Einstein s theory of special relativity with respect to sound waves

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1 Inerninl Jurnl Applied Reerch 17; 3(11): ISSN Prin: ISSN Online: Impc Fcr: 5 IJAR 17; 3(11): wwwllreerchjurnlcm Receied: Acceped: Lendr Meléndez Lug Deprmen de Fíic Iniu Ncinl de Ineigcine Nuclere Crreer Méic-luc /n L Mrque Occc CP 575 CDMX Méic Einein her pecil relii wih repec und we Lendr Meléndez Lug Arc An riginl her pecil relii cuic i deelped Fr hi new her wih cu n und we ined elecrmgneic we new crdine rnrmin h re dieren rm he Lrenz rnrmin which rnrm phericl we rn r he erer re in nher phericl we rn r he erer in unirm relie min re deduced Preiul i h een quniiel demnred h Lrenz rnrmin d n d wh Aler Einein he d In he demnrin h i decried in he preen ud we cnclude h he cnnc he peed ligh uing he Lrenz rnrmin r ne he w erer i n iied Al wihin hi new cuic her new eprein r he Dppler Eec re ined wih dierence in regrd he rdiinl eprein Finll new reliiic cnequence re und h preen dierence wih repec he rm he her Aler Einein Kewrd: Specil relii Lrenz mmer Lrenz rnrmin reliiic cnequence ime dilin Crrepndence Lendr Meléndez Lug Deprmen de Fíic Iniu Ncinl de Ineigcine Nuclere Crreer Méic-luc /n L Mrque Occc CP 575 CDMX Méic 1 Inrducin A pecil relii cuic her Wh genere nher her pecil relii in hi ce n cuic her? h i wih und ined elecrmgneic we hi new rgumen egn in 5 er h w declred he Inerninl Yer Phic cmmemring he iieh nnierr he deh Aler Einein ( ) nd he 1 er he Specil her Relii [1 ] On h ccin erie cnerence he Ninl Aunmu Unieri Meic were rgnied One pecili pke Einein her pecil relii During quein ime he w ked u he piili ppling Aler Einein her where erer wuld cmmunice wih und we ined elecrmgneic we he lecurer repnded heinl: Yu knw h ligh i mehing pecil herere u culd n he wh u re prping During he cee hur er he cnerence dicuin were held wih he lecurer u n cler cncluin w reched in hi regrd Uing Einein her pecil relii i i eile irm h ligh pl ecndr rle [3] h i ligh i nl he medium h llw cmmunicin eween w erer in unirm relie min Neerhele in he riginl wrk Einein in he pule in priculr he ne he ligh i i underd nher hing Aler Einein gie prepndernce nd mre prgnim he cnnc he peed ligh [1-3] In c much he reliiic reul ccrding Einein her he d wih er pecil pec ligh nd epecill he eec h peed cn he n meril die [1-3] In hi her pecil relii cuic mng her iue here re imprn dierence in he reliiic cnequence wih repec Einein riginl her hi i n nl ecue i i uing und u ecue much he cncepuliin (ime nd pce) nd pr he deelpmen (he genei) he her urn u e er dieren priculrl cnidering he Dppler Eec h Einein nl cnempled cllerll [1] Wih repec he Dppler Eec cued he relie elci i cn e irmed h here de n eem e nher eec greer imprnce n he percepin he een he w erer in unirm relie min Mreer will e een he mhemicl rnrmin r hi cuic her urn u e dieren rm he Lrenz rnrmin ~ 411 ~

2 Inerninl Jurnl Applied Reerch I huld e ned in hi pr h he Lrenz rnrmin d n d wh Aler Einein clim he d [1 ] rnrm phericl we rn r n erer re in nher phericl rn r nher erer wih unirm relie min Bere enering in he deil reled hi c i i dile i piin egin deelp pic i i gre imprnce e equipped in dnce wih perinl deiniin n he cncep e ued h llw precie nd deque reerence when i i required Cerinl deiniin mu l e elihed n wh i e underd her relii [5] Wh i pecil relii her? A deiniin cn e: A her pecil relii i e knwledge uull peculie rigin h eplin nd predic he dieren percepin ech een h w erer wih unirm relie min he [5] Under hi deiniin i cn e enured h he genei her pecil relii cn e cniued e mhemicl rnrmin h cmpl wih he philphicl undin he her nd re liner eprein h preere he phicl reli he een (minining gemer nd uniquene) in ddiin eperimenl cnirmin Al rm hi deiniin i cn e irmed h ligh i nl men cmmunicin eween h erer he im hi ud mng her elemen i hw h: he ni-inuiie chrcer Aler Einein her pecil relii (he cnnc he elci ligh nd in he nli) which rm ime ime pper in he d lierure n he ujec [5] h i min rening eped in me ecin he preen pper hi will e underken wih i in he deelpmen he her pecil relii cuic in which undmenl dierence re elihed wih repec he rdiinl reliiic cncluin mde Einein c Spce nd ime deiniin here re eerl cr h genere he ni-inuiie chrcer Einein herie [6] In her relii i i imperie deine pce nd ime n much philphicll u in mewh perie w In c me peple h Einein her pecil relii i icll decripin pce nd ime [7 8] Nrmll in rder deine n jec we ue me i chrceriic hi i wh jec uch pce nd ime eing cmple nd incrprel he eerl deiniin Fr emple ime cn e decried : ime ppruni ccin circumnce en enirnmen clime emperure ree pce lie pern cciden he er h epree he mmen in which he cin i perrmed ec Wh i pce? Wh i ime? [7] Mn he rdiinl deiniin h re gien r hee eniie cn lee ne wih ene dicmr Fr emple: Fr Arile (384-3 BC) ime i he numer he memen ccrding he ere r er René Decre ( ) reled pce wih he eenin die He rgued: Spce i idenicl eenin u eenin i linked die herere pce de n ei wihu die hence here i n emp pce Ic Newn ( ) in chlium dded he Principi h hi decripin ime: Alue ruh nd mhemicl ime iel nd i wn nure lw unirml wihu relin n eernl hing Ern Mch ( ) juiil minined h ime in hi w: I i n rcin which i reched hrugh he chnge hing [9] Aler Einein replcing he legendr eher wih he pce-ime mehw denie de Decre he eience emp pce ince in n ereme ce he meric enr repreening Einein griinl ield culd he lue h d n depend n he crdine nwhere here will e ield [1 ] In hi ene here i n pce wihu ield (emp) r eerhing ill pce-ime he unce which pce-ime i mde eem e ininie Fr Einein he ide ield repreen he rel i repreen reli here i n ree-ield pce Wih repec he nure pce here w gre cnrer eween Gried Leiniz ( ) nd Ic Newn Newn deended he lui cncepin pce: I i hree-dimeninl cniner in which Gd plced he meril uniere Leiniz deended he relinl cncep pce: I i he li pil relinhip eween meril jec Bere here were meril jec pce did n ei Ulimel here we im impli nd implemen rdiinl philph wih repec he jec pce nd ime mre dmeic enirnmen which migh e mre in line wih ur undernding herere uing he Crein pprch pce nd ime re deined : Spce: I i h which medie eween w dieren jec nd in dieren piin ime: I i h which medie eween w diinc nd nn-imulneu een Spce nd ime re n mehing h cn e perceied direcl hrugh ur ene One cnn her mell r uch pce-ime I i newrh h he w cncep which eniie d n elng he hree-dimeninl wrld wh we cll reli re he i he dierenil equin h gern he ime eluin mng her hing he meril die hemele 3 Spce nd ime re dieren eniie Aer hee deiniin pce nd ime ne cn ee gre dierence eween pce nd ime On he ne hnd he eience uniere wihu ime cn e cnceied I wuld e uniere wih iner mer wihu cii wihu chnge n kind h culd gie rie een Wihu he eience een nhing chnge here re nl die wih which pce cn e deined u n ime herere i i pile cnceie he eience uniere where here i pce u here i n ime In hi ce here wuld e n eing k quein wh i pce r wh i ime Cnerel i eem impile cnceie uniere where here i n pce u where here i ime I culd e n emp uniere hence here i n pce u ecue here i n mer een culd n e genered h wuld llw he eience ime hi prduce peciic diereniin eween hee cncep; eween wh i ime nd wh i pce here cn e uniere wih pce u wihu ime u n he ppie uniere wih ime wihu pce cnn ei Wih regrd hi rgumen i i imprn rememer h Aler Einein in hi her relii inrduce pce-ime where he cncep wh i ime nd wh i ~ 41 ~

