of reaiiy is expeed o be expanded beyond is horizon. The aboe eqaion an aso be wrien as / / / / () Hene, in he assia ii ( << ) / / / / () whih is he Gaiean ipiaion of he eoiy by he nber. PROPERTIES OF We wod ike o erify if he expression for, as gien by Eq. (), orrey represens he L-E s gien by Eq. (). The orre represenaion has o hae he foowing properies. (3) Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 9

5 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) Page 3 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE (4) n n ) ( ) ( ) ( (5) ) ( ) ( ) ( (6) n n (7) Where, and n are any nbers and and are eoiies. For exape, Eq. (6) an be wrien, foowing Eq. (), as / / / / / / / / ) ( ) ( (8) ) ( / / / / / / / / (9) Ths, i an be shown ha a ondiions of Eqs. (3) (7) are ffied. Hene, we an onde ha he forais gien in Eq. () orrey represens. RATIVE LEGTH As shown in Fig., he engh beween he wo irrors is. ow in ie, he igh raes wie beween he irrors, () or () As obsered by an obserer in S, he disane (d + ) raeed by igh bea fro irror M o M in he forward direion is d / ()

6 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) Again he disane (d - ) raeed by he sae bea fro M bak o M (i.e. in he bakward direion) is d / (3) Hene, he reaie engh () beween he irrors wi be eqa o he geoeri ean (jsifiaion is gien in Seion 6) of d + and d - expressed as foows: d d (4) Ths, he engh is onraed in he ongidina direion by he faor. RATIVE TIME Frher, as obsered by an obserer in S, he ie aken by he igh bea ( + ) o rae fro M o M (i.e. in he forward direion) is fro Eq. (), d / (5) Again, he ie aken by he sae igh bea ( - ) in raeing fro M bak o M (i.e. in he bakward direion) is fro Eq. (3), d / (6) Hene, he reaie ie (/) aken by he igh bea o rae fro one irror o oher is he geoeri ean of + and -. (7) (8) where, is he oa reaie ie for he igh bea o rae fro M o M and bak o M. The sae reaion an aso be obained fro Eqs. () and (4) as Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 3

7 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) Page 3 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE (9) Ths, ie is onraed in he ongidina direion. JUSTIFICATIO OF USIG GEOMRIC MEA FOR RATIVE QUATITIES Le s onsider a differen eoiy insead of in Eq. (3), ha is d (3) Then, fro Eqs. () and (3), he arihei aerage of d + and d - is d a (3) (3) w G (33) Where, w G (34) is he Gaiean addiion of eoiies. Howeer, we need reaiisi addiion whih an be ahieed as foows. Le s ake he geoeri ean of d + and d - : d d d g (35) (36) (37)

8 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) w r (38) where, w r (39) is he reaiisi addiion of and aording o he Lorenz Einsein aw. When =, wg = wr = and d a (4) d g (4) Hene, we an onde ha geoeri ean is reaiisi and arihei aerage is Gaiean in nare. IMPLICATIOS OF THE RESULTS (I) RATIVE TIME Howeer, aording o Einsein s heory of reaiiy, ie is diaed in he ranserse direion whih an be deried by onsidering he igh bea ok paed perpendiar o he direion of oion. Le s denoe i by (in he sffix, E sands for Einsinian reaiiy and T for ranserse), hen i an be expressed as foows: (4) where, is he ie inera dring a opee rond rip of a pse in a saionary igh bea ok (proper ie). ow, sine Eq. (8) represens onraion of ie in he ongidina direion, we wi denoe he reaie ie by (L sands for ongidina). Heneforh, a reaie qaniies wi be denoed wih sffixes for ongidina and for ranserse aes. Then, Eq. (8) an be wrien as (43) Fro he aboe wo eqaions, we ge Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 33

