Nonstandard Lorentz-Einstein transformations

Transcription

1 Nostadard Loretz-istei trasformatios Berhard Rothestei 1 ad Stefa Popesu 1) Politehia Uiversity of Timisoara, Physis Departmet, Timisoara, Romaia brothestei@gmail.om ) Siemes AG, rlage, Germay stefa.popesu@siemes.om Abstrat. The stadard Loretz trasformatios establish a relatioship betwee the spae-time oordiates of the same evet whe deteted from two iertial referee frames I ad I i the stadard arragemet. This evet is haraterized by the spaetime oordiates (x,t ) ad (x, t! ), t ad t! represetig the readigs of the stadard syhroized loks C(x) ad C (x ) loated i the two frames where the evet takes plae. We obtai the ostadard Loretz trasformatios establishig a physially orret relatioship betwee the readigs of the stadard syhroized loks ad the readigs of other loks (t a, t! a ) of the same iertial referee frames. This relatioship of the type t =f(x,t a ), t! ( x!, t! ) expresses the stadard Loretz a trasformatios as a futio of t a ad t! a respetively. We preset several ases of ostadard Loretz trasformatio (the ase of radar detetio, the ase whe oe referee frame is filled with a ideal trasparet dieletri ad the ase of relativity of the apparet, atual ad syhroized positios of the same movig partile). 1. Itrodutio Leuber, Aufiger ad Krumm 1 state that I view of persistig ofusio i the literature, it is ot surprise at all that the udergraduate ourse of speial relativity passes very swiftly over the questio of lok syhroizatio. So swiftly i fat, that the begier is left with the impressio that there exists a sigle ad essetially uique lok syhroizatio proedure (amely the stadard or istei proedure despite the qualifier ovetio used by the istrutor. As a osequee, studets (ad also may teahers) erroeously idetify the properties of spae-time with the familiar form that ertai syhroizatio depedet quatities assume uder the stadard syhroizatio (legth otratio, time dilatio). Stimulated by this statemet the teaher starts to prepare his ow leture o the subjet. Lookig for referees, he fids out that may approahes to the subjet are based o otios like base vetors 1, the wave equatio i a aisotropi spae. These oepts are ot i a easy reah for begiers ad they make little use of istei s speial relativity theory. A experieed teaher kows that suh approahes fid little audiee i a lass of studets who already kow about istei s speial relativity ad who osider that they are obliged to lear agai the subjet. He also likes to 1

2 avoid usig the oept of two-way speed of light, a oept used by some other approahes to the subjet. 3. Cosider the evets (x,y,z,t ) ad ( x!, y!, z!, t! ) deteted from the iertial referee frames I ad I i the stadard arragemet. The evets are haraterized by their Cartesia spae oordiates (x,y,z) ad (x,y,z ) ad by their time oordiates t ad t! respetively. If the two evets take plae at the same poit i spae ad the orrespodig time oordiates represet the readigs of two loks C(x,y,z) ad C (x,y,z ) loated at the poit where the evet takes plae, eah syhroized i its rest frame with the other loks of that referee frame followig a syhroizatio proedure proposed by istei 6, the the orrespodig spae-time oordiates are related by the stadard Loretz-istei trasformatios x! + t! x = (1) t! + x! t = () y = y! (3) z = z! (4) The iverse Loretz-istei trasformatios are obtaied by simply hagig the orrespodig uprimed physial quatities with primed oes ad hagig the sig of - the relative veloity of I ad I. The stadard Loretz-istei trasformatios are the osequees of the two postulates of istei s speial relativity theory: (I) The priiple of relativity: assertig the existee ad the equivalee of iertial referee frames. (II) The priiple of the ostay of the oe-way speed of light: assertig the ostay of the oe-way speed of light i empty spae relative to these frames. Otherwise stated the light propagates i all diretios of empty spae with the same speed. I order to obtai a ostadard Loretz trasformatio we establish a physially orret relatioship betwee the readig of the stadard syhroized lok ad the readig of aother lok at rest i the same iertial referee frame expressig the Loretz trasformatio as a futio of the readig of the later lok. Followig this strategy we have preseted reetly a simple approah to the osequees of Leuber s everyday lok

