Ehrenfest Paradox, Sagnac Effect, and the Michelson-Morley Experiment

Transcription

1 Ehefes Paadox, Saac Effec, ad he Michelso-Moley Expeime Jaoslav Hyecek 1 Absac I his aicle he esoluio of he famous Ehefes paadox [1] is peseed. The paadox elaes o a spii disk ad he Special Relaiviy Theoy (SRT) applied o i. The paadox esoluio is based o he poposiio ha he paadox esuls fom a icoec applicaio of SRT o a sysem ha is o i a ieial moio. The ceifual ad ceipeal foces esuli fom he oaio ae always pese ad eed o be accoued fo. Usi he auho's peviously deived meic fo he axially symmeic space-ime he effec of ceifual ad ceipeal foces ca be coecly icluded. Whe his is doe o paadox is obaied ad i is show ha he spii disk appeas o have fla space-ime eomey. This fidi also povides he coec iepeaio of he ull esul of Michelso-Moley expeime, he coec explaaio of he Fizeau expeimes, ad a simple ad cosise explaaio of he Saac effec. The heoeical descipios of all hese expeimes should, heefoe, always iclude he effec of he ceifual foce of Eah's oaio. The measued daa fom ohe expeimes coduced o oai sysems ae explaied by he ieial mass icease as coecly descibed by SRT. Key wods: Ehefes Paadox; Mősbaue Effec; Special Relaiviy Theoy; Geeal Relaiviy Theoy; Schwazschild Meic; New Space-Time Meic; Ehefes Paadox; Saac Effec; Michelso-Moley Expeime; Tasvesal Fizeau effec; Oe-way Speed of Lih. 1. Ioducio Thee have bee may papes published o he esoluio of he Ehefes paadox wih vaious deees of success ad wih vaious coclusios [1]. Mos of hem ae ypically aimed a jusifyi he applicaio of oly SRT o his case ad he paadox esoluio is ofe obaied by a vey cooed easoi. The paadox esuls fom applyi he Loez coodiae asfomaio o a spii disk whose cicumfeece should coac while he adius should o sice he moio of he adius is always pepedicula o he disk oai diecio. As a esul he cicumfeece, accodi o SRT, is o loe equal o L o = π, which leads o a ofla space-ime eomey ha is o a domai of SRT. Fom his cosideaio i is clea ha oly he kiemaic appoach o esolve his poblem, as offeed by SRT, is o eouh. SRT deals wih he sysems i ieial moio ad does o accou fo he acceleaio ad ieial foces. I ode o esolve he paadox, i is ecessay o use he meic fom he Geeal Relaiviy Theoy (GRT) o use ohe space-ime meics ha descibe he o-fla space-ime eomey, which may be adoped o iclude he ceifual ad ceipeal foces. The well-kow meic descibi he space-ime aoud a ceally aviai body ha has a mass M is he Schwazschild meic: ds 1 1 R / ( cd ) 1 R / d d si d s s, (1) whee R = κm/c is he Schwazschild adius, κ he aviaioal cosa, ad c he speed of lih i ou local iealacic eihbohood. Howeve, a ew meic fo he axially symmeic space-ime has bee ecely published [], which is moe suiable fo sudyi his case: ds e ( cd ) d e d dz. () 1 jhyecek@escape.e Iseex, Ic

2 The paamee φ is he Newoia aviaioal poeial of a mass cofiuaio wih a axial symmey. The coodiae sysem fo his meic is cylidical wih he symmey axis i he z diecio. A bief deivaio of he meic show i Eq. is ive i he Appedix.. The paadox Resoluio The ew meic ca ow be used o esolve he paadox. A obseve placed o he spii disk cicumfeece, ad fimly holdi oo i, obseves, as is well kow, a ceifual foce. This foce is acually a ieial foce, which is he eacio o he ceipeal foce caused by he ie-aomic cohesio foces of he disk foci he aoms o avel i a cicle wih a adial acceleaio. The obseve is holdi oo hese aoms o say o he disk ad o o fly off. The ceipeal foce ca be modeled by a aviaio-like poeial whose adie is equal o his foce. The poeial will be called he pseudo poeial φ i ode o disiuish i fom he sadad aviaioal poeial φ. I is hus clea ha fom he poi of view of he disk aoms i does o mae if hey ae foced o avel i a cicle by he disk cohesio foces o by he aviaio-like foce deived fom he adie of he pseudo poeial aci diecly o he aomic uclei hemselves. The aviaio-like pseudo poeial ad is adie hus faihfully simulae he acio of he ceipeal foce. I he ex sep i will be cosideed ha accodi o he Riema piciple he moio of paicles