Einstein s Derivation of the Lorentz Transformations in the1905 Paper is Internally Inconsistent

Transcription

1 Einsein s Deriion of he Lorenz Trnsformions in he195 Pper is Inernlly Inonsisen Jon C. Freemn Absr The generl onsensus in lierure onerning Einsein s 195 pper on speil reliiy, is h he independenly deried he Lorenz rnsformions using kinemi mehod. Th is he ws pprenly unwre of he ler work of Lorenz nd ohers while omposing he pper. Here i is demonsred h he deriion is mhemilly inonsisen. There re seerl errors in he deelopmen nd he min one is he inonsisen seps of requiring pril deriie o be zero one poin, bu neessrily finie he ls sep. This deriie n only equl zero if he relie eloiy beween he frmes goes o zero. I is lso shown h he sheme of following ligh bem refleing beween wo moing mirrors (he kinemi mehod) nno orrely rrie he rnsformions. A sudy of mny ebooks on reliiy will reel h his deriion is lmos neer used o inrodue he Lorenz rnsformions. Only one soure ws found h disussed his priulr mehod; nd i ws in n ppendi. Inde Terms Einsein, Hisory of Siene, Speil Theory of Reliiy. T HE firs deriion of he Lorenz rnsformions gien by Einsein ws in he lssi pper published in 195. To he bes of our knowledge his mehod ws neer gien gin in ny of his subsequen publiions. An up-o-de disussion on he pper by A. Mrinez [1] on he oneps nd ehnique is illumining. Briefly, he sys he deriion onins mbiguiies due o boh impreise noion nd possible muliple definiions. Furher inesigion here reels inonsisen mh seps h inlide he proedure. I is remrkble h he inonsisenies he no been noed in he lierure. One my jusifibly sk How n here be ny errors noed now, fer 11 yers sine he publiion? I is speuled h mos reders of he pper pprenly did no emp o fill in he omied seps where he problems rise. There hs been uneriny mong hisorins nd biogrphers bou he quesion if Einsein knew he form of he equions he wned, nd sough o deelop hem; rher hn obining hem direly from he deriion iself wihou prior knowledge. The resuls here nswer he quesion; yes, he mus he known he finl form, bu he seps used o ge hem were impreise. The equions hd been published prior o 195 so i is no surprising h Einsein ws wre of rious forms for he rnsformions. In his pperbk [] M Born, Nobel Luree nd personl friend of

2 Einsein, ge he following quoe in he Inroduion. Reliiy ully ough no be onneed wih single nme or wih single de. I ws in he ir bou 19 nd seerl gre mhemiins nd physiiss- Lrmor, Fizgerld, Lorenz, Poinre, o menion few- were in possession of mny of is onens. In 195 Alber Einsein bsed he heory on ery generl priniples of philosophil hrer, nd few yers ler Hermnn Minkowski ge i finl logil nd mhemil epression. In hper VI Born deeloped Einsein s kinemis nd did no use he ehnique gien in he 195 pper. Insed he used differen pproh o deelop he Lorenz rnsformions. MATH The Lorenz rnsformions re s follows:,, 1 1 ( ) (1) The onsn is gien he symbol in modern noion; bu we re dhering o Einsein s noion. The res frme uses oordines while he frme moing speed o he righ uses (,,,). By definiion nd re independen ribles s re nd. The sheme onsiders wo mirrors sionry in he moing frme. They re prllel o he -is nd sepred by n unspeified disne (whih is hosen s d for he momen). The lower edge of he lef mirror is he origin of he (,) plne. Now onsider hree suessie eens. They re: emission of phoon from he lef mirror, refleion he righ mirror, nd hen reurning o he lef one. Assume he ime insns reorded in he moing frme re, 1,. For n obserer in he moing frme he ime inerls for he phoon o rerse he disne beween he mirrors re equl, hus1 1, hen rerrnge o 1 [ ] 1. Ne ssume he ime insns for my be epressed s funion in erms of nd. Th is: Een,, nd re oordines of Een s mesured in he res frme. Similrly where 1 1,1, e. Ne he ses insed of soling for (,), sole for (,) where will be presribed ler.

