Relativiti Aroah to Cirular Motion and Solution to Sagna Effet Yang-Ho Choi Deartment of Eletrial and Eletroni Engineering Kangwon National Univerity Chunhon, Kangwon-do, -7, South Korea Abtrat: he Sagna effet ould not have been exatly exlained with oniteny under the theory of eial relativity (SR). he onundrum of SR ha been omletely olved fully in the relativiti ontext. Seial relativity i reformulated without the otulate of the relativity rinile and the light veloity ontany, emloying a omlex Eulidean ae (CES), whih i an extenion of Eulidean ae from the real number to the omlex number. In the reformulation, the relativity in the rereentation and the light veloity ontany are obtained a roertie that iotroi ae-time ae have. he oordinate ytem eah have erendiular axe in CES and the relativiti tranformation ha the form of rotation. hee harateriti of the formulation ave the way for the relativiti aroah to irular motion. he relativiti tranformation from the inertial frame to the irular frame i hown to have the ame rereentation a that from the irular to the inertial, whih imlie that irular motion an be deribed relative to linear motion and vie vera. he differene between the arrival time of two light beam in the Sagna exeriment an be exatly found by the irular aroah reented, whih how that the non-relativiti and relativiti analyi reult are the ame within a firt order aroximation. he irular aroah an alo be alied to the analyi of the Hafele Keating (HK) exeriment. he analyi of Hafele and Keating aear to exloit the reult of thi aer, though irular motion were treated a liner motion. he relativiti aroah for irular motion, whih i formulated without any otulate, an lead to roound undertanding of relativity and true ae-time ae. hee iue are examined. (Keyword: Cirular motion, Sagna effet, Seial relativity, rue ae-time ae, Paradoxe, Comlex ae)
I. INRODUCION he theory of eial relativity (SR) [ 5] ha been formulated baed on the two otulate, the rinile of relativity and the ontany of light veloity, whih reult in the Lorent tranformation between inertial frame. he Sagna effet [6 ], whih i a henomenon of interferene enountered on a rotating late, eem to how the inoniteny of SR with the reality. he exeriment reult aear to be in agreement with the non-relativiti analyi, eemingly violating the otulate that the light eed i ontant irreetive of the veloity of an emitting oure. hough ome relativiti exlanation on the Sagna effet [e.g., 7] have been given, they do not reent exat analye with rigorou theoretial derivation. he argument on the inoniteny of SR [e.g., 9] till remain. hi aer omletely olve the Sagna effet in the relativiti ontext, introduing a omlex Eulidean ae (CES) in whih time i rereented a an imaginary number. Seial relativity i reformulated in CES without any otulate exet the iotroy of inertial ae-time ae []. In the reformulation, the relativity, whih mean that relativiti rereentation between inertial frame have the ame form, and the light veloity ontany are obtained a roertie that iotroi aetime ae have. he relativiti tranformation an be regarded a a maing from one time-ae oordinate ytem S to the other S. In the Minowi ae [3], the time and ae axe of one oordinate ytem, ay S, are not erendiular to eah other though thoe of the other, ay S, are o. In CES, eah axi of both oordinate ytem i erendiular to the orreonding one and the relativiti tranformation i formulated in the form of rotation. hee harateriti of the CES formulation hed light on the relativiti aroah to irular motion. In the relativiti aroah to irular motion, whih will alo be alled the irular aroah, four oordinate ytem S, S, S, and S are exloited. he oordinate ytem S and S with a tilde are rotating while the other one with no tilde i fixed. he Lorent tranformation from the inertial to the irular i made in the tilde oordinate ytem o that the unrimed i relativitially onverted to the rimed. he tranformation reult are rereented in a time-indeendent oordinate ytem S. he relativiti tranformation from the irular to the inertial, in whih the rimed i relativitially onverted to the unrimed, ha the ame form a the revere one, from the inertial to the irular. In other word, the relativity between the irular and inertial frame hold in term of the rereentation. Moreover the linear veloitie and have the ame magnitude and ooite diretion, i.e., where r / ( r / in the rimed) with r and ( r and ) denoting a radiu and an angular veloity, reetively, in S ( S ) and denote a light eed in vauum. In the Sgana exeriment, the effet of the differene between the travel time of two light beam i
meaured. he irular aroah allow u to exatly find the time differene een through the Lorent len, whih i hown to be, to a firt-order aroximation, idential with that by the non-relativiti analyi. he irular aroah an alo be alied to the analyi of the Hafele Keating (HK) exeriment [, 3]. he analyi of Hafele and Keating ha been nown to be onitent with the exeriment reult. Surriingly, their analyi, exet for art of general relativity, aear to exloit the reult of the irular aroah, though irular motion were dealt with a linear motion. Inertial frame are aoiated with linear motion. For onveniene, the word linear in lae of inertial i often ued uh a linear frame and linear motion. It i ontrated with irular. he terminology linear inertial frame i alo ued to ditinguih it and irular inertial frame, whih i addreed in Setion IV. Following thi Introdution, Setion II reent the reformulation of eial relativity in CES. In Setion III, baed on the reformulation, the relativiti aroah for irular motion i dealt with and time differene in the Sagna exeriment are analyed. he relativiti aroahe reented are formulated without any otulate exet the iotroy, whih may lead to rofound undertanding of otulate, aradoxe, relativity, and true ae-time ae. Setion IV examine thee iue, and i followed by Conluion. II. SPECIAL RELAIVIY IN COMPLEX EUCLIDEAN SPACE he SR wa derived under the otulate of the relativity rinile and the light veloity ontany []. Here, we reformulate eial relativity without the otulate. o thi end, we introdue a oordinate ytem S with an imaginary time axi of it where / i () and t denote time. he ontant rereent the ratio of time to ae in S. For a oint (, x) in S, the alar quantity d (, x), whih rereent the ditane between the oint and the oordinate origin, i defined a d( x /, x) d( ) ( ). (.) In the Minowi time-ae whih emloy the real time axi, the ditane i exreed a d( ) t / ( x ). An oberver O i moving along the x -axi with a ontant veloity of v with reet to O who i loated at x. When, the two oberver O and O meet. At, O i at (, ) where x iv /. A time ae, O goe along the -axi while O follow x the -line whih i the line roing the oordinate origin and, a hown in Fig.. In fat, the -axi i the et of obervation oint of O, the obervation line of O. he -line i the et of