3 Inerninl Jurnl Applied Reerch pce re preciel equl nd eerhing i he me hing Fr Einein ime urn u e nher pil dimenin (ie i h dimenin pce) [1 ] In mre mdern ime Sephen Hwking e h wh i clled rel ime lw pe rm he p he uure (here i n rrw rel ime) He ge n hi elihe he nl diereniin eween pce (which de n he hi cnrin) nd rel ime [5 1] Hwking l inrduce n imginr cncep ime [5] Imginr ime wuld e cncepulied ccrding Hwking in n imginr pce-ime n phere where he prllel repreen pce nd he meridin wuld mrk he direcin imginr ime In hi repreenin n he phere he imginr ime ehe nher pil dimenin in he ene h we pek circle h r he prllel line nd r he meridin n he phere Fr emple in hi ce he uniere i rn in imginr ime in he Nrh Ple he nl pin A hi ime pe w here wuld e diplcemen wrd he uh n he phere In hi prce he circle liude equidin rm he ple ecme lrger (epnin he uniere) reching mimum ize hen he reurn zer he Suh Ple In priculr hi decripin eek elimine he prlem he ingulri he uniere [5] Wih repec he rrw ime hi cncep i genered when we lk u he cncep ime in relin enrp (rder nd dirder) nd we he he eeling h here i n rrw h repreen ime relling in ne direcin wihu pping cerin peed Apprenl ime i mehing h lw rel rm he p he uure wih me peed [5 1] In h ce wh i he peed ime? I here unierl peed wih which ime pe? I here peed which he rrw ime rel? [1] In irming h mehing rel i huld e cniued mehing meril; i culd e mer r le energ wih we memen Recreing he reli ur enirnmen where eerd lie deelp ering wih enin here eem e nhing h rel u wh cn rel chnge rnrm re he jec whe memen r e re cninull eing reled pre-e ime pern I eem h wh rni (chnge) re he jec wihin cenri where me een elped h i eing ued like pern like men cmprin Nrmll he pern h i ued urn u e he een clled he rin he erh n i wn i Uing er mll pr hi mehing h ei eween w dwn we deine he uni ime h urn u e he ecnd [9] herere wih repec he ene ime in ur dil lie wh cull rel nd me i preciel he plne erh lhugh pprenl ned Arile he un me rund he erh Hw i hi hppening? In emping repnd hi queining i cn e underd h he er cncep peed i er relie ince he peed n mile i in c eing cmpred in ric ene wih he rin he erh which cnin in me w he ecnd cncep h i elihed h i he elci i deined he diplcemen diided he uni ime he ecnd Frm ll he reging eplnin we cn h ricl peking here i n inrinic lue unierl elci which ime culd p he philphicl cnrdicin pper in he c h in urele humn eing we d eem eel hugh we ei in hi ping ime nd cunerpr pce de n ~ 413 ~ pride hi eeling ecue pce uull eem impl e here wihu elping Reurning he rrw ime ce where hi ide he rrw ime w l ued w in h rgumen Sephen Hwking where he uggeed he piili reering he direcin he rrw ime when he epnin he uniere p nce he gri due he erge deni he uniere wuld mnge p he upped unierl epnin h w prduced he Big Bng He prped h in he pping he epnin he rrw ime wuld l p i memen nd h mmen i wuld reere i ene nd rm here een wuld e e ck h i eerhing wuld e lied in reere in he ene he rme mie h re pled ckwrd [5] Hweer hi i n lnger cnidered uch Hwking me pin wihu much preence hd dmi h he w wrng rgue in hi w Hweer i eem h Hwking erin de peri in h he cndiin in he phe cnrcin he uniere i hi ccur wuld n e deque r he eience inelligen eing Here in hi pr i men h hen he ellr uel (hdrgen) unierl leel wuld he een cnumed nd he whle uniere wuld e in e mimum enrp In hi e i eem h lie cnn e prduced Fr he il prcee i i required h here i n incree enrp h here i n rrw ime [5] he rrw ime wuld pin in he direcin in which he enrp mu incree I i i n lnger pile incree enrp neiher ime nr lie eem he piiliie N nl de he ime rrw n reere i ene he end he epnin he uniere u ccrding Hwking he rrw ime dipper [5] In cncluin ime nd pce re dieren here i n ime rrw r nhing repreenie rel n gien peed here i n peed which ime pe he er rin he erh n i wn i i ued pern cmprin clck 4 A ime h eend nd nher dimenined Ju he memen rnlin he erh rund he un nd he rin he erh n i i re ued pern cmprin r n een in he eerd wrld hi new her pecil relii cuic l require pern cmprin In hi her will e w cncep ime h will ener he gme cmprin in rder h w erer in unirm relie min he dieren percepin een One hee ime h d direcl wih he requenc he urce in hi ce he nru ( perid) he her i inled wih he elci und prpgin () I will e deined ccrding he dince relled he we rn Fr phericl we rn prduced he rigin he rdiu he phericl we rn will e prprinl ime he cnn prprinli will e preciel he und elci hi cncep ime dhere Einein cncepin ime muliplied elci pil dimenin [1 ] Hweer in hi new her i will clerl dierenie eween ime nd pce he cncep ime () re mre ched he ide Decre h which medie eween w een which h direc relin he nin perid r requenc rher hn he ide pce hi cncep ime i mre reled he cncep rhhm hn wih

