On Einstein s Non-Simultaneity, Length-Contraction and Time-Dilation. Johan F Prins

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1 On Einsein s Non-Simulaneiy, engh-conraion and Time-Dilaion Johan F Prins CATHODIXX, P. O. Bo 537, Cresa 8, Gaueng, Souh Afia Non-simulaneiy of wo simulaneous eens whih our a differen posiions wihin an inerial referene-frame passing by a a speed ; lengh-onraion of a rod passing by a a speed ; and ime-dilaion aused by a lok passing by a a speed, hae been milesones of Einsein s Speial Theory of Relaiiy for more han years: Here, hese aspes are meiulously deried from he orenz-ransformaion: I is found ha he aual physis responsible for non-simulaneiy of wo simulaneous eens has no been orrely eplained by Einsein. I is also found ha lengh-onraion anno our a all, and ha ime-dilaion does no iolae Newon s onep of absolue ime a eery insananeous posiion in graiyfree spae.. Inroduion Galileo Galilei ) oniningly argued ha wihin an enlosed spae, like a ship s abin wih no porholes o look o he ouside, i is impossible o do any physial measuremen whih would be able o proe wheher he abin is uniquely saionary wihin he unierse or moing along wih a onsan speed relaie o a referene frame ha is really uniquely saionary wihin he unierse. This well-esablished behaiour, aording o whih any moing obje an be onsidered as being uniquely saionary wihin he unierse, has beome known as ineria. All referene-frames wihin our unierse whih moe wih onsan speeds relaie o one anoher are hus, in his sense, eah uniquely saionary; and herefore hey are known as Galilean, or inerial referene-frames. Galileo s logi obiously demands ha he laws of physis mus be he same wihin any inerial referene-frame as if suh a referene-frame is he only uniquelysaionary referene-frame wihin our unierse. If his were no so, one would be able o do an eperimen or a measuremen wihin an inerial referene-frame, whih an proe wheher suh an inerial referene-frame is uniquely saionary, or moing relaie o a really unique referene-frame ha is aually he only saionary referene-frame wihin our unierse. Newon ), in his famous Prinipia, quanified Galileo s ideas by inroduing he oneps of res-mass and momenum: Aording o Newon, he reason why a moing body is saionary wihin is own inerial referene-frame K, is ha he body has a res-mass m: Ineria is hus he resul of his res-mass resising any effor o moe a body ou of is sae of res wihin is own inerial referene-frame K. When, howeer, iewed from anoher inerial referene-frame K, whih is moing wih a speed relaie o K, he body wih mass m (ha is a res wihin K ) is seen o be moing pas a a speed wihin K. This, howeer, sill means ha his mass is a res wihin is own inerial referene frame: This moing ineria wihin K has been defined by Newon as momenum p; whih he alulaed by aking he produ of he res mass m and he speed relaie o K: i.e. p= m. His seond law posulaes ha in order o moe a body wih mass m wihin is own inerial referene frame K, wihin whih i is iniially a res, a fore mus be applied o suh a body. This fore aeleraes he body from is posiion of res, hus

2 ausing suh a body o gain momenum wihin he inerial referene-frame in whih i had been a res before he fore aed: Sine his body piks up speed, i is oninuously generaing new inerial referene-frames wihin whih i would be saionary if he applied fore is suddenly swihed off. As required by Galileo s fundamenal priniple of ineria, Newon s physis anno be used wihin an enlosed inerial referene frame o deermine wheher his referene frame is moing or no moing wih a onsan speed. Waes and heir ineraions had been well-known long before i was onlusiely esablished ha ligh onsiss of waes. Sine all preiously-known waes form wihin a subsane ha deermines he speed wih whih he waes an moe hrough his subsane, i was logially reasoned ha here mus be suh a saionary subsane (ha was alled he eher), whih mus fill he whole unierse so ha ligh waes moe wih a speed hrough his subsane. If here is suh a subsane wihin whih ligh-waes moe wih a speed, his subsane mus be uniquely saionary wihin our whole unierse: Einsein 3) realised ha if his is so, i will hus be possible o do a physial measuremen wihin Galileo s enlosed, inerial referene-frame, whih will allow one o deermine wheher one is moing relaie o his subsane; and o know wheher one is moing or no moing relaie o anoher ouside body (he eher) whih is uniquely saionary wihin he unierse. This an simply be done by measuring he speed of ligh along differen direions wihin Galileo s enlosed inerial referene frame. Einsein hus posulaed ha Galileo s ineria should no jus be alid for Newon s physis, bu also for all physis, inluding Mawell s equaions 4) for eleromagnei waes: This would mean ha one anno use he speed of ligh o deermine wihin an inerial referene frame wheher i is uniquely saionary or moing: i.e. he speed of ligh mus always hae he ea same magniude along any direion wihin any inerial referene frame. This posulae led Einsein o his Speial Theory of Relaiiy. Before Einsein posulaed his heory of relaiiy, Mihelson and Morley 5) ried o use ligh speed o measure he moion of he earh relaie o he eher. For his hey used an inerferomeer wih wo perpendiular arms of equal lengh : An inoming ligh-beam is spli ino wo a he junion where he wo arms mee: One beam moes along an arm (whih will be ermed he horizonal arm) and is refleed bak by a mirror a he end of his arm, while he oher ligh-beam is moing along an arm whih is perpendiular o he horizonal arm and refleed bak by a mirror a he end of his arm. If he whole inerferomeer is moing relaie o he eher wih a speed along he horizonal arm of he inerferomeer, he wo ligh-beams mus epend differen imes o moe along heir respeie arms o he mirrors (a he ends of he arms) and reurn o he junion. The formulas for hese differen imes an be found in any elemenary e book on modern physis. Along he perpendiular arm, he ime is: perp = () In aniipaion of he argumens ha will follow below, he formula for he ime ( hor ) along he horizonal arm, alhough already aailable in elemenary e books on physis for more han years, will again be deried here: In his ase, he horizonal ligh-beam, afer being spli off a he junion, hases he end of he arm