avelli i a fla space-ime ude he ifluece of aviaioal foces ca be descibed by he cuved space-ime i which he paicles avel i a fee ieial-like moio alo eodesic lies wihou bei aced o by ay foces. Usi his cocep i will be possible o coecly iclude he acio of he ceipeal foce io he cosideaios ad esolve he Ehefes paadox. The pseudo poeial descibi he acio of he ceipeal foce is calculaed fom he followi cosideaios: fo he ceifual ieial foce fom he elaivisic Newo s law, as obseved i he laboaoy coodiae sysem, i holds ha: f cf mov /, () 1 v whee m o is he es mass of a es body locaed a he cicumfeece of he disk, ad whee i was cosideed ha he ieial mass depeds o velociy as follows: m i mo. (4) 1 v Fo he compesai ceipeal foce of he disk maeial ha is simulaed by he aviaioallike foce i will be also cosideed ha i depeds o velociy of he obseve locaed a he cicumfeece accodi o he aviaioal mass velociy depedece [] : m mo 1 v. (5) I is ecessay o emphasize hee ha his elaio does o follow he famous Eisei s Weak Equivalece Piciple (WEP) m i = m [4]. The pseudo poeial fo he simulai aviaioallike ceipeal foce is he foud fom he foce equilibium codiio: ' m 0 1 m 0 1. (6) Moe deails abou his equaio ad he deivaio of he fomula i Eq.5 ae also ive i he Appedix. Afe ieaio he pseudo poeial, as obseved i he laboaoy coodiae sysem,

3 is foud o be: d c ' l 1, (7) 1 0 c whee v = ω, ad ω is he disk aula velociy. The Riema cuved space-ime diffeeial meic lie eleme ha descibes he acio of he ceipeal foce o he aomic uclei of he spii disk is he obaied by subsiui he foud poeial i Eq.7 io Eq.: ds (1 cd d d ) (1 dz. (8) ) Sice he paicles i his cuved space-ime ae ow movi alo he eodesic lies wihou expeieci ay acceleaio i is o poblem o use he Loez SRT leh coacio fomula o calculae he cicumfeece leh of he spii disk as follows: L c Lo 1 v. (9) Howeve, accodi o he meic lie eleme i Eq.8 i is easily see, ha afe he subsiuio fo he elaio: ω = v, he cicumfeece leh as obseved by he laboaoy obseve is: L i Lo / 1 v. (10) The effecs hus pecisely cacel each ohe ad o paadox esuls. The laboaoy obseve will see he disk peiphey o coaced. The simila coclusio is obaied fom he meic i Eq.8 also fo ime. The simulaed ceipeal foce pseudo poeial affecs he ime meic coefficie hus causi he ime coacio while he Loez ime dilaio compesaes his effec. The space-ime eomey of he oai disk as viewed by he laboaoy obseve is fla. I hus seems ha all he SRT effecs ae bei compesaed by he cuved space-ime meic ad he oly emaii SRT effec ha is o compesaed fo is he ieial mass icease. Fially, he impoa law o veify fo his meic is whehe he obii es body saisfies he cosevaio of aula momeum. The fis ieals of Eule Laae equaios fo he obial moio deived fom he Laaia coespodi o he meic i Eq.8 ae as follows: 1 d k, (11) (1 ) d d k, (1) (1 ) d whee k ad α ae he abiay cosas of ieaio. Elimiai dτ fom hese equaios, sice dτ is a compued ivaia ad o a obsevable paamee, esuls i he sadad fomula fo he cosevaio of aula momeum: d. (1) d This is oe of he well-ecoized fudameal piciples of physics ha mus always be saisfied ad which he Schwazschild meic of Eisei s GRT supisily does o saisfy. The above deived esuls ae suppoed by he expeimes published elsewhee [5]. Howeve, he mos covici aume i suppo of he peseed Ehefes paadox esoluio comes fom he GPS daa [6]. I is a expeimeal fac ha he ime ae measued aywhee o he

4 Eah's suface is he same. Oly he small diffeeces i he aviaioal poeial affec he suface locaed clock ae. I is impoa o ealize ha he sadad Schwazschild meic does o offe he simila soluio o he Ehefes paadox ad does o suppo he cosevaio of aula momeum as saed i Eq.1. This is a cosequece of he icoec meic coefficie sadi by he aula coodiae. The esoluio of he Ehefes paadox usi he ew meic ad he diffee depedecies of ieial ad aviaioal masses o velociy hus povide a impoa addiioal suppo fo he coecess of hese fomulas. The easoi used i he above deivaio ca also be evesed ad i could be saed as a heoem ha i ode o avoid he Ehefes paadox he meic fo he axially symmeic aviaioal