9 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) (44) Tha is he prod of he ongidina and ranserse reaie ies is eqa o he sqare of he proper ie. The aboe eqaion aso indiaes ha he proper ie is an inarian qaniy and is eqa o he geoeri ean of he reaie ies. (II) RATIVE LEGTH Using Eq. () in he aboe eqaion, we an wrie 4 (45) Hene, sing Eqs. (4) and (43), we obain 4 (46) or (47) or or where, and (48) (49) (5) (5) where, and are ongidina and ranserse enghs respeiey. Eqaion (5) indiaes ha he engh is onraed in he ongidina direion as in Eq. (4). On he onrary, Eq. (5) indiaes ha he engh is diaed in he ranserse direion. Howeer, Eq. (49) shows ha he prod of he ongidina and ranserse enghs is eqa o he Page 34 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE

10 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) sqare of he engh a res. I eans ha he engh a res is an inarian qaniy and is eqa o he geoeri ean of he reaie enghs. (III) RATIVE VOCITY Rearranging he ers, Eq. (46) an aso be wrien as (5) Hene, sing Eq. (), we ge (53) (54) where, and (55) (56) and are he ranserse and ongidina eoiies, respeiey. I is ear fro he aboe eqaions, boh of he beoe eqa o he speed of igh when. Howeer, for < <, > > and < <. Tha is, for inreasing, dereases b inreases fro. Tha is and hange in opposie direions wih inreasing eoiy. So ha for, and. Moreoer, he prod of hese wo eoiies is eqa o he sqare of he speed of igh. Tha is he speed of igh is an inarian qaniy, in onforiy wih he posae of speia reaiiy, and is eqa o he geoeri ean of he reaie eoiies. (IV) RATIVE MOMUM AD MASS Mipying boh sides of Eq. (53) by, we ge (57) or Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 35

11 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) p p p (58) where, p (59) p (6) and p (6) In he aboe eqaions, p is he oen when. p and p are he oens in he ongidina and ranserse direions, respeiey. I is ear fro Eq. (58) ha he prod of he ranserse and ongidina oens is eqa o he sqare of he oen when. I eans he oen p is an inarian qaniy and is eqa o he geoeri ean of he reaie oens. Moreoer, fro Eq. (57), we an wrie (6) or where, and (63) (64) (65) In he aboe eqaions, is he res ass; and are reaie asses in he ranserse and ongidina direions, respeiey. I shod be poined o here ha is he reaie ass presened by Einsein in his heory of speia reaiiy. Moreoer, Eq. (63) shows ha he prod of he reaie asses is eqa o he sqare of he res ass. I eans he res ass is an inarian qaniy and is eqa o he geoeri ean of he reaie asses. The expressions for ongidina ie gien by Eq. (43), ranserse engh gien by Eq. (5), Transerse eoiy gien by Eq. (55), ongidina eoiy gien by Eq. (56) and ongidina ass gien by Eq. (65), respeiey, are he issing eqaions in Einsein s Page 36 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE

12 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) heory of speia reaiiy. Frher, he expressions for he inariane of ie gien by Eq. (44), he inariane of engh gien by Eq. (49), he inariane of he speed of igh gien by Eq. (54) and he inariane of ass gien by Eq. (63) are aso neessary o ake he oa se of eqaions sef-onsisen. Ths, we onde ha hese eqaions aong wih he exising ones ake Einsein s heory of speia reaiiy sef onsisen and opee. The opee se of sef onsisen eqaions for he reaie ass, ie, engh and eoiy in he ase of Einsein s heory of Speia Reaiiy (SR) has been gien in he foowing Tabe. The abe indes boh he exising and issing eqaions of SR. For eah of he reaie qaniies, boh he ongidina and ranserse aes and he ae of heir prod are gien. The reaie aes in he ranserse and ongidina direions are denoed by he sffixes and, respeiey. Tabe : Eqaions of reaie qaniies in Einsein s heory of speia reaiiy Reaie Qaniy Mass Transerse Longidina Prod Tie Lengh Veoiy E Where, he sybos hae heir sa eanings. The eqaions in yeow oor are he issing eqaions. COCLUSIOS Throgh a hogh experien based on L-E aw for he addiion of eoiies, we hae fond: (a) A reaiisi re for ipiaion of a eoiy by a nber. (b) Tha boh he engh and ie onra in he ongidina direion. () Geoeri ean is reaiisi and arihei aerage is Gaiean in nare. Frher, as ipiaions of he ress obained, we hae fond: (a) The reaie ie onras in he ongidina direion b diaes in he ranserse direion. Their prod is eqa o he sqare of he proper ie. Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 37