3 wih a speed (-) and herefore reahes he mirror afer a ime, hor, whih is gien by: = ( ) () hor Afer refleing a he mirror, he ligh moes ino he speed of he inerferomeer, relaie o he eher, wih a speed (+) relaie o he junion, and hus reahes he junion wihin a ime-ineral, hor, whih an be alulaed as: = (+ ) (3) hor Thus, he oal ime for he reurn rip is: The ime inerals, hor = hor+ hor = (4) perp and hor are learly differen: Thus, afer heir respeie journeys, he wo ligh-beams should mee up haing a phase-differene whih an be measured; and from whih he speed relaie o he eher an be alulaed. I is well-known ha his eperimen onsisenly failed o measure a phasedifferene. I migh hus mean ha he earh is saionary relaie o he eher. This is, howeer, unlikely sine he speed-direion of he earh hanges relaie o he sun when i irles he sun and herefore i mus also hange direion relaie o he saionary eher. I was found ha een hough suh an inerferomeer is sensiie enough o measure he earh s moion, when i hanges direion relaie o he eher oer a period of one year, no phase-hange ould be measured. One should noe a his poin ha een hough he earh suffered aeleraion while moing around he sun, his also did no ause any obserable phase hange. I was proposed by boh orenz 6) and Fizgerald 7) ha he null resul migh be aused by a lengh-onraion of he horizonal arm when his arm moes along is lengh wih a speed relaie o he eher. Suh a onraion will anel he phasedifferene and hus he measuremen of he speed. For his o our, an arm whih has an aual lengh, mus onra o a lengh when i is moing wih a speed relaie o he eher: This onraion is gien in erms of he saionary lengh, speed, and ligh speed, as: = (5) By replaing in Eq. 4 wih his epression for, Eq. 4 beomes he same as Eq.. Thus, for suh a onraion here will no be a ime-differene ha an gie a phasedifferene beween he ligh-beams. Equaion 5 has beome known as he orenz- Fizgerald onraion: I indiaes ha he eher migh be isous and ha, herefore, eher-drag migh be responsible for he shorening of any obje when i moes wih a speed hrough he eher. Insead of requiring a lengh-onraion aused by eher-drag, Einsein Speial Theory of Relaiiy eplains he null resul of he Mihelson-Morley eperimen far more simply: Aording o his posulae on he speed of ligh, he wo 3

4 ligh beams mus always be moing wih he same speed along boh arms of any Mihelson-Morley speromeer; no maer a wha speed he speromeer is moing relaie o any oher inerial referene frame wihin he unierse. A lengh-onraion aused by he eher is hus no required o eplain he null resul. This means ha he speed of a body wih mass is no absolue, bu ha he speed of ligh is absolue when measured relaie o any and all moing bodies. Therefore, when flying a an inredible speed in a spaeship, a passenger on suh a spaeship will no obsere ligh o moe a differen speeds when i moes along differen direions. Aording o suh a passenger, he spaeship is for all inen and purposes uniquely saionary owing o boh is mass-inerial behaiour, as well as he onsany of he speed of ligh. This implies ha when wo spaeships moe pas one anoher a a relaie speed, one saionary wihin an inerial referene-frame K and he oher saionary wihin an inerial referene-frame K, he speed of ligh mus always hae he same alue in all direions as measured wihin eiher one of hese spaeships. Owing o his requiremen, i is found ha an een ha ours a a erain posiion and ime wihin a passing inerial referene frame K, does no neessarily our a ealy he same ime wihin a referene frame K relaie o whih i is moing. The moion of K relaie o K auses aual sequenial physis eens ourring wihin K, o be disored in boh posiion and ime wihin K. To disinguish hese disored eens wihin K from he real physis eens whih aually our wihin K he laer physis eens will be alled primary physis-eens. In order o derie he onomian disored physis-eens wihin he referene frame K relaie o whih K is moing wih a speed, one needs o ransform he primary physis-eens ha our wihin he inerial referene-frame K, ino heir onomian disored physis-eens wihin he inerial referene-frame K. Einsein s posulaes led him o a se of oordinae ransformaions from K o K, whih are ealy he same as equaions whih had been disoered preiously by orenz 8) ; bu whih orenz obained by aeping he alidiy of he orenz- FizGerald onraion: They were herefore, iniially, no inerpreed as oordinaeransformaions. Owing o orenz s prioriy in publishing hese equaions, hey are a presen known as he orenz-ransformaion for posiion and ime oordinaes. Aording o Einsein s Speial Theory of Relaiiy, he orenz-ransformaion maps a primary physis-een whih ours a a posiion,y,z a a ime wihin an inerial referene-frame K ono oordinaes,y,z a a ime wihin anoher inerial referene-frame K. In order o ahiee his mapping, a referene poin for ime-measuremen has o be hosen whih is simulaneously alid wihin boh K and K: I is hus assumed ha an ideal, perfe lok wihin K mus be synhronised wih anoher idenial, ideal, perfe lok wihin K a he ery insan in ime ha he wo loks pass eah oher a he same posiion in spae. The general rule is ha he wo loks are siuaed a he wo origins = y = z = and = y= z= of he wo referene-frames K and K respeiely, and hen synhronized so ha = = a he insan when hese origins oinide. Wih his referene poin for ime, he following mahemaial formulas onsiue he orenz-ransformaion from K o K: i.e. = + (6a) 4

5 y= y z. A Mihelson-Morley Speromeer in Moion z= (6b) + = (6) Assume, for he sake of argumen, ha Mihelson and Morley are doing heir measuremens on he passing spaeship whih is saionary wihin referene-frame K : They will ge a null resul sine, aording o Einsein s posulaes, ligh raels wih he same speed along boh arms wihin K. A quesion whih omes o mind is he following: Wih wha speeds does he horizonal ligh-beam moe from he junion o he mirror and hen bak when his primary eperimen wihin K is ransformed ino is disored forma wihin K? Sine he speed of ligh is also in all direions relaie o K, he misake has been made oer he pas years, and is sill being made, o onlude ha, sine he apparaus is moing wih a speed wihin K, he horizonal ligh-beam mus moe wih a speed (-) owards he mirror and hen wih a speed (+) bak o he junion when obsered wihin K. The ime ineral as seen from K mus hen be he same as gien by Eq. 4, and his implies ha here mus now be a phase shif: Howeer, when he wo beams arrie a he same insan a a single poin wihin K, he physis ourring a ha poin mus be he same a he orresponding, disored oordinaes wihin K. This mandaes ha when here is no phase-shif wihin K, here an also no be a phase shif wihin K. I hus seems ompelling ha he only way o ensure ha here will also no be a phase shif wihin K, is o again inoke a orenz-fizgerald onraion of he horizonal arm when he speromeer moes relaie o K; een hough here is now no an unique eher whih an ause his onraion by means of eher-drag. This is wha Einsein onluded ) (see also seion 4 below). Bu, as will now be shown, his onlusion is wrong. The only aepable way o really deermine wha happens wihin K when he speromeer wihin K moes pas wih a speed, is o sar off from, and o use he orenz-ransformaion (Eqs. 6): Wihin K, he speromeer is saionary, and when ligh leaes he junion a posiion = a ime =, i will reah he mirror a he end of he arm when: m = and = m (7a) I will hen refle and reurn o he junion, whih now has he following posiionoordinaes and ime wihin K : j = and j = ( ) (7b) A ime =, Eqs. 6(a) and 6() gies = and = respeiely wihin he ouside referene-frame K. When he beam reahes he mirror, hese same equaions gie, in onjunion wih Eq. 7a, for m and m he following: 5