obervation oint of and an be written a O, the obervation line of O. herefore the -line i the time axi for i t. In Fig., the x -axi i loated erendiular to the -axi in a ounter lowie diretion. Aordingly, the ae axi, are aumed to be iotroi in inertial frame o that x -axe i denoted by S, the rimed one. x -axi, for O O i et. Sae-time ae. he oordinate ytem with the - and he oordinate of an arbitrary oint (, x) an be exreed in vetor form a [, x] where tand for the tranoe. A een in Fig., and rereent the angle between the - and -axe and between the -axi and, reetively. Sine and are omlex number, the angle alo beome omlex number. For a omlex number, the trigonometri funtion i i i i o and in are defined a o ( e e ) / and in ( e e ) /(i). It i traightforward to how that o in. When i given a hown in Fig., the trigonometri funtion an be exreed a o ( / d( )) (.) / ( ) where in ( x / d( )) / i( ) (.3) v /. he vetor an be written in olar form a d ( ). From Fig., the oordinate vetor i rereented in S a [o, in ] where o( ). (.4) in( ) Exloiting the um and differene identitie in trigonometri funtion [4], (.4) i exreed a [] ( ) (.5) where Even if,, x, and L o in ( L ). (.6) in o x are hanged to x, x,, and, reetively, (.5) i the ame, whih imlie that ae and time have the relationhi of duality. he matrix ( ) i the x Lorent tranformation matrix in CES. It i traightforward to ee that ) ( ) and L L( L L ( ) L ( ) I where I i an identity matrix. he relationhi (.5) an be written in differential form a d L ( ) d where d and d are differential vetor. It i eay to ee 3
that d d (.7) whih imlie that the tranformation reerve the ditane. Deribing the motion of O from the eretive of O, the oberver O i moving with a veloity of v along the x -axi in S. In Fig., if the oint for i, the oition of the oint for, whih i obtained a (.5), i different from the oition of ine i a omlex number. For examle, when, tan ), it i rereented in S a ( ( / o, ), whih i loated at a oition different from, a hown in Fig.. Let the oint that i loated between the x - and form Fig. that i written a -axe in Fig. rereent that for. It i eay to ee L ( ). (.8) he omarion of (.5) and (.8) lead to the relativity that the relativiti rereentation between inertial frame have the ame form. In SR, the Lorent tranformation ha been derived in the real number with the otulate of the relativity rinile. In the CES aroah, the relativity i a roerty derived from the ae-time iotroy, whih mean that. When the ae-time ae i of iotroy, the obervation line of oberver in linear motion have the ame harateriti, whih enable inertial frame to have the relativity. In fat, the relativity i one of roertie that iotroi ae-time ae inherently have and the iotroy of ae-time ae i more fundamental than the relativity. Ditane are reerved a in (.7). It i well nown that a moving lo run at a lower rate than a lo at ret, whih i due to the roerty of ditane reervation. he moving lo orreond to the lo on an obervation line. In Fig., the quared ditane between and on the O obervation line i written a d d d where d and d with,,. In thi ae the O lo i the moving lo. he i exreed in [, ] S a and [ o, in ] d i given by d d ( d dx ) where d. From d d, the time interval d i written a d dx / d, whih an be rewritten a d d o o that d i maller than d. In ae and are on the obervation line of O, on the ontrary, d beome maller than d. he differential vetor on an obervation line ha ero atial omonent and thu the abolute value of it norm orreond to the time interval. However, the orreonding one in the other oordinate ytem ha nonero atial omonent. A a reult of it, the lo on an obervation line run lower than the 4
other. he time dilation of lo on obervation line an alo be een diretly from (.5) and (.8). he obervation line of O i rereented in S a (, tan ). Subtituting thi vetor into (.5), the time oordinate of i given by. (.9) o he obervation line of O i rereented in S a (, tan ). Subtituting it into (.8), the time oordinate of i given by. (.) o If either d or d, the other, from (.7), i alo ero. he differential ditane d beome ero when dx / dt, whih rereent a veloity, i equal to. In the iotroi aetime ae, the value of orreond to an invariant eed. It i nown (or otulated) that the light eed i invariant in inertial frame, and then. he loe of the line roing the origin and the oint in Fig. i v tan. he veloity of an objet on the line i given by / i tan, whih i rereented in S, by the um and differene identitie, a v / v / v / i tan( ). (.) v v / It i hown in (.) that v beome when v. he invariant eed i alo one of roertie that the iotroi ae-time ae inherently have. he eed that ha the ame value a beome invariant and it i the light eed. he formulation in CES rovide ome rofound oint in the fundamental onet, whih an be ummaried a follow: ) he eial relativity in CES ha been formulated without any otulate exet the iotroy of ae-time ae. he iotroy mean that the ale ratio of time to ae in ae-time ae with unit are the ame. he inherent value of the ale ratio aear to be one in unit-le time-ae before the manifetation. ) he relativity i obtained a one of the roertie that the iotroi ae-time ae have. It i not idential with the otulation of the relativity rinile in SR. he former mean that the relativiti rereentation between inertial frame have the ame form a (.5) and (.8), but not the equivalene between them. he light veloity ontany i alo obtained a a roerty. he iotroy of ae allow them to have an invariant eed. If they are not iotroi o that, the eed are different even though d when d. he iotroi ae have an invariant eed of 5
