Special relativity with clock synchronization

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1 Speial elativity with lok synhonization Benhad Rothenstein ), Stefan Popesu ) and Geoge J. Spi 3) ) Politehnia Univesity of Timisoaa, Physis Depatment, Timisoaa, Romania, benhad_othenstein@yahoo.om ) Siemens AG, Elangen, Gemany, stefan.popesu@siemens.om 3) BSEE Illinois Institute of Tehnology, USA, gjspi@msn.om Abstat. We pesent an intodution to speial elativity kinematis stessing the pat played by loks synhonized following a poedue poposed by Einstein. A. Speial elativity with lok synhonization A.. Intodution A.. The event and its spae-time oodinates. The onept of event is of fundamental impotane to speial elativity theoy. We define it as any physial phenomenon that take plae at a given point in spae at a given time. We define the point whee the event takes plae by its spae oodinates (Catesian o pola) and the time when it takes plae by the eading of a lok loated at that point when the event takes plae. The physiist who obseves natue by onduting epeiments is by definition an obseve. He knows the laws of natue by heat and he is able to opeate measuing devies. Einstein s obseves wok as a team. At eah point in spae we find suh an obseve. A fist type of obseve ollets infomation about the events that take plae immediately in font of him. Knowing his loation in spae using mete stiks and using a lok he haateizes an event by the spae oodinates defining his loation in spae and by a time oodinate that equates the eading of his lok when the event takes plae. Fo moe laity, let say that the lok defined above is the obseve s wist wath. Anothe type of obseve ollets infomation about the spae-time oodinates of events that take plae emotely at diffeent points in spae fom the light signals that aive at his loation, o fom the light signal that he sends out and eeives bak afte efletion on a mio loated whee the event takes plae 3,4. The obseve woks in a laboatoy. He an be onfined to it o he an be an astonome who ontemplates the sky. The obseve attahes a efeene fame K(XOY) to his laboatoy that epesents its est fame (Figue a). We say that the laboatoy and the efeene fame attahed to it ae inetial if when an objet at est is plaed at an abitay point, it will ontinue to stay thee. The efeene fame enables the obseve to assign

2 spae oodinates to the points whee events take plae. The position of point M is defined, in a two spae dimensions appoah, by its Catesian (,y) and by its pola (, θ ) spae oodinates elated by = osθ (.) y = sinθ (.) + = y (.3) Y M y O θ Figue a. Defining the position of a point M elative to a efeene fame K(XOY) using Catesian and pola oodinates as well. X A.. The elativisti postulates Conside two inetial efeene fames K(XOY) and K (X O Y ). The oesponding aes of the two fames ae paallel to eah othe, the OX(O X ) aes ae ovelapped and K moves with onstant speed in the positive dietion of the OX(O X ) aes. C (, ) is the wist wath of an obseve R (,) at est in K and loated at its oigin O, wheeas (,) is the wist wath of an obseve R (,) at est in K and loated at its oigin O. When the two oigins O and O ae loated at the same point in spae the wathes ae set to ead a zeo time. We say that unde suh onditions the two efeene fames ae in the standad aangement. We an ahieve this by plaing a soue of light in the immediate viinity of the stopped lok C. Clok stopped as well omes along the OX ais. Both loks display a zeo time. A photoeleti devie detets the aival of lok at the loation of lok C, losing the eleti iuit of a light soue that emits light signals pependiula to the XOY plane stating the two loks. Obseves of the two fames say R (, y) and espetively R (, y ), pefom measuements in ode to find out the physial popeties of a physial system. The esult of a physial measuement is a physial quantity and we epess it as a podut of a numeial value and a physial unit

