On Rotating Frames and the Relativistic Contraction of the Radius (The Rotating Disc)

Transcription

1 On Rotating Fames and the Relativisti Contation of the Radius (The Rotating Dis) Pantelis M. Pehlivanides Atlanti IKE, Athens 57, Geee Abstat The elativisti poblem of the otating dis o otating fame is studied. The solution given implies the ontation of the adius and the hange of the value of π depending on the type of obseve. To foms of otation ae onsideed. One is ith onstant angula veloity, independent of the adius, implying a hoizon, the othe is ith exponentially deeasing angula veloity ith espet to the adius and does not imply a hoizon. In all ases the paths of signals emanating fom the oigin of the otating fame advane helially in the positive and negative z dietion, hee they ae onentated, due to the ontation of the adius, and in some ases appea as jets.. Intodution The otating dis has been the subjet of a multitude of papes sine Ehenfest [] published hat is today knon as the Ehenfest Paadox. He noted that sine the peimete of a otating dis ould elativistially ontat, hile the adius emained the same, a defomation of the dis should take plae. Many authos have sine then ontibuted to the undestanding of the poblem by studying the geomety of the otating dis. A histoial evie an be found in Rizzi and Ruggieo [] and in Gøn [3]. The appoah in the pesent study is lose to the idea that thee is a ontation of the adius of the otating dis. Simila ideas have been exploed by Ashoth, Davies and Jennison [4], [5] and by Günbaum and Janis [6], [7]. In patiula, Ashoth, Davies and Jennison sho that the adius of the otating dis ontats aoding to an obseve that is otating ith the dis at a distane fom the ente. Günbaum and Janis ague that an obseve that is not otating ith the dis ill see the adius of the dis ontat by the same elativisti fato as the peimete. The latte appoah is lose to ous although e do not find the same esults beause e allo the value of π to hange fo the non-otating obseve, hen he makes measuements egading the otating fame. The pape is oganized as follos: In setion e state ou basi assumptions that ill help us deive the tansfomations in the folloing setions. In setions 3 e pesent ou notation and knon elativisti esults egading the otating fame. In setion 4 e fomulate and solve the poblem of the ontation of the adius of the dis and the tansfomation of the value of π fo the non otating obseve. In setion 5 e pesent a summay of the esults up to that point. In setion 6 e onside the spae o dis defomation as seen by the non-otating obseve and distinguish beteen to kinds of non- otating obseves: one ithin the hoizon of adius / and one outside. In setion 7 e disuss the esults. In setion 8 e genealize to otating fames in 3 dimensions and

2 plot the signals emanating fom the oigin of the otating fame in the adial dietion, as seen by the non-otating obseves. The signals gadually bend sideays until they eah a 9º degee angle ith espet to the adius, hile they advane in the positive and negative z axis dietion. In setion 9 e examine otation ith slippage so that the angula veloity is assumed to deease exponentially as the adius ineases and as z ineases. We examine both the lose by non-otating obseve and the fa aay non-otating obseve. The signals ae in this ase not limited to a hoizon but gadually bend sideays until they eah a imum defletion fom the adial dietion and then tun asymptotially bak to the adial as the adius ineases, hile they advane in the positive and negative z dietion. In setion e pesent ou onlusions.. Assumptions on the otation of to onenti diss In the folloing e ill make thee assumptions. Assumptions and 3 ae standad in elativity theoy. Only Assumption is ne.. Disussion to justify Assumption Suppose an obseve O is at any plae on dis (dis ) and a seond obseve O that sits at any plae of anothe dis (dis ) paallel and oaxial to dis. O stiks a penil though his dis paallel to the axis of otation ith its point touhing the othe dis (dis ). O does the same thing ith the tip of his penil touhing dis. As thee is elative otation of one dis ith espet to the othe, eah obseve ill ath the peimete of a ile being dan on his on and the othe dis. Eah ill see that the duation of time, aoding to his on lok, fo a omplete ile to be dan on his on dis ill be the same ith the duation it takes fo him to da a omplete ile on the othe dis. This obsevation holds fo half a ile o any fation of a ile. In shot, e say that the to obseves ill agee on epiente angles measued as fations of a ile (not adians). If e denote the magnitude of an angle as fation of a ile and θ the magnitude of the same angle in adians then. Hoeve, e ill not use fo the moment beause e ae not sue the obseves agee on. Imagine no that the same to obseves stand at the ente O of thei dis (one on top of the othe) and let the diss otate ith espet to eah othe. If the fist obseve announes that aoding to his measuements the seond dis is otating ith fequeny, sine the situation is exatly symmetial e ill expet the seond obseve to make the same announement egading the fist dis. But fequeny is defined as evolutions pe unit time o Θ pe unit time. One they agee on Θ and, they have to agee on time ate. In fat, they may even use the same lok, sine they ae olloated. We may summaize in the folloing, Assumption (a)to obseves sitting at to paallel onenti diss otating ith espet to eah othe aound an axis vetial at thei ente, ill agee that the magnitudes of epiente angles taveled by the othe dis, measued as fations of a ile, ae equal. (b) If they also stand at the ommon ente of thei diss, they ill futhe agee that time ates ae equal. Remak: The situation ith the to obseves at the ente of thei diss (one on top of the othe) is symmetial and thee is no point in aguing ho is otating and ho is