. 3) he oordinate ytem S obtained form S aording to (.5) i jut for the oberver other word, there i jut one obervation line, the one for not obervation line. he oordinate ytem onnetion with the view that O ee. O, and any O. In x x ( ) in Fig. are S how how the oberver ee the world, in 4) ime and ae are in the duality relationhi. One i rereented in the real number while the other i rereented in the imaginary number. 5) he CES formulation an deal with an inertial frame moving fater than. When v, the role of ae and time are interhanged. A a matter of fat, the Lorent tranformation with the imaginary time i older than that with the real number and wa already ued by Poinare [3]. However, the old formulation may be nothing more than a mathematial maniulation. he CES aroah i baed on the onet of the obervation line and derive the Lorent tranformation without any otulate exet the iotroy. It enable u to get inight into the fundamental onet, a reented above. III. CIRCULAR MOION AND RELAIVISIC APPROACH Here, we extend the CES to 3-dimenion (3-D), inluding the y - and y -axe. Unrimed and rimed oordinate ytem are related by the Lorent tranformation. In addition to S and S, oordinate ytem S and S are introdued to handle irular motion. he oberver of S, S, S, and S are denoted by O, O, O, and O, reetively. Motion between O and O are dealt with. he veloity diretion of O and axi of S, reetively. O alway orreond to the x -axi of S and x - A. Framewor for Relativiti Aroah he unrimed and rimed oordinate ytem S, S, S, and S are hown in Fig., where the radiu of the here, though art of it i hown, i one. For onveniene, negative art of the - and -axe are dilayed and the x -axi i rotated by with reet to the x -axi. he - and x -axe and the - and x -axe lie on the ame lane a the axe of S and S in Fig. are on the ame lane. he oberver O i moving with a veloity of O. Given a oordinate vetor, v / in the diretion of x -axi when een by [, x y] in S, the Lorent-tranformed vetor 6
, [, x y ] in S i written a where ( L ) i the 3 x 3 Lorent tranformation matrix ( ) (3.) L o in L( ) in o. (3.) A roblem in the rereentation into S and S i that if their axe are rotating it i meaningle to rereent oordinate in them. Hene, it i neeary to rereent oordinate in another oordinate ytem whih are time-indeendent. he oordinate of an be readily onverted into a time-indeendent oordinate ytem S. Conidering the rereentation of into S, (3.) i rewritten a ( )( A( ) A ( )) ( ) A( ) where A ( ) i a rotation matrix, whih given by L L (3.3) A ( ) o in (3.4) in o and and are related by A( ). (3.5) he invere of A ( ) i given by A ( ). Form (3.5), A( ) ( ) L, whih an be rewritten a where A( ) A( ) L L ( )( A( ) A ( ) A( ) ( )) A time-indeendent oordinate ytem in the rimed i denoted by S -oordinate into o that the - and (3.6) A ( ). (3.7) S. In (3.7), A ( ) onvert S -oordinate. It an be aumed without lo of generality that when, x -axe of of S, a in Fig. where the y - and S an be laed in the ame lane a the - and x -axe y -axe are overlaed. If, S orreond to S. he S i the rimed oordinate ytem orreonding to the unrimed S. o find A ( ), we ue Fig.. Reall that the Lorent tranformation ha been already done, and 7
the remaining ta i to find the differene in the orientation between S and S to obtain whih an be a ingle variable or a vetor with multile variable. In Fig., the y -axi lie on the x - y lane and the lou of the x -axi form a one a inreae from ero to. It i obviou that if the Fig. 3 learly how the orientation of x -axi i rotated by on the urfae of the one, S exatly overla u S with reet to the here in Fig. i equal to one, the aimuth angle S. Note that, S.. Sine the radiu of u i idential with the ar length between P and P. o ue Fig. for the alulation of the ar length, we hange the -axi to a real axi, - axi. Aordingly, beome a real number and o i given by In the oordinate ytem S with the -axi, the oordinate of the oint P in Fig., are written a he ar length i equal to Sine relative motion between O and the eretive of o. (3.8) ( ) / ( P, whih ha the ame oition a P x, y, ) (o,, in ). (3.9) x and thu i given by u o. (3.) u O are dealt with, hould be ued in aordane with O a i ued in aordane with the eretive of O in the oordinate onverion of (3.5). Fig. how the oordinate ytem from the eretive of the non-rotation (O and O ). Aording to the eretive of the rotation ( O and O ), S and S are rotated by and reetively. hu i obtained a If u,. For,. o. (3.) u A A ( ) i alo a rotation matrix, A ( ) A( ). Uing thi relationhi, (3.3) and (3.7), we have where LR (, ) (3.) LR (, ) A( ) ( ) A( ). (3.3) L 8
) Linear-to-Cirular B. Cirular/Linear ranformation In the linear-to-irular, motion are deribed relative to O or O. We have etablihed a framewor for the relativiti tranformation of irular motion. Conider that varie linearly with time: where (3.4) i and i a ontant. he x - and y -axe in Fig, are rotating around the / rotating enter C R with an angular veloity. he oberver O i loated at a radiu r in S, and the veloity of O i ir r / in the x -axi diretion. We ue a notation v to denote the -D atial vetor of a ae-time 3-D vetor v, exluding the time omonent. he atial vetor [ x, y] [ x, y] and are related by where A ( ) i a rotation matrix for atial oordinate, A ( ) (3.5) o in A ( ). (3.6) in o he diret tranformation of the unrimed oordinate into the rimed one aording to (3.) an be alied to the trivial ae where i ontant. In that ae, with obtained a (3.), we an rereent into S. However, in ae i varying with time, the relativiti tranformation hould be handled with differential vetor whih an be onidered to be ontant during an infiniteimal time interval d. In other word, the vetor in (3.) and (3.7) hould be relaed with differential vetor in uh a way that d ( ) ( ) d ( ) (3.7) L d ( ) A( ) d ( ) (3.8) ( ) A( ) ( ). (3.9) where d v( ) v( d ) v( ) for a vetor v ( ). Note that the new notation d, not d, ha been introdued. he ( ) an be obtained a (3.9) only after ( ) ha been found. Let u exlain the reaon, whih alo exlain why the differential vetor d ( ), intead of ( ), hould be ued for the Lorent tranformation of (3.7). A S rotate, S alo doe. For imliity, uoe that the angular veloity of S i ontant. Fig. 4 illutrate the rotation of the x -axi with reet to x and the differential angle between the axe i d, whih i an angle hift during d. 9
