The length of a laterally-moving rod

Transcription

1 Th lngth of a latrally-moing rod Johan F Prins CATHODIXX 8 Portland Pla, Northliff t. 15, Johannsburg 195, South Afria johanprins@athodi.om Kywords: Lorntz-transformation, Spial Thory of Rlatiity, lngth-ontration, ohrnt wa motion, simultanity, ltron-wa, d Brogli s walngth, Lorntz-Fitzgrald ontration. Einstin usd th Lorntz-quations to transform th instantanous-simultanous position-oordinats at th bginning (moing-tail) and nd (moing-nos) of a rod, within an inrtial rfrn-fram rlati to whih th rod is moing with a spd, into th moing inrtial rfrn-fram within whih th rod itslf is atually stationary; and laimd that suh a rod ontrats whn it is moing. Hr, th hang in lngth of a rod passing at spd, is drid by Lorntz-transforming th stationary position-oordinats of th bginning and nd of th rod within th moing inrtial rfrn-fram (within whih th rod itslf is stationary) into th stationary inrtial rfrn fram (rlati to whih th rod is moing with spd ): An inras in th lngth of th moing rod is obtaind. It is shown that this lngth-dilation is dmandd by any moing mattr-ntity in ordr for this ntity to ha a d Brogli walngth. 1

2 1. Bakground and introdution 1.1 Einstin s postulat Thr ists no bttr dmonstration of Einstin s gnius than his insight in 1905 that th Lorntz-transformation mandats that th spd of light, masurd rlati to diffrnt bodis moing rlati to on anothr, must ha th sam magnitud ms rlati to any, ah, and all of ths bodis [1]. In trms of Galilan trminology, all bodis whih ar stationary rlati to on anothr, jointly dfins an inrtial rfrn-fram (IRF); whil all bodis moing with th sam loity rlati to ths stationary bodis, also dfin an inrtial rfrn-fram (IRF) within whih th lattr bodis ar stationary. Sin thr ar many bodis moing with many loitis rlati to on anothr, an infinit st of IRF s ists within our Unirs. Th rlati-motion of diffrnt bodis has bn isualisd in an abstrat mannr by th motion of diffrnt IRF s within ah of whih thr ar bodis whih ar stationary; and whr th motion of suh an IRF is mathmatially modlld in trms of a Cartsian oordinat-systm that is moing through Eulidan spa. Th rspti positionoordinats (,y,z ) and (,y,z) within two inrtial rfrn frams IRFK and IRFK, passing on anothr with a rlati spd ar ompard by assuming that th tims on two loks within IRFK and IRFK rsptily ha bn synhronizd to both rad zro whn th origins of th oordinat systms oinidd. Although th Lorntz transformation from on IRF into th othr was known bfor Einstin postulatd that th spd of light must always b th sam rlati to all bodis in th unirs, no mattr with what spd suh a body mos rlati to othr bodis, it is at prsnt aptd that this postulat is rsponsibl for th physis-rality whih dmands th alidity of th Lorntz-quations. 1. Th Lorntz-transformation A primary nt is dfind as an nt within an IRF whih will our at th sam positionoordinats within this IRF if it wr to our at a latr or arlir tim []. If it has to our at diffrnt position oordinats within an IRF at diffrnt tims, it is not a primary nt within this IRF. In othr words to b a primary nt, th aus of th nt must b stationary within th IRF within whih th nt ours. Th Lorntz-transformation (LT) for a primary nt at tim t and position (,y,z ) within IRFK into IRFK, follows as: t γ( t ) (1a) LT LT y y (1b) And LT z z (1)