3 stationay. Hoeve, if they had a ay to measue the entifugal aeleation off thei ente, they ould pobably find that it is diffeent. In that espet e may ightfully all the fame ith zeo entifugal foe the pefeed fame. In hat follos e ill assume that one obseve, usually to be denoted as O (o O ), sits at the ente of a pefeed fame K (o K ), hih oinides ith the laboatoy fame unless otheise speified.. Disussion to justify Assumption Many expeiments have veified the onstany of the speed of light, hih is the basi assumption fo speial elativity theoy. The speed of light is not the same fo obseves unde aeleation o, aoding to the Piniple of Equivalene, gavity. Assumption Light speed does not depend on the speed of the emitting soue and its speed is onstant fo all obseves that ae not unde the influene of aeleation. 3. Disussion to justify Assumption 3 An obseve on a fame ill agee ith anothe obseve on the same fame on the measuements of lengths. This is expeted if measuements ae made not by signals but by atually plaing the measuing stik on the length to be measued and ounting ho many times it fits to it. We state then, Assumption 3 Obseves on the same fame ill agee on measuements of length. 3. Time Rate and the Contation of the Peimete on a Rotating Dis As e talk intehangeably about otating diss and fames, e need to laify that hen e talk of a dis, e imagine it plaed at the x-y pane of the espetive fame ith ente at the oigin. Let to fames K and K ith ylindial oodinates, ommon oigin O and ommon axis of otation Z. Let obseve O sit at the oigin of K and O at the oigin of K. Let K be a non otating fame - laboatoy fame- and let O obseve fame K otate. Let a thid obseve O3 on K sit at a distane fom the ente. When thee is no dange of onfusion e ill ename the thee obseves using O O, O O, O O3 and the fames K K, K K. To help laify ideas e intodue the to subsipt notation, hih e ill abandon shotly afteads fo eonomy: A quantity Q is defined as the quantity measued by obseve i given it is stationay in ij the fame of obseve j Fo example, L is the length measued by obseve fo a line segment on the peimete of the dis that is stationay in the fame of obseve. Also, t is the time inteval that obseve sees that a lok stationay and ith obseve shos fo the duation of an event. 3

4 Note that by Assumption 3, Lii is a onstant fo all i and ill be denoted as Lstat sine all obseves agee on the same segment hen stationay in thei fame. The same is not tue fo tii. In patiula, e expet t t tstat t33, and t t tstat beause obseves and have the same lok, hile obseve 3 is unde the influene of entifugal aeleation, hih e suspet that ill affet t33 in some unknon as yet ay. Theefoe, the lok of obseve and epesents stationay lok time intevals and ae denoted as tstat. Also, t3 t33 beause obseves and 3 ae on the same fame and time intevals measued by obseve looking at the lok of obseve 3, agee ith hat obseve 3 sees looking at his on lok. Distanes in the adial dietion ae measued as ij, hile the angula veloity of a dis is measued by ij. Speifially, 3 is the adial distane as seen be the non otating obseve egadless of the seond subsipt sine both obseves and 3 ae stationay on the otating dis. Similaly, 3 beause the angula veloity of the otating dis as seen by the non otating obseve, using his on lok does not depend on the seond subsipt sine both obseves and 3 ae stationay on the otating dis. Most of the authos (see fo example Mølle [8] pp.-5 on otating dis) stat fom the tansfomation beteen a otating fame K and a non otating fame K and the tansfomation elation of ylindial oodinates, z z, t. In this ase The meti fo the otating fame is ds d d dz () ddt dt () Fo a non-moving point in spae, the spae diffeentials ae null and equating the metis of the otating and non-otating fames e find dt () dt fom hih e find that the lok of the otating system uns sloe, dt dt () The line element d is given by d dx dx hee x takes the values {,, z} and exept fo,, zz. Hene, e find, d d d dz Fo a line segment dl along the peimete this fomula implies that i dl dl (4) Whee ( dl d and dl d ). Wheeas the esult () is ithin ou expetations fom the speial theoy of elativity, the esult (4) is ontay to the expeted esult. Sine the nomal Loentz ontation ould give a ontation of the peimete instead of a lengthening as e have found above. The (3) 4

5 esult (4) is ounteintuitive fo anothe eason also: If e inease the adius, hile deeasing the angula veloity so that that tangential veloity is onstant ( ) then the peimete tends to a staight line and the tansfomation should appoah the Loentz length ontation. Instead, aoding to (4) sine is onstant the lengthening fato emains onstant and thee is no ay of appoahing the Loentz ontation. As the method ith hih the esult is obtained is oet, the only suspet is the fom of tansfomation assumed. We ae motivated, theefoe, to seah fo anothe tansfomation that ill also satisfy the ontation of the peimete aoding to the Loentz length ontation of speial elativity. Ou quest is, theefoe, to find a tansfomation that satisfies in ou notation the folloing, (a) The ate of the lok of O3 ill appea sloe to obseve O : t3 t stat (Reall that 3 and 3 as e mentioned above) (b) Line segments along the peimete ae ontated, L Lstat (6) Obseve that (5) implies that the ate of the lok of O3 ill also appea sloe to O, ho has the same lok as O. Namely, t3 t33 t stat (7) But sine thee is no elative motion beteen O and O 3, O ill think that is due to the entifugal aeleation that O3 feels. Requiements (a), (b) along ith the assumptions,, 3 imply a tansfomation (ontation) on the adial distane as e ill see belo. 4. The Contation of the Radius of the Rotating Dis Refe again to obseves O (o O ) on K (o K ), O (o O ) and O (o O 3 ) on K (o K ), ith K otating ith espet to K ith fequeny v aoding to O (and aoding to O ). Suppose obseve O has a od that extends adially fom the ente O to some point A (see Figue ). The od is hollo mioed inside and infinitesimally thin so that light taveling though it follos a staight line. The od is just an atifat to help imagine things, a statement saying that O sends a light signal adially outad is enough. ObsevesO and O ill agee on time ates ( dt dt ) (Assumption ) and the fequenies of otation they obseve ill be equal ( ) (Assumption ). They ill not agee on angula veloity measued in adians pe unit time, beause they ill in geneal disagee on and, theefoe, e may say that fo obseve O the angula veloity is, ( v ), hile fo O, ( v ). (5) 5

