Derivation of longitudinal Doppler shift equation between two moving bodies in a reference frame at rest using the particle property of photons

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1 Deriaion o longiudinal Doppler shi equaion beween wo moing bodies in a reerene rame a res using he parile propery o phoons Masanori Sao Honda Eleronis Co., d., Oyamazuka, Oiwa-ho, Toyohashi, ihi , Japan bsra: The equaion o he Doppler shi o wo bodies in inerial moion in a reerene rame a res i.e., saionary reerene rame is deried. In his deriaion, he wae-parile dualiy o phoons in he heory o speial relaiiy is onsidered. The dilaion o ime o a moing lok, whih is deried rom he orenz ransormaion ha depends on he eloiy o he moing lok in he reerene rame a res, and waeron ouning by geomerial drawing are used o desribe he longiudinal Doppler shi. We show ha he equaion o he Doppler shi depends on he eloiies o he wo bodies in he reerene rame a res. PCS numbers: 3.3.p Key words: Speial relaiiy, wae-parile dualiy, reerene rame a res, Doppler shi, orenz ransormaion. Inroduion Wae-parile dualiy has been disussed in quanum mehanis, howeer here is no disussion in he heory o speial relaiiy. Sao [] disussed he Mihelson-Morley experimen in erms o single phoons using de roglie-ohm piure: ha is, he Mihelson-Morley experimen showed an inererene ondiion and did no show he simulaneous arrial o wo phoons his is beause here is only single phoon. The Mihelson-Morley experimen shows he wae propery o phoons: he wae propery is assumed o be a nonloal quanum poenial in ohm heory []. I is imporan o poin ou ha wae-parile dualiy should be disussed in he heory o speial relaiiy as well as in quanum mehanis. Mos o he ounerinuiie aspes in he heory o speial relaiiy arise rom wae-parile dualiy. I wae-parile dualiy in he heory o speial relaiiy is disussed learly, a more inuiie inerpreaion o he heory o speial relaiiy is possible. The Doppler shi shows he parile propery o phoons: ha is, he inererene o phoons does no appear. I is raher diiul o disinguish he properies o a wae and parile; a his sage, he wae propery is deined by he inererene experimens and he parile propery is deined by he experimens wihou inererene. Thus, requeny ouning is deined by no wae bu parile properies. In a Doppler shi experimen, he phoon is measured as a parile, whih raels a he

2 speed o ligh in auum regardless o he eloiy o he ligh soure. The Fizeau experimen, in whih a reraie index is aeed by he low o waer he speed o ligh in waer is aeed by he low o waer, is also an inererene experimen; a single phoon inererene experimen is possible. The aberraion o ligh is deeed wihou inererene; a phoon is deeed as a parile. Thus, experimenal resuls should be disussed rom he iewpoin o wae-parile dualiy in he heory o speial relaiiy, whih is summarized in Table. The Mihelson-Morley experimen shows he wae propery o phoons; hus, he inerial moion o he earh was no deeed beause he wae propery o a phoon is nonloal, and he inererene ondiion does no depend on he eloiy o he rame. The parile propery o phoons is loal, whih means ha hey rael a he speed o ligh; hus, he parile propery is aeed by he inerial moion o he experimenal seup. In his paper, using he parile propery o phoons, we show ha he Doppler shi o ligh is deried rom a geomerial drawing and he orenz ransormaion o he reerene ime. We derie he equaion o he longiudinal Doppler shi beween wo bodies moing relaie o eah oher in a reerene rame a res. Furhermore, we show how o ind he reerene rame a res. Table Wae parile dualiy in he heory o speial relaiiy Wae propery inererene Parile propery Mihelson-Morley experimen Doppler shi Fizeau experimen berraion o ligh. Relaie moion o hree bodies Figure shows he reerene rame a res and wo inerial moing objes: O, he reerene rame a res,, a moing obje roke wih he absolue eloiy, and, a moing obje roke wih he absolue eloiy. Rokes and eah hae an aomi lok and a ligh soure ha are preisely adjused wih respe o he reerene rame a res. In his paper, an absolue eloiy deined in he reerene rame a res is adoped, whih will be disussed in seion 5. The absolue eloiies and are deined in he reerene rame a res. e he reerene ime in he reerene rame a res be ; he reerene ime in a moing rame is deined by he orenz ransormaion using. The reerene ime o obje is and ha o obje is :,

