The Relativistic Field of a Rotating Body

Transcription

1 The Relaivisi Field of a Roaing Body Panelis M. Pehlivanides Alani IKE, Ahens 57, Greee ppexl@e .gr Absra Based on he pahs of signals emanaing from a roaing poin body, e find he equaions and properies of he field ha hey form. Depending on he ype of observer he field differs. The field is no enral bu varies in orienaion and magniude ih boh disane and angular veloiy of roaion. The magniude of he field in some ases forms a barrier aay from he origin, hih may be very srong depending on he angular veloiy of roaion. The resuls apply boh o miroosmos (sub-aomi level) and maroosmos (osmi level).. Inroduion This paper is a oninuaion of he paper [] ha sudied he relaivisi roaion of frames and he signals emanaing from a roaing poin body loaed a he origin. A field an be vieed as signals ha do no ross eah oher. These signals are reeived by a body ha is subje o he field and indue i o a in a erain ay. We ill assume ha hese signals behave like ligh signals and ravel he same ay. This ill allo us o use he resuls found for he pahs of he signals emanaing from a roaing body [] o dedue he form and behavior of a field, e all G, due o he mass of he body and is roaion. We examine o ypes of roaion: One is alled roaion ihou slippage, here he angular veloiy of he spae around he roaing body is onsan regardless of disane from he body and he oher is roaion ih slippage, hen roaion of spae has an exponenially dereasing angular veloiy as he disane from he roaing body inreases. In eah ase e ill disinguish beeen o ypes of non roaing observers: One lose o he body and one far aay. The observers are assumed mass-less and no affeing or affeed by he signals emanaing from he roaing body. In he ase of onsan angular veloiy he far aay observer ill noie ha a ylindrial barrier, (rapid inrease of he field and sideay urn of is direion) is formed a radial disane, in ylindrial oordinaes, / from he axis of roaion, hile for he nearby observer no suh barrier is formed. In he ase of exponenially delining angular veloiy, he far aay observer ill see a barrier being formed only hen he angular veloiy is very big and in ha ase, he barrier is formed a a radial disane approximaely inverse o he angular veloiy, leading us o he subaomi disanes (miroosmos) and resembling he non slippage ase. This barrier is sronger as he angular veloiy of roaion inreases. The haraerisi of he barrier is, as in he previous ase, a rapid inrease in he magniude of he field and urn of he direion of he field from he radial direion. Ouside his barrier, he field gradually regains is radial direion and normal magniude as he effe of roaion of he body on spae delines, and reurns bak o he normal Neonian graviaional field. On he oher hand, for small angular

2 veloiy no barrier is formed as is he ase for osmi disanes (maroosmos). Simply he field direion sars radially (in ylindrial oordinaes) from he body and urns gradually sideays ih respe o he radial unil i reahes a maximum defleion and hen urns bak gradually o he radial direion and again ends up looking like a normal Neonian graviy field. The sideays urn is no aompanied by an inrease in he magniude of he field as in he miroosmos ase. This paper is organized as follos: In seion a shor revie of previous heory is presened. In seion he onneion beeen signals and fields is exposed. In seion 4 e find he field, G, for an observer, ho does no roae ih he body, hen he angular veloiy is onsan ih respe o he disane from he body (no slippage ). In seion 5 e reain he no slippage assumpion bu hange o he far aay observer, and alulae he field G ha he sees. In seion 6 e alulae he relaivisi mass of he roaing body. In seion 7 e assume ha he angular veloiy is dereasing exponenially ih he disane from he body (slippage ase) and e alulae and presen he graph of he field G ha he far aay observer sees. I is shon ha a barrier is formed a he miroosmos level. In seion 8 e oninue ih he slippage assumpion and shorly disuss he G field for his ase. Conlusions follo in seion 9. A shor revie of formulas relaed o previous heory on roaing frames and he pah of signals emanaing from a roaing body a he origin. We ill summarize he resuls of he heory [], on hih his paper sands by presening he ransformaion of ylindrial oordinaes for eah ase. A. Roaion ihou slippage (he angular veloiy of roaion of signals is onsan ih respe o he disane from he roaing body). Preession of he roaing body is assumed having a very small ampliude and is hus negleed. A.I Observer O a he origin bu no roaing ih he body. (The ransformaion holds for z ). sin( I,) () v () z z () (4) (5) I(,) v v (6) here,, z,,, v are he radial disane in ylindrial oordinaes, he angle of roaion as fraion of a irle (for example degrees), he z direion ha oinides ih he axis of roaion, ime, he number pi, and he frequeny of roaion respeively for observer O, ho is loaed a he origin and roaes ih he body. And here,, z,,, v are he same quaniies for observer O, ho is loaed a he origin bu no roaing ih he body. The speed of ligh is for boh observers. Furher, here,

