Time dilation and Langevin paradox

Transcription

1 Time diaio ad Lagei paradox A López-Ramos Deparameo de Físia, Uiersidad de Oiedo Edifiio Ese, Campus de Viesques, 333 Gijó, Spai E-mai: ramos@uioi.es Absra. The priipe of iariae of he eoiy of igh is oy aid for he wrog measuremes of ieria obserers who igore heir ow moeme ad osider hemsees a res. The Lagei (or ok) paradox arises whe i is assumed ha he rea eoiy (ad o oy he wrog measured eoiy) of igh i free spae a be he same for differe obserers. Neerheess, i is geeray assumed as rue suh iohere assumpio ha eads o he paradox. To arify he issue is of grea ieres o aayse he physis uderyig he pheomeo of ime diaio (whih is o oy he resu of a mahemaia formua) ad his has bee doe i his work. The exa aue of igh draggig oeffiie is aso deried. PACS umber: 3.3.p Key words: assia ime diaio, reaiisi ime diaio, Lagei paradox, ok paradox, wi paradox.. Iariae of he eoiy of igh We sha begi wih he usua exampe of a ieria obserer O paed a he midde of a osed arriage i a rai moig a uiform eoiy. Le us suppose ha he obserer i he rai ries o es his sae of moeme by sedig a igh siga o he eds of he arriage, where wo mirrors are paed. The arriage does o drag ha siga, beause he eoiy of igh i free spae does o deped o he eoiy of is soure. As, durig he rasmissio, he bak ed of he arriage omes o mee he igh siga, whie he fro ed reedes from i; he wae fro reahes firsy he mirror i he bak ed. Neerheess, obserer O does o kow i ye. He eeds o erify he arrias of he igh sigas o he eds of he arriage; for isae, by ookig a he images of hose arrias. Bu whe omig bak, he reaie eoiies of he sigas are exhaged: ha omig from he bak mirror, has ow o pursue obserer O; who, meawhie, omes o mee he siga proeedig from he fro mirror. Therefore, he reurig ime of eah siga is jus equa o he goig ime of he oher. The whoe go-ad-reur ime is he same for boh pahs; i suh a way ha, despie he fa ha igh reahed firsy he bak mirror, he obserer i he arriage sees boh rays of igh arriig simuaeousy a he arriage eds; ad eeryhig happes as if he eoiy of igh wih respe o him were he same i he direio of his moeme ad i he opposie.. Ligh paria draggig The resu obaied aboe is riia, bu obserer O i he arriage a ry o es, i a more sophisiaed way, wheher he is moig, paig he amp iside a waer ak wih raspare was ad wahig he refeios of he igh omig from he bak mirrors B ad B, ad he fro mirrors, F ad F, respeiey, as show i Fig. Figure. Ligh goig hrough waer ad reurig hrough air.

2 Sie he eoiy of igh i waer is he quoie bewee he eoiy of igh i air ad he refraie idex of waer, he differee i ime bewee he igh rays ough o be arger (i absoue aue) for he goig pahs i waer ha for he reurig pahs i air. This ew aemp, he Masar-Jami experime [], was made i 87, by meas of he obseraio of he ierferee friges produed by he wo rays, usig he Earh (whih is reay moig) as he arriage. Bu i aso fais, as expeed by he auhors of he experime: obserer O pereies agai he rays simuaeousy. I order o expai a simiar experimea resu (he maieae of he aberraio age i eesopes fied wih waer) ad o make he reaiiy of uiform moeme ompaibe wih he uduaory heory of igh; Frese [] had predied, i 88, ha waer arries ou a paria draggig of igh sigas. Le us see, wih Masar, wha he Frese s draggig oeffiie, f, (fraio of he eoiy of waer rasmied o igh sigas) mus be whe he eoiy of he arriage is muh smaer ha ha of igh. Caig he eoiy of igh i air;, he refraie idex of waer,, he eoiy of he rai ad, he arriage egh measured by he obserer O o he groud; oe has, equaizig he ime differees i goig ad reurig igh pahs (Fig. ): Figure. Veoiies of igh sigas i waer ad air. from where: f f ( f) ( f) ; ( ) ( ) ( ) () f f () ad whe << : f f (3) whih is he same aue as ha obaied by Frese (i air ad igh draggig is egigibe). Fizeau [3] ofirmed experimeay, i 85, Eq. (3), by rasmiig igh hrough moig waer. Neerheess, Freses souio shows weak pois. O oe had, i is a ad ho hypohesis o expai a experimea fa; o he oher had, Frese had o assume a hage of eher desiy wihi moig bodies o jusify, physiay, igh paria draggig. Ad his aer is oo ie ompaibe wih he fa ha he refraie idex (ad, wih i, he hage i eher desiy) depeds o igh waeegh. Lorez [] gae, i 89, a heoreia expaaio of he igh draggig oeffiie whe buidig his eero heory. A simpified deduio a be easiy made by usig figures 3 ad. Fig. 3 shows igh rasmissio i quie waer, whih osiss of he reieraio of wo seps: a igh rasmissio i free spae from oe moeue o aoher; ad a ibraio of a eero reahed by igh. If we a he disae bewee wo oseuie moeues of waer;, he ime eeded by igh sigas for heir propagaio i free spae from oe eero o aoher i he ex moeue; ad, he ime i whih sigas are a eero osiaio (ad osequey hey do o moe o); he aerage eoiy, u, of igh sigas i waer a res is u ()

