Electrodynamics in Accelerated Frames as Viewed from an Inertial Frame

Transcription

1 Copriht IJPOT, All Rihts Rsrvd Intrnational Journal of Photonis and Optial Thnolo Vol. 3, Iss. 1, pp: 57-61, Marh 17 ltrodnamis in Alratd Frams as Viwd from an Inrtial Fram Adrian Sfarti Univrsit of California, 387 Soda Hall, UC rl, California, USA (Rivd 3 rd Fbruar, 17; Aptd 13 th Marh, 17; Publishd: st Marh, 17) Abstrat- In th urrnt papr w prsnt a nraliation of th transforms of th ltromanti fild from an inrtial fram into an alratd fram of rfrn. Th solution is of rat intrst for ral tim appliations, baus arth-bound laboratoris ar inrtial onl in approimation. W onlud b drivin th nral form of th rlativisti Dopplr fft and of th rlativisti abrration formulas for th as of alratd motion. K Words: Alratd motion, Gnral oordinat transformations, Alratd partils, Planar ltromanti wavs, Rlativisti Dopplr fft, Rlativisti abrration. 1. INTRODUCTION Ral lif appliations inlud alratin and rotatin frams mor oftn than th idalid as of inrtial frams. Our dail primnts happn in th laboratoris attahd to th rotatin, ontinuousl alratin arth. Man boos and paprs hav bn ddiatd to transformations btwn partiular ass of rtilinar alration and/or rotation [1] and to th appliations of suh formulas [-11]. Thr is rat intrst in produin a nral solution that dals with arbitrar orintation of alration in th as of rtilinar motion. Th main ida of this papr is to nrat a standard bluprint for a nral solution that ivs quivalnt of th Lornt transforms for th as of th transforms btwn an inrtial fram and an alratd fram.. TRANSFORMS TWN TH INRTIAL FRAM AND TH ACCLRATD FRAM Lt S rprsnt an inrtial sstm of oordinats and S an alratd on. Invrtin th S S transforms drivd in a prior papr: = = osh + = osh = = osh = osh + w obtain th S S transforms: (1) = = = osh () = osh + = osh + = osh In matri form: 1 * * osh = τ τ * * osh 1 (3) Th lmnts mard with astriss rprsnt ntitis without an manin. From () w obtain immdiatl: = (4) 3. PLANAR WAV TRANSFORMATION AND SPD OF LIGHT IN A UNIFORMLY ACCLRATD FRAM In this stion w appl th formalism drivd in th prvious pararaph in ordr to obtain th transform of a planar wav from an inrtial fram S into an alratd fram S. Assum that a planar wav is propaatin alon th ais in th inrtial fram S. Th wav has th ltri omponnt and th manti omponnt alon th and as, rsptivl. Th omponnts quations ar (Fi. 1): = os( ωt + φ) = = = os( ωt = = + φ) Aordin to rfrn [1] th transformation for th partiular as of alratd motion alon th -ais from S (τ) into S is: (osh 1) (6) = Ph _ rtilinar + t t (5) Pa 57

2 Intrnational Journal of Photonis and Optial Thnolo Vol. 3, Iss. 1, pp: 57-61, Marh 17 Fi.1.Th planar ltromanti wav osh 1 Ph _ rtilinar = 1 1 osh so: = t = τ + t τ osh + τ (7) (8) W an s from (1) that in th alratd fram S th wav hibits omponnts alon th as and. Ths omponnts ar a dirt fft of th alration. From (11) w obtain: ω ω ω = + + = + = osh (13) W an now alulat th phas liht spd masurd in th alratd fram: ω vp = = (14) Finall, w an alulat th abrration in fram S indud b th alration: os θ = = tanh (15) Th Pontin vtor in th alratd fram is: S = osh = osh + (16) Substitutin (8) into (5) w obtain: = os[( ω osh τ ) t + ( ω τ ) + ϕ + ω τ )] (9) On th othr hand, in fram S, th wav quation is: = os( ω t + φ ) (1) Sin = it follows that: ω = ω osh = ω = = = (11) ω ϕ = ϕ + In th alratd fram, th wav vtor ains a (timvarin) omponnt in th dirtion as a onsqun of th alration. Additionall: = = (1) = = osh 4. GNRAL CAS OF UNIFORM ACCLRATION IN AN ARITRARY DIRCTION In a prior papr w hav shown that th partiular transformation (1) an b nralid for th as of arbitrar dirtion onstant alration = (,, ) to: 1 = ( Tr * Ph _ rtilinar * Tr) + N t t whr: Tr = Rot( ) * Rot( ) * Rot 9 9 ϕ 1 osh Ph _ rtilinar = osh 1 (osh 1) N = (17) (18) Introduin th triplt (,, ) ( a b =,, ) th followin prssions hold: Copriht IJPOT, All Rihts Rsrvd Pa 58