field has o have a fom ive i Eq.. I is also possible o eealize Eq.7 fo ay saic space-ime meic as follows: c l, (14) ad eealize he esul fuhe by elimiai he aviaioal poeial: 0 / 0, (15) whee ad o ae he meic deemias of he meic lie eleme i Eq.8 wih ad wihou oaio. Howeve, he meic deemia o has a slihly diffee meai i a eeal case as has bee explaied elsewhee []. The o is he deemia of he Mikowski fla physical spaceime ha coespods o space-ime of he cuved coodiae sysem wih he deemia. These deivaios, howeve, do o aee wih he soluio of Eisei s field equaios ad he Eisei s WEP, which ulimaely aises a siifica doub abou he accuacy ad coecess of he GRT [7]. Thee have bee may expeimes pefomed i he pas o oai sysems o cofim vaious GRT pheomea, bu as is clea fom he above explaaio oly he SRT ieial mass icease, ad he effecs elaed o he ieial mass icease such as he absopio lie shif i he Mossbaue Fe 57 effec ca be obseved [8]. No GRT effecs elaed o he cuved space-ime eomey ca be measued i hese expeimes. I is also ecessay o udesad i deail he cosucio of he paicula clock used i he expeimes o make sue ha i is he ime wha is measued ad o he ieial mass icease.. Michelso-Moley expeime The pevious secio has povided a adequae backoud fo he coec aalysis of he Michelso-Moley expeime [9] (MM) ha is discussed i his secio. Whe he ypical aalysis of his expeime is peseed he ceifual foce of he Eah's oaio is o cosideed. The Eah's aviaio subsaially oveshadows his foce; heefoe, he ceifual foce is deemed o impoa ad is eleced. I ode o sepaae he acio of he Eah's aviy fom he effecs of oaio i is helpful o imaie fo a mome ha mos of he Eah's mass is o oai ad oly a hi suface shell is oai ad lidi o he coe. Fom he poi of view of a obseve locaed o he Eahs' suface hee would be o oiceable diffeece. I is hus clea ha he ceally aviai mass of Eah is jus ehaci he Eah's suface cohesio ad povides he foce ha holds he objecs o he oud. The Eah's aviy hus does o have o be cosideed ay fuhe whe he oaioal effecs ae sudied. The MM expeimeal appaaus is elaively small i compaiso o he Eah's size ad i is hus clea ha hese ess ae always pefomed o he same aviaioal equipoeial suface. The siuaio is hus 4

5 L esseially ideical o he disk case ad he aalysis of he MM expeimes should always be coduced i he axially symmeic space-ime whe he Eah's oaioal effecs ae o be icluded. The expeimeal aaeme of he Michelso iefeomee is show i Fi.1. To simplify he dawi ad he aalysis oly oe half of he pahs he lih avels is show. I is easily see ha i is ecessay o calculae oly he ime he lih avels alo he pah fom he beam splie, mio M1, o he mio M, which is pepedicula o he moio of he iefeomee, ad o he mio M posiioed alo he pah paallel o he moio. The lih avel ime back fom he mios o he beam splie o ceae he iefeece is easily foud usi he same fomulas whee a simple subsiuio of v v is made. M M1 Beam splie M Lih beam v L v Fi.1: Schemaic diaam of Michelso Iefeomee. The beam splie M1 divides he lih beam io wo compoes, oe coiui hoizoally i he x diecio o he Mio M ad he secod oe efleced i he z diecio o he mio M. Mios he eflec he lih back o he beam splie fo he obsevaio of iefeece. To simplify he cosideaios fuhe, he aalysis of he MM expeime will be divided io wo cases: I he fis case i will be cosideed ha he iefeomee is locaed o he Eah's equao, so ha he axially symmeic meic ca adequaely descibe, i a small equao's eihbohood, he space-ime wihou complicaios of he Eah's aviaioal foce o bei pepedicula o he oaioal axis; ad i he secod case o he Eah's pole whee hee is o ceifual foce ad hus o ceipeal eacio o i. The axial coodiae sysem fo boh cases will be he sysem ceeed o he Eah's oaioal axis ad will o oae wih Eah. The meic descibi he space-ime of he iefeomee fo he equaoial locaio hus follows fom he meic deived i Eq.8. ds cd dx dz, (16) (1 v ) (1 v ) whee v is he velociy of he Eah's suface elaive o he saioay efeece, dx = Rdφ is he diecio alo he equao wih R bei he Eah's adius, ad dz is he meidia diecio pepedicula o he equao. Howeve, as was explaied peviously, he Loez coodiae asfomaio compesaes he meic coefficies effec by is leh coacio ad he ime dilaio, heefoe, he -x-z space-ime of he Eah suface, as viewed fom he o-oai coodiae sysem, is Mikowski fla. I he z diecio, howeve, he Loez leh coacio is o pese, heefoe, he physical leh of he iefeomee am is acually loe equal o: 5

