Electrodynamics in Uniformly Rotating Frames as Viewed from an Inertial Frame

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1 letrodnamis in Uniforml Rotating Frames as Viewed from an Inertial Frame Adrian Sfarti Universit of California, 387 Soda Hall, UC erele, California, USA (Reeived 3 rd Feruar, 7; Aepted 3 th Marh, 7; Pulished: 3 st Marh, 7) Astrat- In the urrent paper we present a generaliation of the transforms of the eletromagneti field from an inertial frame of referene into the frame o-moving with a uniforml rotating partile. The solution is of great interest for real time appliations, eause earth-ound laoratories are inertial onl in approimation. We onlude deriving the general form of the relativisti Doppler effet and of the relativisti aerration formulas for the ase of uniforml rotating frames. Ke Words: Aelerated motion, General oordinate transformations, Aelerated partiles, Planar eletromagneti waves, Relativisti Doppler effet, Relativisti aerration.. INTRODUCTION Real life appliations inlude aelerating and rotating frames more often than the idealied ase of inertial frames. Our dail eperiments happen in the laoratories attahed to the rotating, ontinuousl aelerating arth. Man oos and papers have een dediated to transformations etween partiular ases of retilinear aeleration and/or rotation [] and to the appliations of suh formulas [-]. The main idea of this paper is to generate a standard lueprint for a general solution that gives equivalent of the Lorent transforms for the ase of the transforms etween an inertial frame and a rotating frame.. UNIFORMLY ROTATING MOTION TH TRANSFORMS OF TH LCTROMAGNTIC FILD In this setion we disuss the ase of the partile moving in an aritrar plane, with the normal given the onstant angular veloit ω(a,, ) (Fig. ). Aording to Moller [], the simpler ase when ω is aligned with the -ais produes the transformation etween the rotating frame S (τ) attahed to the partile and an inertial, non-rotating frame S attahed to the enter of rotation: () Ph _ rotation t where Ph _ rotation osαosβ + osαsinβ γsinαosβ where simαosβ γosαsinβ sinαsinβ sinβ os β γ () γ, u rω, α ω / τ, β ωγ τ u In previous papers, we have derived the transformations from the rotating frame S into the inertial frame S: ( γ osα os β + sinα ) os β osα ) + osα os β ) osα sinα + γ ( γ osα os β + sinα ) os β os β osα ) ( γ osα sinα os β ) + osα os β ) osα sinα + γ ( γ osα sinα os β ) os β The aove an e re-ast in matri form os os sin sin sin os os sin u os γ α β+ α β γ α β α β γ β (4) γos α sin αos β γsin α+ os αos β u γsin β os α sin α γ os β os os sin sin sin os os sin γ α β+ α β γ α β α β u sin (5) γ β γos α sin αos β γsin α+ os αos β u γos α u γsin α γ This oservation allows us an eas wa of inverting the transforms in order to otain the transforms from the inertial frame S into the rotating frame S : os os sin sin os sin sin os os γ α β+ α β γ α β α β α (6) γsin αos β os α γsin α+ os αos β sin α os β γ os α os os sin sin os sin sin os γ α β + α β γ α β α β u sin (7) γ α γ sin αos β os α γ sin α + os αos β os β γ A ver nie onsequene of (6) and (7) is that: (3) Page 6