6 6 m m m + = + = (8a) And m m m + = + = (8b) The pah-lengh wihin K, when he ligh-beam is moing from he junion o he mirror, is hus m hor = (Eq. 8a); and onomian ime-ineral is hus m hor = (Eq. 8b). Thus, he speed of he ligh-beam obsered wihin K (when he ligh-beam moes owards he mirror) is hor hor ; whih urns ou o be equal o. I is hus no (-)! The speed of ligh relaie o he mirror remains wihin boh K and K. Afer all, he mirror is in is own righ an inerial referene-frame relaie o whih he speed of ligh mus always hae he same onsan alue! When, afer is refleion and reurn, he beam again reahes he junion, he orenz-ransformaion, in onjunion wih Eq. 7b, gies for he orresponding disane j and ime j wihin K, he following: j j j = + = (9a) And j j j = + = (9b) The pah-lengh, from he mirror o he junion, is gien wihin K by j m hor =, and he reurn ime is m j hor = : Thus from Eqs. 8a and 9a: hor = (a) And from Eqs. 8b and 9b: hor = (b) The ime ineral is now negaie: Can ime run bakwards? Alhough i is adoaed wihin he mainsream physis-lieraure ha his is possible, so ha an eleron an supposedly moe bakwards in ime o, in his manner, eis as a posiron, i is more probable ha i is neer-eer physially possible: No maer wha! In his speifi ase, he negaie sign is mandaed by mahemaial self-onsiseny, sine, when now

7 alulaing he speed hor hor, one obains, as one mus: The ligh is moing ino he opposie direion afer haing been refleed. The simple fa is he following: Wheher a ime-ineral is posiie or negaie is solely deermined by he hoie of he oiniding posiions wihin K and K when synhronizing he loks wihin hese wo inerial referene-frames (see also seion 3 below). The laer is an arbirary hoie and he sign of a ime-ineral does hus no mean ha ime is hanging from he presen ino he pas. If he magniude of he speed of ligh did no remain, he ligh would hae been obsered wihin K o moe on is way bak o he junion a a speed (+), whih would be larger han he speed of ligh relaie o he junion. Suh a relaie speed for ligh is, aording o Einsein s own posulaes, neer possible relaie o any moing obje. Thus, in addiion o he fa ha he speed of ligh remains wihin any inerial referene-frame, i is also impossible for a ligh-beam approahing a moing obje, or speeding away from suh a moing obje, o do so wih any oher speed relaie o he moing obje han he same onsan speed. 3. Simulaneiy and Non-Simulaneiy I is well-known ha aording o Einsein s Speial Theory of Relaiiy wo eens whih our simulaneously a wo differen posiions wihin an inerial referene-frame anno our simulaneously wihin anoher inerial referene-frame relaie o whih he firs referene-frame is moing. Einsein 9) eplained his effe in erms of a rain passing hrough a saion wih a speed a he ery insan ha wo simulaneous lighning-srikes hi he embankmen a posiions A and B, a disane apar parallel o he rain. He hen onsidered wha a person on he rain will obsere when heshe finds himherself midway beween he wo lighning-srikes a ealy he insan when hey our. Quoing Einsein direly:..he is hasening owards he beam of ligh oming from B whils he is riding on ahead of he beam of ligh oming from A. Hene he obserer will see he beam of ligh emied from B earlier han he will see ha emied from A. Obserers who ake he railway rain as heir referene body mus herefore ome o he onlusion ha he lighning flash B ook plae earlier han he lighning flash A. To undersand how Einsein s physis-logi failed him when he used his argumen, we will modernise his eplanaion of simulaneiy by again onsidering wo idenial spae-ships (boh haing he same lengh (wihin heir respeie referene-frames K and K) whih pass eah oher wih a relaie speed along heir lenghs. We onsider he ase when, wihin he spaeship, whih is saionary wihin referene-frame K, a ligh in is ail and a ligh in is nose are swihed on simulaneously. Consider a ligh deeor in he spaeship K whih is siuaed midway beween hese ligh-soures: Relaie o his spaeship, he frons of he ligh-beams oming from he wo lighs mus reah he deeor simulaneously. This mus be so sine he ligh-frons, rushing owards he deeor from he nose and he ail of he spaeship, boh moe a he same onsan speed wihin K, and also oer he same saionary disane ½ o reah he deeor. Aording o Einsein s rain-argumen 9), he lighs will no swih on simulaneously wihin K, sine an obserer in K is moing away from he ligh-fron, rushing owards him from he nose of he moing spaeship wih speed (suh ha he ombined relaie speed beween he ligh and his obserer is -) and he obserer is moing ino he ligh rushing owards him from he ail of he moing spaeship (so ha in his ase he ombined relaie speed beween he ligh and he 7

8 deeor is +): Therefore, Einsein reasoned, he ouside obserer mus see he ligh in he ail swihing on before heshe sees he ligh in he nose swihing on. The firs problem wih his argumen is ha he ouside obserer mus be a a posiion wihin K ha oinides wih he deeor wihin K when he wo lighs swih on in order o jusify he onlusion ha he ligh in he bak swihes on before he ligh in he nose does wihin K. I is, howeer, highly unlikely, and mos probably impossible, ha physis will be deermined by he presene and foruious posiion of an obserer. If he lighs swih on a differen imes wihin K, i should no jus happen when here is an obserer a a speifi aommodaing posiion wihin K; bu een when here is no obserer presen a all. If his is no he ase, one is modelling paranormal meaphysis. All he laws of physis mus be he same wheher here is an obserer presen or no presen a all. The seond, and really major problem wih his argumen an be seen by reisiing he resuls whih hae been deried for he moing Mihelson-Morley inerferomeer in seion aboe: The magniude of he relaie speed of he ligh oming from he ail as well as he nose mus remain equal o he same onsan alue wihin boh K and K, and relaie o he mirror and junion of he speromeer. I mus hus approah he deeor wihin he moing spaeship wih ealy he same speed relaie o he deeor; een when obsered wihin K. The aual reason why he lighs do no swih on simulaneously wihin K (when hey do so wihin K ) will now be direly deried from he orenz-ransformaion gien by Eqs. 6: e us assume ha he lighs swih on a he ery insan when a lok in he ail of he passing spaeship wihin K is synhronised wih a lok wihin K: i.e. a his insan, one has ha = = and = =. Bu a his same insan in ime, he ime N in he nose of he spaeship mus also be zero, or else he ligh in he nose of he spaeship (wihin K ) will no be able o swih on simulaneously wih he ligh in he ail. Relaie o K his happens a posiion N =. The orresponding alues of N and N wihin K hen follow from Eqs. 6 as: And N = (a) N = (b) Thus, jus as in he ase of he moing Mihelson-Morley speromeer, boh he posiion of he nose and he ime ha he ligh in he nose swihes on, hange when iewed wihin K. I is imporan o noe ha, aording o his resul, ime anno be he same a eery posiion ( wihin he inerial referene-frame K ) when his ime is measured wihin K. A he ery insan when an obserer in K synhronises hisher lok wih a lok in he ail of he spaeship, any poin along he spaeship, is siuaed wihin he fuure of he ouside lok. All he eens ourring simulaneously a ha ea insan in ime wihin K, will only manifes a laer imes wihin K: Therefore he ligh in he nose swihes on a a laer ime N wihin K. This sounds quie silly: Afer all, he fron of he spaeship should pass by firs. How an he nose be in he obserer s fuure when he ail only passes him laer? I 8