3 where and are he reerene imes o he moing objes seen rom he reerene rame a res ha is, he dilaion o ime. We anno measure or direly; raher, i an be deeed hrough he Doppler shi requeny. O Reerene ime Fig. Doppler shi o moing bodies in a reerene rame a res O: Reerene rame a res reerene ime : Moing obje roke reerene ime : Moing obje roke reerene ime 3. Deriaion o he longiudinal Doppler shi o ligh The longiudinal Doppler shi o ligh onsiss o he lassial Doppler shi o sound i.e., geomerial drawing and he orenz ransormaion. Figure shows he way in whih he longiudinal Doppler shi is deried; he lassial Doppler shi deermined rom he geomerial drawing and he orenz ransormaion are ombined. The geomerial drawing shows he ouning o waerons. In Fig., obserer ouns he waerons o ligh rom ligh soure. Thereaer, he orenz ransormaion is applied. The requeny o ligh rom soure and he reerene ime o obserer wih whih he requeny is ouned are modiied by he orenz ransormaion. Reerene rame a res O Waeron Obserer igh soure Fig. Deriaion o he longiudinal Doppler shi rom a geomerial drawing and he orenz ransormaion The lassial Doppler shi o a requeny, represened by equaion 3, is deried rom a geomerial drawing, where is he eloiy o he obserer, S is ha o he ligh soure, and is he 3

4 requeny o he ligh soure in a saionary sae. ' ± ± S 3 here, and indiae he direion o he relaie moion o he ligh soure and he obserer. The or sign is seleed aording o he relaie moion beween he ligh soure and he deeor. The requenies are summarized in Table. Table Summary o requenies : Frequeny o lassial Doppler shi deried using geomerial drawing M : Frequeny o moing ligh soure inerse o reerene ime : Frequeny o saionary ligh soure : ongiudinal Doppler shi obje is seen rom iewpoin o obje : ongiudinal Doppler shi obje is seen rom iewpoin o obje In Figs. and, when he obserer a res on earh sees roke he ligh soure leaing, hen he sign is seleed o derease he requeny aording o S. Where S is replaed by, hus, equaion 4 is deried rom equaion 3 as ' M. 4 Then, we apply he orenz ransormaion. The requeny o he moing ligh soure M is deried as he inerse o he reerene ime o equaion, as shown in M. 5 y inroduing equaion 5 ino M in equaion 4, we obain O. 6 Equaion 6 is he Doppler shi deried rom he geomerial drawing and orenz ransormaion. The saionary obserer sees he ligh soure moing a he eloiy. When he moing obserer wih eloiy sees he ligh soure a res, he geomerial drawing is obained by seing S in equaion 3, and onsidering he direion o moemen o he obserer, we sele. Thus, we obain 4

5 '. 7 The reerene ime o he moing obserer beomes long aording o he orenz ransormaion. Thus, he requeny o he ligh soure ha is seen by he moing obserer inreases. This is beause he reerene ime o he moing obserer dereases. Equaion 7 is modiied as O. 8 Wih equaions 6 and 8, we show ha he equaion o he Doppler shi is equialen or he obserer and he ligh soure. This is he ase when eiher he obserer or he ligh soure is saionary. In equaions 6 and 8, he sign o he eloiy, or, should be onsidered; a eloiy denoes ha he disane beween he obserer and he ligh soure inreases, and a eloiy denoes ha he disane beween he obserer and he ligh soure dereases. 4. Deriaion o he Doppler shi equaion o wo moing bodies in he reerene rame a res Here, we disuss he relaie moion o he wo moing bodies in he reerene rame a res. The Doppler shi equaions or roke and roke are deried. We see he ligh soure in roke rom he iewpoin o roke using he lok in roke as a reerene. The obserer is roke and he ligh soure is roke, and he direions o he eloiies o rokes and are hosen as shown in Figs. and. Thereore, he obserer raels away rom he ligh soure and, a he same ime, he ligh soure raels owards he obserer; hus, we hoose in equaion 3. I he obserer is roke and he ligh soure is roke, hen is replased by and S is replased by and, hus, we obain '. 9 Then, we arry ou ime onersion, ha is, ime modiiaion using he reerene imes o rokes and. The reerene imes o rokes and are modiied using he orenz ransormaion. When he obserer in roke sees he ligh soure in roke, he reerene ime o roke in equaion is used. The requeny o he ligh soure is he inerse o equaion. Thus, when he obserer in roke sees he ligh soure in roke, he orreion onsan or he Doppler shi is /. Muliplying equaion 9 by / we obain 5