3 (whee S.I. units ae usually pefeed). The efeene fame elative to whih the physial system is in a state of est epesents its est fame. A pope physial quantity epesents the esult of a measuement pefomed on a physial system by obseves at est elative to it. Speial elativity theoy beomes involved when we ty to establish a elationship between the physial quantities, haateizing the same physial system, as measued by obseves of K and espetively K. An equation that mediates these physial quantities is a tansfomation equation. A physial quantity that has the same magnitude when measued fom diffeent inetial efeene fames epesents a elativisti invaiant. Pefoming measuements on the same physial objet we obtain a vaiety of physial quantities. An equation that mediates a elationship between them epesents a physial law. Established in a given efeene fame, a law mediates the magnitudes of diffeent physial quantities that haateize the physial popeties of the same physial objet measued in that fame. If we pefom the tansfomation of eah physial quantity pesent in a physial law in aodane with its tansfomation equation and the algebai stutue of the law does not hange, we say that it is ovaiant with espet to all the tansfomation equations, whih, in this ase, onstitute a goup. The deivation of a tansfomation equation stats with the statement of Einstein s postulates on whih speial elativity theoy is based: Fist postulate ) Obsevation of physial phenomena by moe then one inetial obseve must esult in ageement between the obseves as to the natue of eality, o the natue of the univese must not hange fo inetial obseves in elative motion. ) Evey physial theoy should look the same mathematially to evey inetial obseve. 3) No popety of the univese will hange if the obseve is in unifom motion. The laws of the univese ae the same egadless of inetial efeene fame. 4) A student who has studied physis in the inetial laboatoy of a given univesity will pass the eam at eah othe laboatoy moving unifomly elative to the fist one. 5) It is impossible, by epeiments pefomed onfined in an inetial efeene fame, to deide if the fame is in a state of est o in a state of unifom motion along a staight line. 3

4 6) If the efeene fame K moves with onstant veloity in the positive dietion of the OX ais, then K moves with onstant veloity elative to K in the negative dietion of the same ais. 7) Distanes measued pependiula to the dietion of elative motion have the same magnitude in all inetial efeene fames in elative motion. 5 The simplest eplanation of this onsists of the fat that, in the ase of the standad aangement, the elative veloity has no omponent in the nomal dietion and so we onside that y = y. (.4) The fist postulate is an essential etension of Galileo s postulate in whih most physiists hadly believe. Seond postulate ) The speed of light in vauum, ommonly denoted as, is the same in all dietions and does not depend on the veloity of the objet emitting the light. As a oollay, we ague that when pefoming an epeiment the physiist is onfonted with the univese. The univese defends itself, peventing the obseves fom obtaining esults beyond etain limits imposed by the auay and apabilities of the available measuement devies, some of these estitions being atually desibed by the Einstein s postulates. Okham 6 teahes us that if two theoies eplain the fats equally well, then the simple theoy is the bette one. Paaphasing Okham, we say that if the same theoy eplains the same fat equally well, then the shote eplanation is the bette one. Ou teahing epeiene onvined us that it is easie to teah those who know nothing about the subjet than those with wong pevious epesentations about it. Fo an easie ompehension of this pape it is essential fo the eade to admit the Einstein s postulates even if they potentially hallenge some instintual belief developed duing pevious study of physis. The aeptane of these postulates is suppoted by many epeiments pefomed in last yeas as well as by the fat that until today no epeiment was possible onduted to ontadit these postulates. 7,8 4

5 A..3 The light lok and the othe loks assoiated with it. Time dilation and length ontation A lok is a physial devie that geneates a peiodi phenomenon of onstant peiod T. The best known lok fo elativisti epeiments is the light lok. 9 We pesent it in its est fame K (Figue b). The lok onsists of two mios M and M paallel to the ommon OX(O X ) aes and loated at a distane d fom eah othe. A light signal oiginating fom mio M is efleted bak by mio M and finally etuns to the loation of M. The time inteval T d = (.5) duing whih the light signal bounes bak and foth between the two mios epesents the peiod of suh a lok. M Y C d π θ ' = M O,S X Figue b. The light lok in its est fame K and the loks (,) and (,) assoiated with it. The invaiane of the speed of light in vauum and of distanes measued pependiula to the dietion of elative motion makes suh a lok vey tempting fo elativisti speulations. Usually the authos fail to mention that we an attah to the epeimental devie desibed above a seond lok 5

6 (,) loated in font of mio M and a thid lok (, d) loated in font of mio M. The light signals that boune between the two mios ould pefom the synhonization of the two additional loks in aodane with the lok synhonization poedue poposed by Einstein. 9 Conside that lok (,) in font of mio eads t = when a soue of light S (,) loated M stats to emit light in all dietions of the X O Y plane. Peviously the lok (, d) is stopped and set to ead d t =. The light signal that popagates in the positive dietion of the O Y ais stats the lok (, d) just when aiving at its loation. Fom this vey moment the two loks display the same unning time. The efleted light signal etuns to lok (, ) when it eads t = and at this vey moment the lok (, d ) eads the same time. Aftewads both loks will ead the same unning time. Speial elativity beomes involved when we detet the synhonization of loks (,) and (, d) fom the efeene fame K(XOY). The light signals emitted by the soue S '(,) o by a soue S (,) at est in K and loated at its oigin O, ould pefom the synhonization of the loks in the efeene fame K(XOY). Let C (,) be a lok of the efeene fame K(XOY) loated at its oigin O(,) and let C ( = osθ, y = sinθ ) be anothe abitay lok of that fame. All loks ead t = t = when the oigins O and O ae loated at the same point in spae. We wath the synhonization of loks (,) and (, d) fom the efeene fame K(XOY). We show the esult in Figue. d 6