3 Intrnational Journal of Photonis and Optial Thnolo Vol. 3, Iss. 1, pp: 57-61, Marh 17 a 1 Rot (19) = a 1 os(9 φ) sin(9 φ) 1 Rot( ) = 9 φ sin(9 φ) os(9 φ) 1 () sinφ osφ = 1 osφ sin φ Rot( ) * Rot alins with. Th sond stp is 9 ϕ omprisd b anothr rotation around th -ais b 9 that alins with (Fi. ): Rot( ) = 1 1 (1) 5. APPLICATION I: TH GNRAL XPRSSIONS FOR ARRATION AND FOR TH DOPPLR FFCT W hav sn in stion 3 that th rlativisti Dopplr fft an b drivd from th fram invarian of th prssion: Ψ = ωt ( + + ) + ϕ (3) W hav sn in stion 4 that th nral oordinat transformation from an alratd fram S into an inrtial fram S is: (osh 1) a11 a1 a13 a14 a1 a a3 a (4) 4 = a a3 a33 a 34 + t a41 a4 a43 a44 t Th subsript rprsnts th dpndn of th matri lmnts a ij =a ij () of th alration =(,,,) btwn frams S and S. Substitutin (4) into (3) w obtain: ω ϕ Ψ = ( a4. + ) ( a1. + (osh 1)) ( a. ) ( a3. ) + = = ( ωa a a a ) t ( a + a + a ωa ) ( a + a + a ωa ) ( a + a + a ωa ) ϕ + ω (osh 1) (5) On th othr hand, in fram S : Ψ = ω t ( + + ) + ϕ = Ψ (6) Comparin (5) and (6) w obtain th nral prssions of th rlativisti Dopplr fft btwn th inrtial fram S and th alratd fram S : Fi. : Gnral alration Tr -1 rvrss all th ffts of Tr. prssion (17) ivs th solution for th nral as, of arbitrar alration dirtion. Thn, from (3) and (17) th matri for th nral transform in th as of fram with arbitrar dirtion alration S into th inrtial fram S is simpl: 1 * * Tr 1 * * * * = Tr * Copriht IJPOT, All Rihts Rsrvd osh osh 1 () ω = ωa a a a = a + a + a ωa = a + a + a ωa = a + a + a ωa τ ϕ = ϕ + ω (osh 1) In matri form: a44 a14 a4 a34 a41 a11 a1 a = a4 a1 a a3 a43 a13 a3 a33 For th rvrs transformations w start with: 1 (osh 1) a11 a1 a13 a 14 a1 a a3 a 4 = a a3 a33 a 34 t a41 a4 a43 a44 t For simpliit, w r-writ (9) as: (7) (8) (9) Pa 59

