Slowing time by stretching the waves in special relativity

Transcription

1 Slowing time by strething the waes in speial relatiity Denis Mihel To ite this ersion: Denis Mihel. Slowing time by strething the waes in speial relatiity: The elusie transerse Doppler effet. 04. <hal > HAL Id: hal Submitted on 7 Jul 07 HAL is a multi-disiplinary open aess arhie for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or priate researh enters. L arhie ouerte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de nieau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou priés.

2 Slowing time by strething the waes in speial relatiity Denis Mihel Uniersite de Rennes-IRSET. Campus de Villejean Rennes Frane. E.mail: Abstrat The orretion of the lassial Doppler formula by the time dilation fator gies the urrently admitted relatiisti Doppler equation. When the soure path is not ollinear to that of the reeier, this orretion gies a transerse wae dilation whose possible detetion was onsidered by Einstein as an ideal mean to proe the time dilation of speial relatiity. It is suggested here that this elusie effet whose measurement remains ontroersial, ould in fat be absent from relatiisti as well as lassial ontexts, but that this absene paradoxially proes the time-dilation of speial relatiity. An intuitie reasoning using parallel Doppler effets is first proposed to oneie why the absene of a wae arrier medium imposes time dilation for eletomagneti waes. A rational reonstrution of the non-ollinear Doppler effet is then presented using only reiproal quantities suh as the soure-reeier distane. This rigorous analysis reeals a irtual equation whose transerse ontration effet is exatly anelled by the relatiisti time dilation. The andidate Doppler equation emerging from this treatment is remarkably elegant, geometrially symmetrial and entered on the losest point between the soure and the reeier. Keywords: Time dilation; Relatiisti Doppler equation; Transerse ontration. Introdution The wae dilation of speial relatiity is a general phenomenon whih gies, when surimposed to the lassial Doppler effet, the relatiisti Doppler formula. A Doppler effet is aused by the eloity of a wae soure relatiely to an obserer. The lassial doppler effet shortens the apparent waelength of an objet approahing at speed suh that = T T where T is the period, giing / =. Conersely it strethes the apparent waelength of a reeding objet suh that / = +. In the general ase, when the eloity etor is not stritly ollinear with the line of sight, these equations should be modified by replaing by a smaller alue. When using radial eloities whih are the orthogonal projetion of the eloity etor on the soure-obserer line ( os θ), there is no transerse effet beause when the soure is at the losest point from the obserer (θ = π/), the radial speed is zero. But this intuitie reasoning using radial eloities makes uses of angles whih are not equialent between the soure and the reeier. A new equation is built using only parameters symmetrial between the soure and the reeier and where the non-reiproal osine eloities will be replaed by the reiproal rate of soure-reeier distane hange. But let us start with thought experiments enlighting the importane of a wae-arrier medium in Doppler effets. Medium-dependent and independent wae propagation The lassial Doppler effet holds for waes arried by the medium, like sound, whereas the relatiisti Doppler effet applies to eletromagneti waes propagating in auum.. Longitudinal Doppler effet A soure and a reeier an moe relatie to one another and in addition, an moe relatie to an hypothetial stati medium supporting wae propagation at speed. The soure and the reeier reede form eah other at speed. At time t E, when spaed from the reeier by D E, the soure emits a wae pulse towards the reeier. Different results are expeted depending on whether this is the soure or the reeier whih moes relatiely to the bakground medium (Fig.). The soure is onsidered immobile relatie to the bakground medium In this ase, the wae is expeted to reah the reeier after rossing a distane D R (middle line of Fig.). The duration of the wae trael is t R t E = D R / (a) At t R, the new spaing between the soure and the reeier has beome D R = D E + (t R t E ) (b)