3 syhroizatio. 1 The aim of this paper is to preset several o-stadard Loretz trasformatios attrative by their simple ad trasparet shape.. Nostadard Loretz-istei trasformatios.1. Radar detetio of the spae-time oordiates of a evet The seario we follow is skethed i Figure 1. It is deteted from the iertial referee frame I i a two-spae dimesios approah. It ivolves the stadard syhroized loks of the I frame C (x y ) loated at the differet poits of the plae defied by the axes of I ad lok C! (0,0) 0 loated at the origi O, all displayig the same ruig time. Y' y'! ' r' O' C o ' S o ', x' X' Figure 1. Seario for the radar detetio of a evet as deteted from I. Whe lok C! 0 reads t! e, the soure of light S! (0,0) 0 loated at the origi O emits light sigals i all diretios. Oe of the light sigals emitted alog a diretio that makes a agle θ with the positive diretio of the O X axis arrives at the loatio of the lok C the positio of whih is defied by the Cartesia spae oordiates (x y ) or by the polar oordiates (r,θ ), r represetig the legth of the positio vetor ad θ the polar agle. Arrivig at lok C, the light sigal geerates the evet " ( x" = r" os!"; y" = r" si!"; t" = t" e + r" / ) haraterized by the spae oordiates x =r osθ ; y =r siθ ad by the time oordiate t! = t! e + r! /. Deteted from the iertial referee frame I, the same evet is haraterized by the spaeoordiates: / (os!" / ) t" x" t" e x = = r" r" / (5) y = y" = r " si! " (6) the legths of the positio vetors trasformig as: 3

4 / [( os!" + / ) + t" e] r = x + y = r" si!" + r " / (7) 1 # / The trasformatio equatios derived above are a typial example of ostadard Loretz-istei trasformatios whih eable observers from I to express the legth of the positio vetor r deteted from I as a futio of its legth deteted from I ad of the ruig time t! e displayed by the lok C! 0 whe the radar sigal is emitted ad ot as a futio of the time oordiate t! displayed by the lok loated where the sigal is reeived. If whe deteted from I the radar sigal propagates alog a diretio θ the whe deteted from I it propagates alog a diretio θ, the two agles beig related by: si! " y ta! = =. (8) x / os! " + + t" e r " / The time oordiates of the ivolved evets trasform as r" t" + x" t" (1 os ) e + +!" t = =. (9) Beause r t = te + (10) we obtai that the emissio times trasform as r" / t" e + (1 + os!") (os!" ) r" + + t si r / e = #!" + ". (11) I the partiular ase whe t!=0 e the trasformatio equatios (7) ad (9) derived above beome os!" r = r" (1) ad 4

5 os!" t = t" = Dt" (13) the trasformatio beig performed via the Doppler fator D defied as os!" D =. (14).. Loretz trasformatios for the ase whe the iertial referee frame I is filled with a ideal trasparet dieletri at rest relative to it. Cosider that the iertial referee frame I is at rest i a trasparet perfet dieletri haraterized by its proper refrative idex > 1. A light sigal propagates there alog the positive diretio of the O X axis with speed =. (15) Cosider that at a poit M (x ) loated o the O X axis we fid the loks C! ( x! ) ad C! ( x! ). The first oe is syhroized with lok 1 C! (0) 0 loated at the origi O followig istei s lok syhroizatio proedure ad we suppose that it displays a time: x! t! =. (16) Clok C! x ( x! ) is syhroized with lok C! (0) 0 usig a syhroizig sigal that propagates with speed =/ displayig a time x! t! =. (17) Combiig (16) ad (17) we obtai x! t! = t! + (1 " ). (18) Performig the Loretz trasformatios to the iertial referee frame I the result is: x! [1 + (1 " )] t! x = + (19) x! (1 ) t! " + t = +. (0) 5

6 Now hoose to be the refratio idex that makes equatio (0) idepedet of x i.e. 1! + = 0 (1) whih solves as = () ad so, i that partiular ase =. (3) Substitutio i (19) ad (0) gives t 1! x = x! " + (4) ad t! t =. (5) The iverse trasformatios obtaied by ombiig (4) ad (5) are x! t x" = (6) 1! t! = t. (7) x x The speeds u = ad u! = are related by t t u! u" = (8) 1! the iverse trasformatio beig (1 u ) = u! " +. (9) We have reovered that way the results obtaied by Selleri 4, Abreu ad Guerra 5. Selleri 4 derives the equatios we have obtaied above startig with the followig assumptios: (i) Spae is homogeeous ad isotropi ad time homogeeous, at list if judged by observers at rest i I: 6

