Derivation of the Relativistic Doppler Effect from the Lorentz Force

Transcription

1 Apeiron, Vol. 1, No. 1, Janary 5 47 Deriation of the Relatiisti Doppler Effet from the Lorentz Fore Nizar Hamdan Department of Physis, Uniersity of Aleppo P.O. Box 183, Aleppo, Syria nhamdan59@hotmail.om I This paper demonstrates that allation and interpretation of the relatiisti Doppler effet is possible sing only the Lorentz fore and relatiity theory. This method eliminates the need for the Lorentz transformation and time dilation. It also demonstrates the proper link between the inariane of the light speed and the Doppler relations, λ λ. Keywords: Doppler Effet, speed of light; time dilation. Introdtion Einstein [1] presented the relatiisti Doppler effet (RDE) in terms of Lorentz transformations (LT) in order to inorporate relatiisti dynamis and time dilation. This led many physiists to attempt eqialent (RDE) theories withot (LT) and its kinematial effets [, 3]. Sine there is no propagating medim in (RDE), only the relatie eloity between the sore and obserer is onsidered. Therefore, 5 C. Roy Keys In.

2 Apeiron, Vol. 1, No. 1, Janary 5 48 the differene between the lassial Doppler effet and (RDE) is the introdtion of time dilation and the transerse Doppler effet (TDE) Assme two moing inertial frames S and S, a sore with freqeny in frame S and an obserer in frame S, Sore and obserer are reeding from eah other with the relatie eloity. We then hae two ases: If the obserer is reeding from the sore, the freqeny obsered is lassially, 1 (1a) Conersely, if the sore is reeding from the obserer, (1b) In either ase, the freqeny derease de to reessional motion is alled the 'lassial red shift'. If sore and obserer are approahing one another, is replaed with in Eqs. (1a) and (1b), 1 + (1) (1d) This is the 'lassial ble shift'. In SRT s formalism, this effet is represented by time dilation. 5 C. Roy Keys In.

3 Apeiron, Vol. 1, No. 1, Janary 5 49 From the obserer s point of iew, if the sore is reeding, its lok rns slower by. This redes the freqeny by Hene, Eq. (1b) beomes (a) This represents the relatiisti red shift freqeny. As in Eq. (1d) (approahing), we replae with in Eq. (a) to proide, (b) This represents the relatiisti ble shift. When the light wae is at an angle θ relatie to the ox axis, then Eqs. (a) and (b) beome (3) osθ If the motion is normal to the line oneting sore and obserer, then θ 9 and osθ, and we hae the red-shifted (TDE) relatie to the obserer; i.e., γ (4) 5 C. Roy Keys In.

4 Apeiron, Vol. 1, No. 1, Janary 5 5 Experiments [4,5] show that the obsered freqeny is reded by γ [Eq.(4)]. This is onsidered eidene of time dilation and proes the alidity of Eq.(4). Therefore, time dilation plays an important part in modern physis. Eq. (4) is onsidered a niqe featre of SRT and is related to the dilation of time only for the moing sore. Therefore it was assmed in some textbooks that TDE shold not or in lassial physis. Howeer, it was reognized that time dilation was essential to (RDE) as withot it, absolte motion old be deteted. These statements are inalid in some representations [6a,6b], and inorret if (TDE) is deried withot resorting to an ether or relatiisti modifiations. This is demonstrated in the following: II Energy, Mass, Momentm, Veloity and Eletromagneti Transformations. As determined in [7] and ontrary to what is often laimed in SRT, relatiity theory was deried from its fndamental postlates and the lassial laws, i.e., F d p dt, d ε F (5) dt This method is sed in the ase of a harged partile q moing with eloity in frame S, sbjet to an eletri field E and a magneti flx density B. We find that Eqs. (5) yield: dp q( ) dt E + B, d ε q E (6) dt The Cartesian omponents of Eqs. (6) in frame S are dpx q( Ex + VyBz VzBy) (7a) dt 5 C. Roy Keys In.

5 Apeiron, Vol. 1, No. 1, Janary 5 51 dp dt y dp dt z ( y z x x z) q E + V B V B (7b) ( z x y y x) q E + V B V B (7) d ε q( ExV + x EyV + y EV z z) (7d) dt Starting from Eqs. (7a,7d), we mltiply Eq. (7d) with and then sbtrat the reslt from Eq. (7a). Following the approah sed in [8], we obtain P x γ Px ε (8a) and E Vy V y Vx γ Vz V z Vx γ γ +, x Ex, By By E z B z γ Bz E y, V dt γ dt 1 x (9b) (9) 5 C. Roy Keys In.