7 C Y C C Y M d C θ O O X, X a Y M Y d C C 3 π-θ O O X, X b Figue. The synhonization of two loks of the K (X O Y ) efeene fame as deteted fom the K(XOY) efeene fame. Figue.a shows the situation when lok (, d), ommoving with mio M, aives in font of a lok C ( = osθ, y = sinθ) of the fame 7

8 K(XOY) whih is synhonized with C (,) by a light signal emitted by S (,) at t = along a dietion that makes an angle θ with the positive dietion of the ommon OX(O X ) aes. When loks (, d) and C ( = osθ, y = sin ) meet eah othe, the fist eads θ seond eads t = d t = while the. Pythagoas theoem applied to Figue.a leads to = d + (.6) whee we have taken into aount that duing the synhonization of loks C and C the lok has advaned with in the positive dietion of the ommon aes. We undeline that (.6) elates only physial quantities measued in the same efeene fame K(XOY). Epessing (.6) as a funtion of the eadings of loks and C when they meet, we obtain that the two eadings ae elated by t t = = γ t (.7) using the shothand notations γ = ; β =. β Figue.b shows the situation when the efleted synhonizing signal is eeived at the loation of the oigin O whih is when lok eads d = t. At this vey moment lok is loated in font of lok C 3 the d fist lok is eading t = while the seond lok is eading t. We undeline that (.7) holds only in the ase when the initial position of the involved loks is defined by a zeo absissa (in ou ase = ). Physiists all the eading of a lok time oodinate. They also use the onept of time inteval that sepaates two events, defined as a diffeene between the eadings of two loks of the same inetial efeene fame loated whee and when the events take plae. Conside the situation when lok meets lok C 3, as shown in Figue 3. 8

9 a Y Y C O, O X, X Y Y C C 3 O O X, X b Figue 3. Clok meets lok C 3. Figue 3.a shows the situation when lok is loated in font of lok C and they both ead t = t =. We onside that at that vey moment the two loks ae synhonized by simple ompaison, as they ae set to ead a zeo time, a poedue that does not involve light signals. Figue 3.b shows the situation when lok aives in font of a lok C 3 (,) of the K fame afte a given time of motion; the fist lok eads t, while the seond lok eads t. Theefoe, we know that = t. In aodane with the esults obtained so fa, we an onside that the two eadings ae elated by (.7), beause t and t epesent lok eadings i.e. time oodinates. We have all the elements neessay to pesent an event as E( = osθ, sinθ, t) whee the spae oodinates define the loation of the point whee it takes plae, wheeas the time oodinate t equates the eading of the lok loated at that point when the event takes plae. We say that event E and event E ( = osθ, y = sinθ, t ) epesent the same event if the two events take plae at the same point in spae, whee t and t epesent the eadings of loks of K and espetively K loated at the point whee the events take plae, both synhonized in thei est fames in aodane with Einstein s lok synhonization poedue. 9

10 The events involved in the epeiment we have just desibed ae E (,,) and E (,, ) assoiated with the fat that the loks C and ae loated at the ovelapped oigins O and O of the two efeene fames and both ead a zeo time and also the events E ( = t,, t) and E (,, t ) assoiated with the fat that lok aives at the loation of lok C ( = t,) when the fist eads t and the seond eads t. Events E and E ae sepaated by a zeo distane and by a zeo time inteval. Events E and E ae sepaated by a time inteval t = t (.8) and by a distane = (.9) obviously elated by = t. (.) The time inteval t is measued as a diffeene between the eadings of two loks C and C loated at two diffeent points in spae. Events E and E ae sepaated by a zeo distane = (.) and by a time inteval t = t (.) whih is measued as a diffeene between the eadings of the same lok in font of whih both events take plae. By definition, this epesents a pope time inteval. In aodane with (.7), the time intevals ae elated by t = γ t (.3) whih elates the pope time inteval t and the non-pope time inteval t. Beause t > t elativists say that a time dilation effet takes plae. So fa, we have onsideed only events that take plae on the ovelapped aes ( y = y = ). Retuning to Figue a when lok C '(, d ) d eads t =, it is loated in font of a lok C ( =, d) whih eads t =. The poblem is to find equations that elate the spae-time oodinates of d events E ( =, y = d, t = ) and E ( = osθ, y = sinθ, t = ) assoiated with the fat that the loks and C ae loated at the same point in spae. We an define the loation of lok using pola oodinates so that C ( = d, θ = 9 ). As we have seen, thei time oodinates ae elated by t = γ t. (.4)

