Illustrating the spacetime coordinates of the events associated with the apparent and the actual position of a light source


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1 Illustting the spetime oointes of the events ssoite with the ppent n the tul position of light soue Benh Rothenstein ), Stefn Popesu ) n Geoge J. Spi 3) ) Politehni Univesity of Timiso, Physis Deptment, Timiso, Romni ) Siemens AG, Elngen, Gemny 3) BSEE Illinois Institute of Tehnology, USA Abstt. We pesent spe time igm tht isplys in tue vlues the spe time oointes of events ssoite with the ppent n tul positions of point like soue moving with onstnt veloity. We use it in oe to onstut the tul shpe of moving luminous pofile o to etemine the eltionship between ppent n tul istnes. We show tht the simple ft tht light popgtes t finite spee hs impotnt onsequenes like length onttion o length iltion the effets being sensitive ginst the ppohing n the eeing hte of the soue eltive to n obseve lote t the oigin of the inetil efeene fme.. Intoution Deissle pesents some physil effets ssoite with the ft tht light popgtes with finite spee. The effets e etete fom the est fme K(XOY) of n obseve R (,) equippe with lok T (,) lote t the oigin O. At eh point of the spe efine by the es of the K efeene fme we fin n obseve R(,y) with his lok T(,y). All the loks isply the sme unning time s esult of lok synhoniztion poeue popose by Einstein. y S S O Figue. Appent position S n tul position S of soue of light tht moves with onstnt veloity = pllel to the OX is t istne pt of it. The two positions e efine by the spe oointes S ( = os, y = sin ) n S ( = os, y = sin ) espetively, using pol n Ctesin oointes s well. In Figue S (, y ) = S ( os, sin ) epesents position of the point like soue. The soue moves with onstnt veloity pllel to n in the
2 positive ietion of the OX is t istne pt of it. A lok T, y ) ( lote t this point es t = when the soue S emits light signl tht will be eeive by R (,) when his lok T (, ) es zeo time. S epesents the ppent position of soue S. On the sme figue S (, y) = S( os, sin ) epesents the tul position of the soue, lote t this point when the obseve R (,) eeives the light signl tht ws peviously emitte fom the ppent position S. At the sme moment the lok T (, y) = T ( os, sin ) lote t the tul position es zeo time too. When R (,) eeives seon light signl emitte fom the tul position S his lok T (, ) will e: t = () The events involve in the thought epeiment esibe bove e: E (, y, ) = E ( os", sin", ) ssoite with the ppent position S, E (, y,) = E( os, sin,) ssoite with the tul position S, E (,,, ) ssoite with the eeption t O of the fist light signl emitte fom the ppent position S n (,, E, ) ssoite with the eeption t O of the seon light signl emitte fom the tul position S. The istne between S n S is, the istne tvelle by the soue between its ppent n tul positions. Fom the point of view of telemety,3 R epesents type of obseve who ollets infomtion bout the spetime oointes of istnt events fom light signls tht ive t his lotion. The pupose of ou ppe is to pesent spetime igm tht isplys in tue vlues the spetime oointes of the events efine bove. Whees Deissle pesents the oointes of the ppent position s funtion of those of the tul one, we pesent hee the oointes of the tul position s funtion of the ppent position. Ou esults e simple n moe tnspent.