6 Apeiron, Vol. 1, No. 1, Janary 5 5 We start one again with Eqs. (7a,7d), bt now mltiply Eq. (7a) with and add it to (7d). Following the approah sed in [8], we get Vx ε γ ( ε P x ) (8b), V x (9a) Vx and E γ E B E γ E + B y ( y z), z ( z y) The salar fator γ an be fixed by applying relatiity theory to Eq. (9b) to get 1 γ 1 (a) or γ 1 (b) (1) Now starting from Eq. (7b) and sing Eq. (1a). Following the approah in [8], we hae P y Py (8) B x Bx In a similar way, if we start with Eq. (7), we hae P z Pz (8d) Now the onentional definition of momentm in frames S and S, i.e., P mand P m, will lead to the tre relatiisti expression for mass and energy. As we know, Eqs. (9) are eqialent to 5 C. Roy Keys In.

7 Apeiron, Vol. 1, No. 1, Janary 5 53 V 1 x 1 V V V x Vx V V (11a), (11b) Mltiplying Eqs. (11) with m, and omparing both Eqs. (11), with (8a) and (8b), we dede and m m m V m V, ε m (1), ε m It is simple to show that Eqs. (8) and ( 1 ) lead to 4 ε P ε P m (13) Based on the aboe formlation, we derie the Doppler relations for light waed. No ether or relatiisti assmptions are reqired [9a,9b]. 5 C. Roy Keys In.

8 III Apeiron, Vol. 1, No. 1, Janary 5 54 Doppler Effet and Aberration The onstany of light speed, i.e., that V in a speifi referene frame, was disoered by Hertz. Using this in the relation V λν, we hae λ (14) Applying relatiity priniple to Eq. (14) means that Eq. (14) is alid for all obserers, i.e., λ (a), λ (b) (15) SRT s assmption of the inariane of light speed is based ompletely on the onept of time dilation. If time is not delayed as presented by SRT, then light speed inariane annot be maintained. Aording to Eqs. (15), one may onlde that the light speed postlate is an ineitable onseqene of the relatiity priniple. Now assme that partile q in frame S, as an ion (sore), emits a light wae (photon) that moes at an angle θ with the positie x axis. Light is reeied by the obserer in frame S at angle θ relatie to the x axis. The momentm of the photon in frame S has omponents P Posθ, P Psinθ, P (16) x y For a photon with rest mass m, we hae from Eq. (13) ε P or ε p (17) ε If we insert P x osθ [Eqs. (16) and (17)] in formla (8b) we find ε γε osθ z 5 C. Roy Keys In.

9 Apeiron, Vol. 1, No. 1, Janary 5 55 For a photon we an let ε h, ε h γ osθ (18a) Bt the onnetion between the two freqenies in frames S and S is also gien by γ osθ (18b) If we onsider frame S to be o-moing with the sore and reeding from /approahing the obserer, (18a) beomes 1) for θ θ.. ) for θ θ 18.. γ 1 (19a) γ 1 + (19b) Eqs. (19a,19b) do not hae relatiisti analoges, bt hae lassial analoges in Eqs. (1a,1). Aording to the relatiity priniple, we an also onsider the frame S to be o-moing with the obserer and reeding/approahing the sore. Then Eq. (18b) old be written, 5 C. Roy Keys In.

10 Apeiron, Vol. 1, No. 1, Janary 5 56 (18) osθ Hene we hae Eqs. (19,19d) whih are, similar to Eqs. (19a,19b): (19) (19d) Eqs. (19,19d) are idential to Eqs. (a,b) in SRT, and the lassial analoges are (1b,1d). If the eloity of the obserer/ sore is perpendilar to the line of sight, then we obtain from Eq. (18a) γ (a) Formla (a) has been onfirmed in Mossbaer experiments [1]. From Eq. (18), we get γ (b) 5 C. Roy Keys In.

11 Apeiron, Vol. 1, No. 1, Janary 5 57 As is known, the relatiisti formalism of (RDE) applied only to a moing sore s. a stationary obserer. The ase of rest sore and moing obserer for (RDE) is absent. For this reason the two expressions, (19a) and (19b) do not hae relatiisti analoges. Aording to the priniple of relatiity, o-moing sore and moing obserer is oneptally the same as a moing sore and omoing obserer i.e., the hange of referene frame (obserer or sore onsidered at rest) does not hange the physis. Therefore, we hae to onsider the ase where the sore is stationary and the obserer is moing. As indiated aboe, if we onsider that the frame S to be omoing with the sore and reeding from/approahing the obserer, we hae Eqs. (19a) and (19b). If we onsider frame S to be omoing with the obserer and reeding/approahing the sore, we are led to the same reslt as Eqs. (19a) and (19b), i.e., Eqs.(19) and (19d). Ths, it is meaningless in the longitdinal Doppler effet to distingish between the ases "stationary sore and moing obserer" and "stationary obserer and moing sore". The lassial Doppler effet for light gies two ales [ 1 +, / 1 ] for the approahing obserer and sore. In (RDE), the obsered Doppler shift is the same for the two formlas bease the (TDE) exatly offsets the differenes. This means, Similarly,, i.e.,. 5 C. Roy Keys In.