4 Inerninl Jurnl Applied Reerch wh culd e he ime h i hndled in he equin h decrie he e em he rhhm clck ( requenc) will e een i rnrmed dierenl cmpred he ime cied wih he dince while relling he diurnce he rdiu he we rn In c he w cncep ime re dieren n nl ccrding heir deiniin u l wihin hi new her will he dieren cr rnrmin One will he lw qudric rnrmin in he relie elci eween erer while he her will he lw liner rnrmin in he relie elci In Einein her relii w cncep ime re l hndled: h cied wih peed nd h which h d wih he rhhm clck Fr ren h re n peciied me pin in he deelpmen Einein her relii hee w cncep ime merge in ne O cure ingle rnrmin lw r hi cncep i l hndled [1 ] 5 Glile nd Lrenz rnrmin Wh i he Dppler Eec? he Aurin mhemicin nd phici Chriin Jhn Dppler ( ) in 184 pulihed ne where he clled enin he chnge in he clur jec nd he chnge in he percepin he requenc und due he relie memen eween he urce nd he erer [1] In 1845 he Duch cieni Chriph Hendrik Diederik Bu Bll ( ) perrmed reliel imple eperimen Uing lcmie pulling n pen crrige wih eerl rumpeer he hwed h he ne und (requenc) emied n pprching urce eem he erer mre cue hn when he urce i ming w r i re [1] Hw imprn i he Dppler Eec? A will e hwn hi mniein i gre imprnce Preciel he relii he percepin h he dieren erer in relie memen cme rm hi eec he reliiic eec decried here in hi new her relii he nhing d wih me meriu r mgicl eni uch he dilin ime he ccur icll due he Dppler Eec h i due he eec he peed n percepin when he urce nd/r he erer me herere in her pecil relii he percepin und will e dieren r w erer wih unirm relie min he m imple nd well-knwn emple pecil relii her i he her pecil relii epunded Glile Glilei ( ) Glile rnrmin re hwn in Fig 1 where i he unirm elci wih which he erer me relie he erer re Fig 1: An erer in reerence rme wih rigin i re he reerence rme wih rigin direcin he i wih repec he medium me wih elci in he piie In rder impli he ehiiin in he igure uull here will e prjecin n he plne lhugh i i rell repreenin in hree dimenin In hi ir ce udied in hi wrk he rme reerence h i in min i igned rile wih premium r emple he cnequence Glile her pecil relii rdiinll in ek re menined inccurel nd incrrecl [13] Since i i n idelied nd er imple ce i i reerred he nn-reliiic ce which uull men h i much mller hn he ignl prpgin elci nd herere i i clled he clicl ce [13] In uding he relii Glile i huld e emphied irl h i i her where ime i lue equl r h erer hi h er imprn cnequence hi c i equilen irming h he peed prpgin he ignl wih which h erer cmmunice i ininie In urn hi implie h here cn e n del in he ignl h culd cue me eec r riin emprli [8] uch i prduced he Dppler Eec hi ler imprn cncluin u he impiili he Dppler Eec in hi her i n nrmll cnidered in ek N nl i he rgumen ided u he eience i irmed h i nnreliiic Dppler Eec i ued in he ce he rnrmin Glile [13] Bere ging in he deil hi rgumen le u lk nher ce pecil relii her A mre cmpliced emple pecil relii her i cniued Aler Einein her relii [ ] In hi riginl pulicin 195 er elihing peculir nchrn nd l ui generi imulnei [1 14 Einein ir deduced he Lrenz rnrmin 15] hi wrk Einein hd n reerence nd cure ge n credi he Irih phici Gerge FizGerld ( ) r he Duch phici Hendrik A Lrenz ( ) wh ere Einein ried eplin he cnrdicr reul he Micheln-Mrle eperimen [4 1] In heir wrk n cnrcin d h rel he eher he l cnidered he riin in he rhhm clck In ddiin i cn e id h Einein wuld n he gie credi Lrenz nd FizGerld ince uppedl in hi wrk he himel deduce hee crdine ~ 414 ~