2 (8) The rotation mies the omponents of the eletromagneti tensor in a wa that is different from the ases of inertial motion or uniforml aelerated motion. 3. PLANAR WAV TRANSFORMATION AND SPD OF LIGHT IN A UNIFORMLY ROTATING FRAM In this setion we appl the formalism derived in the previous paragraph in order to otain the transform of a planar wave. Assume that a planar wave is propagating along the ais in the inertial frame S. The wave has the eletri omponent and the magneti omponent along the and aes, respetivel. The omponents equations are (Fig. ): os( Ωt + θ ) e (9) os( Ωt + θ ) e On the other hand: Fig. : The eletromagneti wave 4 4 () t where: osαosβ + simαosβ γosαsinβ sinβ osαsinβ γsinαosβ sinαsinβ os β γ () Sustituting () into (9) we otain: os[ Ω ( + + t) ( + + t) + θ] os[( Ω ) t ( Ω ) ( Ω ) + θ] os[( Ω ) t ( Ω ) ( Ω ) + θ] () On the other hand, in frame S, the wave equation has two alternative forms, First one: ( γ osα os β + sinα ) + ( os α) } [( γ osα os β + sinα ) + ( os α) ]* *os[( Ω ) t ( Ω ) ( Ω ) + θ ] [( γ sinα os β osα ) + ( sin α ) ]* *os[( Ω ) t ( Ω 4 ) ( Ω 4 ) + θ ] 44 4 u os β γ ( + )* *os[( Ω ) t ( Ω ) ( Ω ) + θ ] Seond one: os( Ω t + θ ) e os( Ω t + θ ) e os( Ω t + θ ) e Comparing (3) and (4) we otain: ( γ osα os β + sinα ) + ( os α) Ω Ω θ θ 44 4 Ω 4 Ω 4 The phase light speed v Ω v p in the rotating frame S is: Ω 44 4 p + ( Ω 4) + ( Ω4 ) (3) (4) (5) (6) So, the light speed in the rotating frame equals the light speed in the inertial frame,. We an now proeed to alulating the amplitude and the phase transformation etween the inertial and the rotating frame: θ θ (7) The general equation of the Doppler effet is: 4 os α uos α Ω Ω 44 4 Ω( 44 ) Ω( γ ) γ( ) Ω γ u (8) u rω α ωγτ β ωγ τ The general equations for aerration are: (sin αos β γ os α ) os β (9) (sin α + γ osα os β + ) In the rotating frame, the wave follows a urved trajetor. It is interesting to oserve that for τ we have so αβ and: u γ ( + ) () Page 63

3 meaning that the wave vetor in frame S starts in the same diretion as the wave vetor in frame S efore it starts to progressivel urve awa as the proper time τ inreases. 4. GNRAL CAS OF ROTATION AOUT AN ARITRARY AXIS In a prior paper we have shown that the partiular transformation () an e generalied for the ase of aritrar diretion. The general ase is treated transforming the prolem into the partiular ase treated in [] through a transformation into the anonial ase, followed an appliation of the transformation from the rotating frame into the inertial frame, ending with the inverse of the first transformation, as shown elow: ( Rr * Ph _ rotation* Rr) () t Rr Rot( e ) * Rot( e ) * Rot () 9 ϕ 9 9 ϕ Rot( e ) * Rot aligns ω with e. The seond step is omprised another rotation around the -ais 9 that aligns ω with e : Rot( ) (3) e 9 pression () gives the solution for the general ase, of aritrar angular veloit diretion. Re-writing (6) and (7) in tensor form we get: * * * * * * * * * * γosαosβ + sinαsinβ γsimαosβ osαsinβ osα γosαsinβ sinαosβ sinα os β γ os β sinβ γ (4) The asteriss represent entries with no partiular phsial meaning, we do not are aout them. Then, the general transform is γosαosβ + sinαsinβ γsimαosβ osαsinβ osα γosαsinβ sinαosβ sinα Rr * os β γ os β sinβ γ * Rr (5) Fig. Uniform rotation with aritrar diretion of angular veloit 5. APPLICATION I: TH GNRAL XPRSSIONS FOR ARRATION AND FOR TH DOPPLR FFCT We have seen in setion 3 that the relativisti Doppler effet an e derived from the frame invariane of the epression: Ψ Ωt ( + + ) + ϕ (6) We have seen in setion 4 that the general oordinate transformation from an aelerated frame S into an inertial frame S is: a a a3 a4 a a a3 a 4 (7) a3 a3 a33 a t a4 a4 a43 a44 ω t The susript ω represents the dependene of the matri elements a ij a ij (ω) of the angular veloit ω (ω, ω, ω ) etween frames S and S. Sustituting (7) into (6) we otain: Ψ Ω( a4. ) ( a. ) ( a. ) ( a3. ) + ϕ (8) ( ωa a a a ) t ( a + a + a ωa ) ( a + a + a3 ωa4 ) ( a3 + a3 + a33 ωa43 ) + ϕ On the other hand, in the rotating frame S : Ψ Ω t ( + + ) + ϕ Ψ (9) Comparing (8) and (9) we otain the general epressions of the relativisti Doppler effet etween the inertial frame S and the aelerated frame S : Ω Ωa a a a a + a + a Ωa 3 4 a + a + a Ωa 3 4 a + a + a Ωa ϕ ϕ In matri form: Ω a44 a4 a4 a Ω a4 a a a 3 a4 a a a3 a43 a3 a3 a33 ω For the reverse transformations we start with: (3) (3) Page 64