3 ih os z z I(,) os d (7) os z os sin, sin, sin, z os here is he z angle of inlinaion of he signal ih respe o he z axis. Angle is he angle of defleion of he signal from he radial as observer O sees i. From he above e an find he ransformaion of he angular veloiy using he formula ( v and v ) and he angle of roaion measured in radians (using and ) as, (9) () () Relaion (5) is obained by requiring ha he speial relaivisi Lorenz onraion of he perimeer holds for all ligh rays (8) () and using he fa ha v and v e find may express () as From his (5) is obained. Noe ha hen z ( 9 ) () beomes, arsinh(). Then e () A.II. Observer Ois he far aay observer ouside he ylindrial volume defined by ) for hih he ransformaion belo holds. () v (4) z z (5) (6) (7) v v (8) (9) Where he double primed quaniies have he same meaning as he single primed above bu refer o observer O. The angle of defleion is, and is given by an( () sin)

4 And he angle of inlinaion of he signal ih respe o he z axis is given by an an ( sin) ( sin) () B. Roaion ih slippage. (The angular veloiy of roaion of signals dereases exponenially ih respe o he disane from he roaing body). This ase has more meaning physially han ase A above, and e also avoid he unnaural boundaries ha appear a z and a. The angular veloiy is given ()( sin z os) by e e ih, and he frequeny of ()( sin z os) roaion is v ve ve. Preession is assumed o have very small ampliude and is negleed oherise mus be replaed by here an an os, here is he angular veloiy of preession. here B.I. Observer O a he origin bu no roaing. sin( I,,,) () ( sin os) ( sin os) v () e ve d ( sin os) () z z (4) (5) () z e (6) I(,,,) os d (7) here is he angle of defleion of he signal from he radial and os e os z e e sin e ( sin os) () z ( sin os) () z (9) and using () ih (6) and he fa ha,, e find he ransformaion of he roaion angle in radians ( sin os) ( sin os) () e () e d ( sin os) () e Also assume ha, he ondiion needed for os o be real for all. (8) B.II Observer O(he far aay no roaing observer) () z e () 4

5 ( sin os) ( sin os) v () e ve d ( sin os) () z z () (4) (5) (6) ( sin os) () d ( sin os) e e (7) ( sin os) The angle of defleion is given by z an ( sin os) sin While he veloiy of signals ( sin) as observer Osees hem ill be () z (( sin os) sin) sin( sin) The inlinaion of he pah of he signal ih respe o he z axis is given by So ha os an (( sin os) sin) an an sin ( sin) (8) (9) (4) os os (4) Signals and Fields Consider a poin in a field. A his poin he field has a magniude and direion. Le a small fla surfae a, hose normal is poining in he same direion as he field. We ill define he field srengh n as he number of signals per uni surfae per uni ime falling perpendiular o he surfae. Le also, v denoe he veloiy of he signals. We ill assume ha he direion of he field is given by he opposie direion o ha of he veloiy of he signals. Then aording o our definiion, G nˆ v (4) Where G is he veor field and ˆvis he uni veor in he direion of he veloiy of he signals. Leing no a beome infiniesimal da e say ha he infiniesimal volume dv raversed by he signals in ime d is dv v da d (4) Also he number of signals dn in he infiniesimal volume dv is, dn n da d (44) dividing e obain, 5

