Relativistic Analysis of Doppler Effect and Aberration based on Vectorial Lorentz Transformations

Transcription

1 Uniersidad Central de Venezuela From the SeletedWorks of Jorge A Frano June, Relatiisti Analysis of Doppler Effet and Aberration based on Vetorial Lorentz Transformations Jorge A Frano, Uniersidad Central de Venezuela Aailable at:

2 Journal of Vetorial Relatiity JVR 6 () 3-9 Relatiisti Analysis of Doppler Effet and Aberration based on Vetorial Lorentz Transformations J A Frano-Rodríguez ABSTRACT: In this paper, we deried more general and orret expressions for the Relatiisti Doppler and Aberration effet. KEYWORDS: Relatiisti Doppler Effet, Relatiisti Aberration. I. INTRODUCTION Einstein in his first paper on Relatiity of June 95 [], in Setion 7, did the analysis of the problem of the Doppler Effet and obtained his relatiisti formula for the frequeny γ ' measured by a moing obserer, relating it to a known alue of the frequeny γ of a retilinear eletromagneti signal oming from a ery far point at rest loated at a distane far apart and forming an angle θ with the diretion of a moing obserer in the X axis. The moing obserer sees and measures the same signal with a frequeny γ '. His obtained formula was:.osθ γ' γ. () We will arrie at Einstein s onfiguration for analyzing Doppler Effet, starting diretly with that one used for the deriation of the Vetorial Lorentz Transformations [], see Fig., whih are defined as: r' k. ( r r ) r' k. r k ; t'. t.. u, for:. r and later transforming it into the Einstein s equialent onfiguration. k, r u. t and r. t II. DEVELOPMENT Firstly, let s suppose that in a moing two-dimensional frame of referene, where for α β, Fig. redues to Fig., an obserer joined to its origin O is loated for t ' at point A, and for t T at point B. Points A and B are loated in a fixed referene frame with origin O (A oinides with this Independent Researher, Caraas, Venezuela, jafranor@yahoo.om 3A June

3 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June origin O of the fixed referene frame). And, he is measuring the frequeny of a light signal sent at t ' t from O, whih arries at point P, fixed relatie to O, at t T, see Fig.. In this way, we hae separated the two instants in whih the moing obserer measures the frequeny of a light signal, t and t T, in two referene frames with relatie motion between them: a referene frame at rest with origin at O, defined oiniding with point A, and a moing referene frame urrently at point B at time t T, that moes at speed relatie to O. Z Z Pulse of Light u B(x,y,z,t) (x,y,z,t ) z r r z A(...) r O m y Y O α β Y X x X x y Fig. General three-dimensional onfiguration for VLT Pulse of Light P Y Y r r A θ B θ x X X x.t.t Fig.. Two-Dimensional VLT s onfiguration When moing origin O oinides with O a signal of light is sent from O to the spae going along with a line that forms an angle θ with X axis. At this preise instant, loks on both referene systems JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 4

4 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June starts ounting the time. Measuring finishes when light displaement reahes a omplete waelength λ. In this way measured time must be the period of time T (inerse of the frequeny γ ), elapsed during the displaement of a omplete waelength λ. Seondly, this onfiguration now is transformed in the equialent one shown in Fig. 3, for alulating Doppler Effet under the equations of VLT, where a star is loated at point P, in the fixed referene frame, far at right of A and B, sending its light signals, whose frequeny a moing obserer is measuring. The Doppler Effet is defined as the measured frequeny γ ' instead of that measured as if the obserer were at rest, for instane, if he were at O. The measured frequeny by an obserer at O is γ while the measured frequeny by the moing obserer at O is γ ' Y Y r Pulse of Light r O θ O θ x X X.t Fig. 3. Two dimensional equialent VLT s onfiguration for measuring Doppler Shift In this last onfiguration we an easily see that if θ ' is less than 9 the moing obserer is approahing to the star beause he is moing from left to right and star is at right. Also, it is easy to see that if θ is greater than 9 moing obserer is moing away of the star. And, if in the referene frame at rest, representing the instant in whih the moing obserer is at O, a monohromati wae of a frequeny γ is measured, then in the moing referene frame O, representing the urrent position of the moing obserer at time t T that moes at right on the X axis at a speed relatie to the former, the frequeny γ ' of a beam of monohromati light is measured. For applying VLT transformations, we hae that beause the moable referene system is moing on the X axis the speed etor has null omponent on the Y axis. So, measurements of distane (waelength) and time (Period) are r' λ' k ( λ.t ) and t ' T', so, γ ', for the γ' T' t' general obtained expression of time in VLT, x t' k. t.osα.os β. x + + t.sinα.os β. y t.sin β. z () k t'. t. x y z.osα.os β. +.sinα.os β. +.sin β. (3) t t t JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 5