6 Obseve O (ho is not unde aeleation beause he sits at the oigin although he is otating ith the dis) sends a light signal fom O toads A though the od. Aoding to him the signal tavels ith veloity (Assumption ) the distane OA = (see Figue ). Until the signal eahes the end of the od, the od ill have moved to position OB. Obseve O ill see the signal tavel a uved path (OCB') ith onstant tangential veloity (Assumption ). At the peimete the dietion of the veloity of the signal, aoding to O, ill make an angle ith espet to the adius OB' hih ill have length OB ( ). A O' O O C θ ' B' B φ Figue The path of the light signal oiginating at O is OCB' aoding to obseveo (the lab obseve) Let : the adius as obseve O measues it (stationay length) (OA=OB) : the adius aoding to obseve O (OB') : the adial omponent of the veloity aoding to obseve O : the omponent of pependiula to the adius aoding to obseve O : the angle beteen and (to be alled angle of defletion fom the adial dietion) We may ite the folloing elationships fo light signals letting fo : Fo a light signal and then os (8) sin (9) () hee is the angula veloity in adians as obseved by O and theefoe, () Fom (9) and() sin () And substituting in (8) 6

7 (3) In ode to satisfy the ondition fo the ontation of the peimete (see (6)) e futhe equie that (4) Solve fo and substitute in () to obtain We aleady defined. Substitute in (5) and solve fo to obtain d d No substitute in (3) noting that and e find dt dt d dt (7) Using t, (7) beomes d dt t (8) and integating ith espet tot e find t ln( t ) t ln( ) t a sinh() t t t t (9) hee, the onstant of integation is zeo beause e equie that fo t. Equivalently, sine t, (9) an take the fom sinh () Using (), (6) beomes, (5) (6) t t () asinh () ln( t ) t t t ()ln( ) () A simila elation () holds also beteen and beause and (see (4) and (5) ) Using () and (6) e find os () and e equie that. We have shon that O (the laboatoy obseve) ill see a ontation of the adius of the otating dis given by (9) o (). Obseves O and O ill not agee on the angula veloity, and on the value of. In fat, obseve O ill peeive and as vaying ith the distane. This situation aises fom the fat that the ontation fato along the t 7

8 peimete is diffeent fom the ontation fato along the adius and the equiement that the speed of the signals is onstant and ageed by all obseves. One emak about angula veloities is useful to laify things. Obseves O, O and O ill agee on epiente angle, measued as fation of a ile (see Assumption ). But aoding to the definition of angula veloity e may ite,, t t, hile fo angles,, t, hee the tildas efe to obseve O ( O 3 ), the pimes to obseve O ( O ), and the plane lettes efe to obseve O ( O ). The oet notation using the subsipt notation of setion 3 is,, t t tstat t t tstat os. But 33 (o ), beause t t33 tstat obseves agee both on adial and on peimete lengths hen stationay on thei fame (Assumption 3). We onlude, theefoe, that (3) t33 (4) 33 t os and t33 (5) t os os 5. Summay of Results The angle of defletion and in patiula os takes many equivalent foms that ae pesented hee fo ease of alulations os sin (7) tan t sinh (8) Whee is the angle of the ile taveled by the signal until it eahes the distane fom the ente (see Figue ). The tansfomation among the obseves O, O and O is summaized belo in Table(a), (b),(). Hoeve, ou inteest in this study ill be foused on the elation beteen obseve O and O. (6) 8

9 Table (a) Tansfomations beteen Obseves O ( O ) and O ( O ) Quantitities Tansfomations Time inteval t t tstat, ( tstat t t t t ) Length segment on peimete L L os, ( L Lstat os ) Radius asinh Angula veloity os asinh() Pi and angles os Table (b) Tansfomations beteen Obseves O ( O ) and O ( O 3 ) Quantitities Tansfomations Time inteval t t t, ( t3 t stat 33 ) os os Length segment on peimete L L, ( L3 L3 Lstat ) Radius Angula veloity t os t Pi and angles, 33 Table () Tansfomations beteen Obseves O ( O ) and O ( O 3 ) Quantitities Tansfomations Time inteval t t t, ( t3 t stat 33 ) os os Length segment on peimete L L os, ( L3 Lstat os ) Radius os asinh os Angula veloity os asinh() os Pi and angles 33t stat 33 os, t 33 9

10 6. Wap o Ripples? Fom (4) and using () e see that os. This implies that a sinh exept fo fo hih equality holds. The deease of implies a aping of the dis o the eation of ipples (see Figue (a) and Figue (b)). In the ase of aping (Figue (a)) obseve O sees the adius as the segment OA ith length, obseve O sees the uved segment OB ith length, hih fo him looks as staight and PB is the theoetial staight line (pojetion of on Eulidean spae) ith length. Hoeve, e may exlude aping, beause of the folloing agument: Conside thee paallel onenti diss. Let the middle one otate ith fequeny and ith espet to the othe to that ae stationay. If indeed aping oued the middle dis ould inteset one of the othe to diss, sine e ae alloed to bing them abitaily lose to the middle dis. This seems unphysial. We ae, theefoe, inlined to exlude aping as a possibility and to onside ipples instead. The ipples fomed on the dis (Figue (b)) make the adius be looked in to diffeent ays. One is the adius touhing the sufae of ipples ( ) (the sufae adius) and anothe is the theoetial staight line disegading ipples ( ) (the staight adius o the pojetion of on the flat plane of otation). The latte one satisfies the equation os and theefoe, os (9) P O '' B ' A '' O ' (a) (b) Figue (a) The otating dis is aped. The adius on the sufae of the aped dis is. The staight line (Eulidean) adius is.the stationay adius is. (b) The otating dis foms ipples. The length of the adius on the ippled sufae is. The staight line (Eulidean) adius of the dis is. The stationay adius is. As e ill see in the disussion belo (setion 7), if is finite but then,,, os. The lab obseve O ill see the sufae adius ( ) tend vey sloly (logaithmially) to infinity and, hile the staight (on the flat Eulidean sufae) adius tends to. Physially, is the adius that an obseve Oon the laboatoy fame ill obseve, ho is loated at a distane geate than fom the axis of otation of the dis. If O, entes into the egion of distane less than fom the axis of otation, then his geomety eases

