Generalized Lorentz Transformation Allowing the Relative Velocity of Inertial Reference Systems Greater Than the Light Velocity

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1 Generlzed Lorentz Trnsformton Allowng the Relte Veloty of Inertl Referene Systems Greter Thn the Lght Veloty Yu-Kun Zheng Memer of the Chnese Soety of Grtton nd Reltst Astrophyss Astrt: Ths pper dsusses theoretlly the posslty of onstrutng spe-tme oordnte trnsformton onnetng two nertl referene systems whh possesses relte mong eloty greter thn tht of lght Our nlyss shows tht the orgn of lght eloty s lmt eloty n the Spel Theory of Reltty s the postulte of onstny of the eloty of lght Hene we me new postulte tht lght eloty depends on the stte of moton of the lght soure Bsed upon t we dere new system of trnsformton equtons Ths new spetme trnsformton ontns onstnt whh theoretlly s undetermned due to the ndonment of the postulte of onstny of the eloty of lght We propose method of eperment usng the optl Doppler Effet to determne ths onstnt The uthor epets tht ths onstnt wll e numer smller thn ut ery lose to In ths se the new trnsformton llows the relte eloty of the nertl referene systems to e equl to or greter thn the lght eloty nd thus the trnsformton oeromes the lmtton of the Spel Theory of Reltty It lso predts new lmt eloty of the relte elotes of ll nertl referene systems The new spe-tme oordnte trnsformton nludes the Lorentz trnsformton nd the Gllen trnsformton s ts two lmt ses Key words: lght soure; lmt eloty; superlumnl; trnsformton group; optl Doppler effet; new lmt eloty; l mong ody; ultmte lmt eloty Introduton The prnple of reltty nd the prnple of onstny of the eloty of lght (CVL) re two ornerstones of the Spel Theory of Reltty (STR) Bsed upon ths foundton Ensten dered the Lorentz trnsformton nd founded the STR [] But the STR ontns lmtton tht t nnot del wth the prolem of moton t eloty equl to or greter thn tht of lght The Lorentz trnsformton ontns the lght eloty s lmt eloty suh tht no eloty of moton of nertl system of referene n eeed tht eloty On the other hnd n reent yers mny superlumnl phenomen he een osered usng people to st the douts on the theoretl ss of the STR The mn prolem here s tht whether the postulte of CVL etly reflets the nture of the lght propgton The purpose of the present pper s to onstrut new spe-tme oordntes trnsformton lnng nertl systems whh llows the relte mong eloty of the nertl systems greter thn the lght eloty n order to re the lght-speed rrer n the Lorentz

2 trnsformton We strt our wor from nestgtng the CVL Our nlyss (n Seton Ⅰ) shows tht the eloty of lght s lmt eloty tully s smply result of the postulte of CVL Therefore for solng the prolem tht the spe-tme oordntes trnsformton lnng two nertl systems possle possessng relte eloty greter thn we must ndon the postulte of CVL Hene we me new postulte tht lght eloty my depend on the moton stte of the lght soure to reple the postulte of CVL Usng ths new postulte together wth some other hypotheses of the STR we dere new spe-tme oordntes trnsformton Ths new trnsformton s generlzton of the Lorentz trnsformton Andonng the postulte of CVL uses the new trnsformton to ontn onstnt whh nnot e determned theoretlly We propose n epermentl method usng the optl Doppler Effet to determne ths onstnt We epet ths onstnt to e numer smller thn ut ery lose to In tht se the otned trnsformton would llow the relte eloty of nertl systems equl to or greter thn tht of lght ⅠThe Lmt Veloty nd the Postulte of Constny of the Veloty of Lght S Let S nd e two nertl systems of referene where s mong wth onstnt eloty relte to S Assume tht there s lght soure L fed n S P s pont fed n S nd L wll go through P when t moes wth S Now suppose tht t the moment when L ondes wth P t emts sgnl n the dreton of moton of S If the lght sgnl trels n S wth the eloty then y the postulte of CVL ths lght sgnl trels on S s lso wth the eloty Therefore the eloty of the lght sgnl relte to the pont P s On the other hnd the mong eloty of the lght soure L relte to P s nd the lght sgnl emtted n the dreton of moton of the soure lwys trels n the front of the lght soure Hene udged from the pont P of system S the lght sgnl lwys trels fster thn the lght soure Tht mens tht s relte eloty of n nert system of referene no mtter how gret t s the nequlty < should lwys hold nmely the lght eloty under the postulte of CVL s n upper lmt of the relte elotes of moton of ll nertl systems of referene The oe nlyss shows tht the lght eloty s lmt eloty s ust orollry of the postulte of CVL Therefore for onstrutng new spe-tme oordnte trnsformton lnng two nertl systems of referene llowng the relte eloty greter thn we should he to ndon the postulte of CVL nd ntrodue new postulte to reple t We propose the followng postultes for derng the new trnsformton: <> All systems of nert re equlent <> Spe-tme oordntes trnsformton lnng nertl systems slner trnsformton <> Lght eloty depends on the moton stte of lght soure <4> Lght eloty s the onstnt only n the referene system relte to whh the soure of lght s t rest (ths s the defnton of n the sequel) Note: The postulte <> s tully the result of theorem nd the deduton of ts lner trnsformtons generlly needs the ondton of CVL [] Thus t seems tht there s n nter-ontrdton etween the postultes <> nd <> But the tul stuton n e unle so We fnd tht the lner trnsformtons n e dedued wthout the ondton of CVL ut only from the nture of nertl system (See Append) S