6 l' l / 1 v. The iefeomee am's leh measueme fom wihi he iefeomee coodiae sysem, howeve, will o show his eloaio, sice he measui sick is also loe i his diecio. This coclusio follows fom he equieme ha boh ams should have he same physical (ivaia) leh ha mus be used i compui he phoo avel ime. Followi he meic i Eq.16, he physical leh of he am i he x diecio is heefoe: l l / 1 v l l'. I pacice, howeve, i may o be possible o make he ams exacly xph zph of he same leh, so he appaaus is oaed by 90 dui he measueme o deec ay effec o he iefeece fom he possible diffeece i he am's lehs. The speed of lih i he efeece sysem i he x diecio is c x = c ad i he z diecio is also c z = c, as his follows diecly fom he fla space-ime meic. The speed of lih i he x-z plae, paallel o he Eah's suface, is hus isoopic. I is ow easy o fid he phoo avel imes o he especive mios ad back o he beam splie. Fo he mio M he avel ime is foud fom he elaio: l ' v c, (17) / ad similaly fo he avel ime o he mio M ad back o he beam splie fom he elaios: l v / c, l v' / c. (18) 6 ' I is hus clea ha he oal phoo avel imes fom he beam splie o he especive mios ad back o he beam splie ae ideical ad equal o: o o lc v, (19) whee fo he oal avel ime i he x diecio i is: o '. The hihe ode ems ha esul fom he oaio iduced slih beam diecio deviaio have bee eleced i his aalysis cosidei ha he codiio l << R is always saisfied. No iefeece fie shif due o he Eah's oaio elaive o he hypoheical saioay medium ha suppos he phoo popaaio (ehe) ca hus be deeced by his expeime. The ull esul of he MM expeime pefomed a he equao, heefoe, does o cofim he special elaiviy heoy, sice all he SRT effecs ae always compesaed by he ceipeal foce ha is added o he Eah's aviy. Fo he veificaio of validiy of fomula i Eq.19 aohe deivaio is ive i secio 5 elaive o he obseve locaed o he Eah suface. A quesio could also be asked wha if he iefeomee wee posiioed veically. Fo his case i is clea fom he meic i Eq.8 ha he coodiaes z ad have he same meic coefficies ad ae hus iechaeable. The esul fo his iefeomee oieaio will be ideical o he hoizoal oieaio case excep pehaps fo some small ifluece fom he veical diffeeces i he Eah's aviaioal poeial. Howeve, fo he MM expeime coduced a he Eah's pole he siuaio is slihly diffee. Thee is o ceipeal foce hee o cacel he Loez coodiae asfomaio ad boh ams of he iefeomee appea o have he same leh whe o movi. Of couse, he expeimeal seup is appoximaely a es elaive o he coodiae sysem ioduced peviously, so o effec is expeced ad he phoo avel imes o he especive mios ad back o he beam splie mus be ideical ad equal o: l c. (0) o o / To coside he ehe dif i is heefoe ecessay o selec aohe coodiae sysem fo he aalysis. The ew coodiae sysem ca be a hypoheical saioay sysem efeeced, fo example, o he Uivese backoud adiaio elaive o which Eah will ow move wih a velociy u i he x diecio. The speed of lih i his coodiae sysem is cosideed also

7 isoopic ad equal o a local c. I his case SRT ad he Loez coodiae asfomaio mus ow be cosideed leadi o he followi expessios fo he phoo avel imes o he especive mios: l u, (1) fo he aival ime o he mio M, ad fo he avel ime o he mio M ad back o he beam splie: l 1 u u/ c, ' l 1 u u' / c. () By solvi hese equaios he oal phoo avel imes o he especive mios ad back o he beam splie ae: o l o '. () c 1u Aai, boh imes ae ideical, o iefeece fie shif due o he moio of Eah elaive o he Cosmic backoud adiaio coodiae sysem ca be deeced hus saisfyi oe of he fudameal ees of SRT. Fo he cofimaio of he deivaio coecess i is easily see ha Eq. ca be obaied diecly fom Eq.0 by simply icludi he Loez coodiae ime dilaio faco. The ull esul of he iefeece fie shif i he MM expeime is cosideed as