9 obiously anno: I is only in he fuure of K a he ery insan in ime ha he ouside lok is synhronised wih a lok siuaed in he ail of he spaeship! When, in onras, he ouside obserer synhronises his lok wih a lok in he nose of he passing spaeship, one obains from he orenz-ransformaion ha he oordinae T and ime T in he ail of he spaeship wihin K are: And T = (a) T = (b) Boh hae he same magniudes as heir ounerpars, N and N in Eq., bu are now negaie quaniies. The disane mus be negaie sine i is now measured along he negaie direion from he origin of K. The negaie ime in he ail of he spaeship, as well as all poins from he ouside lok in K owards he ail of he spaeship (and furher), are, a ha ery insan in ime, in he pas of he ouside lok. When he lighs in he ail and nose of he spaeship now swih on ealy a he insan when he ouside obserer synhronises his lok wih he lok in he nose of he spaeship, his obserer sees he nose-ligh swihing on a ha ery insan in ime: Bu from wihin hisher referene frame K, he ligh in he rear mus already hae swihed on before heshe synhronized hisher lok wih he ligh in he nose of he passing spaeship. Does his imply ha he ligh in he ail of he spaeship aniipaed he fuure? If he ouside obserer is informed by an all-knowing being, who an simulaneously eis wihin boh K and K, ha he wo lighs hae aually swihed on simulaneously on he spaeship wihin K, heshe will be fored o admi ha aording o hisher lok he ligh in he ail did aniipae he fuure. Bu if heshe really beliees ha his is possible, heshe needs o go and see a shrink! Minkowski Oh Minkowski!! The fa is ha, een when synhronising hisher lok wih he lok in he ail of he spaeship, heshe (when belieing he all-knowing being) mus onlude ha he ligh in he ail is aniipaing he fuure. This an obiously no be wha really happens! A lok in he ail and a lok in he nose of he spaeship mus in aual realiy be keeping ealy he same ime for simulaneiy o be possible on he spaeship. For a passenger wihin he moing spaeship, he ime is ealy he same eerywhere wihin he spaeship: In fa, eerywhere wihin he inerial refereneframe K. Time does no ary from poin o poin a all; jus as Newon had assumed years before Einsein ame on he sene. Perfe loks a any and eery posiion wihin he spaeship mus all keep he same idenial ime. There is hus no an aual ime-oordinae whih hanges wih posiion wihin K. Only wihin referene-frame K, an an all-knowing being onlude ha ime is now a funion of posiion along he lengh of he spaeship: i.e. only under hese ondiions does ime form a fourh oordinae aording o suh a being. Bu does i really form a fourh oordinae wihin K? If i ould be possible, he wo inerial referene-frames would no be equialen as Einsein has posulaed ha hey mus be. 9

10 To be equialen, any wo lighs whih are saionary wihin eiher referene-frame K or referene-frame K, mus be able o swih on simulaneously wihin heir respeie inerial referene-frames; no maer a whih posiions he lighs find hemseles. This demands ha he ime mus be ealy he same a all posiions wihin boh K and wihin K. Alhough he ime in Eq. 6 is gien as a funion of posiion and he ime, he orre physis-inerpreaion of his equaion is as follows: When, wihin he referene-frame K, in whih here are idenial loks iking away a he same rae a any and, in priniple, a eery posiion, an een ha ours a posiion,y,z a an insan of ime (as shown by all he loks whih are saionary a any posiion wihin K ) will our wihin he referene-frame K a a posiion,y,z, when all he loks, whih are saionary anywhere wihin K, reahes he ime gien by Eq. 6. To re-emphasise: When he lok a reahes he ime, all he oher loks, a all he oher posiions wihin K, will also show his idenial ime. Thus, ime anno really hange wih posiion a all: No wihin K, no wihin K and no wihin any inerial referene-frame! I is, of ourse possible o swih on he ligh in he nose of he spaeship K before swihing on he ligh in he ail so ha he ime ineral will be suh ha he lighs will swih on simulaneously wihin K. I is easy o derie his ime ineral from he orenz-ransformaion and herefore i will no be done here. Thus, aual non-simulaneous eens an be obsered as ourring simulaneously when iewed from ouside he inerial referene-frame in whih hey aually our non-simulaneously. This does no mean ha ha hese primary eens are aually simulaneous wihin K. 4. engh-onraion Consider again a passing spaeship wih a saionary lengh as measured wihin is own inerial referene-frame K : Wihin K, he ime in he ail of he passing spaeship is ahead of he ime in he nose of he spaeship. This means ha i is neer possible o measure a whih posiions he nose and he ail are a a single insan in ime on any lok whih is saionary wihin K. And his brings us o he mos remarkable blunder ha Einsein has made more han years ago, and whih is sill being augh in physis e books as being orre. Afer Einsein deried he orenz-ransformaion in erms of his posulaes on whih he Speial Theory of Relaiiy is based, he immediaely wen ahead and used he orenz-ransformaion (Eqs. 6) o derie he orenz-fizgerald onraion (Eq. ). In order o do so, he had o assume ha he fron-end (nose) and he bak-end (ail) of a passing obje wih lengh an be simulaneously presen ) wihin K a a single insan in ime on he loks wihin K. As inredible as i may sound, his means ha jus afer Einsein had proed in 95 ha wo spaially separaed eens, whih our simulaneously wihin a passing inerial referene-frame K, an neer be simulaneous on any of he loks wihin he inerial referene-frame K relaie o whih K is moing wih a speed, he used he orenz-ransformaion o map he nose-oordinaes and he ail-oordinaes wihin K as if hey hae simulaneous posiions wihin he referene-frame K. He had o do his in order o derie a lengh wihin he inerial referene-frame K whih is he same as he orenz-fizgerald onraion (see Eq. 5): And his is sill being done in e books o his day. Bu, wihin he referene-frame K of he spaeship, passing by wih a speed, he nose and ail are a any insan in ime, on any lok raelling wih he spaeship,