11 to be Eulidean. He beomes obseve of type O. He is no on the ippled sufae (hih he peeives as flat) his pi is no and the adius of the dis is no given by, hih is not limited by any bounday. Sine otation of the dis does not affet the time ate of thei loks, beause they do not patiipate in the otation, e expet that the time ate fo O and Oafte O osses the bounday to emain equal o that t t tstat, although they annot exhange light signals. If e denote by double pimes the quantities obseved by O, the above implies that t t t (3) Regading lengths (peimete and adial) obseve Osees the lengths of obseve O smalle by a fato of os beause of (9). Hene, L L L os (3) Futhe, epiente angles ae not affeted, (3) Hene, thee is ageement on fequeny of evolution measuements, that is v v v beause v t stat Also fom (4) and (9). Sine obseve O lives in Eulidean spae, his pi denoted as must be equal to the nomal pi o. It follos that. Fom these obsevations one easily dedues using (6) that. Substituting in (6) e find, os (33) This as expeted sine the Loentz ontation of the peimete fo O is given by. Finally, substituting (33) into (9) e obtain, (34) And solving e find (35) Whih as also expeted sine by (6) os 7. Disussion of Results Fist e note that fom (6), (9) and beause lim lim (36) And similaly, lim lim lim() t (37)

12 Also And t lim lim lim() (38) lim (39) lim (4). As ;, and os and and. This says that as the tangential veloity of the otating dis beomes big, the ays at the iumfeene ae almost tangential and thei tangential veloity appoahes the speed of light, hile thei adial veloity tends to zeo. In othe ods, light signals emanating fom the ente, O, ill bend and tun aound in iles expanding vey sloly as they ill be bend almost entiely tangentially.. In patiula if hile (and hene t ) emains finite,, os,,, and. In this ase, hen the angula veloity ( ) beomes big, hile the est adius ( ) emains finite, the adius ( and ) fo the lab obseves O and O shinks to beome vey small (but ), the light signals stating fom the ente, O, bend to tun in iles ( os ) ith tangential veloity and the adial veloity of the light signals tends to zeo ( ) 3. If is finite but then,,, os. Obseve O ill see the sufae adius ( ) tend vey sloly (logaithmially) to infinity (hile ), hile O ill see the adius ( ) tend to. The light signals bend and go aound in tighte and tighte iles ith tangential veloity that tends to ( ), hile the adial veloity dops to zeo. 4. It is staightfoad that hen thee is no otation e end bak in fame K ith, L L L, as expeted. 5. To sho that the light signals ill spial out in tighte and tighte iles e may examine d os (4) d hih is ineasing ith diminishing ate as ineases. Simila obsevations hold fo hee d (4) 3 d

13 6. Angle AOB in Figue is tavesed by the signal hilst it tavels the length, and is given by t hee t o tan sinh. Angle an beome vey big and even be the esult of many evolutions. 7. If the signal oiginates fom the peimete toads the ente, then its path ill be symmeti ith espet to the adius OB' in Figue. 8. A plot of fo ineasing time ill look like the folloing Figue Figue 3 The path of a light signal oiginating fom the ente of a otating fame as seen by the non otating obseve O. Numbes on the axes ae nonessential sine saling hanges ith. 8. Genealization to thee Dimensions An obseve O at the ente O of a otating fame K is otating ith the fame. He aies a od (simila to the one e used in the dimensional ase above) pointing adially but ith an angle ith espet to the z axis. As e said in the -dimensionalase, the od is simply an atifat to help us imagine the situation. It suffies to say that obseve O sends a signal fom the oigin ith an angle to the z axis. The situation is depited in Figue 4 3

14 A E B' E' φ F Β O z C F' G ρ ξ O θ ρ' z D Η Figue 4 The signals oiginate fom O and move along the od OA fo obseve O, ho otates ith the fame. The non-otating obseve O sitting at O ill see the signal tavel a helial path OCB' hile the od tavels fom position OA to OB until the signal taveses the od. The pojetion of the veloity veto B'F of the signal on the plane of otation is B'F', as obseve O peeives it, has magnitude sin. The angle EBˆ F = is the angle of defletion fom the adius OzB =OD= hee OzB is dan fom B pependiula to the z axis. The angle tavesed by the od is GOH= ˆ 8. Non-otating neaby obseve O z-axis Suppose a signal ith veloity oiginates fom O ith angle ith espet to the axis of otation as seen by obseve O and is dieted toads A though the od OA (see Figue 4). The adial (in ylindial oodinates (,,)z ) veloity of the signals fo obseve O is sin and the z omponent is os. Suppose no that O z otates ith the od OA ith fequeny as seen by anothe obseve O that sits on top of O but does not otate itho. Let also be the fequeny that O thinks his fame, K, otates ith espet to K of obseve O. Beause of Assumption, and it has meaning to define the angula veloity of the fame K as. Obseve O ill see the signal tavel the helial path OCB' in the same time that it takes to tavese the od fo obseve O, hile the od moves fom position OA to OB. Fo him light tavels along the helial path ith the same veloity and the z omponent equals that of obseve O ( os ). The veloity veto fo obseve O is B'F and it makes an angle ith the z- 4