4 3 4 a a a a a a a3 a 4 (3) a3 a3 a33 a t a4 a4 a43 a44 t For simpliit, we re-write (3) as: (33) t t ω In the rotating frame S : Ψ Ω t ( + + ) + ϕ () Sustituting (33) into (): Ψ Ω(. ) (. ) (. ) (. ) + ϕ 4 3 ( Ω ) t ( + + Ω ) ( + + Ω ) ( + + Ω ) + ϕ (35) On the other hand, in frame S: Ψ Ωt ( + + ) + ϕ Ψ (36) Comparing (35) and (36) we otain the general epressions of the relativisti Doppler effet etween the inertial frame S and the aelerated frame S: Ω Ω Ω Ω Ω ϕ ϕ In matri form: Ω ω Ω (37) (38) 6. APPLICATION II: TH DOPPLR FFCT FOR ROTATING MITTR AND RCIVR In this setion we treat the general ase, the emitter moves with angular veloit ω and the reeiver moves with aritrar angular veloit ω, oth with respet to the inertial frame S. Aording to (38): Ω Ω 4 3 (39) ω emitter Aording to (3): Ω a44 a4 a4 a a4 a a a 3 a4 a a a3 a a a a reeiver ω Ω (4) From (39) and (4) we otain the general form of Doppler effet and aerration for the ase of the emitter and the reeiver moving with aritrar angular veloities ω and ω with respet to the same inertial referene frame: Ω a44 a4 a4 a a4 a a a3 a a a a 4 3 a a a a reeiver ω (4) Ω ω emitter A qui sanit he shows that for ω ω: Ω Ω * * * (4) * * * * * * reeiver emitter In this ase, there is no Doppler effet ut there is oviousl aerration sine the terms denoted asteriss are not null. 7. CONCLUSIONS We onstruted the general transforms from an inertial frame of referene into an uniforml rotating frame. We onluded deriving the general form of the relativisti Doppler effet and of the relativisti aerration formulas for the ase of uniforml rotating frames. The solution is of great interest for real life appliations, eause our earth-ound laoratories are inertial onl in approimation; in real life, the laoratories are aelerated and rotate. We produed a lueprint for generaliing the solutions for the aritrar ases and we onluded with an appliation that eplains the general ase of planar eletromagneti waves. A ver interesting onsequene is the fat that light speed in vauum in the rotating frames is. A seond interesting onsequene is that rotation indues aerration. RFRNCS [] C. Moller, The Theor of Relativit, Oford Press, 96. [] L. H. Thomas, Motion of the spinning eletron, Nature, vol. 7, no. 945, pp. 54, 96. [3] A. en-menahem, Wigners rotation revisited, Am. J. Phs., vol. 53, no., pp. 6-66, 985. [4] S. en-menahem, The Thomas preession and veloit spae urvature, J. Math. Phs. vol. 7, no. 5, pp , 986. [5] H. Kroemer, The Thomas preession fator in spinorit interation, Am J. Phsis., vol. 7, no., pp. 5-5, 4. [6] J. A. Rhodes, and M. D. Semon, Relativisti veloit spae, Wigner rotation and Thomas preession, Am. J. Phs., vol. 7, no. 7, pp , 4. [7] G.. Malin, Thomas preession: orret and inorret solutions, Phs. Usp., vol. 49, no. 8, pp , 6. Page 65

5 [8] M. I. Krivoruheno, Rotation of the swing plane of Fouaults pendulum and Thomas spin preession: Two faes of one oin, Phs. Usp., vol. 5, no. 8, pp. 8, 9. [9] A. Sfarti, Hperoli Motion Treatment for ell s Spaeship periment, Fiia A, vol. 8, no., pp. 45-5, 9. [] A. Sfarti, A. Coordinate Time Hperoli Motion for ell s Spaeship periment, Fiia A, vol. 9, no. 3, pp. 9,. [] A. Sfarti, Relativit solution for Twin parado : a omprehensive solution, Indian Journal of Phsis, vol. 86, no., pp ,. Page 66