3 . The spetime igm isplys in tue vlues the spetime oointes of events ssoite with the ppent n the tul positions of point like soue. Simple geomety pplie to Figue les to: = + = (os" + ) () y = sin == sin = (3) Fom () n (3) we obtin the following epession fo the tul (imensionless) position / of the soue s funtion of the pol ngle tht efines the ppent position: os + " = (4) sin Figue is plot of s funtion of fo iffeent vlues of. 5 5 β =.3 β =.6 β = [DEG] Figue. A plot of s funtion of the pol ngle fo thee iffeent vlues of the eue veloity = (.3;.6;.9). Fo < < / the soue of light is eeing whees fo / < " < the soue is ppohing the OY is. We obtin tht the lengths of the position vetos tht efine the ppent n the tul positions e elte by: 3
4 = + os" + = + os" + sin". (5) When S ehes the n tul position lok T es: t = = + os" +. (6) The equtions eive bove enble us to onstut spetime igm tht isplys in tue vlues t wellefine sles the oointes of the events ssoite with the ppent n tul positions. We pesent this igm in Figue 3. Its es oinie with the es of the K(XOY) efeene fme. The ile C of ius hving its ente t the oigin O is the geometi lous of events E n the ile C efine by (4) is the geometi lous of events E. The invine of the y oointes enbles us to estblish the position on the igm of two oesponing tul n ppent positions S n S espetively. y.5 y=y C.5 C,t θ θ Figue 3. The spetime igm isplys in tue vlues the oointes of the events ssoite with the ppent position S n with the tul position S. It isplys lso the ile C of ius = hving its ente t the oigin O n the ile C esibe in pmeti epesenttion by () n (3) fo =. The invine of the y(y ) oointes enbles us to fin out the lotion on igm of the ppent n of its oesponing tul position. The igm isplys in tue vlues the Ctesin n the pol oointes of events n s well s the eings t=/ of the loks of K when obseve R (,) eeives the light signl emitte fom the tul position. Consie tht S epesents one of the suessive tul positions. Then epesents the eing of lok T when R eeives the light signl emitte fom tht position. Une suh onitions, if t given sle we hve = t then t the sme sle = t. 4
5 Stting with () we llow fo smll hnges in the vlue of vibles. The esult is: = +. (7) Let t be smll hnge in the eings of the loks lote t the iffeent points of the K fme. By efinition = os epesents the il t omponent of the soue spee t its ppent position, = epesents t its tul spee n = epesents the OX omponent of the soue t spee. With the new nottions (7) les to: =. (8) " os We pesent in Figue 4 plot of s funtion of fo iffeent vlues of =. 8 β = β =.6 β = [DEG] Figue 4. A plot of / s funtion of the pol ngle fo thee iffeent vlues of the eue veloity = /. 5
6 Fo = the soue is eeing n fo " = the soue is ppohing the vetil is. With β efine bove (8) beomes: = (9) m eoveing Deissle s esult. As we see, fo " = / the two veloities e equl to eh othe. In the eeing onitions, >, whees when the soue is ppohing <. 3. The spetime igm t wok Consie tht the geometi lous of the tul positions is stight line pllel to the OY is esibe in pol oointes by: = () os epesenting its istne to the OY is. In one with () n (3) the geometi lous of the oesponing tul positions is esibe by the pmeti equtions: & ( # = $ + () % os' " y = " tg () The ules of hnling the spetime igm enble us to onstut, point by point, the tul shpe of (), s shown in Figue 5. y P P.5 C C.5 3 Figue 5. The spetime igm isplys the bsi iles C n C. The vetil line = (P ) is the geometi lous of ppent positions of point like soues lote on it, the uve (P) epesenting theis tul position. We hve onstute (P) pointbypoint using the ules of hnling ou spetime igm. 6
7 4. Conlusions We hve onstute spetime igm tht isplys in tue vlues the spetime oointes of events ssoite with effets genete by the finite spee of light signls. Epessing the spe oointes of the tul positions s funtion of those of the ppent positions we obtin simple n tnspent esults s ompe with those pesente by Deissle. Ou ppoh illusttes tht the simple ft tht light popgtes with finite veloity eltive to n inetil efeene fme hs nti ommon sense onsequenes like mking net istintion between ppohing o eeing positions of the light soue. Refeenes Robet J. Deissle, The ppene, ppent spee n emovl of optil effets fo eltivistilly moving objets, Am.J.Phys. 73, (5) Ashe Pees, Reltivisti telemety, Am.J.Phys. 55, (987) 3 H. Bltte n T. Gebe, Abetion n Dopple shift. An unommon wy to eltivity, Am.J.Phys. 56, (988) 7
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