3 Replaing the duration in Eq.(b) by the alue gien by Eq.(a), yields a distane ratio orresponding to a lassial Doppler effet D R = D E () assign the relatie moement to either the soure or the obserer, it seems natural to take the geometri mean of the two extreme situations of Fig. (Eqs ()/(b) and Eqs (d)/()). This gies the relatiisti Doppler effet. For the reession: In ase of ollinear approah, the same reasoning gies D R D E = + (d) DR D E = + (3a) and for the approah DR D E = + (3b) Figure. A wae pulse is emitted by a soure (S) when spaed from a reeier (R) by D E. Just like a ball thrown between two players, the wae traels through a stati medium relatiely to whih either S (middle line) or R (bottom line), is onsidered immobile. The reeier is onsidered immobile relatie to the bakground medium The wae reahes the reeier at time t R after rossing a distane D E (bottom line of Fig.). Hene, the duration of the wae trael is t R t E = D E / (a) Replaing the duration in Eq.(b) by its alue gien by Eq.(a), yields D R D E = + (b) Distane inreases in the same ratio that the lassial Doppler effet. In ase of ollinear approah, the same reasoning gies D R = () D E These Doppler effets are suitable for the sound that is arried by physial supports, but not for light traelling in auum. Sine it is impossible in auum to Suh a orrespondene between waelength distortion and soure-reeier distane hanges during light trael has also been formulated by Lemaître in the ontext of the osmologial redshift [].. Parallel Doppler effet The lassial transerse Doppler effet is inexisting for a sound wae emitted by a moing soure loated at the losest point from an immobile reeier. But if the soure and the reeier moe in parallel and in the same diretion, the waelength appears shortened as represented in Fig., with a ontration between S and R preisely orresponding to the inerse of the relatiisti dilation fator. The soure and reeier do not moe relatie to eah other, but both moe relatie to the medium whih works as an apparent wind shifting bak the wae rests. The seond step of the reasoning is to swith from sound waes to eletromagneti waes propagating in auum. One the medium is remoed, it is no longer possible to assert that the soure and the reeier hae an absolute moement. Relatiistially speaking, they belong to same referene frame, so that the preious apparent wind is inexisting. It is therefore neessary to anel the wae ontration between S and R.

4 the hypothenuse, L 0 that of the shortening side and D the onstant side, and H 0 = D + L 0 (4a) (H 0 ht) = D + (L 0 t) (4b) whose substration allows to eliminate D and yields ht = H 0 H0 + (t) L 0 t (4) Figure. Classial Doppler effet predited when the soure (S) and the reeier (R) run in parallel at the same speed. There is no relatie moement of S and R, but both moe relatiely to the medium. The wae between S and R appears stably ontrated by. Now let us remoe the medium and imagine that the wae self propagates in auum. The soure and reeier then learly belong to the same inertial frame and as a onsequene, the preious wae distortion orthogonal to eloity etors and now orresponding to the light path should be anelled by multiplation by /. 3 A new reiproal Doppler treatment The Doppler effet will be realulated using as a speed the rate of distane hange between the soure and the reeier, that is ompletely summetrial. Classial Doppler effets mix longitudinal and transerse effets and range between the two asymptotes and +. The intermediate alues are urrently defined using the angle θ between the motion line and the reeier. In fat, sine θ aries with time and generates ertain problems suh as aberration effets, it seems more rational to skip it and to alulate the Doppler effet diretly as a funtion of ariation of distane between a soure and an obserer that are unique and reiproal. A Doppler-generating speed will be defined using the Pythagorean theorem, whih remains alid in the Eulidean relatiisti spae. Let us define irtual rates of simultaneous shortening of the hypothenuse (rate h) and of one side (rate ), while maintaining the other side onstant and the triangle retangle. h and are related to eah other with a ouple of simple equations. If denoting H 0 the starting lengh of Figure 3. A soure moing at onstant speed starts from a distane H 0 from the immobile obserer. The shortest distane between the soure and the obserer is D. Now let us apply this general result, whih does not orrespond to a speifi physial situation, to the partiular ase represented in Fig.3, of a the soure moing non-ollinearly relatie to an obserer and reahes at speed the losest point from this obserer. The triangle of Fig.3 eoles suh that the hypotenuse redues from H 0 to D while the soure path redues from L 0 to 0. H 0 and L 0 are preisely adjusted suh that the wae front reahes the reeier when the soure reahes the losest point, following a time delay t. In this ase, H 0 and L 0 an be replaed by t and t respetiely and Eq.(4) beomes h = ( t ) ( t) + (t) t t /t (5) When inserted in the lassial Doppler formula, this speed gies the results presented in Fig.4. The signal reeied when the obserer is at right angle to the motion line was emitted at h = /. This alue, giing a Doppler effet of / = is not defined and alulated as a series expansion limit. More interestingly, The signal is emitted at right angle to the obserer at h =, giing a Doppler effet of / =. The speed h may reflet the transerse ontration fator of Voigt s transformations, but the Doppler ef- fet generated by this speed will be alled irtual Doppler effet (VD), beause on the one hand, it is not relatiisti sine it does not take into aount the relatiisti time dilation, but on the other hand it is not more lassial as it is alulated without taking into aount a referene 3