3 t t γ LT t (1d) In ordr to mak physis-sns, only a primary nt within IRFK an b transformd from IRFK into IRFK by mans of ths quations. If a primary nt ours at tim t and a position (,y,z) within IRFK, it an b transformd into IRFK by mans of th rrs Lorntz-transformation, whih is gin by th following quations: LT t γ( t) (a) y LT y z LT z (b) () And t t LT γ t (d) It dos not mak physis-sns whn a primary nt within IRFK, aftr bing Lorntz transformd into IRFK, is transformd bak into IRFK by mans of this rrs Lorntz transformation []. Only an nt whih is a primary nt within IRFK an b transformd into IRFK by mans of th rrs Lorntz transformation. It has bn found that, ontrary to what has bn blid for mor than 100 yars, th tim t LT for a LT-nt within IRFK is not simultanous with th tim t within IRFK, whn th primary nt ours within IRFK []. Whn th primary nt ours within IRFK at th tim t, th lok within IRFK simultanously shows th tim t whr tt. Similarly, whn th transformd nt is obsrd within IRFK at th LT tim t LT, at th position ( LT,y LT,z LT ), th tim on th lok within IRFK shows simultanously th sam tim t LT. Furthrmor, owing to th non-simultanous tims for a primary nt within IRFK aftr bing Lorntz transformd into IRFK, thr is not any Lorntz-Fitzgrald lngthontration. In fat, thr is just th opposit, namly a lngth-dilation. Einstin, howr, drid that a rod (or mtr stik) whih is stationary along th -dirtion within IRFK will ontrat whn it is obsrd within IRFK whil passing by with a spd. In doing so, Einstin transformd non-primary nts within IRFK into IRFK by using th rrs Lorntz-transformation. Hr Einstin s driation is risitd, analysd, modifid and disussd. 3

4 . A passing rod.1 Einstin s driation For a stationary rod of lngth L within IRFK, whih mos past with a spd rlati to IRFK, Einstin ddud a ontration in th rod within IRFK so that it has a lngth L<L. H motiatd this driation as follows [3]: I pla a mtr-rod (Einstin hos th lngth of th rod as unity: In th prsnt as th rod will b assumd to ha a lngth L ) in th -ais of K in suh a mannr that on nd (th bginning) oinids with th point 0, whilst th othr nd (th nd of th rod) oinids with th point L. What is th lngth of th mtrrod rlatily to th systm K? In ordr to larn this, w nd only ask whr th bginning of th rod and th nd of th rod li with rspt to K at a partiular tim t of th systm K. By mans of th first quation of th Lorntz-transformation th alus of ths two points at th tim t0 an b shown to b: * (bginning of th rod) 0 *(nd of th rod)l th distan btwn th points bing L. But th mtr-rod is moing with th loity rlati to K. It thrfor follows that th lngth of a rigid mtr-rod moing in th dirtion of its lngth with a loity, is of a mtr (i.. of th lngth L ). Th rigid rod is thus shortr whn in motion than whn at rst. Einstin s driation rsts on th inhrnt assumption that th bginning of th rod and th nd of th rod ar simultanously prsnt within th IRFK. To mphasiz hr that h mad this assumption, w will rpat part of th quot abo: What is th lngth of th mtr-rod rlatily to th systm K? In ordr to larn this, w nd only ask whr th bginning of th rod and th nd of th rod li with rspt to K at a partiular tim t of th systm K. Einstin thn transformd th bginning ( b 0) and nd ( L) oordinats of this supposdly, instantanous lngth L within IRFK from IRFK into IRFK, by using th rrs Lorntz-tranformation: But, sin th front and nd positions of th rod hang with tim within IRFK, thir instantanous positions ar not primary nts within IRFK, and an thrfor not b transformd into IRFK by mans of th rrs Lorntz transformation. En if this would ha bn physially allowd, ths rrs-transformd oordinats of th bginning and nd of th rod annot b simultanous within IRFK. Thus, thy annot rlat to th atual lngth L of th rod within IRFK whos front and nd oordinats ar simultanously always th sam within IRFK. 4