7 (ii) I the isotropi system I the veloity of light is i all diretios, so that loks a be syhroized i I ad oe-way veloities relative to I a be measured; (iii) The origi of I, observed from I, is see to move with veloity < parallel to the +x axis, that is aordig to the equatio x=t; (iv) The axes of I ad I oiide for t=t =0; (v) The two-way veloity of light is the same i all diretios ad i all iertial systems; (vi) Clok retardatio takes plae with the usual veloity depedet fator whe loks move with respet to the isotropi referee frame I. The fat that our approah preseted above ad the oe proposed by Selleri lead to the same result is due to the fat that I is filled i our ase with a trasparet dieletri ad its loks are beig syhroized by a sigal propagatig with speed =/. Abreu ad Guerra 5 ad obtai similar results due to the fat that they derive equatio (6) as a result of the legth otratio effet ad equatio (7) as a result of the so alled exteral lok syhroizatio..3. Speial relativity ad the apparet, atual ad syhroized positios of the same partile, movig with ostat speed relative to I i the positive diretio of the OX axis. I order to keep the problem as simple as possible osider the oe spae dimesios seario skethed i Figure as deteted from I. Y a S C 0 O x a C a M a x X Y b S C 0 O M a x a X M a x Figure. Seario for the defiitio of apparet positio M a ad of the atual positio of the same partile movig with ostat speed i the positive diretio of the overlapped OX axis. 7

8 I Figure a M a (x a ) represets the loatio of a partile that moves with speed i the positive diretio of the OX axis whe the stadard syhroized loks of I read t=0. At same time a soure of light S(0) loated at the origi O i frame I emits a light sigal i the positive diretio of the OX axis. We all M a (x a ) the apparet positio of the movig partile. I Figure b M(x) represets the loatio of the same partile whe the light sigal arrives at poit M a. M(x) represets what we all the atual positio. The oordiates x a ad x are related by x = xa + xa (30) The evet a (x a,t a =x a /) is assoiated with the arrival of the light sigal at the apparet positio whereas the evet (x=x a (/),t a =x a /) is assoiated with the arrival of the partile at the atual positio ad the arrival of the light sigal at the apparet positio, the two evets beig simultaeous but takig plae at differet poits i spae. X x s x x a WL() D WL() C x a A B O t a Figure 3. A lassial spae-time diagram displays the evets assoiated with the apparet positio B, with the atual positio D ad with the syhroized positio C. Figure 3 illustrates the seario we propose o a lassial spae-time diagram o the axes of whih we measure the spae oordiates x ad the produt t betwee the speed of light i empty spae ad respetively the time oordiate of the ivolved evets. WL() represets the world lie of the light sigal emitted at t=0 from the origi O whereas WL() represets the world lie of the partile movig with speed desribed by (30). M a represets the evet assoiated with the apparet positio whereas M represets the evet assoiated with the atual positio. Performig the t s t 8

9 Loretz trasformatios of the spae-time oordiates of evet to the rest frame I of the movig partile we obtai: ad x! = xa[ " ] (31) 1 t! = ta[ " ]. (3) The itersetio of the world lies WL() ad WL() geerates the evet s haraterized by the spae oordiate x s obtaied from (30) xs = xa + xs (33) i.e. xa xs =. (34) 1! vet s is haraterized by a time oordiate t s =x s /. Performig the Loretz trasformatios to the rest frame of the movig partile I we obtai the ostadard Loretz trasformatios x! = s x a. (34) ta t! =. resultig that u! = us. (35) u! ad u s represetig the speeds of the same partile measured i I ad i I respetively. The loatio M s of the movig partile represets i this ase what we all the syhroized positio. 9

10 3. Colusios The importat otributio of our paper osists i the fat that it shows that the stadard Loretz trasformatios aout for the osequees of ay arbitrary but physially orret lok syhroizatio proedures. Moreover they lead to the orrespodig trasformatio equatios for the spae-time oordiates of the ivolved evets. Referees 1 C.Leuber,K.Aufiger ad P.Krumm, lemetary relativity with everyday lok syhroizatio, ur.j.phys. 13, (199) Abraham A.Ugar, Formalism to deal with Reihebah s speial theory of relativity, Foudatio of Physis, 1, (1991) 3 H.Reihebah, Axiomatizatio of the Theory of Relativity, tras.ad ed.by M.Reihebah, Uiversity of Califoria Press, Berkeley (1969) 4 F.Selleri, No-ivariat oe-way speed of light, Foudatios of Physis, 6, (1996) 5 Rodrigo de Abreu ad aso Guerra, Relativity :istei s lost frame, ]xtra[muros 005 ad referees therei. 10