5 medium. When applied to the lassial Doppler effet with a referene medium (CD), the time dilation fator gies the traditional relatiisti Doppler formula (TRD). Appliation of the same Lorentz fator to the new irtual Doppler effet (VD), turns to exatly ompensates its transerse ontration effet and gies an interesting andidate relatiisti Doppler effet (CRD). Figure 4. Eolution of the Doppler-generating speed h gien by Eq.(5). The time unit t is the trael time of the signal reahing the reeier when the soure is the losest to it. The origin of time t = 0 is entered at this losest position. The Doppler effet generated by light emitted at this position (at right angle to the reeier) is shifted blue by (/). 4 Comparison of the different Doppler approahes 4. Normalization of the different Doppler formulas with respet to distanes To ompare the different Doppler formulas, they should be omparable for any relatie onfiguration of the soure and the obserer. The omparison with the relatiisti equation desribed with angles [] is deliate beause seeral equations are possible depending on the angle used: either the original angle between the eloity etor and the soure-obserer onnetion line (θ) or the reeption angle (θ ). + = os θ = os θ, (6a) the two angles of this identity are related to eah other through the so-alled aberration formula [] os θ = os θ os θ (6b) Table : The aberration effet in speial relatiity is related to the time points of wae emission. t os θ os θ / t / 4

6 Approahes using angles are onfusing beause if one assumes that the transerse effet is obtained when the osinus is 0, the first formula of Eq.(6a) predits a wae dilation, whereas the seond formula gies the inerse wae ontration, beause θ and θ annot be simultaneously equal to π/ [3]. This subtlety is a matter of delay of wae trael t (Table.). The nonollinear Doppler formulas ontain two ariables: the speed and an angle. This angle aries along the wae path and an be expressed as a funtion of time, suh that θ(t) = tan (D/t). Hene, on the one hand os θ(t) = / + (D/t), and on the other hand the distane D an itself be defined as a funtion of t (D = t ), thereby allowing to make the formulas funtions of distanes only. There are other ambiguities in the literature about the sign of the eloity ( and +) in Doppler equations. Table : Doppler effets generated by a wae emitted at the normalized distane d from the losest point and alulated using the different formulas. CD= Classial doppler effet; TRD= traditional relatiisti Doppler effet, VD= irtual doppler formula based on the rate h; CRD= andidate relatiisti formula. The new formulas are built using the lassial Doppler framework /, in whih is replaed by the speed h gien by Eq.(5). The unit of distane is t where t is the time-of-flight of the wae reahing the obserer when loated at the losest point from the soure. The irled indiate the inersion points where the Doppler effets anel. The distane σ orresponds to the inersion point for the TRD, but has no partiular meaning for the other equations. Doppler. d = d = d = σ d = 0 d = + d = + CD TRD VD CRD + d Φ CD Φ + d + d VD with d = t t, Φ = ( d ) and σ = + To eliminate all these soures of onfusion, the equations are omposed here without referring to angles and using speeds always positie, irrespetie of the relatie loation of the obserer, by transferring the sign to the time t ranging from and +. To synhronize the formulas at the time points of wae emission, in the new formula, t should be replaed by t+ t. Finally, a dimensionless normalized distane is defined for all soure paths relatiely to the losest point ( d = t/ t = t/ t). A little algebra satisfying all these requirements gies the equations ompiled in Table.. For d = where the irtual funtion is not defined, the Doppler effet takes the limit alue. These different Doppler equations desribe general ombinations of longitudinal and transerse Doppler effets for any relatie position of the soure and the obserer. As these normalized equations an now be ompared, their profiles as funtions of d are superposed for isualization in Fig.5. 5

7 Figure 5. Comparatie profiles of Doppler effets predited for / = /3, by the different formulas. The swith between the ontration and the dilation ours at the losest point for the CD and CRD, just before the losest point for the TRD (between - and 0, see the text) and at d = + for the VD. Table 3: Arithmeti and geometri means of Doppler effets expressed using either waelengths or frequenies. d and Φ are defined in Table.. Note that for eah type of aeraging of CRD, the same result is obtained for waelengths and frequenies. Mean s ν CD TRD VD CRD Arithmeti Φ d Φ d d ( d ) ν ( Φ ) ) ( + d Φ ) ( + d Φ d Φ d Φ d ( d ) Geometri Φ ( ) ) ( + d + Φ d ν Φ ( ) ) ( + d Φ + d 6

