Red Shift and Blue Shift: A ealisti appoah Benhad Rothenstein Politehnia Uniesity of Timisoaa, Physis Dept., Timisoaa, Romania E-mail: benhad_othenstein@yahoo.om Coina Nafonita Politehnia Uniesity of Timisoaa, Communiations Dept., Timisoaa, Romania Abstat We pesent the geometi lous of the points in spae fom whee moing obsees detet a stationay soue as haing the same Dopple shifted fequeny in a fee of plane wae o ey high fequeny assumptions appoah. 1. Intodution A lagely used Dopple fomula elates the fequeny f of a stationay soue S loated at the oigin O of its est fame K(XOY) to the fequeny f ' attibuted to S by an obsee R at est in a K (X O Y ) efeene fame. The two efeene fames ae in the standad aangement. K moes with onstant eloity elatie to K, in the positie dietion of the oelapped axes OX and O X. The standad deiation of the Dopple fomula is pefomed in the est fame of the soue K(XOY) and inoles its standad synhonized loks loated at its diffeent points (Figue 1). When a lok C 0,0 t 0 ( ) loated at O eads, the soue S emits a light signal along a dietion whih makes an angle θ with the positie dietion of the OX axis. The light signal aies at the loation ( ) C R when it eads t and it is obious that of a lok ( ),θ t = t + (1) Diffeentiating both sides of Eq.(1) with espet to t ( ) and taking into aount that by definition d = = osθ () ( ) dt epesents the adial omponent of the instantaneous eloity of obsee R, we obtain dt = osθ ( ) dt (3) Eq.(3) holds fo eah alue of t een fo ( e t ) = 0. Eq.(3) establishes a elationship 0,0,θ. The eading of a ( ) between the hanges in the eadings of loks C ( ) and ( ) lok o-moing with R undegoes in the expeiment desibed aboe a hange ( ) elated to dt by the time dilation fomula 0 C R ( ) dt'
( ) ( ) dt ' dt = (4) with whih, Eq.(3) beomes osθ dt = (5) ( ) dt ' Eq.(5) beomes a Dopple fomula if we onside that dt epesents the ey small ( ) peiod at whih S emits suessie signals (wae ests) and that dt' epesents the ey small peiod at whih R eeies them. If we define a Dopple fato f ' D = (6) f ( ) ( ) ( ) ( ) 1 e whee f = dt and f ' = dt' epesent the ey small emission and eeption fequenies, espetiely, Eq.(5) beomes osθ D = (7) It is lea now that Eq.(7) holds exatly only in the ase of the ey high fequeny assumption and in the ase of a senaio whih inoles a stationay soue and an unifomly moing obsee. Molle[1] and Jakson[] obtain the same equation, stating with a senaio, whih inoles a plane wae and two inetial obsees in elatie motion with onstant eloity. In that ase, f and f ' epesent the fequenies of the eletomagneti osillations, whih take plae in the wae, as measued by the two obsees, espetiely. The plane wae assumption imposes the ondition that the two obsees ae loated at a ey lage distane fom the soue, whih geneates the plane wae. The onlusion is that Eq.(7) holds in the ase of the ey high fequeny and ey lage soue-obsee distanes assumptions, a fat whih is not always mentioned by those who use it in ode to test speial elatiity.. A ealisti appoah We onside the same senaio as aboe without any assumptions onening how small o how big ae the inoled fequenies and distanes. Let T be the finite peiod at whih S emits suessie signals o wae ests. R 1 (,θ 1 1 ) epesents an instantaneous position of R. He eeies thee a signal emitted at a time T when his lok eads T + 1. R (,θ ) epesents a position of R whee he eeies a seond signal emitted at t = 0 at a time T + 1 + TR. TR epesents the time inteal between the eeption of the fist and of the seond signal. It is obious that
1 = ( T + + TR ) (8) and that 1 = + TR TRosθ (9) Eliminating between Eqs.(8) and (9) we obtain 1 R 1 R (1 os θ ) T T T + + T T + = 0, (10) whee we hae intodued the notations θ = θ and =. Soling Eq.(10) fo we obtain TR A A B T R = (11) T taking into aount the solution with physial meaning ( T R = 0 fo T = 0 ). We hae intodued the notations f A = 1+ osθ (1) and f B = 1+ (13) with f = T. The lok o-moing with R measues between the eeption of two suessie signals a pope time inteal T R ' with whih Eq(11) leads to a haateisti Dopple fato D gien by f ' D = = (14) f A A B The geometi lous of the points fom whee obsees R detet the soue as haing the same fequeny f ' fo a gien alue of its pope fequeny f ( D = onstant) is gien in pola oodinates by D D = (15) osθ Df D We pesent in Figue 3, the geometi loi of the points fom whee the moing obsees eeie the same fequeny, fo 1 = 0.6 and fo a pope fequeny of the
8 soue f = 3 10 Hz. In Figue 3.a we onside the ase of inoming obsees (blue shift), fo D=1.3; 1.5 and 1.8. In Figue 3.b we onside the ase of outgoing obsees (ed shift) fo D=0.5; 0.75 and 1. Fo D=1 the moing obsees do not detet a Dopple shift of the eeied fequeny ( f = f ' ). As we see, in the fee of assumptions appoah the loi of the points, whih oespond to a ed shift, hae positie alues fo x (obsee R eedes the soue). Points haateized by a blue shift hae negatie alues fo x (obsee R appoahes the soue). In the ase when R moes with eloity the minimal alue of the ed shifted 1+ fequeny is wheeas the maximal alue of the blue shifted fequeny is. 1+ Fo 1 = 0.6 the oesponding alues ae 0.5 and espetiely. If the fequeny f of the soue ineases the oodinates x and y of a point on the geometi lous ae edued as x/f and y/f espetiely. 3. Conlusions The fee of assumptions appoah to the Dopple Effet eeals many of its inteesting featues, obsued when we stat with the taditional plane wae o ey high fequeny assumptions. Refeenes [1] C. Molle, The Theoy of Relatiity, Claendon Pess Oxfod 197, Ch..9 and Ch..11 [] John Daid Jakson, Eletodynamis, John Wiley & Sons, In. New Yok-London 196 Ch.11 Y R, C θ S, O, C 0 X Figue 1. Senaio fo deiing the Dopple fomula in the ase of the ey high fequeny assumption.
Y T R R 1 R 1 θ 1 θ O, S X Figue. Senaio fo deiing the fee of assumptions Dopple fomula. Figue 3.a. The geometi lous of the points ( x < 0 ) in the XOY plane, fom whee inoming obsees at est in K (X O Y ) attibute the same fequeny f ' to the soue S(0,0), 8 the pope fequeny of whih is f = 3 10 Hz.
Figue 3.b. The geometi lous of the points ( x > 0 ) in the XOY plane, fom whee outgoing obsees at est in K (X O Y ) attibute the same fequeny f ' to the soue S(0,0), the pope 8 fequeny of whih is f = 3 10 Hz.