5 1. Corrtd driation Th Lorntz-transformd simultanous-oordinats of two stationary nt-positions spad any distan apart within any IRF, annot b simultanous in any othr IRF passing by: This would iolat th rlatiisti non-simultanity of simultanous nts. Sin th rod L is stationary within IRFK, its bginning ( b 0 ) and nd ( L ) oordinats ar at any instant in tim simultanously always th sam within IRFK : Ths positions ar thus primary nts within IRFK : and thrfor Einstin should ha usd th Lorntz-transformation (Eq. 1) from IRFK into IRFK: NOT th rrs transformation (Eq. ). Aording to Eq. 1a h should ha st L and LT LT L in ordr to obtain that: L L γl (3) Einstin should thus ha found that th transformd lngth of th rod boms longr within IRFK; not shortr! Furthrmor, Einstin should ha usd th whol Lorntz-transformation whih gis that th oordinat L at th nd of th rod is not prsnt within IRFK at th sam tim t0 at whih th oordinat 0 is prsnt at th bginning of th rod. Eq. 1d larly dmands that th oordinat L an only b prsnt at a latr tim γ (L ) ; whih is largr than t0. In ordr to atually dri whr th positions of th bginning of th rod and th nd of th rod ar simultanously at a partiular instant in tim within IRFK, Einstin should ha usd th oinidnt oordinats at any tim t, gin in his ampl at t0, for th bginning of b b th rod by 0 and for th nd of th rod by L : Whn subtrating ths position-oordinats, at say tim t0, th sam instantanous lngth for th rod is obtaind within IRFK than th atual lngth of th rod is within IRFK. Thus, at any singl instant in tim ths positions ar simultanous-instantanous atly L apart within both IRFK and IRFK; n though, owing to th Lorntz-transformation, an obsrr at th origin 0 within IRFK annot s this instantanous lngth []. Aording to th Lorntz-transformation suh an obsrr must s a longr lngth, and, in addition, a hang in tim along this lngth. But th quation for th inrasd lngth L dos not ontain any mathmatial trms that rlat to th lattr hang in tim along th rod: Th lngth L is indpndnt of tim. What is th maning of th Lorntz-transformd lngth L>L of th moing rod within IRFK? Could th rod ha a diffrnt lngth if its bginning within IRFK has not bn hosn to b at th origin 0, and if th Lorntz transformation was not don at th tim t 0? 5

6 3. Transforming th rod at any position and instant in tim If th LT-transformd lngth L is a ral-physial inras in th stationary lngth L, it should not hang whn obsrd from diffrnt positions withinh IRFK: i.. th sam lngth L must b obtaind whn LT-transforming a rod with lngth L from a bginning oordinat b 0 and at at any tim t 0. Choosing th position of th bginning of th rod within IRFK to b any oordinat b, so that th oordinat at th nd of th rod must b L, th Lorntz-transformd position-oordinats along th -dirtion at any tim t ar: b And: b t γ(b t ) (4a) LTb t γ(b L t ) (4b) LT Th Lorntz-transformd (LT) lngth L is obtaind as: L L LT LTb γl (5) Thus, th rod an b at any position within IRFK and th transformation an b don at any instant in tim t to obtain th SAME lngth L as in Eq. 3. Th tim t LTb whn th bginning of th rod is at th oordinat LTb, and th tim t LT whn th nd of th rod is at position LT, ar rsptily gin by: And: t b t LTb γ t b (6a) t t LT γ t ( b L ) (6b) Thus, th tim diffrn T rod btwn th position-oordinats at th bginning and th nd of th rod is: 6

7 T t t γl L rod LT LTb (7) Th tim diffrn also dos not hang with tim t: It rmains a onstant alu whih is proportional to th LT lngth L. Th lngth L, and thus also th tim diffrn T rod, ar funtions of only th stationary lngth L and th spd of th rod: i.. Both ar indpndnt of th tim t on all th loks within IRFK within whih th rod of lnghth L is stationary and th synhronous tim tt within IRFK rlati to whih th rod is moing with a spd. Assum now that th nd position of th rod L is at th oordinat 0 and th bginning thus at b L : Th LT position-oordinats at tim t ar thus: And: b t γ( L t ) (8a) LTb t γ(0 t ) (8b) LT By subtrating LTb from LT, on again obtains th rlationship gin by Eq. 3. Th orrsponding tim oordinats t LTb and t LT ar: And: t b t LTb γ t L (9a) t LT γt (9b) t By subtrating t LTb from t LT on again obtains th tim diffrn gin by Eq. 7. Thus whthr th rod is approahing th origin 0 within IRFK, or rding from this origin, it has th sam tim-indpndnt lngth L within IRFK. Now onsidr an obsrr M within th IRFK who has a stop-wath: Whn th nd of th rod (nos) at 0 passs th obsrr at th origin of IRFK, th obsrr starts th stop-wath. Aftr a tim t th distan btwn th origins must b t. Considr th synhronous tim t R t R on th loks within IRFK and IRFK at whih th distan 7