11 We also have = t = γ t (.5) y = y = d (.6) os θ = β (.7) sinθ = γ (.8) tan θ = βγ. (.9) The invaiane of distanes measued pependiula to the dietion of elative motion equies that sin θ = d = y (.) i.e. = γ d. (.) The equations deived above ae tansfomation equations that hold only in the ase when one of the events takes plae on the O Y ais of the efeene fame K. We an hange the senaio we have followed so fa, by onsideing the synhonization of loks C (, ) and C (, d ) of the K fame fom the K fame and by taking into aount that K moves elative to K in the negative dietion of the ommon aes with speed. We obtain the invese tansfomation equations t = γ t (.) = γt (.3) y = y = d (.4) os θ = β (.5) = γ d (.6) whih hold only when only one of the involved events takes plae on the OY ais of the K fame. The equations deived above enable obseves of the two fames to measue the length of a moving od. Conside a od at est in K and loated along the ommon OX(O X ) aes with one of its ends (,) loated at the oigin O. The length of the od, measued by obseves of its est fame K, epesents its pope length L. The position of the seond end of the od is defined by ( L, ). Obseves fom K measue the speed of lok that passes in font of end when lok C loated thee eads t = (Figue 4a) and passes in font of lok C ( L,) loated at end when it eadst = t. By definition the speed of lok is L L = =. (.7) t t whee t epesents a non-pope time inteval.

12 An obseve R (,) loated at the oigin O of his est fame K an use his wistwath (,) in ode to measue the speed of the od. The left end of the od passes in font of him when his wistwath eads t = and the ight end of the od passes in font of him when the same lok eads t = t. By definition the speed of the od is L L = = (.8) t t in whih ase t epesents a pope time inteval, wheeas L is a nonpope length. Combining (.7) and (.8) and taking into aount (.3) we obtain that the pope length L and the measued length L ae elated by L = γ L. (.9) It is impotant to undeline that the time dilation fomula elates a pope time inteval and a non-pope time inteval. It involves synhonized loks only in the efeene fame whee the non-pope time inteval is measued. The time dilation phenomenon loses most of its mystey one we eognize that it is basially the onsequene of ompaing suessive eadings in a given lok with eadings in two diffeent loks. A..4 The Loentz-Einstein tansfomations (LET) Due to the elative haate of some physial quantities, we ae not allowed to epesent physial quantities measued in diffeent inetial efeene fames in elative motion on the same spae diagam. With this in mind we no longe impose the estiting ondition = ( = ). Conside the events E (,, t) and E (,, t ) that take plae somewhee on the ovelapped aes OX(O X ). Figue 4 shows the elative positions of the efeene fames K and K as deteted fom K when the synhonized loks of that fame ead t. A od of pope length L = at est in K is loated along the ommon aes with one of its ends at O. When epesenting it in.4a we should take into aount that its length measued in K is L = ( ). (.3) Adding lengths measued in K we obtain = t + (.3) fom whee we obtain

13 = t. (.3) Y Y C a L O, O X, X Y Y b L O O C X, X Figue 4. The elative position of the efeene fames K and K as deteted fom K when the synhonized loks of that fame ead t. Y a L O X Y L O b Figue 5. The elative position of the efeene fames K and K as deteted fom K when the synhonized loks of that fame ead t. X 3

14 Figue 5 shows the elative position of the K and K fames as deteted fom K when the synhonized loks of that fame ead t. A od of pope length L = at est in K has one of its ends at O. Its length measued in K is L =. (.33) Adding length measued all in K we obtain = + t (.34) fom whih + t =. (.35) Combining (.3) and (.35) we obtain t + t t = (.36) t t =. (.37) We have deived the Loentz-Einstein tansfomations fo the spae-time oodinates of events E and E taking plae at the same point of the ovelapped aes when loks C(,) and C (,) loated at that point ead t and espetively t. By definition they epesent the same event. Fom a patial point of view (.35) and (.36) enable obseves fom K to epess the spae oodinates (, ) of an event that takes plae in K as a funtion of the spae-time oodinates (, t ) of the same event as deteted fom thei est fame K. The patial value of (.3) and (.37) an be eplained in the same way. If the events defined as E (, y, t) in K and E (, y, t ) in K take plae at the same point loated somewhee in the plane defined by the aes of the efeene fames involved then we an add to the tansfomation equations deived above the equation y = y (.38) whih epesses the invaiane of distanes measued pependiula to the dietion of elative motion. 4