5 Inerninl Jurnl Applied Reerch rnrmin (Lrenz rnrmin) [1 ] he Lrenz rnrmin re hwn in Fig [15] nd ere i he unirm elci wih which he erer me relie he erer re he Einein hphei in hi wrk 195 icll cni : An elecrmgneic urce re nd n erer in unirm relie min In h urce nd erer cincide in he rigin A h mmen he urce emi pule whe we rn i cnidered phericl wih rdiu equl he erer ming wrd he piie direcin he i l perceie phericl we rn wih rdiu equl [1 ] Fig hw h percepin he we rn r he c c erer h een ineninll hied in he ericl direcin hwn in he igure in rder in mre clri in he decripin he iuin he Lrenz rnrmin hwn in Fig re uch h he rnrm he we rn z c in he we rn gien z c (1) () (Ligh elci cnnc) Ligh i lw prpged in he emp pce wih deined elci which i independen he e min he emier d [16] I mu e rememered h pule i prpiin whe ruh i dmied wihu pr In n idel iuin her wuld he n pule I wuld uice hing he underling philphicl undin nd premie h i here huld e n need include pule he premie re prpiin whe ruh h lred een preiul pred In n her inrducing new ruh wihu eidence i he crein rik he ruh hu inrduced migh n e he ruh uch A pin cmprin i i meime id h Qunum Mechnic h e i pule I huld e clriied h here i n uniquene in hi emen Hweer uch lrge e pule i riking r her highl pried Qunum Mechnic [17] One ruh lred demnred which i requenl ued in he deelpmen hi new pecil cuicl relii her i he cmmnl knwn herem r principle relie elci [18] Recpiulin: he genei pecil relii her m well e rnrmin crdine A her wih ewer pule nne i pile wuld e mre relile he eperimenl reul i er imprn een wih he e herie Fig : Lrenz rnrmin re id rnigure he phericl we rn z c perceied he erer re z c in he phericl we rn perceied he erer in unirm relie min (he ler i med ericll r clri) Nwihnding he herie mu cmpl wih le w rericin: ) Cnienc wih he eperimen ) Cnienc wih i philphicl undin In he philphicl nd rericie pec h mu he her i i imprnce recnider he pile pule h re inrduced in h her Aler Einein prped w pule in hi pecil relii her [1 ] : (he principle relii) he lw elecrdnmic nd pic will e lid r ll crdine em in which he mechnic equin gern ~ 415 ~ 6 Hw cme Lrenz rnrmin d n d wh Aler Einein he d? Reurning he riginl wrk 195 Einein hei i h he rnrmin he deduce which urn u e he Lrenz rnrmin rnrm phericl we rn z c r he erer re in nher z c phericl rn r he erer in unirm relie min [ ] In Aler Einein wrd: hu ur we i l phericl we wih elci prpgin c when ered in he ming em [1 16] me een hi emen Einein [1 19 ] i eplicle On he ce i he lgeric z c eprein eem e he equin phere In w i i hrd relie h hi eprein i n phericl we rn; ined i i n ellipid reluin ecue he eprein r in Lrenz rnrmin depend n he crdine When king pin n he phericl we rn z c wih dieren lue will l e dieren nd hi h he cnequence h in phere i he rdiu i ried in cerin w reluin ellipid cn e ined hu he Lrenz rnrmin d n rnigure phericl we rn in nher phericl we rn Einein ed [1 19 ] Acull he rnrmin Lrenz rnrm he phericl rn in n ellipid reluin nd hi implie h he peed ligh ccrding he Lrenz rnrmin i n cnn [16] Fr he erer he ignl dnce er in he direcin Hw cn ne ee hi? hn in he direcin nd

6 Inerninl Jurnl Applied Reerch he we rn repreened in equin (1) under he rnrmin Lrenz i rnigured in le u ee in wh: Fr hi eercie migh e herwie i i prped h he relie elci e Frm Fig wih c c 1 he Lrenz rnrmin re z z (6) (7) hi rnrmin i hen pplied eerl pin n he phericl we rn in he crdine em re h r 1 (3) (4) (5) z 1 (8) Such pin n he we rn r he erer re hwn in Fig 3 Fr pin 1 z 1 1 Under Lrenz rnrmin i ecme pin 3 wih crdine 1 (9) in Fig z (1) he pin 3 4 nd 5 n he plne z z he crdine 1 (11) he re rnrmed in pin nd wih crdine z Finll he pin 6 z 1 1 I ecme he pin 6 wih crdine (1) (13) z (14) All hee pin wih heir repecie imge re hwn in Fig 3 I i cler h he inermedie piin hee pin i cnidered wuld cmplee he imge cenred qudric urce which i deiniel n phere A cn e een in Fig 3 he Lrenz rnrmin cing n n elecrmgneic we rn whe gemer i phericl ccrding n erer re d n rnrm hi phericl wern in nher phericl rn u rher 3 in n ellipid reluin Fig ) [1 19 ] Fig 3: Lrenz rnrmin urn he phericl we rn hwn in ) in n ellipidl we rn Sme imge pin prduced he rnrmin re ehiied in ) ' 1 Wih hi eercie i i hwn h ccrding he unirm relie ming erer he prpgin elci he we rn i n he me in ll direcin: I i lrger ckwrd ccrding i perpecie [16] hi pre h he Lrenz rnrmin d n d wh Einein he d: rnrm phericl we rn in nher phericl we rn r he erer nd preere he cnnc he elci ligh [ ] I cn e cncluded h new e mhemicl rnrmin r her pecil relii cuic i required which de rnrm phericl we rn in nher phericl we rn 7 he new crdine rnrmin ce A he ce n erer ming w rm und urce re wih unirm elci Surce: Alw will e und urce mehing h emi pule i wn r me relecin eec I culd e he icking clck r he regulr eing humn her In i i umed h he erer ming wih unirm elci wrd he piie direcin he i cincide he rigin wih nher erer re ne he urce A h inn he urce emi pule wih phericl we rn he erer perceie h phericl we rn cenred n he rigin wih rdiu Here i he peed und Ne: Onl ce wih will e cnidered herwie phicll here wuld e n cmmunicin eween he erer wih unirm relie memen In w hi i equilen ing h here i n mile h i er hn he peed ligh Once he ir pule i prduced he phericl we rn r he erer re i repreened equin (1) ~ 416 ~ ' 6

7 Inerninl Jurnl Applied Reerch I i nw necer elih irl he rnrmin crdine h llw deermine wh rm h he we rn h i perceied he erer in unirm min I i inended h he prcedure e emic h i n lgeric rm i prped r he rnrmin crdine hen he im i mke i perie ccrding he herem relie peed r h erer In Fig 4 prpl r he rnrmin crdine nd repreenin pule emied he urce in whe equin i preciel he eprein z c re ehiied he ericl i n which he pin indiced n he we rn re deermined i he dince in Fig 4 Fig 4: he pule emied he urce nd he prped crdine rnrmin h rnrm hi phericl we rn in nher phericl rn perceied he ming erer re hwn he prmeer e deermined re: 1 1 nd In hi her pecil relii cuic we minl ue herem h h he uncin nrmliing he nli nd he demnrin in ech ce nmel he principle r herem relie elci [18] : I n erer perceie h nher erer i ming w wih peed iˆ hen i urn u h r he erer i i he iˆ erer h me w wih peed [18] hi herem e h (15) pce nd ime Nr i n independen (cnn) erm included ince here i cincidence in h erer he eprein r nd re ineninll ppned ince here will e unepeced eprein ccrding he cndiining h we he preiul receied in ur cdemic preprin Nw im ind he lue h mu he he menined ceicien h hi rnrmin reprduce he percepin he erer z h i he lue crrepnding nd h re in greemen wih he philph nd re much he principle relie peed he pprch he prlem iel In Fig 4 he pin n he phericl we rn rdiu 1 1 in he ir plce re highlighed n he ericl i ping hrugh On he ne hnd hee pin re chrceried hing crdine equl zer On he her hnd he l he he chrceriic h heir crdine i he dince h epre nd nd he ime ccrding he erer herere r hee pin Suiuing hee cndiin hwn in Fig 4 we in h S 1 1 (16) Wih hi eprein r uiued in nd he rnrmin wih hree prmeer e deermined i nw epreed 1 1 (17) herere in prperl chen Crein em ne h h iˆ iˆ he undmenl quein hi ge i: Hw de he erer perceie hi we rn Fig 4 which he erer perceie in phericl rm? he nwer will e prided preciel he crdine rnrmin e elihed A prcedure in he crrepnding crdine rnrmin cni prping he repecie eprein r he rile hwn in Fig 4 In hi igure he cnriuin rm nd he ceicien 1 nd repeciel he me cn e een r r which cnriuin re chrceried he ceicien 1 nd he prped eprein re liner in chrcer ince hmgenei prperie re riued ~ 417 ~ Fig 5: he pule emied he urce nd he rnrmin i nw ehiied in erm nl hree prmeer he pin n he we rn mrked n he ericl i ping hrugh he crdine In Fig 5 he new eprein r nd re hwn Uing he prped herem [18] ccrding hee pin n he ericl i ping hrugh he rigin me w rm i in he direcin he i wih elci ; herere he quien / dening he