3 ⅡThe Derton of the Trnsformton of Spe-tme Coordntes wth Lmt Veloty Greter Thn Let S ( t y z) nd S ( t y z ) e two nertl oordnte systems We ssume tht the three oordnte es n nd re prllel to eh other orrespondngly nd tht S s mong relte to S wth eloty n the dreton of the poste -s Let us moreoer ssume tht the orgn O of S ondes wth the orgn O of S t the tme t = t= Then ordng to the postulte <> nd y the onsderton sed on the postulte <> the spe-tme oordntes trnsformton from S to S possess the followng smplest lner epressons: t = p + qt () = m + nt () y = y () z = z (4) where p q m nd n re onstnts to e determned Applyng the relte moton eloty of the two systems the trnsformton equtons () nd () n e hnged mmedtely nto the followng form: t m( r + t) = m( t) p = ( ) m S r = (5) (6) Thus the oeffents to e determned re redued nto two nmely m nd r Frst we determne m Assume tht there s lght soure fed on the orgn O of S nd t the moment t = t = ths lght soure emts lght sgnl L n the poste dreton of the y -s s shown n F: S Suppose tht the sgnl rehes t pont P on y -s wth y = h t tme t = t Let the orrespondng spe nd tme oordntes of P on system S e = y = h nd t = t Then we he from (5) nd (6) t = m( r + t ) (7) = m( t ) (8) Elmntng from these two epressons ges t = m( r + ) t (9)

4 Note tht the proess of the lght sgnl propgtng from O to s udged from the oordnte system S proess of lght sgnl P t propgtng on the lne onnetng O nd nd s the neessry tme of ths proess mesured n oordnte system S Net we onsder proess symmetr to tht n F s shown n F: P Assume tht there s lght soure fed on the orgn O of S nd t tme t = t emts lght sgnl L n the poste dreton of y - s Suppose tht the sgnl rehes t pont P on y -s wth y = h t tme t = t Let the orrespondng spe nd tme oordnte of P on the oordnte system S e t = t Note tht here the y oordntes of the ponts P nd P possess the sme lue h (ths mens tully tht t the moment t = t = the two ponts P nd P " onde wth one nother) Then we he from (5) t = mt () Here smlr s n F the lne onnetng O nd P s the propgtng proess of the lght sgnl L udged n the oordnte system S nd the tme t s the neessry tme of ths proess mesured n oordnte system S Now sne the pont P n oordnte system S s well s the pont P n oordnte system S possess the sme lue h of the y oordntes nd y the hypothess <4> oth the lght sgnls sme eloty we he L nd L possess the t = t = h () Ths ges t = t () On the other hnd euse the proesses of propgton of the two lght sgnls L nd L n ther orgnl oordnte systems S nd S re the sme y the postulte <> ther orrespondng proesses n the mong oordnte systems S nd S respetely (shown y the olque lnes n F nd F) should e the sme too Hene we he t = t () 4