6 dn n dv v Subsiuing n from (45) ino (4) e find, dn G v (46) dv As an example, le us onsider a field (like he Neonian graviy field) ha sends signals radially from a body of mass m. We assume ha he oal number of signals emied from he body per uni ime is proporional o m, say k m for some onsan k G. Sine he signals are emied spherially hey ill ross he spherial surfae a disane r homogeneously. In order o find he n (number of signals per uni ime per uni surfae rossing he surfae of a sphere a disane r ) e mus divide he oal number of signals per uni ime by he surfae area, kgm n (47) 4 r I follos ha his field may be represened using (4) by, kgm G nvˆ v ˆ (48) 4 r here ˆv is he uni veor in he direion of he veloiy of he signals, hih in his ase oinides ih he radius r. 4 The field G reaed by a roaing body and no slippage (observer Oase A.I) As disussed in [], he signals emied by a roaing poin body look differen o observer O(ase A.II above), ho is ouside he volume defined by (hih desribes a ylinder of radius /) and o observer O (ase A.I) above), ho is ihin his volume (as O defines i). We ill examine he o phenomena separaely saring ih O. Ho ill he field G look o observer O, hen he body m roaes ih angular veloiy? There ill be o effes ha deermine ha O observes. One is ha he roaing mass ill look bigger due o relaiviy, (denoe i m ). This effe, hih appears if e le he roaing body have dimensions insead of being a poin mass, ill be examined in seion 6. The seond effe is ha he signals emanaing from his body ill no follo a radial pah. Le s sar ih he seond effe ha is he signals ha are emied from suh a body and look a Figure. The signals aording o observer O, ho roaes ih he body, ill expand spherially and he field G ill follo equaion (48), he lassial Neonian field. The radially raveling signal OAC for observer O ill be mapped o a helial signal shon as OA'C' for observer O. A small volume dv d dzd (in ylindrial oordinaes as observer O pereives spae) around poin suh as A or C ill be mapped o a small volume dv d dzd in (ylindrial onraed oordinaes as observer O pereives spae) around poin A' or C' respeively. Sill boh volumes ill onain he G (45) 6

7 same number of signals dn. I follos, herefore, ha O ill observe a field G aording o, (46) dn G v (49) dv here v is he veloiy veor for signals as seen byo. Using he hain rule of differeniaion (49) an be rien as dn dv G v (5) dv dv Assuming ha signals ravel ih he same speed for our observers v v, e may rerie (5) as dn dv dn dv dv G v ˆ ˆ ˆ dv v v v dv dv G v dv dv (5) z B O C' A' C A Figure The signal as observer O, ho roaes ih he body, sees i is he sraigh line OCA. For observer O, OCA is mapped o a helial pah shon as O C'A' beause of onraion of radial (in ylindrial oordinaes) disane due o roaion. For he ase of he graviaional field using (5) on (48) i beomes, kgm ˆ dv G v (5) 4 r dv here e have replaed m by m o indiae ha he mass ill look bigger, hen i is roaing. Using ylindrial oordinaes, r z, e obain, G kgm dv ˆ 4 z v dv (5) here vˆ is he uni veor in he direion of he signals as seen by O. 7

8 Wha remains no is o alulae dv dv. Bu dv dv (,,) z ransformaion,. (,,) z For ase A.I equaions () hrough () apply, hih hold for ), he ransformaion is, os here () is given by he Jaobian of he z, (equivalenly sin osd sin( I,) (54) and z z (55) () (56) os The Jaobian of he above ransformaion is os z sin (,,) z (,,) z z z (,,) z J (,,) z z z (57) z z z z Observe ha sine from (54) does no depend on, hih as anyay expeed beause of symmery around he z axis as e ake ino aoun all signals emanaing in all direions from he roaing body. No (58) hile from (5) I (,) (59) Hene, subsiuing (58), (59), (54) in (57) i beomes, I (,)sin(,) sin I sin(,) I sin( I,) (6) or I (,) J sin(,) I z (6). 8