11 simulaneously a disane away from eah oher. Eery ik of wo loks (one in he nose and he oher in he ail of he spaeship) defines wo eens whih our all he ime simulaneously; no maer how small he ime-ineral beween wo onseuie iks is. Only for his reason an he saionary lengh be known a a single insan in ime on he loks wihin he inerial referene-frame K. Bu his is no he ase for he ouside obserer in referene-frame K. When applying he orenz-ransformaion, one finds ha he ransformed lengh of he spaeship aually inreases wihin he referene-frame K of he ouside obserer. In fa, he orre ransformed lengh wihin he inerial referene-frame of he obserer when he spaeship moes pas wih a speed an be deried from Eq. 6a (as also onfirmed by Eq. a) and is found o be: = (3) Thus, if he speed of he spaeship approahes he speed of ligh relaie o he inerial referene-frame of he ouside obserer, will beome ery long indeed: Howeer, he apparen ime-differene T, beween being able o obsere he ail and hen he nose of he spaeship, will also be ery large. By using Eq. 6, i is deried o be: T= (4) Wha does his imply? I an only imply wha i is saying: Tha owing o he relaie moemen of he spaeship haing an aual saionary lengh wihin K, an eniy wih a longer lengh is passing by wihin K, bu his lengh anno be presen a he same insan in ime wihin K, sine differen posiions along his lengh only manifess a differen imes on any and all loks whih are saionary a any posiion wihin K. Bu one migh wan o subbornly argue ha, een hough he fron- and rearends of he spaeship anno be simulaneously presen wihin he referene frame K, he spaeship mus surely sill hae, a any insan in ime wihin K, a unique lengh u wihin he inerial referene-frame K. If one ould a any insan in ime, sop ime, so ha all moemen hals, he fron-end and rear-end of he spaeship mus obiously be found a wo differen spae-posiions wihin K. Bu he fa remains ha he nose and ail anno be simulaneously presen wihin K while here is relaie moion, and while i is impossible o sop ime. There is only one way o deermine he lengh of a passing obje wihin K, and ha is o measure he ime ineral ha i akes for he obje o pass a posiion wihin K: i.e. in he ase of he spaeship, here mus be a sopwah whih sars as soon as he nose passes his poin, and hen sops as soon as he ail passes he same poin. The lengh u mus hen be equal o: u = (5) Bu le us now onsider wo idenial spaeships, eah of lengh, as measured wihin heir respeie inerial referene-frames K and K, whih are passing eah oher wih a relaie speed : As soon as he noses of he wo spaeships reah eah oher, he wo apains sar heir respeie sopwahes, hus synhronizing hem,

12 and hen sop heir sopwahes as soon as he ail of he oher spaeship passes he nose of hisher spae-ship. Now, if he apains measured ime inerals and respeiely, hey will alulae ha he lenghs of eah oher s spaeship are u = and u =, respeiely: Bu owing o he symmery inoled, one mus surely hae ha u = u. Sine his is he ase, i is highly unlikely, and probably oally impossible, ha he lenghs u = u an be anyhing else han he aual lenghs. Thus here is no an aual onraion of lengh when an obje passes wih a speed as is laimed in e books. Furhermore, his onlusion, in urn, demands ha: = (6) The loks of he wo apains mus keep ime a ealy he same rae wihin heir respeie spaeships. 5. Time-dilaion Wha is laimed in he mainsream sienifi lieraure is ha, afer an ouside obserer (in he referene-frame K) has synhronized hisher lok wih a passing lok moing wih a speed (for eample on a passing spaeship), he moing lok aually iks away a a slower rae han he lok wihin K. No o make any errors in he aeped dogma, we will quoe Einsein 9) direly on his issue: e us now onsider a seonds lok whih is permanenly siuaed a he origin ( =) of K. = and =I are wo suessie iks of his lok. The firs and fourh equaion of he orenz-ransformaion (see Eqs. 6a and ) gie for hese wo iks: and = (7a) = I (7b) As judged from K, he lok is moing wih he eloiy ; as judged from his referene body, he ime whih elapses beween wo srokes of he lok is no one seond, bu I seonds, i.e. a somewha larger ime. As a onsequene of is moion he lok goes more slowly han when a res. Bu his means ha when a ime-ineral is reorded by he lok in K, hen a ime ineral passes on he lok wihin K; so ha: = (8) Bu should his no be he same as in Eq. 6? The same lok an surely no run simulaneously a wo differen raes! I is well-known ha he ime-dilaion formula (Eq. 8) an be deried by omparing he differen perspeies wihin K and K when a ligh-beam is swihed on a ime = wihin K direed along he y -ais. Wihin K he ligh will reah a heigh H =, afer a ime ineral, as illusraed by he erial arrow in Fig. (a). Wihin K, howeer, he origin of K has moed hrough a disane = while he ligh moed erially wihin K. The ligh-beam hus moes a an angle o he -

13 ais wihin K: Sine he ligh mus also, along his pah, moe a a speed, he disane hrough whih he ligh moes wihin K is. The siuaion is as illusraed wihin Fig. (b). I is lear ha by using he heorem of Pyhagoras one obains Eq. 8. (a) K () K (b) K (d) K Figure : Illusraion of ime-dilaion: (a) A ligh-beam (erial arrow) is swihed on wihin he inerial referene-frame K whih is moing wih a horizonal speed relaie o he inerial referene-frame K. (b) The ligh-beam (inlined arrow) as obsered wihin K. () The ligh-beam (inlined arrow) as obsered wihin K when a ligh-beam is swihed on in he erial direion wihin K whih is moing wih a speed along he negaie horizonal direion relaie o K. () The erial ligh-beam wihin K. I is, howeer, equally possible o swih on a erial ligh-beam wihin K a he same insan in ime = = in he direion of he y-ais. Wihin K, he ligh will reah a heigh H=, afer a ime ineral, as illusraed in Fig. (d). Wihin K, howeer, he origin of K has moed hrough a disane =. The ligh-beam hus moes a an angle o he negaie -ais. Sine he ligh mus also moe a a speed wihin K, he disane hrough whih i moes wihin K is. The siuaion is as illusraed wihin Fig. (). I is lear ha when using he heorem of Pyhagoras one now obains ha: = (9) Whih one of he wo equaions (Eq. 8 and Eq. 9) is orre? Is he lok wihin K slower, or is he lok wihin K slower? The onlusion is ompelling: Cloks do no really keep ime a differen raes. A lok only keeps ime a slower rae wihin anoher inerial referene frame wihin whih i is NOT saionary. Bu, in fa, boh loks are saionary wihin heir referene frames and MUST hus keep ealy he same ime wihin heir respeie referene frames. Eq. 8 applies when he ligh is swihed on wihin K, while Eq. 9 applies when he ligh is swihed on wihin K. Bu an one no argue ha he same ligh has been swihed on simulaneously wihin K and K a he ery insan == when he 3