5 Susttutng () nd () nto () ges t = mt (4) Susttutng (4) nto (9) ges t ( ) = m r + t (5) Elmntng t from (5) we get the followng equton out m : = m ( r + ) (6) Solng for m from (6) we otn m + r Thus we he found the lue of m whh s epressed y r nd The trnsformton whh we re now seeng for possesses the followng form: t = m( r + t) (8) = m( t) (9) m + r Now we he to determne the oeffent r In the Lorentz trnsformton ths oeffent s determned s r = / usng the postulte of CVL Here sne we he ndoned ths postulte of ourse r nnot e determned dretly In the followng we nlyze the struture of the oeffent r For ths purpose we requre the tegory of trnsformtons whh we re seeng for to form group of trnsformtons Let us onsder thrd nertl oordntes system S" ( t" " y" z") whh s mong long the poste dreton of -s wth unform eloty relte to the oordnte system S Smlr to the trnsformton from S to S n (8)-() the trnsformton from S to S" should he the followng form of trnsformton equtons: t " = m ( r + t ) () " = m ( t ) () m + r () Susttutng (8)-(9) nto ()-() fter smplfton we get the reltons etween t " " nd t s the follows: t " = m" ( r" + q" t) (4) " = m" ( " t) (5) where m" = mm ( r ) (6) + " = r (7) r+ r r" = r (8) q" = r r (9) By the nture of group of trnsformtons the two trnsformtons omned together should e lso trnsformton nmely the equtons (4)-(5) should represent spe-tme oordntes trnsformton from S to S" Therefore y omprng (4)-(5) wth (8)-(9) nd ()-() we fnd tht the quntty q" n (4) defned n (9) should stsfy r q " = r = () Ths leds to r r = () The equlty () shows tht the two rtos r / nd r / must e ommon onstnt ndependent of oth nd denotng ths onstnt y w we he: 5

6 r r = = () Hene we get the reltons etween r nd s well s r nd : r = w r = w () Now we susttute r nd r epressed y w n () nto ll the relted epressons dsussed oe Thus susttutng the r of () nto (8)-() ges the trnsformton from S to S s t = m( t + w) (4) = m( t) (5) m = (6) w + w Susttutng the r of () nto ()-() ges the trnsformton from S to S" s t " = m ( t + w ) (7) " = m ( t ) (8) m = (9) + w r of () nto (7)-(8) ges = (4) Further susttutng the r nd + " r" w w + = w ( ) = w" (4) Susttutng (6) (9) nd the r of () nto (6) nd usng (4)-(4) fter smplfton we otn m" = (4) + w" Fnlly susttutng the q " of () nd the r " of (4) nto (4) ges t " = m" ( t + w" ) (4) Referrng to (4) (5) nd (4) now we he the followng equtons onnetng the spe-tme oordntes ( t ) ) nd ( t" ") : t " = m" ( t + w" ) (44) " = m" ( " t) (45) m" (46) + w" Comprng (44)-(46) wth (4)-(6) nd (7)-(9) we see tht the equtons (44)-(46) (together wth y " = y nd z " = z ) omned wth the trnsformtons (4)-(6) nd (7)-(9) re relly spe-tme oordntes trnsformton from system S to system S " where system S" moes unformly long the poste dreton of -s of system S wth the eloty " defned n (4) Ths shows tht the lue q" thus determned n () s resonle; t mes the tegory of the trnsformtons whh we re seeng for group of trnsformton Now let us turn to the smpler oeffent to e determned the onstnt w (ndependent of ) In the followng we wll me use of the mn postulte n ths pper nmely the postulte <> tht lght eloty depends on the moton stte of lght soure to get the struture of the onstnt w Assume tht there s lght soure fed on some pont of - s of the oordnte system S nd t some moment t emts lght sgnl towrd the poste dreton of -s The propgton of ths sgnl n the oordnte systems S nd S s desred y the equtons = ( t ) nd = (t) respetely The spe-tme oordntes nd t re onneted wth nd t y the trnsformtons (4)-(6) Aordng to the postulte 6