8 5 Properties of the new Doppler equations 5. Comparatie symmetry The mean alues of the Doppler effets generated at symmetrial distanes from the losest point, depend on the modes of aeraging, whih are, when expressed using waelengths, the arithmeti mean: and the geometri mean: ( mo ( d) ( mo ( d) + mo (+d) (+d) ) ). The appropriate tool is logially the geometri mean, beause it is expeted to hold both for periods and frequenies suh that T ( d), T (+d) = / ν( d), ν (+d) and satisfies the rule of olor refletane fusion. The new formulas display a perfet geometri symmetry, in suh a way that the geometri mean of the Doppler effets before and after the midpoint are independent of time and always for the VD and for the CRD (Table 3). 5. Swithing points The four different Doppler equations ompared in Fig.5, hae different swithing points between ontrated and dilated waes: for the CD, at d = 0, for the VD at d = (or t = + t), for the TRD, the traditional relatiisti formula yields the most ompliated result, lose to the transerse line d = ( ) (7) for the CRD, at the lostest point d = 0. We get rid of the weird result of the TRD. 5.3 Comparatie angle-dependene The time-dependent profiles of the different formulas hae been ompared in Fig.5. It is also interesting to ompare their shape using an angular representation. The angle θ of Fig.6 will be used while keeping the speed always positie, suh that The TRD is ot(π θ) = t t + ot θ = ot θ + ot θ (8a) multiplying by sin θ allows to reoer the traditional, but no longer modulo-π, formula and the CRD beomes = multiplying by sin θ, = = os θ + ot θ ot θ ot θ sin θ os θ os θ (8b) (9a) (9b) As shown in Fig.6, the profiles drawn to Eqs.(8a) and (9a) strikingly differ by their slopes, those of the new formula being loser to the tangent funtion. 7

9 Figure 6. Comparatie profiles of Doppler effets predited for / = /3, by the lassial (dotted lines), and relatiisti (plain lines) Doppler equations, either traditional or modified here, as funtions of the emission angle θ shown in the inset. 6 Correspondene between time and wae distortion effets. Seeral experiments hae been onduted to test the theory of speial relatiity through the relatiisti Doppler formula of Einstein [4], following a suggestion of Einstein himself [5]. But the present analysis suggests that the time dilation is in fat responsible for the absene of transerse Doppler effet. Time and wae distortions are two faets of the same priniple, illustrated in Appendix A and eidened by a remarkable obseration of supernoae, unfortunately too reent for Einstein to be aware of. The time window of brightness is relatiely onstant for omparable supernoae, but astronomers made an expeted but neertheless striking finding: the apparent time window of brightness depends on the distane of the supernoa, in exatly the same proportion that their redshift. For instane, a distant supernoa with a redshift of app / =.5, has preisely a.5-fold longer duration of brightness [6]. Hene, time dilation orresponds exatly to waelength inrease, or equialently frequeny derease, whereas apparent time ontration orresponds to waelength shortening and frequeny inrease. Reiproally, spaing waelength rests automatially ause apparent time slowing, in the same manner that projeting at frames per seond a moie sheduled for 4 frames per seond, shows abnormally slow senes. 7 Test of the new Doppler equations 7. Longitudinal effets The elebrated experiment of Ies and Stilwell [7] and its desendants [8] foused on the longitudinal Doppler effet. Ies and Stilwell reoered the time dilation fator by measuring the arithmeti means of the shifted waelengths in front and behind moing atoms [7]. This result is howeer not disrimining sine it is also obtained with the andidate formula. 7. Transerse effets The existene of the transerse Doppler effet has been supported in a single study [9]. But more reently, using mirowaes whih exhibit a high spatial purity and a preise polarization plane, a sensitie apparatus failed to detet any transerse effet [0], whih led the author to question the priniple of dilation of speial relatiity. It is suggested here that this seemingly negatie result ould be indiretly onsidered as a erifiation of time dilation. 7.3 Releane of the CRD equation The test of the new andidate equation CRD is problemati onsidering the notorious diffiulty of transerse tests and beause it satisfies the Ies-Stilwell type longitudinal tests as well as the preious equation. In fat, its best asset for now is its elegane, that is often a sign of auray in siene, and its fundamental symmetry illustrated by the geometri means of the non-ollinear paths. 8