Let vetor be ero at. During a time interval, where N d, the differential d hange from d ( d ) to d ( ) and the rotation angle inreae from to where N d. Aordingly, for examle, the differential d ( d ) and d ( ) hould be onverted in uh a way that d ( d ) A( d ) d ( d ) and d ( ) A( ) d( ). If ( ) i alulated a ( ) A( ) ( ) after firt integrating d ( ) to obtain ( ) or after diretly finding ( ) from ( ) without uing differential vetor, all differential d ( d ) for N are inorretly onverted to the S -oordinate. herefore, differential vetor hould be ued and onverted into S a (3.8). Hereafter, for imliity, we will dro the argument indiating time deendeny in notation, if not neeary. he motion of O een in S an be deribed a r [in, o] d. (3.) he tilde vetor for i obtained by inerting (3.) into (3.5) a [, r]. (3.) Equation (3.) indiate that O i at ret on the y -axi of S. Reall that t. he veloity vetor of i given by d dt r[o, in]. (3.) Rereenting in S, it i exreed by ubtituting (3.) into (3.5) a [ r, ]. (3.3) he angular veloity of O i in the x -axi diretion, a een in (3.3), and O rotate in the x - axi diretion that i erendiular to the y -axi on whih O i loated. he Lorent tranformation in (3.7) hould be made under the ondition that the veloity ha the ame diretion a the x -axi. Note that the veloity of O i in the x -axi diretion, a required. Given [, together with the initial ondition that when, the element of ] [, are obtained a and ] i related to by o (3.4) r [in, o ] (3.5)
where with (3.6) r o r (3.7) o o (3.8) o i. In (3.5), the initial ondition that the hae of i / at wa / ued. For derivation, refer to Aendix A. In ae, any relativiti thing mut not our. It i not diffiult to ee that when, i redued to. he oordinate vetor [, ], where i given by (3.), rereent the time axi of O. Equation (3.4) imlie that the lo of O, whih orreond to a moving lo in the Lorent tranformation of (3.7) ine rereent the time axi of O, run lower than that of O. In the rimed, i written a i r r /. From (3.7) and (3.8), whih lead to he and linear eed are the ame. he r and are linear veloitie. hough the angular veloitie in r r (3.9). (3.3) S and S are different, the are both funtion of r and. It i eay to ee that f ( ) and f ( ) i a monotoni inreaing funtion where f ( ) o / o and. Hene r r and, whih indiate the radiu inreae and the angular eed dereae in S. he deendeny of r on the angular veloity, the diretion of whih i erendiular to the radial diretion, the y -axi diretion in S, may be ointed out. he radial omonent doe not hange in the Lorent tranformation roe, but ome to hange due to the oordinate onverion of form S to he atial vetor given by S. For detail, refer to Aendix A. and Equation (3.3) indiate that are related by O i at ret on the d A ( ). he oition vetor of O i r ] y -axi of S. he veloity [,. (3.3) of i
the ame a (3.) with r, and hanged to rimed one and it i rereented in S a [ r, ]. he veloity i in the x -axi diretion. Aordingly O rotate in the x -axi diretion that i erendiular to the y -axi on whih O i loated, whih i onitent with the requirement of the x -diretion motion in the unrimed. In general, an be exreed a r[ in( ), o( )] r[ in ( ), o ( )] (3.3) where and. he atial vetor (3.3) ha a hae of / at. Comaring (3.3) and (3.), the latter i rotated by relative to the former. he tilde vetor for (3.3) i written a where. he veloity A ( )( d / dt) i given a (3.3). he relativiti A ( ) [, r] (3.33) tranformation of [, ] yield o (3.34) r [in( ), o( )] (3.35) where o. he radiu r and the angular veloity are equal to (3.7) and (3.8), reetively. In (3.34), the initial ondition wa ued that when, and in (3.35) that the hae of i / when with o. he derivation are reented in Aendix B. he vetor i rereented in S a A r] ( ) [,. (3.36) ) Cirular-to-Linear Motion are deribed relative to O i aumed to be oitive. From the oint of view of loated at a radiu r in O or O, a hown in Fig. 5 where the linear veloity in O who i at ret in S, an oberver S of S i rotating with an angular veloity of, whih lead to r /. herefore, S and S beome rotating oordinate ytem wherea fixed. he Lorent tranformation i erformed from S to S for the motion in the O S and S are x -axi diretion and the reulting differential vetor i onverted into S. he relativiti tranformation
roe from the rimed to the unrimed i a follow: where i related to In (3.4), the by d ( ) d (3.37) L -axi i a real axi that the imaginary the ame a (3.8) with relae by oberver O and i equal to he motion of. O an be deribed in d A( ) d (3.38) A( ) (3.39) o. (3.4) -axi of S i hanged to, and o i. Fig. 5 i drawn from the oint of view of the rimed S a where r ]. he veloity vetor of i exreed in [in, o (3.4) d dt S a A( ) [ r, ] (3.4) whih imlie that the angular veloity i in the x -axi diretion. he oordinate vetor of O i given by [,, r ]. With the, the relativiti tranformation from the irular to the linear yield o (3.43) r [in, o ] (3.44) where (3.45) (3.46) ro r (3.47) o o. (3.48) o Note that the linear eed doe not vary. Comaring (3.43) (3.48) for the irular-to-linear with the orreonding one for the linear-to-irular, they have the ame form. In that ene, the relativity between the irular and linear motion hold. Now, we have reared tool for the relativiti aroah to irular motion. Let go to tale the onundrum, the Sagna effet. 3
C. Analyi of Sagna Effet In the Sagna exeriment, the oordinate ytem S rereent the one for a laboratory oberver O. A irular late i rotating around it enter with an angular veloity and the light detetor O i loated at a radiu r when een by O. he motion of O in S an be deribed a (3.) with t where i. At, two light beam leave a light oure, / whih i loated at the ame lae a the detetor, and begin to travel on a irumferene in different diretion. he light ignal traveling in the ame diretion a the late rotation and in the ooite diretion are denoted by and, reetively. Aording to the non-relativiti analyi, the light beam and arrive at the detetor at the intant [9] r t (3.49) ( ) where t and t denote the arrival time of differene in the arrival time i alulated a and, reetively, and r /. he 4 r t d ( t t ). (3.5) ( ) he time differene (3.5) orreond to the one een in S under the aumtion that the eed of the light ignal and Let u find the time differene are in S. d oberved through the relativiti len. We ue the tranformation for the irular-to-linear of Subetion III.B.. In S, the rotated angle for d i given by. (3.5) d hen, d i exreed, from (3.4), a o d. (3.5) Inerting (3.48) and (3.5) in (3.5), we have td o td td o. (3.53) In a different manner, the time differene an be diretly found from (3.43). Relaement in (3.43) by itd / reult in (3.53). he time differene i alulated by ubtituting (3.5) into (3.53) a t d 4