7 <4> the eloty of ths lght sgnl on the oordnte system S s Hene we he d dt = (47) On the other hnd y the postulte <> the eloty of ths lght sgnl on the oordnte system S s no longer the onstnt ut some other qunttes dfferent from we denote t y * Therefore we he d dt = * (48) Dfferenttng (4)-(5) nd usng (47) nd (48) we get * = + w* (49) The oe equlty ontns two unnown qunttes w nd * we wll use the nture of w eng rrelent of to determne w nd * Solng w from (49) we he * w = * (5) Note tht w nd re ndependent of whle * depends on Dfferenttng the oe equlty wth respet to we otn dfferentl equton for * : d* ( * ) * d = ( + ) (5) Ths equton possesses two solutons (eept the trl soluton * = ): * = (5) nd ( + = ) (5) * + The soluton (5) orresponds ust to the se of the Ensten postulte we should dopt the soluton of (5) other thn t tht s the soluton (5) where s n ntegrton onstnt Susttutng (5) nto (5) ges w = (54) Thus we he otned the detled epressons for * nd w showng n (5) nd (54) respetely oth these two qunttes ontn the onstnt The epresson for * n (5) lso ontns ths shows the nfluene of the moton of lght soure on the eloty of lght The epresson for w n (54) should e ts fnl form As for the ntegrton onstnt lter we wll propose method of eperment of the optl Doppler Effet to determne t Susttutng (54) nto (4)-(6) we rre t the fnl form of the spe-tme trnsformton of two nertl systems whh we were seeng for: t t = (55) = (56) t y = y (57) z = z (58) Note In solng the funton * from the dfferentl equton (5) we he seleted the ntegrton onstnt s spel epresson of the onstnt to ensure tht the oeffent of the quntty under the squre root of the equtons (55)-(56) eng negte numer otherwse the orrespondng trnsformton wll e n ordnry orthogonl trnsformton of the t -plne or Gllen trnsformton these re not the se we onsder here Note In determnng the onstnt spel dreton the dreton of w of (54) we he seleted -s s the dreton for emttng 7

8 the lght sgnl Atully we ould selet ny dreton to e the dreton of emsson the result for w s the sme s tht of (54) only mng n pproprte seleton of the ntegrton onstnt From (55)-(58) we see tht our trnsformton s dfferent formlly from the Lorentz trnsformton y onstnt When = t eomes the Lorentz trnsformton; when = t eomes the Gllen trnsformton Therefore (55)-(58) nludes the Lorentz trnsformton nd Gllen trnsformton s ts two spel ses Now the prolem s whether the otned spe-tme trnsformton (55)-(58) n stsfy our orgnl purpose nmely whether t n permt the eloty of moton of system S relte to system S to e equl to or greter thn the lght eloty Ths depends mnly on the lue of the onstnt If the lue of stsfes the nequlty < < (59) nd let = (6) then when stsfes (59) we wll he > (6) nmely s eloty greter thn On the other hnd the trnsformton (55)-(58) shows tht the eloty of the nertl system S should stsfy < < (6) ths mens tht s lmt eloty n (55)-(58) Consequently under the ondton (59) the permtted rnge of hs een etended gong eyond the lmtton of the lght eloty nd (55)-(58) s spe-tme oordntes trnsformton whh permts the relted nertl systems to he superlumnl relte eloty Thus under the ondton (59) the trnsformton (55)-(58) stsfes the purpose stted n the egnnng of ths pper Smultneously we n otn new lmt eloty for ll elotes of moton of the nertl systems ⅢThe Trnsformton of Veloty For the determnton of the onstnt t s neessry to now the trnsformton of eloty under the spe-tme oordntes trnsformton (55)-(58) (ⅰ)Trnsformton of eloty of prtle Let prtle moe n the oordnte system S wth the eloty d dy dz ( u u u ) = ( ) dt dt dt (6) nd n the oordnte system S wth the eloty d dy dz ( u u u ) = ( ) dt dt dt (64) We wnt to fnd the relton etween these two elotes From the trnsformton (55)-(58) we he the nerse trnsformton s t t + = + t (66) = (65) y = y (67) z = z (68) Dfferenttng (65)-(68) nd pplyng (6) nd (64) we he 8