8 Inerninl Jurnl Applied Reerch cmpnen he elci wih which hee pin me ccrding urn u e / / 1 (18) Where we he ken he quien eprein (17) r Frm equin (18) we in h 1 (19) In he me Fig 5 i cn e een n he we rn h he pin erer me w rm lwl h i wih peed wih peed u r he hi me pin me w rm i mre herere ince he crdine r hi pin i he quien / nw urn u e / / 1 In hi eprein () cncelling righ lim nd 1 nd rm hi i i cncluded h in h lim ne h () he quien he (1) 1 () Nw in he upper righ pr in Fig 6 he rnrmin crdine in erm nl he prmeer i preened And we re lking r he eprein r Fr hi purpe he remenined herem r principle pplied r emple he highlighed pin n he upper pr Fig 6 [18] i ued I cn e een h he relie elci wih repec hi pin meured i nd z he elci upper pr Fig 6 ccrding he remenined herem [18] mu e equl minu he peed he pin wih repec he indiced pin in he indiced wih repec Ning h rm he her pin iew he peed he indiced pin meured p i iˆ ˆj I cn herere e umed h ccrding (4) r he pin in quein he cmpnen he elci in he direcin i / (5) nd in he percepin ccrding equin (3) nd (4) we he h he elci cmpnen me ie / / i ecl he (6) Likewie we cn elih he crrepnding elci cmpnen wih n enirel imilr rening r he cmpnen Fr ll hi he crdine rnrmin i ugh i hen z z ĵ z (7) Wih hi crdine rnrmin (7) uiued in he eprein r he phericl we rn (1) hi we rn perceied he erer z i rnigured in iˆ ˆj p (3) where nd repeciel î ĵ re uni ecr in he nd direcin Fig 6: w we rn re hwn he we rn perceied he erer pper wih rdiu nd he phericl we rn perceied wih rdiu Eprein he rnrmin prped r z nd he equin h percepin re ehiied ~ 418 ~ z (8) hi equin repreen phere wih i cenre diplced he dince he le Ne h he cenre hi phericl we rn equin (8) i neiher he rigin nr he rigin unlike Einein cncepin he we rn h i cenred n he rigin A will e een hi dince urn u e greer hn In ddiin ince he elci he we rn pin A r he erer i he ime mu e greer hn herere he ceicien mu e greer hn 1 I huld e preciel he quien he elciie perceied h erer ince in h prprin will e greer hn S / (9)

9 Inerninl Jurnl Applied Reerch In Fig 6 mng her hing i hwn he dierence in he percepin h h erer he wih repec he peed prpgin he cuic ignl Fr he erer he und we prpge wih peed in ll direcin including he pin crrepnding Where r he erer he riu pin he we rn he dieren peed prpgin hi i nl pr hi percepin he prpgin peed i he me wih repec he cenre he we rn h i lced he le Fr emple he pin n he i ccrding he peed which h pin i diplced i hen he crdine rnrmin h reprduce he imge h perceie rm he percepin cn inll e wrien z z A (3) An imprn pin h h lred een ned i h h nd re repreened he rdii he we rn perceied nd repeciel hi urn u e rile h in principle re nn-unded unlike he perid urce he icking clck r he eing her which re unded nd he re well-deined ime inerl gien he chrceriic he urce h re deermined One cnequence h cn e immediel niciped rm eprein (9) nd (3) i h mu e greer hn In her wrd prpgin he we rn he erer in unirm min eem e deled r i ke lnger me hwn in Fig 6 I llw h when i i ime inerl eween w een in hi ce he perid he lw rnrmin r h inerl dier rm he urh eprein equin (3) A In Fig 7 w we rn cenred n he rigin hwn he lrge rdiu emied re repreen he ir pule he mller rdi we rn repreen he ecnd pule emied where i he perid he urce ccrding he percepin erer he ime lue hwn in he igure i he ime he we rn reche erer which Fig 7: w cnecuie pule emied he urce he ime re hwn he ir emied in nd he ecnd Pr he crdine rnrmin nd he relinhip eween he elci he we rn perceied nd h perceied re l hwn h i ccrding he erer er he ecnd pule i emied me ime elpe r h ecnd pule rech he erer wh i ming w rm he urce unirm elci hu in he reerence rme he erer he ime ken relling he ecnd pule crrepnding we rn rel he dince llw he (31) he dince relled he ecnd pule i equl he dince relled he erer ince he cincidence h erer i in he rigin up he mmen he encuner Frm hi l eprein (31) we in h 8 Dppler Eec in ce A Ne he Dppler Eec i udied in he deelpmen hi ir ce reing urce nd ming erer A ecnd pule emied he urce; he erer will l ee i ke lnger dnce wih repec he percepin erer hi i ecue he erer phicll me w wih peed A he ecnd pule rel he ming erer will e dince which he ecnd pule will ke me ddiinl ime rel ee Fig 7 Accrding hi circumnce n rhhm r emple he eing her he erer will eem lwer mre leiurel ~ 419 ~ / (3) Hweer deermine he encuner ime ccrding he eprein (3) mu e muliplied hi i due he urh eprein (3) ie (33) In hi eprein repreen he lue he mmen he we rn mke cnc wih erer nd cure i mu e he perid he urce ccrding