9 here e have used sin z Fous on he erm I(,) in (6). and observe ha e may rie I (,) I*(os,) sine I (,) an be regarded as a funion of os insead of. Hene, e may rie, os I(,) I*(os,) *(os I,) *(os I,) os (6) No alulae erms one by one, 4 os 4 os I *(os,) d d os os sin os( sin) (6) and leing 4 U (,) d (64) os( sin) 4 I *(os,) os(,) U os Also os z sin os z z os sin z z sin Subsiuing (64), (65), (66), (67), (68) bak in (6) and hen in (6) e find, I (,) 4 os sin os sin U (,) z I (,) here e have used he fa ha os Finally, subsiuing in (6) and using (67) i beomes, 4 os sin(,) U sin os J os I(,)(,) I An alernaive form for (7) is obained by realling ha z, sin z, os z z (65) (66) (67) (68) (69) (7) 9

10 4 z z U (,) os J ()(,)() z I z I (,) z By similar manipulaions e obain anoher form of (7) in erms of and J sin os(,) U sin os dv No e may use J in (5) sine and find he field, dv J 4 os I(,)(,) I sin (7) (7) k m k m G v ˆ v ˆ G G 4() z J 4 z U (,) os z 4 z I(,)()(,) z I (7) The formula above expresses G in erms of, a quaniy observed by observer O. If e ish o express i in erms of ha observer O sees, e mus subsiue aording sin( I,) o (54). Namely, sin( I,)(,) I The pariular ase hen z implies ha d I(,)(,) I arsinh arsinh z (74) Using (74) ino (7) and seing z, e find he field a he horizonal plane a he origin, hih forms a dis as he signals a he horizonal plane roae and expand ouard logarihmially: kgm()arsinh() kgm G vˆ v ˆ (75) z 4 4 here e have used he fa ha os z The direion of he field G G sin os, G G sin sin v ˆ ( v, v,)(sin v z os,sin sin,os) Figure G is given by is ylindrial omponens: G G os, or he uni veor ˆ v is given by. A plo of he field srengh appears in z

11 Figure G he same general shape. versus and z. The effe of is o derease he sale as i inreases keeping 5 The G field reaed by a body roaing ihou slippage (far aay observer O ase A.II) From () o (9),hih apply for ase A.II above, e have, (76) z z (77) (78) The angle of defleion is an( sin) (79) And he angle of inlinaion is Sine and () an an ( sin) ( sin), he Jaobian is given by J (8) () Using he same argumens as in seion 4 e may express he G field as in (5) G kgm dv ˆ 4 z v dv (8) dv Where dv J and m m. The equaliy m m is shon in seion 6, here aq alulaion of m is also presened. Noe also, ha vˆ is he uni veor in he direion of he field. (8)

12 Then (8) beomes No using (76) e find ha Subsiuing his ino (8) e express G in erms of, kgm() G v ˆ (8) 4 z (84) km G v ˆ (85) 4(()) z Seing in (8) e obain he field along he z axis hih is he simple Neonian graviaional field, exep for he inreased mass m due o roaion. The direion of he field in ylindrial oordinaes is given by he uni veor v ˆ (sin os,sin sin, os). A / he field blos o infiniy. This orresponds o and o an or ha he field has urned 9 degrees ih respe o he radial direion hus a barrier is formed a /. Figure shos ho G looks Figure The graph of G vs. on he lef appears as a onour plo on he righ. We may imagine he minimum lous of Figure 4 plaed on he onour plo as ell as he lines a G, here he minimum in he direion of he signal pah ours. The magniude of is dereasing in he direion unil i reahes a minimum and hen inreases and blos o infiniy. In he z direion i dereases as / z No ake he derivaive kgm G 4() z ( z ) (86)