3 Now he proedure for deeloping ; sr wih,, [ {1 1 { 1 { [1 1 ] 1 { [ ] 1 { [ ] ], reple wih ( + ), hen These mnipulions he rerrnged hings so h only nd pper on he righ side. Cn we sy his is (,)? If we do, n we lso sy nd n hen be reed s independen ribles? Sine = -, i ppers mus depend on. Howeer if one ssumes he lues for re only hose h keep onsn, hen (being onsn) is independen of. Thus we emp o jusify he independene of nd by sing: we re now following poin fied in he moing sysem. Le 1, hen he ls line of () is ely he epression for (,) s gien by Einsein. Thus,,, where differen inerpreion mus be gien for eh side of he equion. On he lef nd re independen wheres on he righ nd re reed s independen s we re following fied poin in he moing sysem. The funion (,) ws gien s he soluion of he following equion., [ ], (3) The seps o rrie (3) were no gien nd n emp will be mde o proide hem ler. Now he imes of fligh for phoon moing beween he mirrors s pereied by he res obserer will be deeloped. For he res obserer he mirror seprion is no ssumed o be d bu some lue o be deermined. For now le he seprion be L s mesured in he res sysem. A he insn of emission he righ mirror is L unis wy. During he fligh i moes he disne r where r is he fligh ime. The phoon mus rel he ol disne L + r. The phoon lwys moes speed, so he ol disne reled is r. Equing disnes we find r L. For he reurn rip fer refleion, he ol disne is now L - L. Agin he phoon s speed is, hus he ol fligh ime while moing o he lef is L L. The denominors in he preious epressions re he losing speeds beween moing mirror nd phoon. Ne Einsein fills ou he equion for he hree ime insns,,, [,,, (1 ){ [,,, ] ] A self-onsisen inerpreion is s follows. The righ side indies he firs slo is for. The () (4)

4 ls slo is for he s mesured by he res obserer. The firs erm on he lef mens he phoon emission ours some rbirry ime, whih is een. The righ side is een 1, nd he phoon is he disne from he lef mirror nd he ime is now + (-), where he seond erm is r found erlier. The seond erm on he lef is he reurn wih he ime now being he preious lue wih L dded. The phoon is gin he lef mirror so is disne from h mirror is zero (he firs slo). Thus eens nd 1 he phoon s disne from he firs mirror is he pproprie lue for. Therefore mus be he phoon s disne from he lef mirror differen insns s mesured in he res frme. Thus he mirrors re sepred he disne in he res frme. The ne sep is o show he seps o moe from (4) o (3). These seps were no gien by Einsein, nd only one soure hs been found h shows some of hem [3]. In his ppendi Prokhonik sys ke he pril wih respe o of (4), nd he gies 1 {1 1 [ [ ] 1 [ ] (5) This equion my be deeloped s follows. Sr wih he firs erm on he lef side of (4),,,,,,,. Is pril is where he subsrip mens elue he pril deriie =. The seond erm on he lef of (4) is,,, (6) For noionl oneniene wrie his s ( =,u) where u is he rgumen in he ls slo. Then u u u where we he pplied he hin rule. From (6), u so u 1 1, sine, s nd re ssumed o be independen. Then he lef hnd side is 1 { u 1 1 (7) Working on he righ side of (4); here le w w 1. Assembling ll erms we he 1 { 1 1 u [ w][1 ], hen using he hin rule gin we find Inspeion of (5) nd (8) shows hey n be equl under he following ondiions. Firs u = w =. This will be he se s pprohes zero. Howeer, he seond neessry ondiion (9) (8)

5 uses mjor problem. By inspeion of (), using he form where (1) =, ] [ {, (1) We see h (9) n only be sisfied if. This is he fl error whih shows he enire deriion is inlid. I is ineresing h will llow nd o be independen, for hen is jus. By inspeion, rerrngemen of (5) resuls in (3). Sine he preious seps do no ge us o he desired resul wihou inernl inonsisenies, le us ry noher ph. For smll, perhps Tylor epnsion migh work. Sr wih,,,,,,,,, Wrie s,,, Rell for Tylor series b b y f b y f b f y f,,,, Now wrie 1 s,,, 1 Colleing ll erms for (4) yields,,,,,,,,, { 1 (11) Obsere (,,,) nels whih lees 1 { Sine is rbirry, hoose i s zero, hen 1 { Therefore we re epnding bou he poin,. Then if we resri, hen we my diide i ou of he boe nd rrie 1 1 {1 1 Whih we obsere is (5) of Prokhonik. Therefore he sme problem menioned erlier is sill

