Illustrating the space-time coordinates of the events associated with the apparent and the actual position of a light source

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1 Illustting the spe-time 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 spe-time oointes of istnt events fom light signls tht ive t his lotion. The pupose of ou ppe is to pesent spe-time igm tht isplys in tue vlues the spe-time 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 spe-time igm isplys in tue vlues the spe-time 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 spe-time igm tht isplys in tue vlues t well-efine 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 spe-time 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 spe-time 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 spe-time 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 spe-time 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) point-by-point using the ules of hnling ou spe-time igm. 6

7 4. Conlusions We hve onstute spe-time igm tht isplys in tue vlues the spe-time 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