a poof of SRT ad he Loez coodiae asfomaio. Howeve, as is commo wih all he ull esul expeimes, he poof is weak, sice hee ae usually ohe assumpios leadi o he same coclusios as was clealy demosaed above. Eq.19 is diffee fom he classical esul peseed i mos exbooks, sice hey do o coside he ceifual foce ad hus deive oly fomula i Eq., which descibes a effec ha is acually o obsevable by he obseve o Eah, ad which is o compaible wih he sadad Saac effec fomula ha is discussed ex. 4. Saac effec The Saac effec [10] is cosideed by may oppoes of SRT as a poof of is ivalidiy. O he ohe had may suppoes deived SRT equaios fo i. The explaaio ha is cofimed by he GPS daa is simple o obai. As i was show peviously he space-ime i he -x-z coodiaes is fla due o he ceifual foce effec, so he ime diffeece fo he sials ha popaae alo he equao i paallel wih he diecio of Eah's oaio ad aais i o equivalely i small poable oai sysems ae easily deived fom he followi equaios: R v / c, R v / c. (4) 7 The ime diffeece eeded fo he evaluaio of he iefeece fie shif is he equal o: 4 Rv v. (5) This fomula is well kow i he idusy ha builds he yoscopes based o his effec. Fo Eah a he equao ad a he sea level his diffeece is: Δ e = 41.6s, which he GPS has veified. I is also impoa o oe ha his esul is cosise wih he fomula deived i Eq.19, sice he udelyi physics is he same. The coefficies appeai i he especive deomiaos mus be ideical, which poves he coecess of he axially symmeic meic used i he aalysis. Aai, o SRT effecs ca be deeced i his expeime, sice he Loez coodiae asfomaio is compesaed by he cuved space-ime meic ha esuls fom he modeli of he ceipeal foce. The oppoes cii his expeime as a poof ha SRT is wo ae o coec, sice he SRT effecs ae o pese hee. Fuhemoe he suppoes of

8 SRT who modify Eq.5 o iclude he elaivisic effecs ae also wo fo he same easo, sice hee ae o SRT effecs i he Saac expeime. 5. MM expeime elaive o he obseve o Eah I mih be possible o aise a objecio o he above peseed deivaios of he iefeece fie shif i he MM expeime ad he Saac effec, sice he deivaios wee pefomed elaive o a o-oai efeece fame. The expeimes ae always obseved fom he suface of he oai Eah, so a diffeece i ime bewee he oai ad o-oai coodiae sysems mih be expeced accodi o he classical elaivisic poi of view. Howeve, as i was clealy show, he Eah's suface meic as viewed fom he o-oai coodiae sysem is fla ad o ime diffeece is obseved bewee he clock aes o he equao ad o he poles as is also cofimed by he GPS [6], he fomulas fo he fie shif should heefoe be ideical. Sice he space-ime of he expeimes is fla ad he Loez coodiae asfom does o hold, he velociies, icludi he speed of lih, mus be addi classically. Fo he speed of lih obseved o Eah i he x diecio i mus heefoe hold ue ha: c x c v, (6) ad similaly fo he speed of lih i he z-diecio i mus hold ue ha: c z c v. (7) The Eah's ceipeal foce oehe wih he foce of aviy heefoe compesae fo he elaivisic effecs ad he Loez coodiae asfomaio of objecs ha ae suppoed by he Eah's suface, o move alo he suface o expeiece he foce. The oly diffeece, as peviously meioed, is he diffeece i he physical leh of he iefeomee am i he z diecio caused by he meic ive i Eq.16. The z diecio iefeomee am leh is: l' l / 1 v. (8) The ime of he phoo flih fom he beam splie o he mio M ad back as obseved o Eah is he simply as follows: ad o he mio M equal o: l v, l v 8 ', (9) l 1 l c ( 1 v ) ( c v ) ( c v ). (0) The oal phoo avel imes ae heefoe ideical ad equal o: o o ' lc v. (1) I is impoa o oe ha his is he same fomula as deived i Eq.19 wih o Loez faco fo he ime asfomaio bewee he o-oai coodiae sysems ad he Eah's suface coodiae sysem as explaied peviously. This also cofims he classical velociy addiio i oai sysems caused by he ceipeal foce of oai Eah icludi he "supelumial" velociy c = c+v [11]. Fially, i is also clea ha fo he same easo he Saac fomula deived i he o-oai efeece coodiae sysem is he same as i he oai sysem, so he obseve oai wih he disc sees he same fie shif as he obseve i he o-oai coodiae sysem.