3 We obai he aue of from he refraie idex of his medium. Sie i resus: u (5) (6) ( ) ( ) Ligh rasmissio whe waer moes is show i Fig., where he egh oraio of he disae bewee waer moeues is o ake io aou, beause we are ow ieresed i he o-reaiisi ase. Figure 3. Ligh rasmissio i quie waer. Figure. Ligh rasmissio i moig waer. The siga se by moeue A a A (beig B a B ) reahes moeue B a B ad is re-emied by B a B 3 posiio. Thus, he aerage eoiy of igh sigas i moig waer wih << is u (7) The ime ow eapsed i he firs sep,, is obaied from Fig. : (8) whie. We he hae, whe he eoiy of waer is muh smaer ha ha of igh: u ( ) ( ) (9) beig i his way heoreiay dedued he draggig oeffiie. Lorezs hypohesis aso jusifies he hage of he refraie idex wih frequey, as far as he eero osiaio depeds o ha frequey. 3. Time diaio The heory oerig he priipe of reaiiy of uiform moeme was deeoped whe he mirosopi osiuio of maer was ie kow. Bu he mehaism aig i he rasmissio of igh sigas hrough waer is raher geera. Chemia reaios (iudig iig beigs meaboi proesses), osious or uosious erous impuses (suh as hose whih oro he hearbeas), rasmissios of srehigs or dispaemes i a soid (suh as he rasmissio of eripea aeeraio bewee ex 3