13 This quaniy ill be non posiive for and hene G ill be minimum for z or for z (87) min z (88) The lous of pairs (, z ), here he minimum in he direion ours represens an ellipse and he value of he field a a minimum is found by subsiuing (88) ino (8) 7kGm z G (89) min 8 To see ho he minimum looks o observer O e mus find he minimum for G ih respe o. We ake G G and sine he minimums ih respe o found above, orrespond o he minimums ih respe o and hene subsiuing (84) ino (88) e find ha 4 z z (9) This equaion hih is symmeri in, z represens he lous of pairs of (, z ) here G is minimum in he direion for eah pariular z. The lous appears in Figure 4 z Figure 4 The shape depis he pair of poins (, z ) here G direion. The graph is symmeri in, z is minimum in he If e are ineresed in he derivaive of G along he pah of a signal ih inlinaion, e ill ake he derivaive ih respe o. In his ase kgm k Gm ( sin) 4 G 4 sin( sin ) and he minimum is reahed a (9)

14 This lous (as varies) is a ylinder of radius o. min sin (9), hih orresponds using (76) 6 The roaing mass Unil no e have onsidered bodies as poin masses. If e allo hem o have dimensions he roaion makes hem suffer a relaivisi inrease in mass. We ill here examine he hange in he mass from he res mass msa o he mass ha O (or O) observes. dmsa dm Le and be he densiy of he mass for observer O and O dv dv respeively. Where V and V is he volume for he o observers respeively. Also e ill aep ha a poin mass ha is a he disane and has angular veloiy aording o O, ill appear o him as having mass aording o he ransformaion of m speial relaiviy, sa msa m and herefore, for a small mass dm dv ha revolves ih angular veloiy a disane from he axis, dv dm dv dm. Bu and subsiuing for e obain, m dv (9) V dv m dv dv dv (94) V Assume no ha he saionary mass has uniform densiy, m dv (95) V As an example assume ha he mass is spherial. Changing o spherial oordinaes and inegraing (see Appendix A) e find, r r m () arsin() r r (96) r hih is he same as V 4

15 5 i i 4( r) r r m () r (97) i i i here r is he saionary radius of m sa. We noe from (96) ha as, m. 4 r hile, from (97) e see ha for, m msa as expeed. The argumens are similar for observer O: And using () e find ha m m. m m sa (98) 7 The G B.II ) field for roaion ih slippage (observer O ase When he roaion is ih slippage he angular veloiy of spae around he roaing body dereases exponenially ih disane from he body. In fa e assume ha he () angular veloiy is given by e z. In his ase hih is sudied in [] he signals sar radially from he origin, hen urn sideays unil hey reah a maximum of sideays urn and hen gradually reurn bak o he radial direion. Hene, he G field ha is reaed is differen from he no slippage ase alhough i keeps he basi haraerisi of he sideays urn and having a barrier hen no slippage as e ill see. For he far aay observer Ohe field is denoed as is big as in he ase of G In his ase he ransformaion is given by ase B.II above (reall () and folloing), Where sin os, e () z () e (99) e d () z z () sin an an sin () z () an z ( sin) an an sin ( sin) The Jaobian of he ransformaion is, (,,) z J (,,) z () (4) 5

16 Bu, and some manipulaion gives Hene, and e () z () e () e e () e () z () z () e J () e () z () z () () z () z () z ˆ G v G () () z z J z e, hih afer (5) (6) k m k m e G v ˆ (7) 4() 4()() For he direion of he field e need he uni veor in ylindrial oordinaes, hih as usual is given by vˆ ( v, v,)(sin v z os,sin sin,os) (8) The minimums and maximums of G are diffiul o alulae. Hoever, i is possible o make he folloing observaions, 7. Observaions on he exremes of G (a) Reall ha haever e find for has a orresponding value for sine by (5), hih is posiive for all, e kno ha is monoonially inreasing in. The derivaive of G ih respe o is () z () z() z G kgm () ()() e e e () z () z 4()()()() z e z e () z () z () z () z () e ()( ) e e (9) Seing i equal o zero and simplifying e obain, ()() ()()() e e e z e () z () () z z () z () z () z e ()( )() e z () The lef hand side of he equaion ells us ha he firs erm is alays negaive he seond erm sars posiive and a urns negaive, he hird erm sars negaive 6