4 r t d. (3.54) / ( ) Comaring (3.5) with (3.54), they are equal within the firt-order aroximation of. IV. BEYOND POSULAES AND PARADOXES A. Relativim in Cirular Motion Debate on relativity in irular motion have a long hitory, inluding the famou Newton buet. he ore of the debate in Galileo relativity i whether or not the motion of O, who i on a late rotating relative to O at ret in S, i an abolute one. In Eintein relativity, the rinile of relativity, together with the light veloity ontany, i alied that the law of hyi are the ame in all inertial frame of referene, whih will be alled the onventional rinile of relativity (CPR). A the abolute motion (or the ether) ould not be found the CPR ha led to the equivalene of all inertial frame and to the eretion that there are no referred frame. he relativiti aroah for irular motion reult in the relativity in the ene that the relativiti rereentation between the irular and linear motion have the ame form. Some relativit may ay, he relativity rinile an alo be alied to irular motion. It wa validated again. Probably, it would not. Intead, the irular aroah may lead to dee undertanding of the relativity. Let u tell a tory, whih i a fition. Suoe that we don t now any theoretial thing of relative motion between irular and linear frame, though ome exeriment reult uh a time dilation were nown. o exlain the reult, we made ome otulate: ) he rinile of relativity (We annot tell whih one, eemingly irular or eemingly linear motion, i atually rotating. Motion are relative, and the irular and linear frame are equivalent.) and ) he linear eed ontany, ay (3.3). With the otulate, we etablihed the theory of irular relativity, whih erfetly and beautifully exlained the exeriment reult. It ha been very onitent with almot every exeriment. Furthermore, an abundane of atonihing new ientifi fat ha been diovered from it. Great advane in iene and tehnology have been ahieved baed on our theory. We, irular relativit, now have big ower and big authority. For a few exeriment reult whih eem to ontradit our theory, we exlain if we an, or neglet if we annot. Nobody an dare to hallenge u. he relativity world loo magi. We, on a rotating late, did handhae with our friend for ten eond and one minute later, we hugged them for ten eond. A linear inertial oberver P ay, You did handhae for ten minute and one hour later, you hugged for ten minute. Another eron, who i in linear motion relative to P, tell another tory. Who i right? he otulation-baed irular relativity may ay that the ten-minute handhae and ten-minute hugging een by P are equivalent to our true ation. 5
In the Sagna exeriment the time differene an be obtained by (3.4) or (3.43). If the otulate that the irular and linear frame are equivalent aording to the relativity rinile i alied, ome roblem are aued. Suoe that the time differene d i obtained from d in aordane with (3.4). Aording to the relativity rinile, (3.43) alo mut be able to be equally alied. he o i the ame irreetive of the ue of (3.4) or (3.43) ine. With d d o, (3.43) give d d o d. A the alulation reeat ontinuouly, the time differene tend to infinity! On the ontrary, in ae the time differene d i obtained from d in aordane with (3.43), d d / o. hen, d i omuted, by ubtituting the d into (3.4), a / o. A the alulation ontinue, d it goe to ero! Similar roblem our in liner inertial frame a well. Let loo into the twin aradox. d d and Our twin, B. win Paradoxe O, left u for a ae tri when. Suoe that the oordinate ytem S S are related by (.5) or (.8), taing no aount of the aeleration. After the ae tri, ome ba to the Earth intantaneouly from a oint in Fig. 6 that the lane of S overla the lane of O, (, ), in Fig. 6. It hould be noted S, and for examle, the oint i denoted a in S and a in S. From the eretive of O, we, O, are moving with a veloity of. Aordingly our oordinate in S are given by, tan ), whih orreond to ( (, ), the oordinate of the oint n in S, where / o. In ontrat, from our n eretive, O i moving with a veloity of and O i urrently at, tan ), the ( oint in S. Whih oint, (, ), n, or other oint, doe O return to? Let u aume that O move to, following the eretive of O. he time oordinate and are then related by o. Aording to the CPR, inertial frame are equivalent o that both (.9) and (.) mut be able to be equally alied. If an objet at move intantaneouly to S, it mut move to the oint in S. Otherwie, if it move to, the equivalene between S and S i violated, the ymmetry being broen. he time oordinate of i o ( o ). A uh intantaneou movement ontinue, it tend to infinity. On the other hand, in ae O move to n, the ontinuation of the intantaneou movement mae it aroah the oordinate origin. he equivalene between the rimed and the unrimed, whih 6
rohibit one frame from deending on the other, eem to aue the aradox and to be elfontraditory. he CPR i inonitent with the Lorent tranformation if what it doe mean i that inertial frame are equivalent. If the frame of O and O in the twin aradox are equivalent, their lo mut be exatly in the ame ondition. he ame law of hyi mut be equally alied to the lo of O a well a that of O alo mut indiate O. If the lo of O indiate -time aing after the dearture, the lo of -time aing o that. Suoe that at ", O and O alo meet another oberver O " who i moving with a different linear veloity ". Aording to the equivalene of CPR, ". heir lo rate are indeendent of motion, whih imlie Galileo relativity. herefore, the CPR mut mean that the law of hyi an be rereented in the ame form in all inertial frame, whih, however, are not equivalent. win aradoxe exit alo between the irular and linear frame when the relativity rinile in the fition i enfored. Let u introdue another verion of twin aradox, nonlinear, irular one. Sadly, our twin A and B have gone to another galaxy, a lanet at it edge. Fortunately we ould alo tae a tri loe to the galaxy. We admire the magnifient enery of the giant galaxy beautifully rotating around it enter with an angular veloity. We immediately found that