6 6, where indiaes he requeny a whih he obserer in roke sees he ligh soure in roke. Equaion depends on and, no on he relaie eloiy. s disuss in seion 5., i he boh eloiies and are no zero, he represenaion o he relaie eloiy is no orre: he relaiisi eloiy addiion law should be adoped. Figure 3 shows he obserer in roke seeing he ligh soure in roke, i.e., he obserer is roke and he ligh soure is roke. Thus, he obserer moes away rom he ligh soure and he ligh soure moes owards he obserer. When we replae by and S by, and areully hoose and in equaion 3, we obain ± ± S '. Then, Obserer Waeron Reerene rame a res O igh soure Fig. 3 Deriaion o longiudinal Doppler shi rom geomerial drawing and orenz ransormaion. Obserer and ligh soure are hanged rom hose in Fig.

7 7. From equaions and, we obain. 3 We noe ha he longiudinal Doppler shi depends on he absolue eloiies o he obserer and he ligh soure raher han he relaie eloiy o he obserer and he ligh soure. We onsider equaion ; i we se we obain he longiudinal Doppler shi o equaion 6. Under he ondiion ha is, he relaie eloiy is zero, he obserers in boh rokes see he same requeny, ha is,. 4 Equaion shows no only he generalizaion o equaion 6 bu also he neessiy o he absolue eloiy. This paper poins ou he neessiy o he absolue eloiy in he Doppler shi equaion. 5. Disussion 5. Represenaion using relaiisi eloiy addiion law Insead o he relaie eloiy -, i he relaiisi eloiy addiion law is used as ollows, wo eloiies are and -. u, 5 equaions and are represened as u u. 6 The eloiy u in equaion 6 onains absolue eloiies and as represened in equaion 5. Howeer, equaion 6 is in good agreemen wih he orhodox represenaion o he Doppler shi requeny. I one o he eloiy or is zero, equaions 6 or 8 an be deried. The posulae o an absolue reerene rame is ompaible wih he essene o he Doppler shi represenaion in he heory o speial relaiiy.

8 5. ppliaion o he Doppler shi equaion We use he Doppler shi whih ouns waerons direly. The inererene o he wae is no used, ha is, he parile propery o wae-parile dualiy is aken ino onsideraion. Figure 4 shows he ligh soure S moing wih an absolue eloiy o 3 km/s, roke a 34 km/s, and roke a 6 km/s. ording o equaion 3, we obain as ollows,, S S and, generally, S S S S. From Fig. 4, roke ouns more waerons han roke does; i is easy o predi ha. Subsiuing 3, km/s he speed o ligh, S 3 km/s, 34 km/s, and 6 S S km/s ino equaion, we obain, S S. 3, 7 S and.4. Thus, he dierene beween he Doppler shi requenies is S S S Waeron 6 km/s S 3 km/s 34 km/s Fig. 4 Relaie eloiies represened using equaion -, howeer his illusraion does no show he orre relaie eloiies. The orre relaie eloiy should be alulaed using he equaion o Doppler shi or equaion 5. The disussion o Doppler shi whih uses he relaie eloiy represened as - is no orre. We anno illusrae he relaie eloiy as shown in Fig. 4. s disussed in seion 5., he relaie eloiy should be represened as equaion 5, ha is, i we se S 3 km/s, 34 km/s, he 8