5 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June This is equialent to the following simpler expression in funtion of the etors and u : k t'. t..u (4) for: x y. u.osα.os β. +.sinα.os β. +.sin β t t z. t In this way, the general formula for a three-dimensional (or n-dimensional) Doppler Effet under VLT (see Fig. 4), with eloities (moing obserer), or u (any signal or projetile) with any diretion or magnitude, beomes: γ γ ' (5) k. t..u For the ase of a light signal, where u, beause the speed is less than the speed of light, for an angle θ ' between these etors less than 9 degrees, the radio-etor of the light front wae measured by the moing obserer at O, say the waelength λ ', is less than that measured by another fixed obserer at O, i.e. the waelength λ. Then, if λ' λ, we should obtain that γ' γ for θ < 9, thus, for k we hae:: γ' γ. From (5), we hae:.os θ + Expression (6) for θ, redues to:.sinθ γ..os θ +.sinθ γ' γ (6) osθ γ' γ (7) Obsere that in this ase of θ, and for obserer approahing to the star, relatiisti result oming from VLT analysis is that γ' γ and for, γ'. Value gien by lassial analysis for an obserer approahing the star is γ' γ + JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 6

6 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June Obsere in the expression shown below in (8) that for θ 9, or for θ 7, and for ±, γ' : γ' γ (8) This is a logial result. Why? - Always speed is less than and the hypotenuse (proportional to the waelength, measured by moing obserer) + is always greater than the side. So, measured frequeny in the moing system is less than that measured in the fixed origin O, say γ' γ as indiated by equation (8). If the speed of the moing system, on the X axis, approahes the speed of light signal on the Y axis, hypotenuse reahes its maximum. Thus, frequeny measured at the moing system reahes its minimum alue. So, obtained result in (8) equaling zero, agrees this analysis. Contraditorily, obsere that in equation () that for θ 9, or for θ 7, Einstein expression of γ ', for any alue of, gies surprisingly γ' γ (!), and for ± gies an infinite alue!, i.e.:. os θ γ' γ.., θ 9 On the other side, for θ 8, our equation gies: γ' γ γ (9) In this ase the moing system goes away of star towards left (Fig. or Fig. 3), and radio-etor from the moing system s origin to star, proportional to the waelength of the signal light, will be greater than that measured from the fixed origin O, say, λ' λ. Thus, for θ 8, frequenyγ' γ. For this ase, if, γ'. Value gien by lassial analysis for an obserer going away of the star is γ' γ. Obsere the differene between our formula (6) and that of Einstein s (): they are positiely distint. RELATIVISTIC ABERRATION As before we will present the relatiisti aberration under a etor presentation, based on the VLT: in priniple it is defined as the relationship between angle θ and the angle θ, with the same meanings preiously used in Figs. 7 and 8. The Einstein s formula for aberration gien in 95 was: JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 7

7 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June os θ.os θ By following Einstein, we will alulate the transitory trigonometri relationship: the Cosθ (it also ould be Sinθ or Tanθ ) in order to arrie at its inerse transformation for angle. For that, looking at Fig. 7 or 8, and applying etorial transformations (VLT) we hae for a twodimensional onfiguration: x' k( x. t) Cosθ ' () r' k( r. t) ( x. t) u.os θ ( x. t) + y ( uos θ. t) + ( usinθ ) u.os θ () u +. u..os θ This relationship applied to the signal light beomes: os θ..os θ (3) π If θ, the equation beomes: Cosθ ' (4) Equation (3) for two dimensions, equialent to the more general equation (), seems to express the law of aberration in its most general form, gien that angle θ depends only on the diretion of eloity of light and on the eloity of moing system, whih are independent of the oordinate systems. Let s hek the generality of these equations. The more general etorial form is that when the moing obserer moes on a generi line m and the angles θ ' and θ are formed by radioetors r' and r with this line m, respetiely. The osine relationship beomes gien by the projetion of the radioetor r' on the line m, by taking in aount that u and u, are unit etors (see Fig. 4). Gien that time t and the magnitude of the speed of light are always positie, we hae ( ) ( ) osθ x' r u. t t.. u u. t (5) r' ( r. t) (. u. t. u. t) u u () JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 8

8 J A Frano-Rodríguez: Relatiisti Analysis of Doppler Effet and Aberration based on V L T June Pulse of Light Y Y r r m θ y y A(,,) O O β α θ 9α O x X X x Fig. 4. General VLT s onfiguration for measuring Relatiisti Aberration Gien that: u u u u u u.. u u +. u u..osθ + In general, os θ os θ (6) u u os θ Gien that we hae found in a general way in (6) the same result obtained for a partiular ase in (3), independent of angles α and β, we hae shown that both equations (5) and (6), alid for any number of dimensions, expresses the law of aberration in its most general and onsistent form only dependent on the angle between the eloity of the moing obserer and that of the starlight. III. CONCLUSION In the present work, we hae obtained general expressions for the Doppler and Aberration effet different to those obtained by Einstein in 95 and gien for grant until now. Howeer, for those skeptis persons, only preision and auray in the measurements is needed for obtaining results that show firstly, the possible inorretness of Einstein s formulas () and () and similarly in a rigorous manner, the orretness of our formulas (6) and (6). REFERENCES [] A. Einstein On the Eletrodynamis of Moing Bodies. Setion 7, June 3,95. [] J A Frano-Rodríguez Time is not a Vetor: Corretions to the Artile Vetorial Relatiity ersus Speial or General Relatiity?. Volume 5, Issue, JVR, Marh. JVR 6 () 3-9 Journal of Vetorial Relatiity 3A 9