4 aoms of ok gears), e., are arried ou by meas of eeromagei ieraios bewee eeros ad bewee uei ad eeros. Sie he eoiy of rasmissio of eeromagei sigas i he spae free of maer does o deped o he eoiy of heir soure, sigas raeig bewee paries beogig o a moig body hae o pursue he reeiig paries, whih hage heir posiio durig he siga rasmissio. Ad, ahough is rue ha he reeiig parie wi ome owards he siga may imes, i yi proesses he whoe ime ioed ireases, as he ime saed by paries moig o he sigas wi o ompesae ha os by paries moig away from he sigas [5]; jus as i happes wih he oa imes of igh rays i he we-kow Miheso-Morey experime. Thus, a immediae osequee of subaomi sruure ad of he fa ha he eoiy of eeromagei waes i free spae does o deped o he eoiy of heir soure is ha proesses i a moig body ru a a sower rae ha whe i is a res. This pheomeo (ime diaio) a aso be expaied wih oher words: i is due o he fa ha moig bodies drag heir eeros, bu o heir phoos, so ha hese as eed, i aerage, more ime o go from oe eero o aoher. I a hypoheia marosopi sysem moig wih he eoiy of igh, a of is aura proesses woud sop. Amog a he paries arryig ou a physia or hemia proess, hose paed behid aog he direio of moeme woud o be abe o oiue he proess: eeromagei sigas se o he oher paries, as quik as he sigas hemsees, woud eer reah heir arge. Proesses akig pae i suh a sysem woud oy be oisios agais exera paries or waes foud i is way. I mus be emphasized ha his pheomeo is a osequee of assia physis (uaoidabe wihou a ompee draggig of eeromagei sigas). Therefore, i is reay surprisig o read from a auhor [6] of seera exbooks o reaiiy ha ime diaio was a shok for assia meaiy. The oey ha he priipe of reaiiy irodues abou ime diaio is ha he derease i he proesses rae is o deeabe wihi a sysem a uiform moeme beause i is quaiaiey uique for a pheomea akig pae here. Our rai obserer ao reaize, for isae, he derease i his hearbeas rae beause i is aso he same as he derease i his ok rae. This uiqueess is due o wo faors: by oe side, o egh oraio, whih equaizes he deay for sigas moig parae or raserse o he direio of moeme; ad, by aoher side, o mass irease, whih equaizes he deay for a kid of proesses. Larmor [7] was he firs, i 9, who spoke abou boh ime diaio pheomeo ad is quaiaie uiqueess. Lookig for he reaioships eeded o keep he aidiy of Maxwe equaios wih he measuremes made by differe ieria obserers, he foud ha a iera proesses akig pae i a sysem wih uiform moeme irease heir duraio aordig o: () where is heir duraio whe he sysem is a res, whih is aed proper duraio. This aows us o fid he exa aue for he eoiy of igh i moig raspare media, wihou usig he approximaio <<. I a soid, or i a fuid surrouded by a soid; oe has, beause of egh oraio, ha /. O he oher had, he eero ibraio sep aso suffers ime diaio, due o he irease of ieria, ad is duraio is o from Eq. (6), bu Iroduig hese orreios io Eq. (9), i resus: ( ) ( ) ( ) () u ( ) / / ( ) ( ) () whih is he same resu as ha gie by he reaiisi formua for he rasformaio o S of he eoiy / i S, as i ough o be.

5 I ur, we a ow aso dedue he exa aue of he draggig oeffiie, isead of he approximae aue gie by Frese. From: u f (3) we hae: f f () beig, fiay, his oeffiie depedig o he aue of.. Lagei paradox If a spaeship eaes Earh wih uiform eoiy ad hages is eoiy o he opposie direio whe i is a a disae (measured by he Earh), due o ime diaio of moig oks, here wi be a differee T ( ) (5) bewee he readigs of wo oks preiousy syhroized (a he begiig of he jourey), paed o he Earh ad o he spaeship, whe hey mee agai (he ok o Earh beig ahead). Lagei or ok paradox arises if he spaeship assumes isef a res whe i is a ieria obserer. I suh a ase i woud be he Earhs ok ha rus sower boh whe goig ad whe reurig. To aompish Eq. (5) i is eessary o assume a jump (or a exremey quik ariaio) i he readigs of he Earhs ok jus a he mome i whih he ship hages he direio of is moeme. There woud be aso a disoiuiy i egh a ha isa: he disae o Earh hages from / o (whe he reaie eoiy is zero) ad agai o /, as show i Fig. 5. The aeeraio sages eed, he, o be deisie for he souio of he paradox. Figure 5. Lagei paradox. Bu higs are reay ee worse: as some auhors [8, 9] remark, he ime readigs i he Earhs ok woud hae o drop bak if he ship ireases is eoiy before he reersa of is moio, beause his impies ha he ees happeig i he Earh ha ow beome simuaeous for he oyager are i he pas of hose whih were simuaeous before he hage of eoiy. Wha is more paradoxia regardig Lagei paradox is ha he fas desribed i he preious paragraphs are assumed o be rue; ad i has bee said ha hey are a osequee of he pseudo- Euidea geomery of he four-dimesioa spae-ime maifod a ariae wih he eeryday oeps of spae ad ime []. O he orary, suh impausibe resus ome from he iompaibe assumpios made by he oyager, of beig himsef a res whe goig ad aso whe reurig. If he assumed himsef a res whe goig, he is obiged o assume as his ow eoiy whe reurig he oe ha he has ow i he same ieria frame where he was a res before. As he Earh is moig wih speed i ha frame ad he ship reurs wih a reaie eoiy i he Earhs measuremes, he oyager mus ow assume ha, o aou of he formua for he rasformaio of eoiies, his rue eoiy is: 5