the lanet i loated at a radiu r from the enter. And we aw that a oon a they arrived at the lanet, one of our twin, twin A, left with an entity, who i very lovely and very benevolent and i a ilot of an identified flying objet (IFO), in the revere diretion of the galaxy rotation. When A returned, we, O, aw our twin again. Who i the younget? he angular veloitie of A and B are a, ia / and, reetively. he b, ib / travel time of A and B from the eretive of O an be alulated a (3.4) with and r /, o ( where r / and a, b. hen the time / ) ratio are given by ratio are given by / o. However, when the travel time are alulated a (3.43), the / o. Whih one are right? Some irular relativit in the fition may ay, here are no referred frame, even between eemingly rotating and eemingly linear inertial frame. hey are equivalent aording to the relativity rinile. he time ratio are all right. he onfuion reult from the ignorane about the relativity of imultaneity. Simultaneou event are valid only in eah frame. hen, it beome another aradox, irular verion. he mehanim auing the aradox between the irular and linear frame i exatly the ame a that between the inertial frame. he aradox in the latter i aued by the equivalene of (.9) and (.) and in the former by the equivalene of (3.4) and (3.43). In SR, elaed time in the unrimed 7
and in the rimed mut not be omared ine both (.9) and (.) annot be atified at the ame time, whih led SR to introdue the o alled relativity of imultaneity. We need more aradoxe, do we? Here i another one, not aoiated with imultaneity. he oberver O and O met at. From the eretive of O who thin that O i in irular motion, the radiu and the angular veloity of O are given by (3.7) and (3.8), reetively. he O thin that O i in irular motion. Alying the relativity rinile, the radiu and the veloity in S are written, by ubtituting (3.7) and (3.8) into (3.47) and (3.48) reetively, a of O are r and r (o / o ) and (o /o ) reetively, though the meaurement. Whih one are right? he otulate eem to reate aradoxe and ontradition, doen t it? he relativiti tranformation for irular motion ha been formulated without any otulate, whih may lead u to reolve the aradoxe. C. rue Sae-ime Sae In SR, dee root of aradoxe and ontradition lie in obervation line or mionetion of the oordinate ytem S. Let u reviit the reformulation of eial relativity in CES. Only the -axi in S i an obervation line, on whih obervation line. When S i given, O ee event. Any other line S how how x x ( ) are not O ee the world from S. In SR, all x x eem to be regarded a obervation line. Let every x x in Fig. be an obervation line (aording to the view of SR). At, if O meet O, any other objet in S annot meet any objet in S and only one objet i allowed to meet. hi iue i alo aoiated with imultaneity, and it i jutified in SR by the relativity of imultaneity. he ituation, however, i different between the irular and linear motion. Objet that are at ret on a rotating late belong to the ame inertial frame, whih i termed a irular inertial frame. Conider N objet loated at a radiu r that are at ret on a rotating late, a in Fig. 7. Let the irular inertial frame S for the objet be a true loal ae-time ae that onit of obervation line, whih will be diued later. he oition vetor of the th objet O an be written a, r [ in, o ] where tranformation from the irular to the linear, aording to the eretive of i a ontant. he relativiti O, an be erformed uing (3.37) (3.39). he atial vetor, i then given by, r [in, o ], a hown in Aendix C, where o. When O meet O at, thee objet O in 8
S, whih belong to the ame irular inertial frame, an alo meet reetive objet O in the linear inertial frame, the atial vetor of whih are equal to,. Reall that the minu ign in the equation o reult from the fat that O (or O ) ee S rotating in the revere diretion of the rotation of S that O (or O ) ee. Fig, 7 ha been drawn from the eretive of O o that the hae of, and have the ame ign in the figure. hough in the irular, motion any number of objet in the rimed an meet reetive one in the unrimed at the ame time, the number ha to be only one in SR. If r i very large, the objet O are eentially on a line and in uniform linear motion relative to O. hey an be moving even very fat a r i very large, even if the angular veloity i near ero. A exlained, thee objet an meet reetive objet in the ame linear inertial frame at the ame time. However, the equivalene of CPR doe not allow it. Let u uoe that a linear inertial frame S rereent a true loal ae-time ae and that multile objet in the rimed linear inertial frame atually meet reetive one in S at the ame time. hough the objet in the rimed meet the orreonding one in S at, eah ee differently. Let rimed objet O and O loated at x and x x meet unrimed one O and O loated at x and x x, reetively, a in Fig. 8. Even if S rereent a true ae-time ae, the rimed oordinate an be obtained by the Lorent tranformation. he oordinate ytem S of the unrimed oordinate vetor O i related to the oordinate ytem S of O by (.5) or (.8) with ( ) [ (), x ()] and the rimed one x ] ( ) [ (), () where ( ). he oordinate vetor of O i exreed a ( ) [ (), x ] in S and that of O a ( ) [ (), x ] in alulated a S. he rimed vetor when O meet O at i () x x ] ( ) [ ( ), ] L( )[,. (4.) Equation (4.) i rewritten a in and x o, whih imly that it eem to O ( ) x x that when ( ) x in, O loated at x ( ) x o met O, a hown in Fig. 8, though they atually met at. In ontrat, O ee atually meet at (. Only the -axi of O ee the world from S. ) O meet O at x in though they S i the obervation line, and S how how Aording to Feynman, the rinile of relativity i deribed a [5, 5]: if a ae hi i drifting along at a uniform eed, all exeriment erformed in the ae hi will aear the ame a if the 9