8 btwn th origins is L t t []: Th LT oordinat-positions of th rod ar at this instant in tim, aording to Eq. 8: R R γ( L t ) 0 (10a) LTb R Eatly what on pts that it should b aftr th tim t R L. But: LT γ(0 tr ) γl (10b) Whih is again th sam as Eq. 3. Although M s stopwath pros that th distan btwn th origins must b qual to th stationary lngth L of th rod, th LT-lngth of th rod is longr than L! Thus th instantanous lngth of th rod rmains L within both IRFK and IRKK, but th rlatiisti lngth, whih dtrmins th physis within IRFK, is L. On is thus ford to apt that in th as of a passing rod th prmannt lngth of th rod within IRFK must b L γl, AND also that within th rod thr is a prmannt tim diffrn T rod btwn th bginning and th nd of th rod. Tim inrass with lngth from th bginning of th rod to th nd of th rod. 4. Mass-nrgy of a moing rod En though tim inrass along th rod of lnghth L, th ddution that L is a onstant physially-ral lngth of th passing rod within IRFK, is supportd by th fat that, aording to Einstin s famous formula Em, th rst mass (say m 0 ) of th rod must inras to add dynami-mass whn it mos at a spd, so that th total mass boms m; whr on has for m in trms of th rst-mass m 0 that: m m0 (11) If th rod has a ross-stional ara S, its rst-mass olum must b SL, and its dnsity ρ rod m 0 (SL ) : Assuming that th inras in lngth, gin by Eq. a, is a ral inras in mattr-nrgy, th rod s inras in mass, gin by Eq. 11, mandats that th mass-dnsity of th rod rmains ρ rod for any spd : i.. m0 m ρ rod (1) SL SL This sms to b a rasonabl rsult. If, in ontrast, th rod had atually ontratd in lngth as Einstin had argud, its mattr-nrgy-dnsity would ha had to inras to aommodat this inras in mass. It is mor rasonabl to assum that if th mass-nrgy dnsity 8

9 has a rtain alu whn th rod is stationary, it must also ha th sam dnsity whn th rod mos and boms longr. Whn th rod is stationary with lngth L, th onstitunts of th rod ar atoms bondd by ltrons: Dos this man that a moing rod, whih nlongats, onsists of mor atoms and mor ltrons than it dos whn it is stationary? On pts that whn a singl atom is moing, this atom should also bom longr in th dirtion along whih it mos to aommodat its own inras in mass-nrgy: This dos not nssarily man that suh a singl atom must sprout tra atoms to form a row of atoms whih is moing with th spd. If this ould happn, it would rquir that th moing atom must inras its mass-nrgy by quantum-stps whn inrasing its spd! On, howr, pts that th inras in mass-nrgy should b ontinuous for any body with mass whn th spd of th atom inrass ontinuously. This mans that th inrasd nrgy must b ontinuoiusly distributd within th inrasd olum of th moing mattr-ntity. 5. A solitary moing ltron 5.1 Lngth-inras An ltron s mass-nrgy must also inras with spd: Thrfor, on pts that an ltron should also inras in lngth along th dirtion in whih it is moing. It is, also in this as, unlikly that it will sprout tra ltrons to inras it s amount of mattr-nrgy by forming a string of ltrons: Th lattr snario would rquir an inras in harg, whih has not bn obsrd for fast-moing ltrons. Furthrmor, th ltrons forming suh a string will plod away from on anothr. It sms omplling to onlud that it must b th atual mattr-nrgy onstituting th singl ltron that inrass. Sin an ltron has not bn found (so far) to b diisabl into smallr sparat omponnts, and sin th ltron s olum is ptd to inras whn its spd inrass (owing to its onomitant lngth inras), this mattrnrgy must b ontinuously distributd within a onfind spa that dlinats th siz of th ltron: Thus, thr must ist distributd mass-nrgy within th olum of th ltron. If this ddution is orrt, a moing ltron must b a moing, distributd nrgy-fild in its own right. And sin a moing nrgy-fild is a wa, it implis that a moing ltron must b an atual wa; and nothing ls but an atual wa. 5. Tim-diffrn Th tim diffrn aross a rod gin by Eq. 6, only applis whn thr is a stationary mattr-ntity with lnghth L within a moing IRFK passing by with a spd within IRFK. If th Lorntz-transformd inrasd lngth L is ral, th tim-diffrn might, and probably dos rlat to a proprty of th mattr-ntity whih hangs whn it mos. Sin a moing ohrnt-wa has a hanging phas-angl at ry point along th lngth of th wa, on might ntur to assign a onomitant phas-tim whih hangs along th lngth of th wa. Thus th hang in tim along th inrasd lngth might b 9