13 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) (b) The reaie engh onras in he ongidina direion b diaes in he ranserse direion. Their prod is eqa o he sqare of he proper engh. (d) The reaie ass inreases in he ranserse direion b dereases in he ongidina direion. Their prod is eqa o he sqare of he res ass. (e) The reaie eoiy dereases in he ongidina direion b inreases in he ranserse direion. Their prod is eqa o he sqare of he speed of igh. REFERECES A. Einsein, Annaen der Physik, 7, 89 (95). H.A Lorenz, KAW, Proeedings, Aserda, 6, 89 (94). L. Sarori, A. J. Phys., 63, 8 (995). M. Ahad and M.O.G. Takder, Phys. Essays, 4, 593 (). M. Ahad and M.O.G. Takder, Sen for pbiaion in Phys. Essays (). M. Ahad, J. of S. Researh, 7 (9). DOI:.339/jsr.i.875 M. Ahad, Phys. Essays., 44 (9) M. S. Greenwood, A. J. Phys., 5, 56 (98)..D. Merin, A. J. Phys., 5, 9 (984). W.. Mahews Jr., A. J. Phys., 73, 45 (5). APPEDI MULTIPLICATIO RULES FOR THE RATIVE QUATITIES: A. Prod Re Eqaions (44), (49), (54) and (63) an be expressed as he foowing genera re: or (A) (A) where, is any reaie qaniy wih being is ae a res; and are is aes in he ongidina and ranserse direions, respeiey. B. Mipiaion by a ber (i) Reaie Veoiy Ping = in Eq. (), we ge Page 38 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE

14 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) ow, fro Eq. (54), we an wrie (B) (B) Using he aboe ae of / in Eq. (B), we ge (ii) Reaie engh Fro Eq. (), we an wrie (B3) (B4) Ping his ae of in Eq. (59), we ge (B5) (B6) Hene, fro Eqs. (B) and (B6), we an wrie (B7) Siiary, i an be shown ha / = /. Ping his ae in Eq. (B3), we an wrie Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 39

15 Takder and Ahad: Reaiisi Re of Mipiaion of Veoiies Consisen wih Lorenz Einsein Law of Addiion and Deriaion (6-4) (B8) C. Reaie ie Eqaion (B5) an aso be expressed as foows: (B9) (B) Hene, fro Eqs. (B) and (B), we an wrie (B) Siiary, i an be shown ha / = / and hene Eq. (B3) an be wrien as D. Reaie ass ow, ipying boh sides of Eq. (56) by, we ge (B) (B3) (B4) Hene, sing Eq. (B4) in Eq. (B), we an wrie Page 4 Copyrigh CC-BY-C, Asian Bsiness Consori AJASE

16 Asian Jorna of Appied Siene and Engineering, Voe, o /3 ISS 35-95(p); (e) Copyrigh CC-BY-C, Asian Bsiness Consori AJASE Page 4 (B5) Siiary, i an be shown ha / = / and hene Eq. (B3) an be wrien as E E E E (B6) Hene, he genera expressions for he ipiaions of reaie qaniies by any nber an be wrien as: (B7) and (B8) where, is any reaie qaniy wih being is ae a res;, are is reaie aes in he ongidina and ranserse direions, respeiey.