15 axis. The pojetion B'F' of the veloity veto B'F (tangential to the helial path fo obseve O ) on the plane of otation is denoted as and poj sin (43) The angle beteen it and the adial (in ylindial oodinates) B'E' fo Obseve O is. This angle is alled the angle of defletion of the veloity veto of the signal fom the adial dietion. Let us denote the veloity in the adial dietion (in ylindial oodinates) as obseve O sees it by. Then os and theefoe, poj poj sin os (44) The tangential omponent of the light signal fo obseve O on the plane of otation ill be pependiula to and ill be given by sin sin (45) Also, by the definition of angula veloity, (46) As usual, the pimed quantities poj,,,, ae as obseve O peeives them. Fo the same easons (Loentz ontation of peimete) as in the to dimensional ase e equie that (4) holds. Namely, (47) Solving fo and substituting in v e find (48) hee v and e note that (48) is the same as (6), as expeted. Finally, d (49) dt Equations (44), (45), (46), (48), (49) ae five equations in five unknons:,,,, given,,. Fom (45) and (46) and hene, sin (5) sin os (5) sin ith the ondition sin. And using (48) os ()sin Sine t sin and z t os, e may ite the above elation as, os t os z t sin (5) (53) 5

16 ith the ondition that t os (o z ) (o sin ) (Note that os eithe hen z o hen goes to ) Substituting in (44) d dt d and sine, dt dt dt sin os sin sin t os t sin (54) t t os dt (55) t sin It is onvenient to epesent the integal in the RHS of (55) as a funtion of and t. So e define, Then e may eite (55) as t t os (56) t sin I(,) t dt sin( I,) t (57) Note that fo, (57) beomes t dt asinh t asinh, hih is hat e found fo the to t dimensional ase (eall (9)). Belo e summaize the folloing equations that ae useful fo alulations, (see (48)) t sin (58) sin sin t sin z t os os sin t sin t tan t os sin z sin os os z 8.. Plot of signals fo obseve O t (59) (6) (6) I(,) t os dt (6) Unde the ondition that t os (hih is equied fo os to be a eal numbe) and, and e ant to alulate I (,) t Make the substitution 6

17 And Whee and then (56) beomes k i ot (63) x it sin (64) I(,) t dx isin x k x (65) x But the integal on the RHS is an inomplete Ellipti integal of the seond kind denoted as E( k,) x. Theefoe, (65) an be itten as I(,)( t ot, sin) E i isin it (66) hih an be used fo alulations. If e take the definite integal of (55) fo t os (the limit alloed by the ondition t os ) e find the value that an take fo eah patiula and. Namely, os t os () sin dt sin(,) I (67) sin os m t This says that fo any fixed, the signals in the adial dietion ae bounded (see Figue 5). The signals bend and otate until they eah z at tm, the angle of os defletion beomes 9º degees, the adial veloity diminishes to zeo and the adial distane of the signal beomes. () sin() I (68) m hee, e denote I m ()(,) I os Hoeve, fo e ome bak to the to dimensional ase and the signal is not bounded. It expands sloly all the time logaithmially and again the adial veloity tends to zeo. We said that all signals, hen they eah m sin() Im, they ae at zm. Afte that, the signals ae not obsevable by O Futhe, beause of (5) sin sin sin and fo m, hee sin, e have m m sin (69) and hene at m the angula veloity of the signals m fo obseve O is m (7) m In ode fo the nomal Loentz ontation of the peimete to hold e must satisfy (47) and (48) hih lead to m 7

18 The effet of is to sale don distanes as it ineases. The faste the body otates, the faste the signals bend making tighte evolutions lose to the body. The signals ae not limited in the adial dietion but ove the hole spae alloed by z, as angle vaies fom to In Figue 5 e plot the path of the signals in the egion z fo seveal signals ith diffeent values of.the paths ae fo the egion z ae symmeti ith espet to the plane of otation. (7) Figue 5 Plot of signals emanating fom O fo obseve O as they otate ith adius hile advaning in the z dietion fo the egion z / fo diffeent values of. The numbes on the axes ae not essential sine they depend on the value of. 8. Intodution of peession Up till no e talked of K as otating fame. Hoeve, as e plan late to plae a point mass at the oigin and imagine the signals it sends out, it is useful to add peession to, K sine otating bodies also peess as they otate unless they ae symmetial in thei axes of inetia. It is also possible to add nutation as ell (if the otating body is subjet to an extenal foe). Hoeve, e ill not onside nutation in this study as it does not add muh to hat e ant to demonstate. The effet of peession (and nutation) is the same as if obseve O osillates his od, though hih the light ay tavels, up and don aound an angle fom the z axis. The addition of peession (and nutation) to the ase of obseve O ill make him see a avy and uved path fo the signal, instead of only uved. The peession thus also justifies ou pefeene fo ipples in the to 8

19 dimensional ase that as studied above. Refeing to Figue 6, a point body at O is otating aound the axis OC. The axes OC otates aound the z axis ith the angula veloity of peession. The angle of inlination of OC ith the z axis is. The pojetion of AC on AB hih is dan paallel to the x axis is AD. Hene, AD AC AD AC os t. It follos that os t o tan tan os t hee OA OA AOC and AOD z A D C B y O x Figue 6 A point body at O otates ith angula veloity aound axis OC hih itself otates aound the z axis ith inlination AOC and angula veloity. If e let AOD then tan tan os t. A signal that is emitted by the point body at angle fom the z axis hen the body does not peess (hen ), hen it peesses it ill have an angle to the z axis. The aveage of angle ove time is. In this fomulation the poblem fo obseve O is given by (7) sin()sin (73) d sin()os dt (74) t sin() (75) v (76) z t os() (77) v (78) Solving the same ay e did fo the no peession ase e find, 9

