Potogaping a time inteval Benad Rotenstein and Ioan Damian Politennia Univesity of imisoaa Depatment of Pysis imisoaa Romania benad_otenstein@yaoo.om ijdamian@yaoo.om Abstat A metod of measuing time intevals by a single obseve poposed by Cowell 3 is extended to te moe geneal ase wen te events sepaated by te time inteval take plae at two points aateized by te same yy spae oodinates. We sow tat time dilation and time ontation an take plae o even an invesion in te time suession an be deteted.. Intodution Weinstein sows ow a single obseve ould measue te lengt of a moving od by simultaneously deteting te ligt signals tat ave left te two ends of te od at diffeent times. He onsides only te ase in wi te obseve is vey lose to te od. Matews pesents te same poblem in te geneal ase and stesses tat beause te ligt signals aise fom diffeent distanes to fom te obseve te measued lengt of te od will be vey diffeent fom tat given by te oentz ontation. He onsides tat te measuement poedue is assoiated wit te Dopple ffet as well. We sow tat te involved obseves ae onened wit te Dopple ffet only in te ase wen te fequeny and te ligt soue-obseve distane ae vey ig. Cowell 3 sows tat tis agument an be applied to te obsevation of a time inteval by a single obseve. He onsides te same simple ase as Weinstein did. e pupose of ou pape is to genealize te appoa poposed by Cowell 3.. Potogaping a time inteval Conside te senaio pesented in Figue. It involves a od of pope lengt at est in te K (X O Y ) efeene fame paallel to te O X axis. e spae oodinates of its edges ae ( x os y sin ) and ( x os y sin ) using bot Catesian and pola oodinates. At a time t an obseve R () (te potogape in K ) loated at te oigin O eeives te ligt signal tat as left edge at a time t and te ligt signal tat as left edge at a timet. e time t is displayed by a lok C ( x y) weeas a lok C ( x y ) displays te time t. All te loks of te K (X O Y ) efeene fame ae synonized following te lok synonization poedue poposed by instein 4. e events x y t and assoiated wit te fat tat te ligt signals ave left te edges of te od ae ( ) ( x y t ) espetively. Fom Figue we obtain () os os sin sin () fom wi we obtain + ( os. (3)
e time sepaation between te two events as measued in K is + os) ( + sin ( sin ( (4) sin( V In Figue we pesent te vaiation of wit fo. 6 and diffeent values of sin wee epesents te distane fom te points wee te events take plae to te ommon axes. As we see depending on te elative position of te two points wee te events take plae te time inteval an be positive if bot points ae outgoing ( > ) but it an be negative as well if bot of tem ae inoming ( < ) o even equate a zeo value wen i.e. wen x and x. Fo and fo we ave ± beause in tat ase te ligt signals oiginating fom and espetively popagate along te line tat joins te two events. As involving pysial quantities measued in te same inetial efeene fame te deivation of (4) involves speial elativity by te onstany of te veloity of ligt postulate. Conside te expeiment desibed above fom te K(XOY) efeene fame. e axes of te two efeene fames ae paallel to ea ote te OX(O X ) axes ae ommon and K moves wit onstant veloity V elative to K in te positive dietion of te ovelapped axes. At te oigin of time in te two efeene fames (tt ) te oigins of te two fames ae loated at te same point in spae. Deteted fom K te spae-time oodinates of te two events defined above ae x os ( y sin( t and x os ( y sin( t. In aodane wit te oentz-instein tansfomation fo te spae-time oodinates of te same event we ave 5 ( ) os (5) ( os ). (6) eefoe te two events ae sepaated in K by te time inteval given by * * os os ( ( (7) ) ) o * +. (8) ( + os)
We an expess (8) as a funtion of ) sin( *. (9) + sin ( sin ( An inspetion of (9) sows tat te time inteval measued in te K efeene fame is te esult of a dilation of te time inteval measued in te stationay efeene fame K and of an exta tem detemined by te fat tat te two events take plae in K at diffeent points in spae. We an expess (9) as a funtion of te angle measued in K given by 6 os os () os In Figue 3 we pesent te vaiation of wit fo diffeent values of. is enables us to evaluate te otations intodued by te detetion poedue we ave used. If te points and ae vey lose to te OX(O X ) axis and bot ae outgoing ( ) equation (9) beomes ) ( ) () ( weeas if bot points ae inoming ( ) it leads to ( + ). () In ode to illustate te esults obtained above we in Figue 4 we pesent te vaiation of wit as pedited by (9) fo diffeent values of and.6. In te expeimental onditions desibed above te time inteval is not a pope time inteval. ee is a patiula situation wen pesented in Figue 5 wen os (3) and + ; +. e two events ae sepaated in K by a time inteval V. (4) As in te ase wen te two events take plae on te ovelapped axes OX(O X ). If potogape R () takes a snapsot of a single event say ( ) ten e measues a time inteval. Deteted fom K it is ( os ). (5) We also ave ( os ). (6) We an onside tat and ae multiples of te wavelengt ( ) and tat and ae peiods so tat (5) and (6) desibe te Dopple ffet in oblique inidene. In tat ase and 3
ae pope time intevals 4. All ote ases studied above ave noting in ommon wit te Dopple ffet as elating non-pope time intevals. 3. Conlusions If te time dilation effet in its ommon use is te esult of a ompaison between a pope time inteval measued in te est fame of one of te loks involved and a non-pope time inteval measued in a efeene fame elative to wi te lok moves te time dilation and time ontation effets evealed above ae te esult of te two elativisti effets mentioned above. In ode to avoid onfusion it is advisable wen we speak about te magnitude of a pysial quantity to mention te obseve (obseves) wo measue it te measuing devies used and wen and wee e (tey) pefoms te measuement. Refeenes R.Weinstein Obsevation of lengt by a single obseve Am.J.Pys. 8 67-6 (96) Robet J. Deissle e appeaane appaent speed and emoval of optial effets fo elativistially moving objets Am.J.Pys. 73 663-669 (5) 3 A.D.Cowell Obsevation of time inteval by a single obseve Am.J.Pys. 9 37-37 (96) 4 P.K.Hsiung R.H.ybadeau C.B.Cox R.H.P. Dunn (99) ime dilation visualization in elativity Supeomputing 9 Poeedings 835-84 Figue aptions Figue. e senaio involves an obseve R () loated at te oigin O of its est fame and an obseve R() loated at te oigin O of its est fame K (X O Y ). At te ommon oigin of time in te two fames tey eeive two ligt signals wi ave left at diffeent times two diffeent points aateized by te same distane to te ovelapped axes and sepaated by te pope lengt in te K (X O Y ) efeene fame. Wat we ompae ae te time intevals and in K and K espetively. Figue. We pesent te vaiation of wit angle fo. 6 and diffeent values of te aateisti paamete a. Figue 3. We pesent te vaiation of te angle wit fo diffeent values of as pedited by te abeation of ligt effet. Figue 4. We pesent te vaiation of wit fo. 6 and diffeent values of a. Figua 5. A senaio in wi obseve R () detets two simultaneous events. 4
Figues Y è è O R X Fig...6.. outgoing inoming - / ( a d ) Fig.. 5
( a d ) d e g ) a d ) â.9 â.6 â.3 / ( Fig. 3 5 5.6. 5-5 6 7 8 9 ( Fig. 4. 6
Y O R () X Fig. 5. 7