9 6. Fizeau expeimes Amo he emaii ess ha should also be meioed as bei possibly affeced by he Eah's oaio is he oe ha is famous fo bei amo he fis o veify SRT. This is he Fizeau expeime whee he speed of lih was measued i a movi wae medium [1]. Fo he es appaaus locaed o he Eah's pole hee will be o diffeece fom he usual elaivisic eame usi he velociy addiio fomula: c / v c 1 c ' v1, () 1 v whee is he idex of efacio of he movi medium ad v is is velociy elaive o he saioay obseve. The expessio i paehesis i Eq. is he well-kow Fesel lih dai coefficie. Afe cosidei he lih eui pah i he diecio opposie o he media flow he phoo flih ime diffeece fo ceai he iefeece will be: 4lv( 1) 4lv p ( 1). () c v c O he equao, howeve, he velociies add classically ouside of he medium ad divided by iside of he medium, so he esuli speed of lih as obseved by he laboaoy obseve is: c v c 1 c' v v1. (4) The secod em i he paehesis ca hus be cosideed o be he ew lih dai coefficie. Similaly, fo he phoo flih ime diffeece a he iefeece mio he esul is: 4lv( 1) 4lv e ( 1). (5) c v ( 1) c The elaivisic fomula hus pedics a slihly lae iefeece fie shif a he Eah's pole. Usi he oiial expeimeal seup paamees foud i he published lieaue [1] ad he coeced value fo he wae flow due o he velociy pofile acoss he ube as published elsewhee [1], he fomula i Eq. pedics he esul: Δλ p /λ = ad he fomula fom Eq.5 he esul: Δλ e /λ = , whee: Δλ =cδ. The acual measued value of he iefeece fie shif was: Δλ/λ = The oiial Fizeau expeime did o coside he wae flow velociy pofile acoss he ube diamee, hus ioduci a cosideable eo io he measueme. I is also possible ha he idex of efacio may have chaed depedi o he wae pessue. This should have bee veified. Vaious ohe objecios abou he deails of he wae flow alo he Eah's suface ad is espose o he aviaioal ad ceifual foces could also be aised, sice wae is o a solid objec. All hese complicaios make his es o a paiculaly useful ad covici fo he poof. A moe compelli evidece fo he classical Galilea velociy addiio o oai plafoms, howeve, comes fom he ece measuemes of he asvesal Fizeau effec [14], whee fo he laeal imae displaceme Δ due o he souce oaio as obseved houh a saioay disk of a hickess L i was expeimeally deemied ha: Δ = L(-1)v/c. This fomula is easily deived fo he case of he oai disk ad a saioay souce usi he ew lih dai coefficie fom Eq.4. Cosidei ha fo he fis ode appoximaio fo a elaively slow oaioal velociy i compaiso o c he phoo avel ime acoss he disk hickess is sill: L = L/c, he laeal imae displaceme is equal o: 9

10 1 v L v1 L 1. (6) c A ieesi aspec of his calculaio is ha his esul aees wih he case whe he lih souce is movi ad he lass disk is saioay hus havi o ceifual foce i i. A advaae of such a expeimeal cofiuaio is ha he disk is fee of ay oaio caused disoios ad possible sess iduced idex of efacio chaes. The imae displaceme due o he lih avel ime i he lass fo his case is he: v v v L L L ( 1). (7) c c c The em Lv/c epeses he imae shif whe o lass medium is pese, which eeds o be subaced. The effec is hus pefecly symmeical saisfyi he souce-obseve elaiviy as expeced. I hus seems ha he imae of he ceifual foce effec o he addiio of velociies i oai plafoms is absoluely ecessay i ode o saisfy he basic ee of elaiviy. The expeimeal daa cofimi his esul wee also peseed a a cofeece i Rochese New Yok i Jue 1, 007 [15]. Fially, his esul suess ha whe Maxwell's equaios ae used o descibe he EM fields ad he wave popaaio i solid oai media, i is also