hi were not moving, rovided, of oure, that one doe not loo outide. hi i the meaning of the rinile of relativity. A long a one doe not loo outide, inertial frame may be equivalent in the retrited ene that the law of hyi are equally alied to every inertial frame. However, return of twin orreond to looing outide. If one i looing outide, in other word, if the unrimed and rimed obervation line are omared, the equivalene between them doe not hold any more. hen, the twin aradox no longer beome a aradox. In linear motion it may not be neeary to loo outide beaue two oberver go far away from eah other. However, irular motion may fore the oberver to loo outide beaue they an ee the ame outide again. We do not have to worry about loing the elegant Lorent tranformation even if the equivalene between inertial frame doe not hold and true ae-time ae are introdued. Of oure, the irular aroah i alo baed on the Lorent tranformation in CES. We an ue it, but with exat meaning. What i the true ae-time ae? An obervation line inlude a et of event that atually ourred to the orreonding oberver. he event on obervation line are true event that atually our. A true ae-time ae i a olletion of obervation line. One obervation line i a true ae-time ae retrited to the owner of it. Obervation line are lie et of imrint arved on ae-time ae regardle of motion. hey annot be hanged by the obervation or the rereentation in other oordinate ytem. he rereentation of true event in a oordinate ytem diaear a oon a it oberver leave the oordinate ytem. However, the true event do not diaear. If either or rereent a true event, the other an be deendently given by the Lorent relationhi. he word roer ha often been ued uh a roer time, whih are found by following o alled world line. he world line, whih robably orreond to the obervation line in bai meaning, an be alled the roer line. We an relae roer with true, an t we? hough it eem to admit referred frame and to violate the relativity rinile in term of the equivalene. Are there any other roer line for a roer one? Exeriment of twin aradox have already been done on rotating late, the Hafele Keating exeriment. he analyi of Hafele and Keating ha been nown to be in good agreement with the exeriment reult [, 3]. he analyi, exet for art of general relativity, aear to exloit the reult of thi aer, though irular motion were treated a linear one. In the HK exeriment, three time meaurement were obtained, one on the urfae of Earth and the other from the lane traveling in the Earth rotation diretion and in the ooite. We ue the reult of Subetion III.B. for the analyi of the HK exeriment reult. Sine the flight altitude are negligible a omared with the Earth radiu, the three oberver, O, O, and O, an be onidered to be rotating at an equal radiu r with different veloitie where O denote the Earth urfae oberver, and O, and O the lane oberver in the ame and the ooite diretion, reetively. he ubrit number
will be ued to indiate the oberver with the ame number o that for examle,,,,, denote the angular veloity of O. In the HK analyi, the unrimed time i emloyed for omarion. he unrimed frame S i a ret frame with reet to the Earth rotation. From (3.43), the time differene for the travel are written a t l tl t, l (o l o ) tl, l, (4.) where t l, t, l and t l denote the time meaured, reetively, by O l, O and O, an oberver / of S, during the travel of O. In (4.), ) o ( ( ) where r,,, l /. he unrimed time were not meaured, and t l eem to have been alulated a t r / v where v i the ground eed of l l l O. Four oberver aear in the HK analyi. hu, there are twelve relationhi of relativity between them. he exeriment reult learly how that there are no equivalene between them. he HK analyi for the time dilation only due to the irular motion with angular veloitie indiate that t and t [], whih are alo een from (4.) by the irular aroah. If (3.4), intead l of (3.43), for the linear-to-irular i ued, their ign hange, i.e., t and t. he oberved time in (4.) are true one, reorded on the reetive obervation line, whih an be rereented in other oordinate ytem. he true event on the time axe of the oordinate ytem S,,,, an be oberved in S, and t l in (4.) indiate the travel time of O l een by O. A far a the travel time are onerned, t l deend on t l. he exeriment how that (3.43) only i valid and o t l annot be obtained from t l by alying (3.4). A mentioned above, if either or i a true event, the other an be deendently given. What if both and rereent true event? In twin aradoxe, both of them rereent true event. hee ae an be reolved by diovering whoe lo i the moving lo, whih an be nown through exeriment. ime dilation, whih imly that moving lo run low, have been well validated through quite a number of related exeriment inluding the HK one [6]. However, the time dilation i inomatible with the equivalene of inertial frame ine one lo i in motion relative to the other. he inomatibility aue the twin aradox, whih i about whoe lo i the atually moving lo with reet to the other. It i the lo that how a time dilation relative to the other when elaed time interval are omared. he HK exeriment learly how whoe lo are the atually moving lo relative to other. A mentioned in Setion II, a moving lo i the one on an obervation line. he oordinate on an obervation line have ero atial omonent, or it differential atial vetor i ero o that the
relativiti tranformation for the differential vetor i written in -D a L ( ) [ d, or d ] L ( ) [ d,, whih lead to d o d ] d or d d o. In the irular aroah alo, differential time are given by the ame equation a in linear inertial frame. In the HK exeriment, the travel time when only onidering the effet of eial relativity an be obtained from the ame equation a well. Hene, equivalently the obervation line of O, O, and O an be deited a in Fig. 9. he ( ) -axi,,,, i the time axi for O. he lane oberver O l, l,, who i at the oint ( l ), intantaneouly move to O. If their relative motion are equivalent, the elaed time interval mut be all equal. Even though the travel of O and O are ymmetri with reet to O, and are different. he roblem of the twin aradox i where, the oint or in Fig. 9, O i on the obervation line at the intant that eah O l doe intantaneou movement. he movement to the oint (oint ) imlie that the lo of O l ran at a fater (lower) rate than the lo of O. Aording to the HK exeriment, O move to the oint while O move to the oint. If an objet on the -axi intantaneouly move to the (l ) -axi, it doe to the oint ( l ). here i no aradox. Inertial frame are equivalent, rovided that they do not loo outide. However, when looing outide, for examle, omaring time aing, the inertial frame are not equivalent a exeriment how different elaed time interval. In general relativity, the aeleration i deribed a with reet to omething, ay aeleration with reet to a tangent lane. In eial relativity alo, relative