14 origins of K and K oinided? Obiously, his anno be possible: A single ligh soure anno be simulaneously saionary wihin boh K and K. The soure an only be saionary wihin a single inerial referene-frame (see also he disussion of Minkowski s ) spae-ime in seion 6 below). There hae been eperimens repored where aomi loks hae been flown around he world and hen ompared wih a lok lef behind on earh ) : I is laimed ha hese resuls proe ha ime-dilaion, as deried from he Speial Theory of Relaiiy, aually ours on he loks whih had been flown around he earh. I is herefore laimed in e books ha a lok on he spaeship moing a a speed does aually ik a a slower rae, and herefore a person on he spaeship will aually age a a slower rae han an ouside obserer on earh. Bu, in iew of Eq. 6, i mus be impossible ha his an be so! The simples logial reason why his anno be so, is ha, if he wo loks do no ik a ealy he same rae, he wo referene-frames would no be equialen: The laws of physis mus hen be differen wihin eah one of hem. This ompleely iolaes Galileo s posulae of ineria, and hus also Einsein s posulaes on whih he based his Speial Theory of Relaiiy. The laer would also imply ha we anno alloae a single lifeime o he age of our unierse. Relaie o whih moing, inerial referene-frame mus his be done? These inerial referene-frames mus all gie he same lifeime, or else hey anno be equialen. Here on earh, we hae dedued ha he age of he unierse is jus under 4 billion years. Aording o a lok on a faraway plane haing he same mass as he earh bu forming par of a faraway galay (moing wih nearly he speed of ligh relaie o us) mus hen imply ha he age of he unierse is far less: Only 7 days perhaps? I is highly unlikely ha our plane required billions of years o form while anoher similar plane whih has sine is inepion moed a a high speed relaie o us, required only 7 days. One is hus fored o onlude ha a lok whih moes relaie o an obserer is no aually iking a a slower rae a all wihin is own inerial referene frame. Noneheless, alhough ime-dilaion does no really our wihin eiher K or K, i anno be ignored as being irrelean: For eample, i is found ha he lifeime of a muon, formed wihin osmi rays ha are moing a a high speed relaie o earh, is longer when measured by a lok on earh, han when measured by he same lok for anoher muon ha is generaed wihin a laboraory on earh and hus moing a a negligible speed. This does no mean ha he lok moing wih he muon iks a a slower rae wihin he inerial referene frame of he muon. The deduion ha a win leaing he earh wih a speed will age a a slower rae han a win saying behind, an hus no be orre. The win saying behind will hae he perepion ha ime is running slower on he spaeship, jus as he win on he spaeship will hae he perepion ha he ime on earh is running slower han hisher ime on he spaeship. In realiy, ignoring graiaional effes, ime mus be iking away a ealy he same rae on he spaeship as i does on earh. I mus hus be oally wrong when Sephen Hawking 3) wries in his book A Brief Hisory of Time he following: In oher words he heory of relaiiy pus an end o he idea of absolue ime! I appeared ha eah obserer mus hae his own measure of ime, as reorded by a lok arried by him, and ha idenial loks arried by differen obserers would no neessarily agree. Can his be orre in he ase of he Speial Theory of Relaiiy whih Hawking is addressing? Alhough eah obserer has o deal wih he perepion ha he lok of anoher obserer, moing relaie o himher, does no ik a he same rae as hisher lok, he wo loks 4

15 (proided hey are perfe loks) mus keep ime a ealy he same rae; or else heir inerial referene-frames anno be equialen. Similarly in his book eniled Blak Holes and Time Warps, Kip Thorne 4) wries abou a fuure spae rip o he enre of our galay as follows: The enire rip of 3, ligh years disane will require 3, years as measured on Earh; bu as measured on he sarship i will require only years. In aordane wih Einsein s laws of speial relaiiy, your ship s high speed will ause ime, as measured on he ship, o dilae and his ime-dilaion (or ime warp) in effe, will make he sarship behae like a ime mahine, projeing you far ino he Earh s fuure while you age only a modes amoun. Wha he does no menion is ha he relaie speed beween he earh and he sarship will dilae he ime on earh relaie o he ime on he sarship, hus ausing earh o behae like a ime mahine relaie o he rew on he sarship. So who will be in whose fuure? Anoher eample is by ord Professor Marin Rees 5), Asronomer Royal and Presiden of he Royal Soiey of ondon: In his book Jus Si Numbers, he wroe: The speed of ligh urns ou o hae ery speial signifiane: i an be approahed, bu neer eeeded. Bu his osmi speed limi imposes no bounds o how far you an rael in your lifeime, beause loks run slower (and on board ime is dilaed) as a spaeship aeleraes owards he speed of ligh. So wha abou he resuls of he aomi loks whih had been flown around he earh )? Firsly he unerainies in he resuls were ery large, and seondly for he flying loks he graiaional fore hanged wih heigh aboe he earh s surfae. Thus he lok on earh was no ompared wih loks whih moed wih a onsan, linear speed ouside a graiy-field. I an hus no be onluded ha he aual dilaion of ime laimed o hae been measured for hese eperimens, alidaes he ime-dilaion, deried from he Speial Theory of Relaiiy, as being real wihin he loks inerial referene frame. Maybe he resuls ha were found hae been wha he eperimeners waned o find? 6. Disussion There is somehing peuliar abou moion: ong before Galileo appeared on he sene, his has been noed by he Greek philosopher Zeno 6). Zeno posed paradoes; all of whih led him o onlude ha he moion of an obje mus be an illusion. If moion is an illusion, i is, howeer, no an ephemeral illusion, sine i has real physisonsequenes. The laer realiy an be painfully esed by jumping in fron of a moing rain. Bu when a person is on a hoer-board (a lá Bak o he Fuure ) moing wih he same eloiy as he rain, nohing aderse happens when he hoerboard is suddenly manoeured ino a posiion in fron of he rain. Thus, he physial effe of moion is deermined by he inerial referene-frame wihin whih i is being obsered and measured. Following Zeno, he moion of a body wih mass will be lassified as being a relaiisi-illusion. This iewpoin doeails nealy wih Galileo s reasoning ha a moing obje is in essene saionary. I should be noed ha momenum and kinei-energy an be similarly lassified as relaiisi-illusions. They do no manifes for a body wih mass wihin is own inerial referene-frame, bu only when his body is iewed from anoher inerial referene-frame. We will lassify relaiisi-illusions whih do no our wihin he primary inerial referene-frame, bu only when iewed wihin a refereneframe moing relaie o suh a primary referene-frame, as Type I relaiisiillusions. Moion, momenum and kinei-energy are hus, aording o his lassifiaion, Type I relaiisi-illusions. 5