9 u u u + u u = (69) + u u + = (7) u u + = (7) These re the eloty trnsformton formuls whh trnsform the eloty of prtle mong n the oordnte system S nto ts eloty n the oordnte system S If the prtle moes long the -s then we he u = nd u = By (69)-(7) we he u u + = (7) + u nd u = u = Let u = u + (7) + u u u = u + (74) + u u Susttutng (69)-(7) nto (7) nd usng (74) we get u u ( )( ) ( ) / u = + (75) (ⅱ)Trnsformton of lght eloty We he otned erler n (5) n epresson of eloty * of lght emtted y mong soure The soure s mong wth eloty long the -s of oordnte system S nd * s mesured n S Atully ths s ust the se of the generl formul of eloty trnsformton (7) wth u = nd u = * Our postulte <> uses the lght eloty now s no longer n spel poston s tht t s n the STR nd ts trnsformton s ust s the generl eloty trnsformton (69)-(7) only we te (64) s the eloty of lght emtted y soure fed on the oordnte system S Thus for lght whose soue s fed n S f we denote the omponents of ts eloty n S y ( ) (76) nd denote the orrespondng omponents of eloty of ths lght n S y ( * * *) (77) then we he y (69)-(7) + * = (78) + + * = (79) * + Besdes we he = (8) = + (8) + ( *) + ( *) ( ) * = + (8) * then y susttutng (78)-(8) nto (8) nd usng (8) we get 9

10 ( )( ) ( ) / * = + (8) Ths s the generlzton of the epresson of lght eloty * n (5) for f we susttute y n (8) then (8) wll turn to (5) Ⅳ Determnng the Constnt y mens of eperment of the optl Doppler effet We now dsuss how to determne the onstnt epermentlly y usng the Doppler effet Let λ e the welength of lght we emtted y the lght soure n ts rest frme λ e the welength of the Doppler lue shft when the lght soure moes towrd the oserer nd λ r e the welength of the Doppler red shft when the lght soure moes wy from the oserer We shll fnd formul whh epresses the lue of n terms of λ λ nd λ r We ontnue to use the nertl oordnte systems S nd S defned n the egnnng of seton Ⅱ Assume tht there s lght soure Q fed t some pont on the -s of the referene system S (F) So Q s mong reltely to S n the poste dreton of the -s wth the eloty Let M e n oserton ϕ nstrument tht s used for detetng the Doppler effet of the reeed lght we fed t some pont on the y -plne of the referene system S Assume tht the lght we emtted y Q s y plne we when t rehes M Let ϕ e the ngle etween the propgton dreton of the we nd the -s n S Then y (55)-(57) we otn the generl epresson for the Doppler effet s follows: * + osϕ λ = λ (84) where * s the solute lue of the lght eloty defned n (8) wth = os λ s the welength of the lght we reeed y M ϕ

11 To otn the epresson for the onstnt let us onsder some spel ses of the Doppler Effet Let M e loted on some pont on - s of the oordnte system S Then M wll reee lght wes orrespondng to When M s loted n the poston suh tht Q ϕ = nd ϕ = π M wll reee the lght we emtted y Q = λ of the moes towrd t then n the dreton ϕ nd peree the lue shft welength lght we By (84) we he + λ = λ (85) When M s loted n the poston suh tht moes wy from t then M wll reee the lght we emtted y Q n the dreton ϕ = π nd peree the red shft welength λ r By (84) we he λ = λ (86) r Further we ssume tht n the oe two eperments the lght soures possess the sme mong eloty (t s no neessry to now the lue of ths wll redue the dffulty of the eperments) then we n elmnte the quntty from (85) nd (86) By ths mens we get the followng formul of : ( λr λ ) λ = (87) ( λr + λ ) λ 4λλr The epresson (87) s the formul for epressng the lue of n terms of λ λ nd λ r Therefore f we n perform the Doppler effet eperments stsfyng the oe mentoned ondtons nd suessfully otn mesurements of these welengths then the lue of wll e determned Mesurements of ths sort do not pper n the lterture We hope tht physst of eperment nd stronomer ould rrnge suh eperments or osertons of stronomy n order to ge udgement for the onstnt Suh eperments would e qute sgnfnt for f we n determne tht the onstnt s smller thn then the lmtton of the STR hs een suessfully oerome wth suh lue of the oordnte trnsformton (55)-(58) res the lght-speed rrer s dsussed t the end of Seton Ⅱ On the ontrry f s determned to e etly equl to then Enstens postulte tht lght eloty s ndependent of the stte of moton of the lght soure s erfed (It seems tht the onstnt nnot e greter thn sne ths shows tht the lmt eloty n (6) for ll moton of the nertl referene systems s een smller thn ) The followng re some onse results from (87): = λ λ = λr = λ = λ λr = Gllen trnsformton (88) Lorentz trnsformton (89) We ntpte tht stsfes the nequlty (59) orrespondngly we he < < λ r > λ > λλr Trnsformton (55)-(58) (9) Sne the onstnt n (55)-(56) hrterzes the deton of the new trnsformton (55)-(58) from the Lorentz trnsformton nd the ltter hs een erfed to ery hgh degree of ury y numerous eperments the lue of must e lose to It s mportnt howeer to determne how lose Note: Reently the Europen Centre for Nuler Reserh delred tht they hd deteted the neutrnos trel n speed of m/se Ths speed s greter thn the lght speed (= m/se) Ths Q