9 relaie eloiy is no alulaed exaly 4 km/s bu 4.45 km/s, and is no exaly6 km/s bu km/s. The relaie eloiy should be deined by he Doppler requeny. I we dee S S, he relaie eloiies beween roke S and rokes and are equal. 5.3 To ind he reerene rame a res by he sing around mehod From he disussion in 5., we ind ha he relaie eloiy an be experimenally deeed using Doppler shi. In his seion, a simple mehod whih an be used o ind he reerene rame a res is desribed. Insead o he Doppler shi, he sing around mehod, whih was preiously ried by Galileo using lanerns on op o wo mounains, is applied. In he ase o aousi waes, or he measuremen o sound speed, he sing around mehod is used. The sing around experimenal seup using a ligh soure [, 3], whih uses wo pairs o ligh soures and deeors as shown in Fig. 5, an be onsrued, where a pulsed signal a lash o ligh is ransmied by ligh soure and deeed by deeor. er is deeion, a new pulsed signal is ransmied by ligh soure, deeed by deeor, ransmied again by ligh soure, and so on. igh soures are onsrued using EDs, and he deeors are by phoo diodes. igh soure Flash o ligh Going pah D Deeor D Reurning pah Deeor igh soure Fig. 5 Sing around experimenal seup using ligh lash o ligh rom ligh soure is deeed by deeor. er is deeion, a new lash o ligh is ransmied by ligh soure, deeed by deeor, ransmied again by ligh soure, and so on. This experimen is modern ersion o Galileo s experimen using lanerns on op o wo mounains. I we use he alue shown in Fig. 4 he dierenes are ery small. We assume ha he wo rokes are raeling a he speeds o 4% roke S and 6% roke o he speed o ligh as shown in Fig ording o equaion 5 we obain u. Roke S ransmis a lash o ligh o rokes and 9 9

10 hen hey lash bak o roke S. We alulae ' using he equaion o Doppler shi requeny or equaion 5. We obain ' 3 so ha S S. 85 Roke Roke S Roked x 3 ' 85 S.4.6.6, roke S.4, roke S ' 3, roke 85. Fig. 6 Sing around mehod o a ligh soure: rokes and are launhed rom roke S in opposie direion in order o dee he same Doppler requeny. The speed o ligh, is assumed o be onsan regardless o he eloiy o he ligh soure. The ligh pahs are drawn in he igure. The Doppler requenies o rokes and shown rom roke S is he same represened as S S. The relaie eloiies beween roke S and rokes and are equal. Howeer, he arrial ime o he reurn pulses deeed a roke S gradually si. This indiaes ha roke S is no in he reerene rame a res, and roke S should be deeleraed. In his disussion, only Einsein s assumpion, in whih a phoon raels a he speed o ligh,, in auum regardless o he eloiy o he ligh soure, is used; hus, he hree rokes will dee he lashes o ligh as shown in Fig. 6. I rokes and are launhed in opposie direions rom roke S, he lash paern o ligh ha roke S dees is shown as he poins on he line.4 in Fig. 6. Thus,

11 we know he driing direion o he roke S in he reerene rame a res. Thereore, we an know how o deelerae roke S o be lose o he reerene rame a res. The sing around mehod does no need he orenz ransormaion o he reerene ime: he dilaion o ime is no aken ino onsideraion. I requires only a geomerial drawing; hereore, he disussion beomes lear and simple. We propose a new mehod ha is dieren rom he Mihelson-Morley experimen o dee he moion in he reerene rame a res. Experimenal proedure o ind he reerene rame a res is: Rokes are sared in all direions rom roke S and aeleraed so as o dee he same Doppler shi requeny: S S, i he same Doppler shi requeny is deeed all rokes are in he same relaie eloiy o roke S. Dee he ime ineral o sing around pulses. Rokes S is moing oward he roke whih shows he shores ime ineral o sing-around ligh pulses. Then, deelerae roke S. 3 Repea he proedure and so as o dee he same ime ineral o sing around pulses or all rokes. We hae wo mehods o deine he eloiy: one is he Doppler shi requeny and he oher is he repeiion requeny o sing around ligh pluses. Thus we an dee he reerene rame a res. The meri o his represenaion is ha he absolue reerene rame an be deeed. We know he absolue eloiies o he wo rokes so ha here is no win paradox. 6. Conlusion The longiudinal Doppler shi o wo moing bodies in a reerene rame a res was deried onsidering he parile propery o phoons in he wae-parile dualiy. The equaion o he Doppler shi depends on he absolue eloiies o he moing bodies. I he relaiisi eloiy addiion law is used or he relaie eloiy beween he obserer and he ligh soure, we an show ha he longiudinal Doppler shi equaion 6 is in good agreemen wih he orhodox heory o speial relaiiy. The meri o his represenaion is ha he absolue reerene rame an be deeed. Reerenes M. Sao, "Proposal o Mihelson-Morley experimen ia single phoon inereromeer: Inerpreaion o Mihelson-Morley experimenal resuls using de roglie-ohm piure," physis/47, 4. D. ohm and. Hiley, The undiided unierse, Rouledge, ondon, M. Sao, "Proposal o aomi lok in moion: Time in moing lok," physis/4, 4.