16 Zeno has been asue when he prolaimed ha moion is an illusion. For eample, when an aeroplane, whih is saionary wihin an inerial referene-frame K, flies oerhead wih a speed relaie o an inerial referene-frame K on earh, and drops a bomb, he bomb will fall sraigh down wihin K, bu i follows a paraboli pah wihin K; as if i had been projeed by a horizonal launh-fore: Suh a fore would hae been needed if he bomb were launhed by a hoering helioper ha is saionary wihin K. In he ase of he aeroplane, he paraboli pah ensues wihin K, bu no wihin he aeroplane s inerial referene-frame K : The paraboli pah is hus learly a Type I relaiisi-illusion wihin K. In onras, he horizonal launh-fore whih is physially mandaed by Newon s seond law for he paraboli moion o iniiae wihin a primary, inerial referene-frame, does no manifes a all wihin eiher K or K: I will herefore be lassified as a Type II relaiisi-illusion. Using he laer lassifiaion of relaiisi effes, i is argued here ha i has all along been wrong o onlude ha, aording o Einsein s Speial Theory of Relaiiy, ime-dilaion aually ours on any lok, oher han in he form of a Type II relaiisi-illusion. Bu his illusion an ause Type I relaiisi-illusions, like he lenghening in he lifeime of a muon when i moes a a high speed relaie o earh (see seion 5 aboe). This is similar o he ype II relaiisi launh-fore being responsible for he Type I paraboli pah. I is hus oally wrong o onlude ha a he insan in ime a whih an ouside obserer in K synhronises hisher lok wih a lok wihin K, any of he loks wihin eiher K or K show differen imes when hey are a differen posiions wihin heir respeie inerial referene-frames. Time does no aually hange from posiion o posiion as if i is a fourh oordinae wihin eiher K or K: The apparen fourh oordinae is a Type II relaiisi-illusion whih does no manifes direly a all. Bu, no surprisingly, i does ause Type I relaiisi-illusions: For eample, when wo lighs swih on simulaneously wihin K bu do no swih on simulaneously wihin K (see also he disussion of he de Broglie waelengh below). By posulaing ha wo separaed eens wihin an inerial referene-frame an our simulaneously, Einsein unwiingly aeped ha, wihin suh a refereneframe, ime mus be he same a all posiions: Jus as Newon had assumed in he 6 s. Time mus be absolue wihin any, and all inerial referene-frames wihin our unierse. And sine all inerial referene-frames are equialen, all perfe loks, wheher hey are a differen posiions in graiy-free spae, or wheher hey are moing wihin graiy-free spae relaie o one anoher wih a onsan speed, mus keep ime a ealy he same rae. The loks of wo persons, moing relaie o one anoher, do hus no aually hae differen ime raes a all. The orenz-fizparik lengh-onraion is een more misguided: This is so sine he deriaion of his supposedly real effe has all along been jus plain wrong. I is nonsensial o use he orenz-ransformaion in order o map wo simulaneous posiions along a rod whih is saionary wihin K, ino wo simulaneous posiions wihin K. As long as K moes wih a speed relaie o K, he wo simulaneous posiions wihin K mus always map ino wo non-simulaneous posiions wihin K whih are spaed furher apar han wihin K. The orenz- Fizgerald lengh-onraion does hus no our: No in realiy and also no een as a Type I or a Type II relaiisi-illusion. Conraion jus does no happen a all! The orenz-ransformaion ransforms primary-physis, whih is allowed aording o Galileo s priniple of relaiiy wihin an enlosed, inerial refereneframe, ino relaiisi- illusionary physis wihin anoher, passing, inerial referene-frame: Howeer, owing o he symmery inheren in he Speial Theory of 6

17 Relaiiy, one an also ransform primary physis-eens wihin he inerial referene-frame K, o find ou how hey will be obsered wihin he inerial referene frame K. An obserer wihin K an obiously also look ouside ino K. The orenzransformaion, when ransforming a primary physis-een,y,z a ime wihin K, ino he onomian oordinaes,y,z in K a ime, are he following: = (a) y = y and z = z (b) = () Mahemaially hese equaions are symmeri o hose in Eq. 6: For eample, when swihing on wo lighs wihin K a posiions = and = N a respeie imes = and = N ; and hoosing N and N as he alues gien by Eqs. a and b respeiely, one obains ha = N and = N : Sine hese are he oordinaes wihin K when he ligh in he nose of he spaeship swihes on wihin K, whih are hen ransformed ino he alues N and N wihin K (gien by Eq. ), i seems as if primary eens wihin K an be ransformed ino K and ha hese resulan relaiisially-disored eens an hen be ransformed bak ino K. Bu his does no make any physis-sense: In order o ransfer hese eens bak, one requires a ligh whih is saionary wihin K a he posiion N. If here is no suh a ligh wihin K, he bak-ransformaion from K o K, alhough mahemaially allowed, is physially meaningless. In e books he Minkowski spae-ime manifold is obained by equaing he following wo epressions whih an be alid wihin K and K respeiely, sine eah epression desribes he posiions of a spherial wae-fron around he spaial origins wihin K and K, respeiely: Wihin K one an wrie for he spae-oordinaes,y,z and he ime ha: And wihin K, ( ) () + (y ) + (z ) ( ) = () + (y) + (z) () = () When one uses he orenz ransformaion (Eq. 6) and replaes,y, z and in Eq., one obains Eq..; and when replaing,y,z and in Eq. by using he orenz ransformaion (Eq. ), one obains Eq.. For his reason i is posulaed wihin he presenly aeped mainsream physis lieraure, ha one an wrie ha: ( ) + (y ) + (z ) ( ) = () + (y) + (z) () (3) I is, howeer, dangerous o equae wo epressions whih are boh zero; as is done in Eq. 3. Generally when doing his, he resul obained is illogial-nonsense. I is hus highly doubful ha i an be orre in his insane. Furhermore, his symmery is based on mahemaial symmery; and no on he fa ha in erms of aual physis 7