15 Conside a patile that stats to move at t = fom the oigin O of the efeene fame K along a dietion that makes an angle θ with the positive dietion of the ommon aes. The speed of this patile elative to K is u ( u, u y ). Afte a time of motion t, the patile geneates the event E ( = u t, = u t osθ, y = u yt sinθ, t ). In aodane with the tansfomation equations deived above the spae-time oodinates of the same event as deteted fom fame K ae = γ ( + t ) = γ t ( u + ) (.39) o = γ ( + ) = γ ( + ) (.4) u u and u t = γ ( t + ) = γ t ( + ) (.4) o t = γ ( + ) = γ ( + u ). (.4) u Beause the patile stated to move at t = t = when the oigins of the two fames wee loated at the same point in spae, the omponents of thei speeds elative to K ae + u u + u = = = (.43) t t u u + + y y γ γ u y u y = = =. (.44) t t u u + + Deteted fom K the patile moves along a dietion θ elative to the positive dietion of the ommon aes given by y u y γ sinθ tanθ = = =. (.45) u osθ + u We tansfom the magnitude of the speed as (osθ + ) + ( )sin θ u u = u + u y = u. (.46) u osθ + and the lengths of the position veto as 5

16 = + y = γ (osθ + ) + ( ) sin θ. u (.47) In many appliations of speial elativity we look fo tansfomation equations that elate the spae-time oodinates of events geneated by light signals. We obtain them fom the tansfomation equations deived above by taking into aount that in this ase, in aodane with the seond postulate, u = u =. The esult is γ sinθ tanθ = (.48) osθ + = γ ( + osθ ) (.49) t = t γ ( + osθ ) (.5) = + (.5) The Loentz-Einstein tansfomations deived above involve synhonized loks in the inetial efeene fames involved, K and espetively K. Using diffeent teahing stategies, authos deive the fomulas that desibe time dilation and length ontation stating with thought epeiments, as we did, wheeas othes pesent them as onsequenes of the Loentz-Einstein tansfomations. We onside that the single well defined physial quantities with whih speial elativity opeates ae the pope length and the pope time inteval. All the othe onepts like measued length and measued time inteval depend on the measuement poedue and so we ould detet not only length ontation but also length dilation and not only time dilation but also time ontation. A..5 Conlusions Undelining the pat played by synhonized loks of the Einstein type of obseves we have deived the fundamental equations of elativisti kinematis in a ompat way and fom a little numbe of senaios. 6

17 Refeenes H.Blatte and T.Gebe, Abeation and Dopple shift: An unommon way to elativity, Am.J.Phys. 56, (988) U.Kaus, Bightness and olo of apidly moving objets: The visual appeaane of a lage sphee evisited, Am.J.Phys. 68, 56 () 3 C.W.Shewin, Regading the obsevation of the Loentz ontation on a pulsed ada system, Am.J.Phys. 9, 67 (96) 4 Bue C. Eaton, Edito and S.A.Pollak, Optial ada fo eduational demonstations, Am.J.Phys. 4, 75 (97) 5 A. Shadowitz, Speial Relativity (W.B. Saundes Company, Philadelphia, London, Toonto 969) pp Spade, Paul inent, William of Okham The Stanfod Enylopaedia of Philosophy (Fall Edition), Edwad N. Zalta (ed) 7 Yuan Zhong Zhang Advaned Seies of Theoetial Physial Siene ol.4, Speial Relativity and its Epeimental Foundation (Wold Sientifi 997) 8 John S. Reid, Why we believe in speial elativity: Epeimental suppot fo Einstein s theoy, Repot on publi meeting held in Abedeen on Mah Hans C. Ohanian, Speial Relativity : A Moden Intodution, (Physis Cuiulum&Instution Lakevill USA ) pp W.N. Mathews, Relativisti veloity and aeleation tansfomation fom thought epeiments, Am.J.Phys. 73, 45-5 (5) 7