10 Inerninl Jurnl Applied Reerch he percepin erer In he rnrmin (33) he urce perid perceied ming erer i cn e een h hi i nher lw rnrmin dieren rm h he urh eprein he em (3) he rhhm clck r he inerl ime eween he e her urn u e nher cncep ime h i er dieren he ime h deine he equin he rnrmin crdine Nw we ind he eprein r he rnrmin requencie h h erer perceie king he reciprcl in h memer he eprein (33) we he (34) In hi equin nd dene he requencie perceied nd repeciel hi reul indice h wheneer n erer me w rm und urce re he requenc he perceie i mller hn h which he pprecie when he i l re he rdiinl Dppler cr in ek i wrien wih he rcke he ir pwer ie [1] (35) Here he requenc h een clculed uing he peed und ined ccrding he knwn mnner wih he peed ligh Frm hi eprein we cn ee h in he rdiinl deelpmen he ud h i underken n he Dppler Eec diereniin eween he w cncep ime h h preiul een elihed de n pper herere hi new her pecil relii cuic predic n imprn mdiicin he preiu Dppler cr In Fig 8 re he grph r he Dppler Fcr crrepnding h erin he rdiinl grph nd he her grph h predic hi new her he dierence in he igure i imprn Fig 8: he rdiinl Dppler cr i he p he igure he Dppler cr h predic he new her i he m he righ line i he rdiinl eprein nd he cure i he new / predicin On he hriznl i ~ 4 ~ 9 he new crdine rnrmin ce B he ce und urce nd erer h in unirm relie min wih In hi ce hw de he percepin h erer chnge? Emiin cenre Swed : When he und urce ming wih elci emi pule he pin emiin whereer i i he diurnce i plned nd prpged he medium wih elci h i independen he peed he urce he ecnd ce e udied cni hree erer: n erer h i upped e re wih repec he medium nher erer unirm relie min wih elci erer ne und urce wih nd hird h i l in unirm relie min wih elci Bh nd re meured wih repec he medium Ne h ucrip re nw ued r ming em he cenri decripin i llw: In i i umed h he erer lng wih he ming und urce cincide he rigin wih he her w erer nd Bh erer nd re ming in he piie direcin he i he reerence rme re in which he rigin ume he preence he erer A he pin cincidence he hree erer he urce emi pule hi pule ccrding he reing erer genere phericl we rn he erer perceie h phericl we rn cenred n he rigin wih rdiu Here ere i he peed und i i he ignl elci Agin ne cn phicll he he cndiin In hi ce he -clled hck we wuld ccur Hweer here l nl he iuin wih he cndiin re cnidered Nw in rder emie he eplnin he ide hird erer h een inrduced A lred eplined here re nw w erer in min nd he hird i re wih repec he medium he c hing he und urce which nw h i wn min culd rie cncern u he llwing quein De he unirm memen he urce ec he prpgin he ignl? Hw de he we rn prpge nw h he urce i in min? I cn e een h Aler Einein included in ne hi w pule he rgumen h he elci he emier h n eec n he prpgin elci he elecrmgneic we [1 ] Here in hi new her i i n necer include uch pule I i well-knwn c h he prpgin und priculrl i elci de n depend n he elci he emier In hi w i repnd he preiu queining nd i will e pile ee in he deelpmen he new her h hi c de he implicin I de ec he predicin he new her in cmprin wih he h reul rm he cnnc he peed ligh in Einein her

11 Inerninl Jurnl Applied Reerch In w i cn e een h he urce when ming wih unirm peed prduce he imprein h i i wing he emiing cenre in dieren plce Frm ech hee plce prpgin egin prperl I he requenc plning i cnn hee emiing cenre re plned epred ied dince All hee cr re picl we memen Uing he e reul nce he urce emi pule n he ir emied pule i eeded he rigin herere he emied we rn r he erer re wih repec he medium urn u e z (36) Fr equin (36) i huld e kep in mind h he rile wihu inde re in he reerence rme re wih repec he medium ke ccun he memen he urce i i necer cu n he ecnd pule emied he z ming urce ee Fig 9 he equin he ecnd pule ered hwn in he upper righ-hnd pr Fig 9: i Frm hi l eprein equin (37) in he percepin h rel lng wih he urce: i ined h i replced in z he erer (39) hi i he equin phere cenred n he pin mrked wih n erik hied he le in Fig 9 he peed he we rn r he erer re i nd r he erer wh rel wih he urce he rhe pr perceied rel w wih elci hi ren he ignl will ke le ime rel nd huld e mller hn S Fr (4) z (37) Nw in he percepin he erer rm he ecnd pule emied he urce l in memen wih repec he medium we cnider he me mechnic llwed here he min pr he rnrmin i wrien (41) Fig 9: he ir pule emied eeded he rigin wih rdiu nd he ecnd pule emied eeded he pin mrked wih n erik re h hwn On he righ ide he equin he ecnd pule perceied he erer nd pr he crdine rnrmin h rnigure h percepin in he we rn h perceie whe equin i hwn in he lwer pr re ehiied he erer percepin i ed n he we rn (37) In hi percepin he ming erer inluence He i ju nher erer h n Wih he prcedure elihed in he ce wih relie ee ne cn ind he m imprn pr he rnrmin h rnigure he percepin in he percepin : A (38) ~ 41 ~ In hee eprein he ucrip crdine z re in he rme reerence he erer in min wih repec he medium O hi ler eprein (41) i clered nd replced in equin (37) he we rn equin perceied he ming erer urn u e z (4) hi eprein i he equin phere wih i cenre hied he righ he erer erik in Fig 9 Nw he peed perceied he erer mrked wih n rm he pr he we rn h i direced wrd him i herere he ime hn ie in equin (41) mu e mller (43) Ne h equin (4) nd (43) re n equl in hi ce