18 he orenz-ransformaion from K o K is no he same as he orenz-ransformaion from K o K. The argumen used in e books o jusify Eq. 3, is o posulae ha a spherial ligh-wae is simulaneously emied wihin boh K and K, a he insan ==, by a single ligh soure a he oiniding origins of he wo referene-frames. Inherenly i is hus assumed ha his ligh soure is simulaneously saionary wihin boh K and K. As already poined ou in seion 5, his is no physially possible. Eq. 3 is ne inoked o onlude ha here eiss an aual spae-ime infiniesimal disane (ds) whih an be obained from he following equaion: ( ds) = (d ) + (dy ) + (dz ) (d ) = (d) + (dy) + (dz) (d) (4) This disane supposedly defines a Minkowski spae-ime manifold whih is hen supposedly he physially-real spae-ime wihin whih we really find ourseles when here is no graiy-field presen. Bu, all ha Einsein added wih his Speial Theory of Relaiiy, is he posulae ha he speed of ligh mus be he same alue relaie o any and all inerial referene-frames; no maer wih wha speed eah is moing relaie o he oher. This did no make any of hese referene-frames non-newonian as far as he onep of absolue ime is onerned: I is sill rue ha for an obserer wihin any inerial referene-frame, eah een ours a a speifi poin,y,z a a speifi absolue ime-insan, whih is ealy he same a all posiions wihin he inerial referene-frame when his een ours. An inerial referene-frame an hus no be a physially-real Minkowski spae-ime ) in whih ime is an aual fourh oordinae whih has aual differen alues a differen posiions. The orenz-ransformaion does no ransform four aual oordinaes,y,z,, from a spae-ime manifold K ino anoher equialen spae-ime manifold K wih aual oordinaes,y,z,. I ransforms primary physiseens wihin a Newonian, inerial referene-frame K, in whih ime is he same a eah and eery posiion, ino anoher passing, Newonian, inerial referene-frame K, in whih ime is also he same a eah and eery posiion; and where he ime-raes wihin K and K are ealy idenial. All ha is obained is ha simulaneous eens wihin K anno be obsered simulaneously wihin K. To posulae ha his defines an aual spae-ime manifold is daf. If Einsein would hae been alie oday, he migh hae had mied feelings afer reading he argumens in his manusrip: Firsly, he migh hae been delighed: For many years (from abou 95 o a leas 9) he opposed he idea of a real Minkowski spae-ime. I probably did no help muh ha Minkowski alled Einsein a lazy dog when Einsein sudied mahemais under him in Zurih. Bu eenually, Einsein rereaed and deided o base his argumens for deeloping his General Theory of Relaiiy, by aeping ha Minkowski s spae-ime manifold really manifess physially. Einsein should hae known ha his firs insin old him orrely ha mahemais does no deermine physis, bu ha mahemais is only he language used o model physis: Mahemais an lie and herefore i is aually physis ha deermines he mahemais ha should be used. I is dangerous o onlude ha when he mahemais is beauiful and self-onsisen, he physis i supposedly desribes mus be orre. In addiion, as proed aboe, lengh-onraion does no our (no een as a relaiisi-illusion) and ime is absoluely hanging a he same rae wihin all possible inerial referene-frames: This mandaes ha, many, if no all, of 8

19 Einsein s argumens, whih he used o erapolae from Minkowski s spae-ime o reah he framework of non-eulidean ured spae-ime, are wrong. For eample, afer inoking his priniple ha graiaional mass and inerial mass are equialen, and eending he priniple of relaiiy o inlude bodies of referene whih are aeleraed wih respe o eah oher, Einsein onsidered he behaiour of loks and measuring rods on a roaing body 7). He onsidered he ase where he referene-frame K is a dis whih is roaing around a enral ais as iewed from anoher inerial referene-frame K. The ais of he dis K does no moe in any direion wihin K. An obserer on he dis will eperiene a enrifugal fore along he radius from he enre of he dis owards is periphery. Aording o Einsein s priniple of equialene, his obserer will inerpre his dis as being saionary and will hus hae o asribe his fore o a graiy-fore being presen wihin hisher referene-frame; een hough his fore of graiy aras him from he enre owards he periphery of his unierse. We will again quoe Einsein direly: The obserer performs eperimens on his irular dis wih loks and measuring rods. In doing so i is his inenion o arrie a ea definiions for he signifiaion of ime- and spae-daa wih referene o he irular dis K, hese definiions being based on his obseraions. Wha will be his eperiene in his enerprise? To sar wih, he plaes one of wo idenially onsrued loks a he enre of he irular dis, and he oher on he edge of he dis, so ha hey are a res relaie o i (he dis). We now ask ourseles wheher boh loks go a he same rae from he sandpoin of he non-roaing Galilean referene body K. As judged from his body, he lok a he enre of he dis has no eloiy. Whereas he lok a he edge of he dis is in moion relaie o K in onsequene of he roaion. Aording o a resul obained in Seion XII (Einsein refers here o he formula for ime-dilaion), i follows ha he laer lok goes a a rae permanenly slower han ha of he lok a he enre of he roaing dis, i.e. as obsered from K. I is obious ha he same effe would be noied by an obserer whom we will imagine siing alongside his lok a he enre of he dis. Thus on our irular dis, or, o make he ase more general, in eery graiaional field, a lok will go more slowly or less quikly, aording o he posiion in whih he lok is siuaed (a res). For his reason i is no possible o obain a reasonable definiion of ime wih he aid of loks whih are arranged a res wih respe o he body of referene. Moreoer, a his sage he definiion of he spae oordinaes also presens insurmounable diffiulies. If he obserer applies his sandard measuring rod (a rod whih is shor as ompared wih he radius of he dis) angenially o he edge of he dis, hen as judged from he Galilean sysem, he lengh of he rod will be less han, sine, aording o Seion XII (Einsein here refers o he formula for lenghonraion) moing bodies suffer a shorening in he direion of moion. On he oher hand, he measuring-rod will no eperiene a shorening in lengh, as judged from K, when i is applied o he dis in he direion of he radius..this proes ha he proposiions of Eulidean geomery anno hold ealy on he roaing dis, nor in general in a graiaional field.. Bu are hese argumens orre physis?: Firsly, een if one ould use he formulas for ime-dilaion and lengh-onraion, hese formulas had been deried for wo inerial referene-frames moing linearly relaie o one anoher: In fa, only in his ase do he equaions of he orenz-ransformaion aually apply. In onras, he periphery of he roaing dis K does no follow a linear pah wihin K. I is hus 9