17 and a urns posiive. Therefore, for he inerval for, he derivaive is alays negaive. (b) The derivaive of G ih respe o ill give us more informaion, G () z () z () z e ()() e e kgm () z e () z () 4()() z e () Seing i equal o zero and simplifying e obain, ()() () z () z e e () And solving () () z e () For he derivaive () sars a ih posiive value. Therefore, as e sar inreasing, G is inreasing bu no maximum is reahed as long as beause () is impossible. (A posiive lef hand side anno equal o a negaive righ hand side) and hene () anno hold and no maximum is reahed. In oher ords, in he inerval, G inreases indefiniely as inreases hile in he inerval. This means ha by inreasing enough I an make max [ G ] exeed G for all values of, suh ha. In shor for big enough he maximum of G over is alays in he inerval. Also his maximum is no in he neighbourhood of hen z, beause a hese poins G has a finie value independen of. Combining no his resul ih our observaion (a), (ha in he inerval for, he derivaive is alays negaive), e may resri his inerval even furher so ha he maximum of G over, o be denoed as G max, is alays in he inerval G max. Sill e have no shon heher here is only one or more loal max in. Hoever, he graph of G (see Figure 5a) indiaes ha here is 7

18 only one inernal maximum in he direion and ha i appears for big enough, oherise, hen is no big enough, he graph of G is monoonially dereasing in boh and z (see Figure 5b). In Figure 5 he plo is agains no bu reall ha here is a monooni relaion beeen and and hene he exisene and number of exremes are he same. Only due o he onraion of he maximum of he field a beomes very aue and looks like a all o observer O. G max z z G G (a) (b) G Figure 5. () G versus and z (a) for big ( rad/s ) and (b) for small 7 8

19 The graph shos he srengh of he field, no is direion, hih hanges aording o he angle of defleion.when is big (a), he field rises sharply in he radial direion forming a barrier ha diminishes in he z direion. (b) For small enough, he field has no suh inerior maximum and drops gradually boh in and z, hile os does no diminish as muh. In () a variaion of ase (a) appears here by hoie of and he barrier appears o iden in he z axis direion. 7. Observaions on From (99) e have, (4) () z e I is obvious ha. For large, beomes as small as e like and e are alking abou he miroosmos (sub aomi level). On he oher hand, small keeps a a magniude omparable o, hih e all he maroosmos. More formally, --If z kgm e hen and G 4() z, hih is he usual Neonian field. This for example holds for osmi level (maroosmos), here is small, hile is very big. -- If z e (This is ahieved for he miroosmos level here is big, hile remains small), hen e z () z e (5) In pariular, for z e see ha (6) a he miroosmos level. I is remarkable ha his is also he limi for for he no slippage ase, here he value of G blos o infiniy. I is ineresing o noe ha he radius in his ase is inversely proporional o he angular veloiy.the fa ha in (6) is independen of means ha he signals of he field for a range of values of are mapped and onenraed on very good harmony ih plos of and is, herefore, a maximum for G, here indeed he maximum appears a G max. This is also in agreemen ih he requiremen of seion 7. ha G max sine i is implied by he requiremen z. G. This is in 9

20 7. The maximum defleion of he direion of G The maximum of he defleion angle, is no easy o alulae hoever, e expe i o happen very lose o he maximum of or he minimum of os, in he direion of he signal pah is found in [] hen e ake os. This ours a max sin os (7) Where is he angle of inlinaion of he signal from he z axis. Corresponding o his maximum are sin max (8) sin os And os z max (9) sin os And he lous of poins here he max ours is a rhombus by revoluion hih in he firs quadran follos he equaion z () Using (4) e find max max sin max max max () max z max maxe maxe ()sin os e z --For he maroosmos ase here e or beause of () max e, e may approximae () by sin max max sin os o () (Reall ha also G max G max from seion 7. bu in he ase of maroosmos i is meaningless sine here is no inerior maximum for he G field and hene G max and G max do no exis.). max --For he miroosmos level here e e may approximae () by max e max () max e Comparing his ih G max, as e disussed in seion 7., e see ha he max defleion (hih e used o approximae he maximum defleion of ) of he field ()