12 nformton supports the ntpton mde from the result of ths pper e there my est new lmt speed n the nture If we te ths neutrno s speed s the new lmt speed defned n (59)-(6) then the onstnt whh we hope to determne here my e dedued s = Ths lue tlles well wth our ntpton n degree of mgntude: lue smller thn ut ery lose to Ⅴ Deton of * from under < < We now estmte the deton of * the lght eloty ffeted y the moton of lght soure from Let Δ = * (9) pplyng (5) we he ( ) + ( ) Δ = + (9) + ges Mng n epnson n terms of ( ) ( ) ( ) Δ + ( ) + = + + (9) Mng further epnson n terms of / ( < < ) nd tng ppromton to the seond order of / we otn Δ ( )( ) (94) Ths s n estmte for the deton Δ wth n ury up to the seond nd / order of oth the smll qunttes ( ) Ⅵ Superlumnl moton of lght soure S Assume tht the oordnte system wth eloty stsfyng < S moes relte to oordnte system < (95) where stsfes (59) nd lght soure s fed on the orgn of S tht s the lght soure s mong relte to S wth eloty greter thn Let us onsder the lght sgnl emtted y the lght soure towrd the negte dreton of -s Aordng to the hypothess <4> the eloty of ths lght sgnl n S s On the other hnd the eloty of ths lght sgnl n S ordng to (5) s ( + ) ( ) * = = (96) + Beuse of nd stsfy (59) nd (95) respetely the quntty * n (96) s poste numer Thus udged n oordnte system S ths lght sgnl s trellng wy from the orgn O Ths mens tht f lght soure s mong wy from the oserer wth eloty fster thn tht of lght then the lght sgnl emtted towrd the oserer neer rehes the oserer A ody mong wy t superlumnl eloty nnot e seen Smlrly to the onept of l hole n the Generl Theory of Reltty we ould ll suh mong ody l mong ody Ⅶ Some Other Propertes of the Trnsformton (55)-(58) ()The lmt eloty = s nrnt y the trnsformton of eloty (7)

13 (ⅱ)Let P ( t y z) nd P ( t + dt + d y + dy z + dz) e two neghorng ponts n the spe-tme oordntes system S The quntty ds defned y the equton ds = ( ) dt d dy dz (97) s nrnt y the trnsformton (55)-(58) ( ⅲ )The Mwell eletrodynms equtons re not ornt under the trnsformton (55)-(58) Ths shows tht our result onflts wth the etnt theory of physs Howeer f eperment n proe tht the onstnt s not equl to nmely the lght eloty s dependent on the moton stte of lght soure then Mwell s equtons should he to e resed ⅧThe Ultmte Lmt Veloty As we n see the Lorentz trnsformton nd our trnsformton (55)- (58) show tht the spe-tme oordntes trnsformtons etween the nertl systems lwys ontn eloty s lmt eloty of ll the relte elotes of nertl system The orgn of ths result s the equlene of nertl systems whh leds to tht n the determnton of the oeffents of the trnsformton the ommon ftor possessed y ll the oeffents of the trnsformton should stsfy seond degree equton s shown n (6) The Spel Theory of Reltty sserts tht the lmt eloty s the lght eloty whle our trnsformton (55)- (58) shows tht the lmt eloty s the quntty defned n (6) whh s lrger thn nd wll supplnt the poston of Now nturl queston s whether ths new lmt eloty (f t ests nd s greter thn ) s n ultmte lmt eloty The lmt elotes nd relted here re generted usng lght we s tool for trnsmttng messges We n epet tht f there s n oet possessng eloty greter thn tht of lght n e used s tool for trnsmttng messges then t s possle to generte new lmt eloty s we he done here nd the new lmt eloty wll e een greter thn APPENDIX Proe of the Lnerty of the Trnsformton Lnng Two Inertl Frms y Usng Sngly the Mehnl Property of the Inertl Systems Let S nd S e two systems of nert mong relte to eh other Aordng to the property of nertl systems the property of moton to e unform nd retlner must e presered n gong from one nertl system to nother In the lterture [] wth ths sngle property of nertl systems (nmely no neessry of the ondton CVL) there hs eer proed tht the oordnte trnsformton lnng two systems of nert s trnsformton of lner frtons ll wth the sme denomntor Let us denote ths trnsformton s αβ β β = α = (A) + β β = β where the rles α nd β ( α β = ) re the orrespondng spetme oordntes of spe pont n the system S nd S respetely We