ecessay o accou fo he ceifual foces ad he esuli cuved space-ime meic. 7. Coclusios The ew space-ime meic was used i esolvi he Ehefes paadox ad he elaed expeimeal veificaios. The meic allowed fo a coec iclusio of he ceipeal foce ad is pseudo-poeial io he cosideaios, which compesaes fo he ime dilaio ad he leh coacio effecs of SRT. The ieial mass icease, howeve, is o compesaed fo, which explais he published expeimeal esuls [7]. The space-ime eomey esuli fom he ew meic also clealy explaied he Saac effec ad he failue o deec he ehe wid i he Michelso-Moley iefeomee expeimes. The esoluio of he Ehefes paadox, he explaaio of he Saac effec, ad he explaaio of he ull esul of Michelso-Moley expeimes hus validae he ew meic coecess ad he diffee depedecies of ieial ad aviaioal masses o velociy. Fially, i was clealy show, suppoed by he GPS daa [6], ad also suppoed by he ece expeimes o aspae cylides ad oai lih souces [14, 15], ha he velociies i oai sysems, due o he ceipeal foce, add classically icludi he speed of lih. This is a ieesi fac ad a fudameally vey impoa fidi ha is coay o he sadad SRT poi of view. The cuved space-ime effecs caused by ceipeal foce hus mus be cosideed whe evaluai he EM fields ad he wave popaaio usi Maxwell's equaios i oai solid body plafoms. Appedix: Meic fo he space-ime wih he axial symmey The deail meic deivaio fo his space-ime is available elsewhee []. The deivaio peseed hee uses a slihly diffee appoach. The eeal fom of he meic lie eleme fo a saic axially symmeic space-ime wih he aviai axis posiioed alo he z diecio is as follows: ds whee he meic coefficies ca deped oly o. cd d d dz zz, (A1) I he ex seps he meic coefficies,, ad zz will be foud by aalyzi moio of a small es body i he -z plae. The Laaia descibi such a moio is equal o: 10

11 cd d dz L zz. (A) d d d Sice he Laaia iself is also he fis ieal (L = c ), ad sice fo he fis ieals of Eule- Laae (E-L) equaios coespodi o he ime ad z coodiaes holds ha: d d, (A) dz zz k, (A4) d whee k is a abiay cosa of ieaio, i is possible o wie he followi equaio ha he es body moio mus saisfy: d d zz dz d c c By diffeeiai Eq.A5 wih espec o τ he followi esul is obaied: d d d d zz 1 dz d zz dz d z d d d d 1 c. (A5) 1 1. (A6) I his equaio i is assumed ha he meic coefficies ae fucios of he Newo aviaioal poeial. I is also clea ha he acceleaio i he diecio cao deped o velociy, paiculaly o he velociy i he z diecio, ad sice hee is o aviaioal field i ha diecio he paameeized acceleaio (z diffeeiaed wice wih espec o τ) mus also be zeo. Boh ems i he squae paehesis hus mus be equal o zeo. Cosidei also fom Eq.A4, ha: dz/dτ = k/ zz, he followi codiios mus be saisfied:, 1. (A7) zz The foud meic coefficies simplify Eq.A6 o ead: d d c 1 zz. (A8) This equaio has he expeced simple fom, sice he paameeized acceleaio i a saic axially symmeic space-ime ca deped oly o he adie of φ (). The em i he paehesis, howeve, mus be equal o uiy i ode o saisfy he well-kow ad may imes expeimeally veified Eisei's equivalece piciple (he Eisei elevao) wih he acceleaio equal o foce of aviy: d. (A9) d This equaio also saisfies he well-kow covaiace piciple of eso calculus wih he oal of covaia quaiies equal o boh sides. This leads o he followi codiio: c 1 1. (A10) 11