12 Inerninl Jurnl Applied Reerch Nw he crdine rnrmin h rnigure he percepin erer in he percepin erer i e und In her wrd we im ind he rmul h rnrm equin (4) in equin (39) Wih relie ee ne cn ee h he rnrmin i wrien Finding u (44) rm equin (44) nd uiuing in equin (4) he percepin he erer (39) i und he erer we rn cle he erer erer greer peed perceie; herere he ime hen equin perceie h he pr he rel mee hi hn wh erer mu e mller hn (45) One w eri equin (45) i replce i in equin (44) nd hen uiue rm equin (41) he reul i equin (38) which prill check he prcedure 1 Dppler Eec ce B Ne we im ud he Dppler Eec in hi ce erer nd und urce h in relie unirm min wih repec he medium In Fig 1 he imge Fig 9 r n erlier ime ie r e i hwn when he encuner he ecnd we rn wih he ming erer ccur Nw he ne imprn een in hi nli i udied he mmen e h he ecnd pule emied in encuner he erer l in min Accrding Fig 1 r he erer re he dince he ecnd pule i relling i Fr hi erer he ime e e deermine he mmen in which he ecnd pule i rugh in cnc wih he ming erer Cnequenl hi dince mu e uch h dded he dnce he erer mu equl he dince h he urce h hied unil he mmen he emiin he ecnd pule h i e e Frm eprein (46) we in e (46) (47) Fcring equin (47) rm equin (43) we in he relin eween hee ime perceied he erer e (48) In hi eprein pper which i he rue perid he urce perceied he erer Mking i cler he erer re perceie h in ime he urce emi he ecnd pule hi ime r him i he perid he urce Appling he rnrmin ind he percepin he erer gie he ime h pper in equin (48) Wh i hi ime n he perid he urce h perceie? he nwer i h unil ler ime e hi erer Fig 1: he ir pule emied in e wih rdiu hwn In he ecnd pule Fig 9 i emied when he urce w he pin mrked wih n erik Bh we rn r he mmen he ecnd rn mee he erer ehiied i re l ~ 4 ~ encuner he pule herere hi ime i he rue perid he urce he perceie In hi ce he wing he pule i n he rigin u dince mrked wih he erik in Fig 9 nd 1 hi dince cue he cincidence he erer e l wih he eeding pin he ecnd pule which empl cerin ime mee he erer erer e in he me rme reerence he wh rel lng wih he urce perceie he perid he urce i h were re Hence rm equin (38) nd (44) (49)

13 Inerninl Jurnl Applied Reerch hen equin (48) i wrien e (5) Replcing rm equin (45) e (51) hi eprein indice h he wn perid he urce nw in memen een nher erer l in memen i perceied er dierenl Accrding he iniil cndiin r hi ce i cn e een h he perid he urce ccrding he ming erer eem e greer hn he perid he urce On he her hnd he eprein (47) r i n reduced he ir ce A ecue nw he ming erer encuner he ignl rm he ppie ide king he reciprcl h memer equin (51) we in h he crrepnding requencie re l reled in mre cmple w h i (5) he Dppler cr hwn here crrepnding he ce h erer nd urce in min indice h he requenc perceied he erer in hi ce i mller hn he requenc h he wuld pprecie eing ne he urce h re wih repec he medium I huld e ned here h ccrding hee eprein in he ereme ce h erer nd urce in unirm min wih repec he medium wih h i when he relie memen eween urce nd erer end zer he requenc perceied he erer end l e he requenc h wuld he wuld perceie i h were re wih repec he men In Fig 11 he grph r he Dppler cr equin (5) when i hwn he Dppler cr ccrding he cure hwn indice h he requenc emiin pule eem mller he ming erer when cmpred he requenc he urce In Fig 11 he crrepnding lue n he i he ci i / Fig 11: he cure r he rdiinl nd he new Dppler cr r he ce where re hwn he lwer grph repreen he rdiinl Dppler cr Belw h i predicin rm he new her wih he me nlicl eprein u qured 11 Cncluin On Einein cncepin: An elecrmgneic urce re nd n erer in unirm relie min; h urce nd erer cincide in he rigin A h mmen he urce emi pule whe we rn i cnidered phericl wih rdiu equl he erer ming wrd he piie direcin he c i l c perceie phericl we rn wih rdiu equl Accrding equin () hi we rn wuld e cenred he rigin hi decripin implie h he ming erer cn cpure he phicl phenmenn nd crr i wih him Phicll hi cnn e he min he erer cue he we rn lg ehind him nd in he end hi priculr percepin i phericl ne h i diplced ckwrd wih i cenre lighl he le he crdine rigin hwn Fig 6 On he cnnc he peed ligh: In he her he pecil relii Aler Einein he Lrenz mmer i undmenl pr A cn e hwn wih relie ee he Lrenz inrin i iled Smehw i wuld e umed h he percepin n erer re wuld e reprduced hrugh he Lrenz rnrmin he percepin nher erer in unirm relie min I i n h he Lrenz rnrmin re rell incrrec u he d n d wh Einein he d: rnrm phericl we rn h perceie he erer re in nher phericl we rn r he erer in unirm relie min he unirm ming erer de n perceie phericl we rn hi reul i diicul cncluin: Wih he Lrenz rnrmin he peed ligh i n he me r h erer Fr ne he erer he peed ligh i n equl in ll direcin hwn wih he ellipid in Fig 3 hi cerinl h rgic cnequence r Einein her pecil relii Sund: An erer ming w rm und urce r relecing die which re re wih repec he medium; will ere h hi uniere i uncining lwer pce hn hi her He will nice h eerhing hppen wih del nd he greer i unirm peed wih repec he medium he greer hi del eec will e ~ 43 ~