4 Intrnational Journal of Photonis and Optial Thnolo Vol. 3, Iss. 1, pp: 57-61, Marh 17 a44 a14 a4 a34 (osh 1) b11 b1 b13 b 14 a41 a11 a1 a (37) b1 b b3 b 4 (3) = = a4 a1 a a3 b b3 b33 b 34 rivr a43 a13 a3 a33 t b41 b4 b43 b44 t From (36) and (37) w obtain th nral form of Dopplr fft and abrration for th as of th mittr and th Th subsript rprsnts th dpndn of th matri rivr movin with arbitrar alrations 1 and with lmnts a ij =a ij () of th alration =(,,,) btwn rspt to th sam inrtial rfrn fram: frams S and S. In th alratd fram S : ω a44 a14 a4 a34 Ψ = ω t ( + + ) + ϕ () a41 a11 a1 a = Substitutin (3) into (): a a a a Ψ = ω ( b4. ) ( b1. (osh 1)) ( b. ) ( b3. ) + ϕ rivr = a43 a13 a3 a33 (38) b44 b14 b4 b34 ω = ( ω b44 b14 b4 b34 ) t ( b11 + b1 + b ω b41) b41 b11 b1 b ( b1 + b + b3 ω b4 ) ( b13 + b3 + b33 ω b43 ) + b b b b b43 b13 b3 b33 1 mittr + ϕ ω + (osh 1) (3) 7. CONCLUSION On th othr hand, in fram S: W onstrutd th nral transforms from th inrtial Ψ = ωt ( + + ) + ϕ = Ψ (33) fram S into th alratd fram of rfrn S. W drivd th nral form of th rlativisti Dopplr fft and Comparin (3) and (33) w obtain th nral prssions of th rlativisti abrration formulas for th as of of th rlativisti Dopplr fft btwn th inrtial fram S and th alratd fram S: alratd motion. Th solution is of rat intrst for ral lif appliations, baus our arth-bound laboratoris ar ω = ω b44 b14 b4 b34 inrtial onl in approimation; in ral lif, th laboratoris ar = b11 + b1 + b ω b alratd. W produd a bluprint for nraliin th 41 (34) solutions for th arbitrar ass and w onludd with an = b1 + b + b3 ω b4 appliation that plains th nral as of planar = b13 + b3 + b33 ω b ltromanti wavs. A vr intrstin onsqun is th 43 fat that liht spd in vauum in th alratd frams is. A sond intrstin onsqun is that alration indus abrration. ϕ = ϕ ω + (osh 1) In matri form: b44 b14 b4 b34 b41 b11 b1 b = b4 b1 b b3 b43 b13 b3 b33 (35) 6. APPLICATION II: TH DOPPLR FFCT FOR MITTR AND RCIVR ACCLRATD AT DIFFRNT RATS In this stion w trat th nral as, th mittr movs with arbitrar alration 1 and th rivr movs with arbitrar alration, both with rspt to th inrtial fram S. Aordin to (35): b44 b14 b4 b34 b41 b11 b1 b (36) = b4 b1 b b3 b b b b mittr Aordin to (8): RFRNCS [1] C. Mollr, Th Thor of Rlativit, Oford Prss, 196. [] L. H. Thomas, Motion of th spinnin ltron, Natur, vol. 117, no. 945, pp. 514, 196. [3] A. n-mnahm, Winrs rotation rvisitd, Am. J. Phs., vol. 53, no. 1, pp. 6-66, [4] S. n-mnahm, Th Thomas prssion and vloit spa urvatur, J. Math. Phs. vol. 7, no. 5, pp , [5] H. Kromr, Th Thomas prssion fator in spinorbit intration, Am J. Phsis., vol. 7, no. 1, pp. 51-5, 4. [6] J. A. Rhods, and M. D. Smon, Rlativisti vloit spa, Winr rotation and Thomas prssion, Am. J. Phs., vol. 7, no. 7, pp , 4. [7] G.. Malin, Thomas prssion: orrt and inorrt solutions, Phs. Usp., vol. 49, no. 8, pp , 6. [8] M. I. Krivoruhno, Rotation of th swin plan of Fouaults pndulum and Thomas spin prssion: Two fas of on oin, Phs. Usp., vol. 5, no. 8, pp. 81, 9. Copriht IJPOT, All Rihts Rsrvd Pa 6

5 Intrnational Journal of Photonis and Optial Thnolo Vol. 3, Iss. 1, pp: 57-61, Marh 17 [9] A. Sfarti, Hprboli Motion Tratmnt for ll s Spaship primnt, Fiia A, vol. 18, no., pp. 45-5, 9. [1] A. Sfarti, A. Coordinat Tim Hprboli Motion for ll s Spaship primnt, Fiia A, vol. 19, no. 3, pp. 119, 1. [11] A. Sfarti, Rlativit solution for Twin parado : a omprhnsiv solution, Indian Journal of Phsis, vol. 86, no. 1, pp , 1. Copriht IJPOT, All Rihts Rsrvd Pa 61