10 furthr idn that th fr motion of any mattr-ntity is always nothing ls than ohrnt wa-motion. Th simplst objt with mass is obiously an ltron: Assuming that a stationary ltron has a sphrial shap with diamtr L, an ltron moing with spd (and thus momntum p m ; whr m is th sum of th rst-mass and dynami mass) should ha a lngth L gin by Eq. 4. Thus, if it has a walngth λ th numbr n of walnghths within its lngth L must b gin by: L n (13) λ If th wa has a frquny ν th diffrn in phas tim T aross th lngth of th ltron is gin by: n L T (14) ν ν λ Stting L qual to L within Eq. 7, and quating Trod with T, gi: m λ ν (15) m If on now rplas m with th Plank formula hν, Eq. 15 boms: h h λ (16) m p Whih is d Brogli s formula for th walngth of an ltron-wa. 6. Disussion In iw of th driations abo, th onpt of wa-partil duality is suspt: Th rsults abo imply that moing mattr onsists of ltromagnti-nrgy whih mos at a spd that is lss than th spd of light. Sin th distributd ltromagnti-nrgy (whih is th ltron) is mass-nrgy, th ltron-wa must ha a ntr-of-mass whih mos lik a point -partil. Thus, it an b argud that th wa and partil bhaiours ar not two mutually lusi attributs of an ltron whih ar omplntary : Both bhaiours ar a dirt rsult of th fat that th ltron itslf is ltromagnti fild-nrgy and that its intnsity rlats to this fild-nrgy and not to a probability-distribution. Sin thr ar many IRF s within whih th ltron simultanously mo with diffrnt spds, this dmands that th ltron simultanously onsists of diffrnt sizs and shaps to aomodat diffrnt amounts of mass-nrgy within ths diffrnt IRF s. It is 10

11 thus in prinipl possibl that within an IRF, rlati to whih an ltron mos with a spd nar th spd of light, th ltron-fild might b abl to strth aross parss; whil, in ontrast, th sam ltron might only b a fw µm long within anothr IRF rlati to whih it is moing ry slowly. This must man that whn a fast-moing ltron is slowd down, its wa intnsity must ollaps into a smallr olum. In th as whr th stopping of th ltron is nar-instantanous, th ollaps of th wa s olum must also b narinstantaous. Th spd of ollaps is not limitd by th spd of light. On thus pts that whn an ltron impings at high spd into a matrial, it will ha a long lngth along its dirtion of motion bfor intrating with th matrial. Whn ntring th matrial two typs of intrations ar possibl: (1) If th ltron is rapidly slowd down, it will ollaps into a smallr olum and ollid lik a loalisd ntity with a ntr-of-mass (a partil ). () If th atoms within th matrial form a suitabl priodi array, th ltron-wa might rathr difffrat. Nonthlss, in both ass th ltron IS and REMAINS a singl holistiwa with its intnsity qual to its distributd mass-nrgy. 7. Conlusion It is omplling to onlud that Einstin s driation that a moing body with mass will ontrat in lngth is not orrt. Th nd for lngth-ontration has bn rmod by Einstin s own postulat that th spd of light must ha th sam alu rlati to any moing body. Bfor this postulat, th Lorntz-Fitzgrald ontration was rquird to dri th Lorntz transformation by ombining this ontration with th Galilan transformation. Sin this ontration is not rquird whn driing th Lorntz-transformation in trms of th onstany of th spd of light, it has bom irrlant. Th driation abo implis that Einstin s Spial Thory of Rlatiity in ssn prditd d Brogli s walngth mor than two dads bfor d Brogli postulatd this walngth. It also implis that moing mattr onsists of ltromagnti-nrgy whih mos at a spd that is lss than th spd of light. This in turn implis that mattr, whih is stationary within an inrtial rfrn fram, might b nothing ls but a stationary ltromagnti fild; whih mos lik an ltromagnti-wa within all th othr inrtial rfrn frams within whih th mattr-ntity is not stationary. Rfrns [1] A. Einstin, Zur Elktrodynamik bwgtr Körpr, Annaln dr Physik 17 (1905) [] J. F. Prins, Dirtional missions from a moing light sour: Coinidn and simultanity, Physis Essays (013) submittd. [3] A. Einstin, Rlatiity: Spial and Gnral Thory, Crown Publishrs, Nw York (195) p