20 And os t os() t sin() t (79) sin()os dt (8) 8.3 Non-otating fa aay obseve O Assuming e have peession, the fa aay obseve O does not see ipples, o even if he obseves them he aes about the staight line aveage fo the adius. This aveage path is given by a signal that has inlination but no peession, as if it oiginates fom a signal ith inlination and in the old of obseve O. The light signal stating fom the ente ith adial dietion that ill follo a avy and uved path (as if the spae has ipples) ith veloity aoding to obseve O, ill appea to obseve Oas uved but non avy having veloity and inlination. The equations desibing the poblem of obseve Oae: Loentz ontation of the peimete, (8) The tangential veloity must equal, sin sin (8) The ate of hange of must equal the adial veloity, d sin os (83) dt The distane taveled in the z dietion is equal fo obseves O and O, os os (84) Whee t sin() and the aveage ove time is given by t sin (85) Sine the aveage of. Solving (8) and denoting fo eonomy as simply e find No solving (86) fo e find that Fom (87) e see that and as, (86) (87). This is also obvious by setting t sin in (86) and letting t thus egadless of the value of. Fom (86) using (85) and taking the deivative ith espet to t e find

21 d sin 3 dt ( t sin) (88) Fom (8) and (83) e have, tan d (89) dt And using (86) and (88) e find tan( t sin) t (9) Dividing (8) by (84) e find tan sin os (9) And using (86) and (9) e obtain t sin os t sin tan ot tan Finally fom (84) and using os e find tan ( t sin) t 3 ( t sin) os tan(( t sin)) t os 3 os( sin) t (9) (93) This equation allos to take values geate and smalle than. The signals aoding to obseve Oae limited in the adial dietion to / hih they appoah asymptotially, as e emaked in (87) and theefoe lie ithin the ylindial sufae of adius / ithout being limited in the z dietion sine z z due to (84). 9. Rotation ith Slippage 9. The to-dimensional Case fo obseve O If e allo the angula veloity to vay as a funtion of the adius it is like having a dis that onsists of ings ith small idth that slide one afte the othe. Let fo example f () (94) hee, () f and non-ineasing in In this setup thee is a multitude of O type obseves, ho stand at the oigin O, one fo eah patiula value of the adius, ho otate ith the angula veloity of that patiula ing. Obseve O is standing on the ente on top of O but is not otating ith the dis. The loks of the to types of obseves ill un at the same ate. Again equations () to (8) ontinue to hold. Substituting (94) into (8) e find, d os (95) dt () t f and

22 If e let f () e o os e find f () () t f t d dt t e (96) (97) dt onst (98) t e To see ho behaves take the deivative of os t d d te () t os os (99) 3 d dt t () t e At t the deivative is negative then as t ineases the deivative ineases and at t it beomes zeo and then positive and finally fo big t it tends to zeo. A plot of os appeas in Figue 7 os Figue 7 The osine of defletion angle φ vesus. The defletion angle stats at zeo (osº=). Then it ineases eahing almost 9º degees (fo big enough ) at. Then it falls again to zeo (os9º=) asymptotially. The defletion angle initially at º ineases appoahing 9º degees (lose to 9º fo highe ) and then dops again to zeo. This behavio is simila to the behavio e have examined fo the otation ithout slippage. Namely, as the angle of defletion ineases the signals stat otating in tighte iles until they eah. Then the signals otate in less and less tight iles until they ae dieted asymptotially adially outad ( ). The hoie of slippage aoding to the ule f () e an be justified by the assumption that fo a dis onsisting of slipping ings eah ings slips ith espet to the pevious by the same popotion in angula veloity. Conside fo example a idth of n layes. The

23 n fist has veloity, the seond, the thid () n n the nth hee n. Eah laye slips ith espet to the pevious by popotion foming a geometi seies. Theefoe, letting n the ing at adius ill have angula veloity. Fo the ase that, hee e ae inteested, e may substitute e hee and then e o f () e. 9. The thee-dimensional Case fo obseve O Fo the thee dimensional ase e assume that thee is slippage both in the adial and the z dietion. To ahieve this e assume that z t ( sin os) e e () hee,,. We disegad peession (assuming the amplitude of peession is vey small) otheise must be eplaed by ompliating the poblem sine is a funtion of t. Relations (44) to (49) ontinue to hold, ith z t ( sin os) e e hee beause hee d e t os sin os sin dt e t sin os ( t sin os) ( t sin os) e z e t os e e t sin () z ( sin t os) () z ( sin t os) () () z t os and t sin. Theefoe, sin( I,, t,) (3) Fo 9 e have, t e t os I(, t,,) dt (4) e sin ( t sin os) ( t sin os) t t I(, t,,) dt (5) t e t Obseve that () is the same as (59), hee e z. The same is tue fo (58) (6), (6). The equied ondition fo () to be eal, is t( sin os) te (6) os It is obvious that fo t lose to and t vey big the above ondition is satisfied, hile the left hand side has only one imum. Theefoe, fo eah, it is eithe satisfied fo all t o thee ae positive t and t ith t t so that it is satisfied fo t ( t,) t and not satisfied ithin the inteval ( t ( t,) t ). 3

24 Given a, if it is satisfied fo all t, then the angle of defletion does not beome 9º degees (exept pehaps at a single point). Theefoe, the signal is alloed to inease its adius fo all t and asymptotially beome adial. If it is not satisfied fo an inteval ( t ( t,) t ), it means that at t the signal has eahed os (angle of defletion 9º degees) and annot inease its adius anymoe. So afte t and until it eahes t the signal is not obsevable in the inteval ( t,) t. Afte t, the signal is again alloed to inease its adius, inease os and beome asymptotially adial. What is the ondition so that given, (6) is satisfied fo all t? It is that the imum of t( sin os) te is less than o equal to. And hat is the imum? Taking the os d ( sin os)( sin os) deivative te t e t (( t sin os)), and imum ous dt at t. So substituting in (6), if ( sin os) os e sin os (7) ondition (6) is satisfied fo all t and thee is no t, t In the opposite ase, hen os e sin os (8) In ode to find t and t e solve (6) t( sin os) ( sin os) ( sin os) te and using the Lambet funtion os ( sin os) ( W (.) ) e obtain ( sin os)() t W and finally os ( sin os) t W () (9) ( sin os) os This, by the theoy on Lambet funtions, gives to solutions, hen ( sin os), hih by the ay is the same as ondition (8) as expeted. os e ( sin os) In patiula, t W () and ( sin os) os ( sin os) t W() heew (.) is the solution nea the ( sin os) os oigin and W (.) is the solution futhe aay fom - on the negative banh of the Lambet funtion. Looking no at os e take the deivative ith espet to t to find its inteio minimum (that oesponds to a imum of ) along the path of the signal. Afte some staight foad manipulation e find, 4