6 presen. This pproh lso shows n ddiionl problem. Here we mus epnd bou bu we lso mus diide by! Een if his problem n be rgued wy by limiing eplnion, he issue of requiring nno be oerome. The primry onlusion is herefore; one nno rrie (3) sring from (4). This is he ph sed in he pper; hus he deriion is inlid. If, howeer, one jus srs wih (1) nd kes prils ssuming nd o be independen, hen Muliply he firs line by nd hen dd his line o he seond one nd rrie (3). Howeer, () nd (1) were deeloped by sring wih he known epression for. Conlusion The resuls of he nlysis show h he deriion gien in he 195 pper is inlid. I is flwed for seerl resons. The mos serious is he onrdiory requiremen h he pril of wih respe o mus be zero o rrie he pril differenil equion for deermining. Bu he soluion for from h equion does no llow his o be he se. The only wy i ould be sisfied is for o be zero. Anoher problem is h wih Tylor epnsion emping o ge he defining pril differenil equion, one mus diide by while sring wih equl o zero. A hird problem is he inonsisen ressignmen of independene nd dependene beween he hree ribles,, nd. The deriion srs wih nd s independen, hen defines s being dependen on boh. Then ler res nd s independen so is hen dependen. Ler fer deermining (,) he wries in is originl form nd goes bk o nd being independen. The reson for his swihing bk nd forh is supposedly jusified by sing h in some ples in he deriion one is following fied poin, while in ohers one is no longer doing h. Anoher problem is he inerpreion of. If i is indeed he disne from he lef mirror s pereied by he res obserer, hen we he pprenly se o uniy (sine ). I () is hen brough bk in he undeermined onsn fer ily ssuming direions perpendiulr o he rnslionl moion re he sme for boh obserers. Then for phoon moing erilly long he surfe of he lef mirror in he moing frme, he res obserer pereies. Also he phoon is pereied o moe in srigh line sloped upwrd nd o he righ. The lengh is, while he horizonl displemen is. The eril disne is s boh obserers gree on his lengh o be he sme. Using his righ ringle one finds he relionship beween nd for he se when. This hen shows 1, whih we knew lredy fer (). I is hoped h he hisory of he deelopmen of speil reliiy will be somewh lrified by onsidering he boe resuls. In [4] i ws sed h he definiie hisory of speil reliiy ws ye o be wrien; so some new informion here my be helpful. I seems h Einsein knew he Lorenz rnsformions nd ws emping o deelop hem using kinemi deriion. Howeer, he emp ws flwed. As he pper ge no referenes, mny subsequen uhors he inorrely ssumed he probbly ws no wre of mny of he ides h were in he ir s sed by Born.

7 APPENDIX Professor A. A. Mrinez hs deeloped n equion h will reple (4) when only nd re used ( no used). 1 {,,,,,,,,, (A-1) Noie h he funion (, ) is ssumed o be liner in boh ribles, nd in he boe equion he funion is h of ribles in erms of boh nd. Under he liner ssumpion he following relionships re rue., j k, j, k, b j kb, b j,, b kb b, Where j nd k re onsns. Afer epnding (A-1) in Tylor series, long bu srighforwrd lulion using he boe relionships will show h one mus ssume o rrie he finl equion (A-) Whih one n esily show o be he equion sisfied by (1). Inspeion of (1) shows. This neessies gin h one needs. Thus he mehod of following he phoon beween moing mirrors nno deelop he Lorenz rnsformions. ACKNOWLEDGMENT Thnks re due o Professor A. A. Mrinez s rile [1] for enlighenmen on he mbiguiies in he 195 pper. His pper sred he inquiry on he subje h led o his effor. REFERENCES [1] A. A. Mrinez, Kinemi subleies in Einsein s firs deriion of he Lorenz rnsformions, Am. J. Phys., ol. 7, no. 5, pp. 1-9, My, 4. [] M. Born, Einsein s Theory of Reliiy, 1965 ed. New York, Doer, Inroduion, p.1. [3] S. J. Prokhonik, The Logi of Speil Reliiy, New Souh Wles Uniersiy Press, [4] H. M. Shwrz, Inroduion o Speil Reliiy, MGrw-Hill, 1968, p. 3. Jon C. Freemn is reired from he NASA-Glenn Reserh Cener, in Cleelnd, OH He lies in Behwood, OH USA (e-mil: freem31@umn.edu)