10 Appendies For eletromagneti waes, the so-alled time-dilation effet should be taken into aount to yield the relatiisti Doppler effet. This is not a Doppler effet stritly speaking, but another type of a wae strething naturally obtained when the soure and the reeier belong to different inertial frames. It has been learly and entirely dedued from the inariane of light eloity by Einstein in his seminal paper of 905 on speial relatiity []. This dilation an be een more intuitiely oneied using the light lok of Einstein desribed below. A The light lok of Einstein The beat of the light lok of Einstein is the rebound of a photon between faing mirrors. This lok is plaed ertially in a wagon rolling at onstant speed. For an external obserer (with an exellent iew!), the light path appears oblique when the train moes, whereas for an obserer loated inside the train, it appears ertial (Fig.A.). The referene time interal t orresponds to the frame omoing with the light lok. lok appears immobile, and that of an ouside obserer for whom there is an additional translation. Hene, giing s = ( t) = ( t mo ) x mo ( and finally t t mo (A.a) ) ( ) x = mo (A.b) t mo t t mo = (A.) In spaetime diagrams like the triangular sheme of Fig.A., the proper time of the lok is obtained when the distane to be rossed by light appears minimal. In Fig.A.., this is the ertial path ( t), whereas for the external obserer moing at speed relatiely to the lok this path appears strethed by t mo / t = / (/). The light lok of Einstein is illuminating in that there is obiously no relatie moement betwen the faing mirrors. In this respet, the waelengh dilation pereied by the external obserer is not a Doppler effet but just an interframe perspetie effet. B Time dilation by wae strething Figure A. The famous triangular time-spae diagram of the moing Einstein s lok. The horizontal sale is artifiially strethed relatiely to the ertial one for better isualization. With respet to the angle θ of Fig.A., elementary trigonometry says and sin θ = t t mo = t t mo os θ = tmo t mo = (A.a) (A.b) whih immediately gies, using the relationship sin θ + os θ =, t mo = (A.) t Another way to obtain this result is to use the straightforward geometrial tool of speial relatiity: the Minkowski spaetime. The oblique (hypotenuse) and ertial paths of light start from and arrie to the same points. This ommon spaetime interal should reonile the point of iew of an obserer in the train, for whom the In the light lok experiment, the distintion between moing and immobile frames is irreleant as they an be permuted. The train and the station platform are two equialent systems of referene and the situation is simply inerted if the lok is put on the platform and if the obserer inside the train onsiders that it is the platform that moes relatie to the train in the opposite diretion. A perspetie effet is naturally reiproal. The best known perspetie effet is the apparent ontration of the size of a person standing far away from us, ompared to a person standing lose to us. We are ery austomed to this familiar effet and easily understand that this person should hae exatly the inerse pereption. Distane is a symmetrial notion, as is uniform motion. The apparent size redution effet is so well integrated in our mind that it is unonsiously orreted. Moreoer, it is used inersely to estimate the distane. The same operation an be applied to time dilation to dedue the relatie frame eloity from the degree of waelength strething. Eq.(A.a) an be modified as follows: On the one hand, = /T and on the other hand, the time interals an be replaed by a gien number (n) of periods t = n/. The inariane of this number from any iewpoint allows to rewrite Eq.(A.a) as x mo t mo = = (B.a) 9

11 whih an be transformed into = (B.b) Eq.(B.b) is typially the equation of a Doppler effet, showing that time flows follows wae frequenies. If the suessie images of a film are two-fold spaed, the film would naturally run times slower. Referenes [] LEMAÎTRE G., Un Uniers homogène de masse onstante et de rayon roissant rendant ompte de la itesse radiale des nbuleuses extra-galatiques. Annal. So. Si. Bruxelles, A47 (97) [] EINSTEIN A., Zur Elektrodynamik bewegter Körper (On the eletrodynamis of moing bodies) Annal. Phys. 7 (905) [3] RESNICK R., Introdution to speial relatiity. Wiley [4] MANDELBERG H. I. and WITTEN, L., Experimental erifiation of the relatiisti Doppler effet. J. Opt. So. Am. 5 (96) [5] EINSTEIN A., Über die Möglihkeit einer neuen Prüfung des Relatiitätsprinzips (On the possibility of a new test of the relatiity priniple) Annal. Phys. 38 (907) [6] PERLMUTTER S. Supernoae, dark energy, and the aelerating unierse: The status of the osmologial parameters. Proeedings of the XIX International Symposium on Lepton and Photon Interations at High Energies. Stanford, California, 999. [7] IVES H. E. and STILWELL G. R., An experimental study of the rate of a moing atomi lok. J. Opt. So. Am. 8 (938) 5-6. [8] BOTERMAN B., BING D., GEPPERT C., GWIN- NER G., HÄNSCH T. W., HUBER G., KARPUK S., KRIEGER A., KÜHL T., NÖRTERSHÄUSER W., NOVOTNY C., REINHARDT S., SÁNCHEZ R., SCHWALM D., STÖHLKER T., WOLF A. and SAATHOFF G., Test of time dilation using stored Li+ ions as loks at relatiisti speed. Phys. Re. Lett. 3 (04) [9] HASSELKAMP D., MONDRY, E. and SCHAR- MANN, A., Diret obseration of the transersal Doppler-shift. Z. Phys. A 89 (979) [0] THIM, H. W., Absene of the relatiisti transerse Doppler shift at mirowae frequenies. IEEE Trans. Instr. Measur. 5 (003)