25 ( t sin os) te (( t sin os)) ( t sin os) d e t sin os dt ( t sin os) os sin e t os ( t sin os) ( sin t os) e t os e t sin () The seond fato, in big paentheses, is positive. Theefoe, looking at the fist fato e see that it stats negative fo t and then hanges sign at t z ( sin os)() z () This oesponds to sin t sin sin os () os z t os sin os (3) Equations () and (3) an be ombined using the definition of os and sin in a single equation giving the lous of points hee attains its imum along the path of a signal, z (4) hih is a staight line. The value of os at the minimum is in (,)z spae, ( sin os) e os os os fo ( sin os) os e tt ( sin os) e sin (5) hih is the same as e z os (6) e The ondition in (5) is a ondition on, os e z (7) sin os This by the ay is the same ondition as (7) that is equied fo the non existene of the solutions t, t. The paameti plot of z vesus is a staight line given by (4). The plot appeas in Figue 8 hee to ases ae shon. In Figue 8 (a) the ondition (7) is satisfied fo all z and all. In ode fo this to be tue, e equie the imum ove of the left hand side of (7) to be less o equal to the ight hand side. This imum ous at and at this point ondition (7) beomes e (8) 5

26 In this ase the minimum of os in the dietion of the path of the signal ous on the hombus by evolution ABCD fo all. In Figue 8 (b), (8) does not hold. This means that fo some (7) is valid and fo the est it does not hold. In patiula, talking about the fist quadant, beause the same hold fo the est by symmety, fo GOB ˆ it does not hold, but it holds fo GOB ˆ. So fo GOB ˆ, os does not attain a minimum on the hombus, beause it beomes zeo befoe eahing it, as it enountes the uve AB (maked as t ), hih is the solution of t. This solution as ell as t (the uve beteen AB maked as t ) exist, hen ondition (7), hih the same as (7), holds. Beteen t and t the signal is not obsevable by O, beause os beomes imaginay, unless e allo O to obseve signals tavelling ith speed geate than. Afte t the adius stats to inease again and the angle of defletion os e etuns asymptotially to zeo. Obseve that at A and B sin os. In the opposite ase, hen GOB ˆ, the loal minimum of os is attained on the emaining of the hombus BCD and EFA. Afte that the signal etuns sloly to the adial dietion. Note that in Figue 8 e plot z vs. But obseve O sees instead of, hih is ontated ith espet to. Theefoe, the shape that obseve O ill see ill not be a hombus but ill be defomed sine it ill be ontated in the dietion. e/ o z A E /μ -/λ D ξ B ρ /λ C -/μ J (a) -e/ o 6

27 K z t t L /μ -/λ A G t B ξ F O C ρ -/μ E t H D /λ M N (b) Figue 8 In (a) / e /. The hombus ABCD (bold lines), desibes the lous of points, hee os is minimum. In (b) / e /. In this ase fo GOB ˆ, the signals eah os, hen they aive at the uve maked t. So the defletion angle has eahed 9º degees and annot inease any moe. Beteen t and t the signal is not obsevable by obseve O. Afte that, the defletion stats deeasing and asymptotially beomes zeo, thus the signal etuns to the adial dietion. If on the othe hand, GOB ˆ, then the signal attains its imum defletion on the line of the est of the hombus BCD and EFA and afte that it deeases asymptotially toads zeo etuning to the adial dietion. The diagams sho z vs, hile obseve O sees. So e must imagine a ontation in the dietion to eflet hat obseve O sees, and then the hombus ill be defomed aodingly. In Figue 9 e plot os as a funtion of hen () does hold. In (a) fo a small value of and in (b) fo a big value of. min osφ t min osφ.9 t (a) ξ.5..5 (b) ξ 7

28 Figue 9 Plot of os as a funtion of the inlination of the signal,, as it vaies fom to / hen ondition (8) is valid. (a) oesponds to small value of hile (b) to a bigge one. In (b) the defletion of the signal in the z dietion is smallest and it ineases (the osine deeases) gadually as e appoah the adial dietion. 9.3 Rotation ith Slippage fo Obseve O Obseve O is the fa aay obseve. Fo him the poblem is desibed by equations (8) to (87) ith the only diffeene that the angula veloity is no not onstant but vaies z t ( sin os) ith the distane fom the oigin aoding to e e Fom (86) and using t sin (eall fom (85) that e denote hih is the aveage of by simply ) t sin (9) () z ( sin t os) e t sin e Taking the time deivative and equating to the adial veloity using equation (83) e obtain, 3 (( sin os) sin) t sin sin os 3 () ( t sin) By dividing (8) by () e find t sin tan t () 3 t sin Whee sin os Dividing (8) by (84) and using (86) and () e find t ( t sin) tan tan os sin( sin) t sin Using (84) e obtain 3 t t () os( sin) t t os t sin( sin) t t (3) 3 os tan os tan The atio / tends to hen t o t. Also fo t,. (Reall that is the time hee attains its imum fo obseve O as e found in ()). Sine / stats at and tends at at infinity and fo t it is, it must eithe be flat equal to o have at least one imum fo t. 8