motion hould be deribed a with reet to omething. If O time, it imlie the motion of () O time run lower than O with reet to O. A a matter of fat, the formulation of eial relativity in Setion II, whih ue Fig. drawn from the eretive of O, imliitly imlie the motion of O with reet to O, though (.5) ha the ame form a (.8) o that their motion aear to be equivalent. A the aeleration in an aelerated motion tend to ero, it redue to a uniform linear motion. When alying with reet to omething to eial relativity, the inoniteny in deribing the aelerated and the uniform motion diaear. Relative motion an be rereented in the ame form, even for irular motion, but they are not equivalent beaue their elaed time interval are different. On the ontrary, if they are the ame, the eial relativity redue to Galileo relativity. A true ae-time ae i a olletion of obervation line. An obervation line i a et of true event. Preiely eaing, a true ae-time ae i a olletion of et of event whih ourred,
our, and will our on time axe of oberver. We an imagine the true ae of a linear inertial frame S a follow. An oberver O loated at x i at ret in S. Let an event on the obervation line of O at time be e(, x). he event e(, x) i the event at (, x) in the true ae S. We an alo imagine a true ae in irular motion. Oberver who are at ret on a rotating late belong to the ame irular inertial frame S. An oberver O i loated at a atial oordinate ( r, ) in S where i an aimuth angle. If an event on the obervation line of O at time i e (, r, ), it i the event at (, r, ) in the true ae S. If all atial oint of S belong to a et, ay {( r, ) r r r, }, S i a true ae over. A true ae an have jut one entry. In the HK exeriment, the obervation line of O, O, and O are all reetive true ae. heir relative motion an alo be deribed a with reet to, aording to the HK exeriment reult, whih indiate the motion of O, O and O with reet to O, the motion of O with reet to O and o on. he eron P ay, You did handhae for ten minute and one hour later, you hugged for ten minute. We exre our warm reet for P oinion beaue P i jut aying what wa een, it true, and beaue we are one. he thing een are P true event, belong to P obervation line, and the ation of the handhae and hugging are our true event, not P, whih belong to our obervation line and whih annot be hanged by other obervation. We eah are waling on eah one own obervation line on the Earth, engraving eah life on eah obervation line. he Earth lane, whih i a olletion of thee obervation line, i a true loal ae-time ae irreetive of the motion of our Earth, Solar Sytem, and Mily Way. We live in the true ae. V. CONCLUSIONS he elebrated Sagna exeriment may have hidden reiou treaure that an ave the way for relativiti aroahe to irular motion and rofound undertanding of relativity. he reented formulation for irular motion, whih have been obtained without any otulate, an allow u to deal with relativity roblem without any retrition. Relativity between irular and linear frame a well a between linear frame hold, but a a roerty derived, not a otulate. Motion are relative and they an be rereented in the ame form, even for irular motion. Relativity hold in term of the rereentation. However, relative motion are not equivalent ine their elaed time interval are different, a hown in the HK exeriment. he reformulation of eial relativity in CES i more fundamental than in the real number. Beide the oint reented in Setion II, it ha hed light on the relativiti aroah to irular motion. Now 3
it time to be free from, to go beyond the otulate and to loo into them a roertie. In the reformulation, the relativity between linear inertial frame i obtained a a roerty derived from the iotroy of the ae-time ae, i.e., the ame, the ratio of time to ae in a manifeted univere. When a eed i the ame a the value of, it beome invariant from frame to frame. he relativity and the light veloity ontany are roertie that the iotroi ae-time ae have. If they are otulate, they mut be unonditionally aeted. A a reult, aradoxe and ontradition may have been reated in SR. he aliation of SR may be limited under the otulate, maybe with aradoxe, ine it ha been derived from them. However, the CES formulation an be alied without the limitation. A true ae-time ae i a olletion of obervation line, whih are et of true event. rue event annot be hanged by the obervation or the rereentation of other frame though they are differently een. In irular motion, a true ae onit of obervation line of objet at ret on a rotating late. he bai motion in univere, marooi and mirooi, i irular. he relativiti aroah for irular motion may rovide a very imortant bae for dee undertanding of them. It an allow u to ee the Solar Sytem and galaxie with the eye of an oberver at ret with reet to their motion. hough we are alo in variou irular motion, we live in the true ae-time ae, maybe oming into a irular age from the linear age. With APPENDIX A [in, o given, the tilde vetor r ] beome a ero vetor o that d ] Reall that. Subtituting (A.) into (3.7), it follow that i written a r] [,. hen [ d,,. (A.) d d o (A.) d in d. (A.3) Under the initial ondition that when, the integration of (A.) lead to (3.4). Note that the y -omonent of d i ero, whih imlie that the Lorent tranformation doe d not hange the radial omonent of d. From and (3.), we have d d. (A.4) o he differential vetor d i obtained, by uing (A.3), (A.4) and (3.8), a 4
where r i given a (3.7), with the ue of d in d o o d in do r din (A.5) in / i o /( i / r) ro. Integrating (A.5) with reet to and uing the initial ondition that the hae of i / at, we have (3.5). hough the Lorent tranformation doe not hange the radial omonent, the oordinate onverion from S to S reult in the hange of it. Uing (A.) and (A.4), we have hen do d. (A.6) o d o. (A.7) d o APPENDIX B Let be. he variable ( ) an be exreed a he vetor hen a i written a. (B.) [ in, o and the tilde vetor r ] d beome a ero vetor. It i obviou that d ] i given by (3.33). d d. he differential vetor an be written [ d,,. (B.) Subtituting (B.) in (3.7), it follow that d d o (B.3) he integration of d in d. (B.4) d under the initial ondition that when lead to (3.34). We introdue a variable defined a where o. It i lear that d d and d d. he differential d an be written from (3.) a From (B.) and (B.5) d d o. (B.5) 5