12 By ieai his esul usi he bouday codiio a ifiiy whee he poeial is se o zeo, he meic coefficie is foud equal o: 1 e. (A11) The meic coefficie sadi by he aula coodiae is foud by cosidei aai a small es body obial moio i he space-ime defied by he followi meic lie eleme: The Laaia descibi his moio is he: cd d d dz ds. (A1) cd d d dz L. (A1) d d d d The fis ieal of E-L equaio coespodi o he aula coodiae is: d, (A14) d whee he suiable ieaio cosas was used. The fis ieal fo he ime coodiae is he same as i Eq.A. I is well kow ad expeimeally cofimed ha he obial moio mus saisfy he cosevaio of aula momeum. Fom Eq.A ad Eq.A14 he follows ha: d, (A15) d ad fom his esul he also follows ha fo he aula meic coefficie i is:, (A16) sice he meic coefficie sadi by he adial coodiae is uiy. The adial coodiae disace is equal o he adial physical disace i his case. Subsiui hese esuls io Eq.A1 he meic lie eleme used i Eq. is obaied: ds e cd d e d dz. (A17) Fially a sho comme is ecessay elaed o Eq.6: The ih had side of Eq.A ca be facoed ou io he wo ideical compoes: d d, (A18) which makes i possible o explicily show he compaibiliy wih he Loez coodiae asfomaio. This is accomplished by subsiui fo oe of he ems he expessio fo obaied fom Eq.A5, fo dz/dτ = 0, which ca be eaaed ad ewie usi Eq.A as: whee c deoes he local adial speed of lih v 1, (A19) c c, ad whee v = d/d. Eq.A18 ca he be eealized fo ay diecio of moio icludi he disc cicula moio as follows: d d 1 v. (A0) I is ow obvious ha his fomula is compaible wih he Loez ime coodiae asfomaio fo he fla space-ime whe = 1 ad also wih he ime coodiae asfomaio whe he es

13 body is saioay i a aviaioal field wih v = 0. Muliplyi boh sides of Eq.A9 by he es body es mass m o ad subsiui fo he ivaia dτ fom Eq.A0, he esul is: d d v m o 1 v m o 1 v. (A1) Fom his esul he follow he depedecies of he ieial ad aviaioal masses o velociy: mo mi, (A) 1 v m mo 1 v. (A) Sice fo he peviously ioduced simulai aviaioal-like poeial o he pseudo poeial φ i is o impoa whehe he foces ae eeaed by he mass locaed alo he z axis o by a oaio aoud he z axis, he simulaed aviaioal-like foce equilibium wih he ieial ceifual foce should, heefoe, follow Eq.A1 ad be moe eeally wie as: m o 1 1 ' m o 1, (A4) whee dv/d = v / ad v = ω fo he cicula obis. Howeve, all he ems coaii cacel ou maki Eq.6 coec also. The deivaio of hese eeal fomulas fo he ieial mass ad he aviaioal mass depedece o velociy ca be also foud elsewhee [16, 17, 18]. Refeeces 1. hp://l.wikipedia.o/wiki/ehefes_paadox.. J. Hyecek, Physics Essays, vol., No., (009) p J. Hyecek, Physics Essays, v 0, No., (007) p J. Hyecek, Apeio, Vol. 16, No. 1, Jauay H. W. Thim, IEEE Tasacios o Isumeaio ad Measueme, vol. 5, No.5, Ocobe B. W. Pakiso, J. J. Spilke J., Global Posiioi Sysem: Theoy ad Applicaios Volume I. Published by: Ameica Isiue of Aeoauics ad Asoauics, Ic., 1996, p J. Hyecek, Physics Essays, vol. 4, No., (011) p H. J. Hay, J.P. Schiffe, T. E. Cashaw, P.A. Eelsaff, Physical Review Lees, vol. 4, Issue 4, pp , Feb H. Mulle, S. Hema, C. Baxmaie, S. Schille, A. Pees, Phys. Rev. Lees 91, (00). 10. R. Wa, Y. Zhe, A. Yao, Phys. Rev. Lees (004). 11. S. J. G. Gif, Applied Physics Reseach, Vol., No. 1; May A. A. Michelso, E. W. Moley, Ameica Joual of Sciece, d, Vol. 1, No. 185, May M. H. Fizeau, Phil. Ma. S. 4. Vol. 19. No. 17. Apil J. Leach, A.J. Wih, J.B. Göe, J.M. Giki, L. Alle, S. Fake-Aold S.M. Bae, ad M.J. Pade, Physical Review Lees, 15 Apil M. Pade, J. Leach, J. Göe, S. Fake-Aold, L. Alle, A. Wih, Cofeece o Coheece ad Quaum Opics (CQO), Rochese, New Yok, Jue 1, 007, Oal Sessio III, pape CME1.pdf (available olie). 16. J. Hyecek, Physics Essays, v, No 4, (009) p hp:// 18. hp://vixa.o/auho/jaoslav_hyecek 1