29 The plot of tan vs t appeas in Figue (a) and shos that it has a imum that oesponds to a fo. In Figue (b) e plot / fo small angula veloity and in Figue () / fo big. tan V C V C t (a) t (b) t () Figue In (a) is the plot of tan vs t. In (b) and () is the atio / vs t, fo small small (about ad/se) and fo big (about 3* 5 ad/se) espetively. Taking d e see that it is positive: d d d () e 3 () z 3 () z e and hene lim unless in hih ase lim esult fo the no slippage ase. Futhe, fo t big,., hih agees ith the (4) The plot of the signal as a funtion of time, t, as it is given by (9), hile z advanes aoding to z t os and given by, t t ( sin os)( sin os) t e dt e ( sin os) appeas in Figue, hee the inlination is a paamete. (5) 9

30 (a) (b) Figue The signal path as it advanes in time upad in the z dietion, hile evolving aound the z axis at ineasing in time adial distane. (a) A single signal path. (b) Many signal paths fo the same time inteval ith diffeent. The signals paths toads the positive z semi axis only ae dan. To omplete the pitue one must imagine the same jet of signals toads the negative z dietion. We see ho most of the signal exept those lose to 9 tavel tight to eah othe in the z dietion until they beak up to etun asymptotially to the adial dietion. The jet like fomation is symmeti ith espet to the plane of otation and anothe jet emanates toads the negative z dietion. The imum of tan (o fo ) is had to alulate although manipulation of the gaph that appeas in Figue (a) shos lealy that a imum ous vey lose to t, the minimum of os (o of ) as e found in (), ( sin os) hen e studied the ase of obseve O. It is logial that they must be vey lose sine obseve Osees an aveage of hat O sees. We ill, theefoe, poeed to a fist appoximation as if they ae the same so that t t (6) ( sin os) This, as e mentioned in () to (4) implies that ou imum tan is vey lose to the lous of points (, z ), hee the minimum of os is attained, namely, the hombus desibed by z (7) Whee

31 sin t sin sin os (8) os z t os sin os (9) Then e alulate hat shape obseve Osees as the hombus hee is imized. Fo this e use (9) and alulate it at to find t sin ()sin os e e (3) Using (3) along ith (8) and (9) e plot the line o lous of points (, z ), (eall this is an appoximation as e said above) hee attains its imum, in obseve s O spae (,)z as a funtion of. This plot appeas in Figue and it foms a iga like lous but it may be ide lose to a etangle ith ounded ones depending on the values of and.the idth of the lous deeases ith ineasing. Obseve that OA=OB (substitute (8) into (3) and set / ) and that the lous e osses the z axis at and at. 4 z /μ A O 5 5 B 4 -/μ 3

32 Figue The lous of points (,)z, (hee attains its imum) as paamete vaies fom to /. The othe quadants ae obtained by symmety. The idth naos damatially as ineases. The shape and oundness also vaies ith paametes and. Fo 9 the distane OA=OB. e The defletion angle at its appoximate imum at t t / ( sin os) Assumes the appoximate value t ( sin os) e tan ( sin os)( sin os) (3). Conlusion Stating fom the assumption that to obseves otating ith espet to the othe aound a ommon axis ill agee on epiente angles as fations of a ile but not neessaily on the value of π, e find the length of the adius of a otating dis as seen by the non otating obseve. The adius ill be ontated but not ith the same fato as the peimete beause e allo the value of π to hange fo the non otating obseve ith egads to his measuements on the otating dis. We agued that e have to onside to types of non otating obseves. One ithin the adius / and one outside. Fo the non otating obseve O ithin /, a light signal stating adially fom the oigin of the otating dis (fame) that otates ith the fame at an angle 9 fom the z axis, ill gadually tun sideays foming tighte iles until it asymptotially eahes 9º degees defletion fom the adial as the adius tends to infinity. His spae is distoted and π is diffeent. If a signal is emitted fom the oigin at an angle 9 fom the z axis it again expands otating in eve tighte to one anothe iles, until it eahes z. Afte that the signal is not obsevable by obseve O. Fo the outside obseve O, ho stands outside the ylinde ith adius /, the light ays follo helial paths ith ineasing adius, hih asymptotially tends to /. Next e examined otation ith slippage (otation is not unifom but deeases exponentially as the adius and z ineases ith slippage paametes and espetively). In this ase, the spae is not divided by a ylinde of adius /, but still thee is ontation of the adial distanes. The light ays oiginating fom the oigin at an angle fom the z axis hange thei initial dietion sideays until they eah a imum defletion fom the adial dietion and then asymptotially tun bak to the adial. The lous of points, hee the imum defletion ous is detemined. Fo an obseve O ithin o nea this lous it (the lous) looks lose to a defomed hombus by evolution 3

33 ith the adial distane ontated, hile fo an obseve Oloated fa aay it looks like a iga depending on the slippage paametes. Fo the slippage ase and fo obseve O, it is oth noting the jet like fomations, in the dietion of the positive and negative z axis, of the signals that emanate fom the oigin of the otating fame. Simila jet effets, but less ponouned, should appea fo obseve O fo the slippage ase.. Refeenes. Ehenfest P., Unifom Rotation of Rigid Bodies and the Theoy of Relativity. Physikalishes Zeisthift, :98, 99. Rizzi G., Ruggieo M. L., Spae Geomety of Rotating Platfoms :An Opeational Appoah, axiv:g-q/74v Sep. 3. Gøn O. Spae Geomety in Rotating Fames: A Histoial Appaisal, Fund. Theoies of Physis vol 35, pp85-333, 4 4. Ashoth D. G., Davies P. A., Tansfomations Beteen Inetial and Rotating Fames of Refeene, J. Phys A: Math Gen., Vol, No 9, 45-44, Ashoth D. G., Jennison R. C. Suveying in Rotating Systems, J. Phys. A:Math. Gen. Vol. 9, No, 35-43, Günbaum A., Janis A., The Geomety of the Rotating Disk in the Speial Theoy of Relativity, Synthese 34, 8-99, Günbaum A., Janis A, The Rotating Disk: Reply to Gøn, Found. of Physis, Vol., Nos 5/6, , Mølle C., The Theoy of Relativity, Oxfod, 95 33