14 Inerninl Jurnl Applied Reerch In her wrd her re will eem lz hi ming erer cmpred heir wn her A preiul eplined hee phenmen re direc cnequence he Dppler eec nd i cn e irmed h hi dierenied percepin ime r de n inle n nhing meriu r hidden Nne hee eec will e permnen r n he ming erer p he rhhm he w her will nw e idenicl N nl h nce recered he w her will he dierenie in her hing hee percepin hee dierence i eiing rm he eginning re permnen nd pprenl here wuld nl e n eec percepin when n he w her me Wh culd e reli i minined he eec he memen eem genere nl priculr percepin nd i wuld e mer pprche hi percepin he qudric Dppler cr: Amng he unepeced cnequence hi new cuic her he -clled Dppler cr which nrmlie he relin eween he requencie perceied erer wih unirm relie min nd in generl memen wih repec he medium; in hi new her i i und h here re l dierence eween he pre-elihed Dppler cr nd he ined wih hee new rmulin hi will unduedl he cnequence when uing he Dppler Eec r emple in medicine underwer ciiie cmlgicl clculin nd mn her pplicin in dieren ield knwledge Prl hee dierence und wih repec rdiinl knwledge he n preiul een deeced ecue renl he relie memen re prduced er lw peed r he urce nd he erer in memen Nwihnding he crrepnding eperimen wuld he e crried u eri ccrding he Scieniic Mehd he dieren rmulin elihed in hi new her On he rnrmin crdine: A w ered during he deelpmen he dieren rmulin in ech he ce he rnrmin crdine urn u e mhemicl uncin whe dmin i he e pin n phericl we rn Oherwie n rm her hn phericl wuld e inling phenmen nirp nd inhmgenei pce nd ime which huld e priculrl ned he crdine rnrmin i uncin h mp phericl we rn n nher phericl we rn herere ppling hi uncin rirr pin in pce m n gie cngruen reul I huld e rememered h he pin which re ujec he rnrmin he emprl crdine h i inimel linked he rdiu he we rn A pin inide he we rn wuld mehw impl he p ju pin n he uide he we rn wuld e reled he uure ( lnger ime) Wh wuld e he mening pin wih crdine z rnrmed in he pin nd wuld i mke n ene? hi pin nd i rnigurin d n i he equin r he we rn perceied he erer: In relin hi rgumen i mu e rememered h he rnrmin in ech reed ce in he new her w preciel ined rm he c h i rnrm phericl we rn nrmlied he principle relie elci I w und h hi rn ecme nher phericl we rn herere i cn e cnidered h i pin de n r m n elng we rn hen i i er likel e uide he crdine rnrmin I i ~ 44 ~ l er imprn ne h he mhemicl eprein he rnrmin rnigure ime linked he rdiu he we rn nd n he perid A reliiic eec wih repec imulnei: I w cmmened preiul wih repec imulnei h he piin he erer in he me rme reerence i imprn wheher i i in min r n Accrding he piin he und urce which culd e he remenined rumpeer n dicered mile plrm he piin he erer wh i l in min i deerminn I he erer i ming in he ck he plrm me dince ckwrd he will idell eperience lrger requenc hn h perceied he erer in he rn he plrm he c i h he we rn plned he rumpeer prpging hrugh he ir r rech he rn erer (mll requenc) while he ckwrd erer wh i cle we rn receie greer elci (lrger requenc) In her wrd een hugh he urce nd he erer d n he relie min chnge requenc i prduced due he min h urce nd erer wih repec he medium Reliiic Cnequence: Energ: Fr he energ in hi new her he c uing undmenll he principle relie peed in he deelpmen dieren rmulin urn u e deiniie: In rder deermine he dieren percepin he erer in relie memen r energ ne mu ue hi principle nd rememer h he energ deni h i rnmied wih he diurnce depend n he requenc he we min M (ineri): here i ill much peculin u he rigin hi prper mer which i ineri In hi new her unil nw he cncep m h n een cnidered herere i i n epeced h w erer wih unirm relie min culd perceie dieren lue r he m he me d Lengh: In Einein her relii here ei r emple he cnrcin die ered rm rme reerence in min In hi new her lengh cn e repreened men w epred und urce epred preciel he dince h i i inended repreen I i l pile repreen dince uing he welengh und urce In h ce wih ne r w und urce he lengh perceied n erer in unirm relie min mu e he elihed phicl relinhip eween peed nd requenc ie: I huld e ned h ming erer perceie dieren lue r he prpgin elci dieren pr he me we rn h i in me ce i will he herere he lengh repreened preciel he welengh nd will he e in ccrdnce wih hee phicl lw ime: Cnidering he dieren eprein h he een und in he deelpmen hi new her wih repec ime i cn e irmed h he relie memen h nhing d wih he ging r rejuenin reing win nd relling win An percepin h ime ge dieren elci urn u e preciel relie eec h i nl percepin nd n phenmenn h cn rncend he circumnce In cncluin he unirm relie min de n prduce n rce h cn mdi he phicl r chemicl prcee he culd e he inled in he melim me erer A preiul

15 Inerninl Jurnl Applied Reerch rgued here i n uch hing ime dilin h de prperl ei nd h cn inluence he il prcee he ping eerd lie i h i i cniued memen r chnge in mer which he heir wn peed chnge hee chnge cnn rell e inluenced mehing h we cll ime Accrding hee cniderin i cn e irmed h unlike wheer meime hppen in cience icin i i n pile rel rirril in ime Furhermre hwn he mhemicl crdine rnrmin nl rnrm he ime reled he prpgin elci he ignl he d n direcl rnigure he urce perid ime nd pce re dieren h cncepull nd philphicll Wh de i men when in n he rmulin ined? I men h ming erer perceie h he we rn ue greer ime in rriing me pre-elihed piin ince him he peed eem dieren ' Wh de i men when in me he udied ce? I men h he urce r he ming erer wrk lwer r emi pule er in i priculr ce An mechnim deigned l in perin r peciied numer ccle r me rgnim uch he l which i ued in reerch wuld eem l lnger r lie lnger ccrding he percepin priculr erer D he rell l lnger? Or i i ju mer pprche? I wuld eem e he ler nl! 13 Kiel Chrle Knigh Wler D nd Rudermn Mlin A Mecánic (Berkele Phic Cure-V I Ediril Reeré Buen Aire Argenin 1968pp 3 14 Selleri F Fund Ph 1996; 6: Lé Lelnd Jen Mrc he imple deriin he Lrenz rnrmin Americn J Ph 1976; 44(3): Mgueij J Rep Prg Ph 3; 66: Aler Glind iire Pedr Pcul De Sn Mecánic Cuánic Eudem Unieridd ISBN: Mrcel Aln Edwrd Finn J Fundmenl Unieri Phic (Mechnic) Addin-Wele Serie in Phic 1967; 1: Bluhm R Ph Wrld 4; 17:41-46 Müller H Brmier C Hermnn S Peer A Elecrmgneic ciie nd Lrenz Inrince ilin Ph Re D Aln Mrcel Finn Edwrd J Fundmenl Unieri Phic (Field nd We) Addin- Wele Serie in Phic 1967; II:78 1 Acknwledgemen he uppr r he elrin hi wrk he Ninl Sem Ineigr Cnc Méic hrugh Grn 1793 i greull cknwledged 13 Reerence 1 Einein A Ann Ph 195; 17: Schel J Einein mirculu er Fie Pper h Chnged he Fce Phic Princen Unieri Pre Brer GJF Einein l luz el epci-iemp l cun Arr 15; 191[775]:66 di: hp://ddirg/13989/rr15775n55 4 Micheln AA Mrle EW Am J Sci 1887; 34: Hwking Sephen A rie hir ime (Frm he ig ng lck hle) Lirr Cngre Clgingin-Pulicin D Brilluin L Relii Reemined (Acdemic New Yrk) C Brin Frhw Je Pr qué E=mc? ( Y pr qué deerí imprrn?) Rndm Hue Mnddri SA de CV Méic DF Lämmerzhl C Ann Ph 5; 14: Grneu Peer Grneu Nel In he grip he din uniere (he cience ineri) Wrld Scieniic Pulihing C Pe Ld Hwking Sephen El unier en un cácr de nuez Ediril Plne SA Brceln Críic SL Brceln 3 11 Lighmn Aln Luz Anigu (Nuer cmine iión del unier) Ediril André Bell Brceln Renick Rer Hllid Did Fíic Cmpñí Ediril Cninenl SA Méic ~ 45 ~