12 Apeiron, Vol. 1, No. 1, Janary 5 58, i.e.,. From these relations we hae the identity In the other ase we hae also an identity De to the identity, the (RDE) formla for a moing obserer an be also written in the form sed for a moing sore. Finally, we an desribe the aberration of light by sing both Eqs. (9a, 9b) and ( 16). The eloity omponents for the emitted photon in frame S are px Vx osθ (1a) ε py Vy sinθ (1b) ε Aording to Eqs. (15), the eloity omponents in frame S are 5 C. Roy Keys In.

13 Apeiron, Vol. 1, No. 1, Janary 5 59 p x V x osθ ε p y V y sinθ ε From Eq. (9a) and (9b) we obtain osθ osθ osθ sinθ sinθ γ osθ 5 C. Roy Keys In. (a) (b) (3a) (3b) Angle θ is related to the angle θ ia the well known expression, Eq. (3a). Howeer, Eq. (3a) is sally qoted as eqation (3b), both of whih are deried from the (LT) in (SRT s) formalism. Eq. (3a) desribes the aberration of light. (SRT) explains the Doppler shift for light as being ased by the motion of the light sore relatie to the obserer: the ble/red shift is ased by a hange in spae/ time de to that motion. So the Doppler priniple in (SRT) is intrinsially kinemati, desribed throgh Maxwell s theory and (LT), bt the laking the intrinsi strtre of light. (SRT) left ot the important fat that the freqeny shift throgh motion is ased diretly by the ariation in energy ( ε ) between light and the obserer. Therefore we deried the Doppler priniple starting with Eq. (8b), whih means that energy, and onseqently the freqeny is relatie. The relatie eloity is a etor, meaning that

14 Apeiron, Vol. 1, No. 1, Janary 5 6 reslts are dependent on diretion [Eqs. (19)], and also dependent on the speed, whih diretion is disregarded. Therefore in normal motion, the radial omponent is zero, r, and sine t + r t We also obtain a hange in freqeny as in Eqation (). The modifiations of relatiity are onsidered symmetrial between sore and obserer. Therefore we hae two formlas of (TDE), Eqs. () and not one [Eq.(4)] related to time dilation only. Conlsion The reslt of Mihelson-Morley s experiment raised two qestions: 1 - Does the ether exist? - If not, then to what is the speed of light relatie? (SRT) proided a dbios answer: Regardless of the natre of light and the existene of ether; all physial laws of natre shold onform to LT. Einstein s approah may hae eliminated dobts abot the inariane of light speed, bt the following qestion remains: If the spae time dependene of photons is: dx dx (4) dt dt Then the modifiation of spae time with relatie motion is reqired to maintain a onstant light speed as Eq. (4) states. In or proposal, (LT) and preferred referene frames are not reqired. The intrinsi properties of light (, k) may be the possible ase of the Doppler shift whih satisfies the relation, λ λ This means that freqeny aries to maintain a onstant speed of. 5 C. Roy Keys In.

15 Referenes Apeiron, Vol. 1, No. 1, Janary 5 61 [1] A. Einstein, On the Eletrodynamis of Moing Bodies, Ann. Phys. 17 (195) 891. [] M. Wolff, Relatiisti Mass Inrease and Doppler shift withot SRT, Galilean Eletrodynamis 9 (1998) [3] L. B. Boldyrea, N. B. Sotina, The Possibility of Deeloping a Theory of light withot Speial Relatiity, Galilean Eletrodynamis 13 () [4] H. E. Ies and G. R. Stilwell, An Experimental stdy of the rate of a moing atomi lok, J. Opt. So. Am. 8 (1938) [5] H. E. Ies and G. R. Stilwell, An Experimental stdy of the rate of a moing atomi lok II, J. Opt. So. Am. 31 (1941) [6a] E. Baird, Transerse Red Shift Effets withot Speial Relatiity, http : //xxx. lanl. go / abs / physis/ 174. [6b] P. Parshin, Theoretial and Exerimental Inestigation of the Relatiisti Doppler effet, Galilean Eletrodynamis 1 (1) 19-. [7] N. Hamdan, Newton s Seond Law is a Relatiisti law withot Einstein s Relatiity, to appear in Galilean Eletrodynamis. [8] N. Hamdan, Abandoning the Ideas of Length Contration and time dilation, Galilean Eletrodynamis 14 (3) [9a] Thomas F Jordan, Photons and Doppler shifts in Einstein s Deriation of Mass Energy, Am. J. phys. 5 (198) [9b] W. Engelhardt, Relatiisti Doppler Effet and the Priniple of Relatiity, Apeiron 1 No.4 (3) [1] H.J. Hay, Alt, Measrement of the Red shift in an Aelerated System sing the Mossbaer Effet in F 57, Phys. Re. Letters 4 (196) C. Roy Keys In.