21 ours a a disane ha is furher aay from he maximum magniude of he field by a faor of e. From () e find ha an (4) max e Sine max. Thus for big enough, 9 degrees 8 The G field for roaion ih slippage (observer O, ase B.I) This ase, here he observer O is affeed by he field, is more ompliaed in he formulas and e ill jus presen i briefly, The ransformaion is sin( I,,,) (5) ( sin os) () e d (6) z z (7) No he Jaobian of he ransformaion is J (8) () z sin( I,,,) () z e e Then e may find proeeding in a manner similar o ha for he no slippage ase. We ill hen find ha os os os( U,) J os (9) I (,,,)(,, I,) () z e here ( sin os) e os U (,) d ( sin os) os e sin () Finally, as usual G kgm 4() z J () I is diffiul o plo G bu from he physis of he problem e expe a maximum o our near he lous here he defleion of he signals is maximized. This lous hih has he shape of a rhombus by revoluion in he (,)z spae, is sudied in [] and presened briefly and used above in seion 7. 9 Summary of he resuls To find he G field (for he nearby observer O ) and G field (for he far aay observer Oand deermine heir properies, e sared by assuming ha a field onsiss

22 of signals and ha he srengh of a field is given by he number of signals per uni volume. We furher assumed ha hese signals ravel ih he speed of ligh or hey behave like ligh. This alloed us o apply he resuls for ligh signals for roaing frames o he ase of graviaional signals emied by a roaing body of mass m o find he form and srengh of he G and G fields i reaes. We also alulaed he relaivisi mass m of a roaing body from is saionary non roaing mass m sa. G and G are fields ha inlude a bending and even inding of signals and an beome very srong. Also depending on he value of he angular veloiy of roaion he field applies o boh he miroosmos (subaomi level) and osmologial sale or maroosmos. We examined o ases: (a) When roaion is ihou slippage (he angular veloiy of roaion is onsan independen of disane from he origin), he G field (of observer O ) exiss for z /. The field looks o ha observer unbounded in he radial direion and resried in he z direion o z /. For observer O, hoever, ho is ouside z /, he field he pereives is G and is resried in he radial by. (b) When roaion is ih slippage ( e z ) he G field is no resried o ihin and. We may disinguish o ases (i) The maroosmos level z for hih e. This ase holds, for example, for small and big, and e have small onraion of spae and he field behaves muh like he usual field, exep ha here is a sideay omponen due o he roaion ha r makes os, or he defleion angle beome posiive, ih is maximum approximaed a max and hen gradually reurn o zero. (ii) The z miroosmos level for hih e. Here, here is boh a maximum of he magniude of he field ha rises sharply and an be very srong and ours a /, and a maximum in he angle of defleion ha ours a G max e approximaely max. This laer phenomenon e all a barrier. () No muh an be said abou G, due o he omplexiy of he formulas. Hoever, from he sudy on he signals emied by he roaing body e expe a similar behavior o ha of G. Referenes. Pehlivanides P. On Roaing Frames and he Relaivisi Conraion of he Radius (The Roaing Dis), hp://vixra.org/abs/4.9, 7

23 Appendix A The spinning mass m dv V (A.) Changing o spherial oordinaes r 4 m r r os osd dr (A.) or r 4 m r r r sin r osd dr (A.) r r 4 r 4() r r m dr arsin() rdr (A.4) r I r The firs erm is and he seond erm is r r I arsin()() d r (A.5) r Inegraing by pars r r I () arsin() r r dr r (A.6) And finally, r r r I () arsin r (A.7) r 6 Therefore, r r r m () aran r (A.8) Using he expansion of 5 7 r r r r r r arsin aran... r 5 7 r hih onverges for, e may rie m as 5 i i 4( r) r r m () r i i i (A.9) (A.)