6 6 V (6) Thus, whe reurig, he oyager mus osider himsef faser ha he Earh, deduig ha his ok rus sower ha he Earhs ok ad i mus be remarked ha he oyager aso kows ha he eoiy of igh wih respe o him ao be agai. The auaios whe he spaeship assumes isef a res durig he firs sage of he oyage are dedued from Fig. 6. Figure 6. Souio of Lagei paradox. I he firs sage, he spaeship is a res ad i wi o begi is moeme ui he disae o Earh is i he Earh s measuremes. Due o egh oraio of he moig frame where he Earh is a res, his disae is / for he ship, ad he ime eapsed i his sage, (whih oiides wih he adae i he readig of he ok o he ship, ) is (7) whie he orrespodig adae i he readig of he moig ok o Earh is, o aou of ime diaio: ( ) (8) I he seod sage, he spaeship moes wih a eoiy V pursuig he Earh, whih goes o moig a eoiy. Therefore, as dedued from Fig. 6: ( ) 3 V (9) The adae i he readig of he (ow moig) ok o he spaeship resus: ( ) V () whie he orrespodig adae i he readig of he Earh s ok is ( ) ()

7 Durig he whoe jourey, he adae i readig of he ok o he ship is gie by he sum of Eqs. (7) ad (): () whie ha of he Earh s ok is gie by he sum of Eqs. (8) ad (): (3) Thus, he differee bewee Eqs. (3) ad () eads o Eq. (5) wihou ay disoiuiy boh i egh or ime, as i mus be; ad he ifuee of he aeeraio sages is egigibe. Ahough hose as auaios were aready iuded i referee [], where he fia readigs of he oks are dedued for differe frames, hey are o he auaios gie i [] for he ship frame, whih are agai hose based o he iosisey of osiderig ha he ship is a res i he wo mai sages of is moeme. Of ourse, as he oyager (who suffers he aeeraios) kows perfey ha he is o a ieria obserer, he a beer orre his measuremes from he begiig ad aep ha his iiia referee frame before he jourey (he Earh frame) is ha i whih he eoiy of igh is isoropi, as has bee doe i referee [9]. I his ase, his auaios are he same as hose of obserers o Earh. 5. Cousios Whe aaysig he Lagei paradox, i is eessary o ake io aou ha he oyager hages from a ieria referee frame o aoher, ad ha he eoiy of igh wih regard o himsef ao be isoropi wie (whe goig ad whe reurig). The priipe of reaiiy says ha he eoiy of igh a be assumed as isoropi for ay ieria referee frame, bu oy for oe. To assume ha he eoiy of igh is isoropi for wo differe ieria frames is a misudersadig ha eads, preisey, o he paradox. This misake omes from he so-aed speia reaiiy heory, whih ress o a ofusio bewee he resus of measuremes ad he rea aues of physia magiudes. O he oher had, i has bee remarked ha ime diaio is a pheomeo iuded i assia physis, beig is quaiaie uiqueess i a ieria moig frame wha he priipe of reaiiy adds abou i. Referees [] Masar, E 87 Modifiaios qu éproue a umière par suie du moueme de a soure umieuse e du moueme de obseraeur A. Si. Éoe Norm. Sup.() [] Frese, A J 88 Sur ifuee du moueme erresre das queques phéomèes dopique A. Chim. Phys [3] Fizeau, A H 85 Sur es hypohèses reaies à eher umieux, e sur ue experiee qui parâi démorer que e moueme des orps hage a iesse ae aquee a umière se propage das eur iérieur C. R. Aad. Si. Paris [] Lorez, H A 89 La héorie éeromagéique de Maxwe e so appiaio aux orps mouas Arh. Néer [5] López-Ramos, A L 998 Teorías de a reaiidad de moimieo uiforme pp 5-6 (Oiedo: Pubiaioes de a Uiersidad de Oiedo) [6] Rider, W 97 Eisei s prioriy i reogizig ime diaio physiay Am. J. Phys [7] Larmor, J 9 Aeher ad Maer p 7 (Cambridge: Cambridge Uiersiy Press) [8] Paaios, J 96 Reaiidad. Ua Nuea Teoría pp 56-7 (Madrid: Espasa-Cape) [9] Va Fader, T 3 Wha he goba posiioig sysem es us abou he wis paradox Apeiro [] Tereskii, Y P 968 Paradoxes i he Theory of Reaiiy p (New York: Peu Press) [] Buider, G 959 Resouio of he ok paradox Am. J. Phys