14 set = t ( lso = t ) nd suppose tht the oth orgns of S nd S ust onde t = nd = Another property of two nertl systems mong relte to eh other s tht ll the ponts of ny one of the two systems moe relte to the other system wth the sme eloty s tht of the system relte to whh these ponts re t rest We wll use ths property to proe tht the denomntor n (A) must e onstnt nmely ll the lues of β ( β = ) must e zero Let the eloty of S mong relte to S e ( ) (A) Let P ( ) ( = ) e n rtrry fed pont of S nd P( ) e the orrespondng mong pont n S then we he d = ( = ) (A) d d = ( = ) (A4) Dfferenttng the logrthm of oth sdes of (A) nd usng (A) - (A4) ges + = = = β = β β + + β = β β ( = ) (A5) After smplfton we he + + ( ) + = = + + = = = = = (A6) Now we ssume tht the seleted fed pont n S to e the orgn P() nd onsder ts moton n S t = the orrespondng pont of P() wll e the orgn P() of S t ths moment Under ths ondton equton (A6) ges + = (A7) = nd now (A6) eomes ( ) + = = = = (A8) Further we ssume tht the seleted fed pont n S to e pont other thn the orgn P () ut t wll pss the orgn P() of S durng ts moton Oously when ths pont ondes wth P() the tme n S wll e n ths stuton(a8)ges ( ) = = (A9) From (A7) nd (A9) we get 4

15 + = (A) = We then onsder the eloty of n rtrry pont P( α ) fed n S mong relte to S Let the eloty of S mong relte to S e ) (A) ( nd the orrespondng mong pont of P ) n S e P ( ) Then we he d d = ( = ) (A) nd d = ( = ) (A) Dfferenttng (A) nd pplyng (A) nd (A) ges + β β β β d β = β = = = d + β β β β β = β = ( = ) (A4) After smplfton we he ( α ( ) [ ( ) ( ) ] = ( = ) = P( α α (A5) Beuse the pont ) n e seleted rtrrly we n otn the followng two equltes from (A5) = ( = ) (A6) nd ( ) = (for ll spe ponts ( ) = ) (A7) Equlty (A7) mples the followng two possltes: = (A8) or ( ) = (for ll spe ponts ( ) = If (A8) holds then fter susttutng t nto (A) we get = ) (A9) = (A) Due to the eloty omponent n e ny lue the epresson (A) should led to = ( = ) (A) The results of (A8) nd (A) show tht (A) must e lner trnsformton = ( α = ) (A) α β = αβ β The equlty (A9) nnot e ld Sne f we pply t nd (A6) to (A) we wll otn the followng trnsformton equtons 5

16 + β = = = ( = ) (A) + β β + + β = = = β β (A4) Ths trnsformton trnsforms ll the ponts of S nto strght lne n S : = ( = ) (A5) Nmely the trnsformton (A) - (A4) s degenerte trnsformton tht s oously unresonle Therefore (A) s the only possle result ACKNOWLEDGMENT The uthor thns Dr M Mrshll for hs refully redng nd resng ths pper nd for hs lule suggestons The uthor lso thns Prof L Lu Prof SL Co Prof J Luo Dr HS Lu nd Prof MC Chu for helpful dsussons The uthor wshes espelly to thn Mr L Tsu for hs deeply onerned t ths wor REFERENCES [] HA Lorentz A Ensten H Mnows nd H Weyl The Prnple of Reltty (Doer Pultons In 9) 5-65 [] V Fo The Theory of Spe Tme nd Grtton Append A Pergmon Press London 959 6