THE SPECIAL THEORY OF RELATIVITY - A CLASSICAL APPROACH PETER G. BASS

Transcription

1 THE SPECIAL THEORY O RELATIVITY A CLASSICAL APPROACH PETER G. BASS P.G.Bass Deebe R Vesion.4. Otobe 8

2 ABSTRACT The uose of this ae is to esent a new silifie aoah to the atheatial foulation of Albet Einstein's Seial Theoy of Relativity. Initially, a new eesentation of Heann Minkowski's PseuoEuliean SaeTie "Wol" is efine, within whih a new onet, Existene Veloity, is inooate. This then enables the silifie eveloent, along lassial analytial lines, of the Seial Theoy's elativisti kineati an kineti elationshis. P.G.Bass i R Vesion.4. Otobe 8

3 MATHEMATICAL NOMENCLATURE Matheatial esentation is as follows: (i) All atheatial haates, ae in Itali Ties New Roan font. i.e., x. (ii) Axis esignatos ae in aitalise nonitali bol, i.e., X. Unit Vetos ae in itali Bol, i.e. n (iii) All seaate satial an teoal veto haates, ae in itali bol, i.e., v, exet whee a Geek haate is use whee it is then ove bae. (iv) All obine satialteoal veto haates ae in nonitali bol haates of a iffeent font, (TianiHeavy), i.e S. (v) Diffeentials ae eesente by eithe the usual, (fun)/(va) o soeties fo laity by the ot fo,, fo the fist iffeential an, fo the seon. The latte is howeve, only use fo iffeentials with eset to a tie vaiable, whee the funtion iffeentiate is not itself a funtion of tie. P.G.Bass ii R Vesion.4. Otobe 8

4 CONTENTS. Intoution. The SaeTie Doain D. Definition. Existene Within D 3. The Mehanis of Sile Satial Retilinea Motion in D 3. Mass an the Equation of Motion 3. Enegy 4. CuviLinea Motion in D Piay Equations 5. Plana Obital Motion Kineatis in D 6. Conluing Reaks APPENDICES A. Equivalene of D with PseuoEuliean SaeTie B. Reution of Selete Relativisti Equations to thei Classial Equivalents. A BELATED ACKNOWLEGMENT. REERENCES P.G.Bass iii R Vesion.4. Otobe 8

5 The Seial Theoy of Relativity A Classial Aoah. Intoution In 95 Albet Einstein ublishe his ae on the Seial Theoy of Relativity, whih, as is well known, is onene with the haateistis of sae, tie an atte when a ass ossesses a onstant veloity in Pseuo Euliean Sae Tie. Subsequently, Heann Minkowski showe that Pseuo Euliean Sae Tie oul be eesente by a fou iensional `Wol', in whih thee iensions wee satial in natue, an the fouth, teoal. A oint within suh a `Wol' was sai to exist at the ooinate ositions eesenting its loation. Minkowski's eveloent subsequently le to the atheatial foulation of the Seial Theoy using suh tools as the Tenso Calulus. By utilising Minkowski's eesentation of saetie in a new way e.g. as a linea olex satial/teoal anifol, in whih the teoal iension is eesente as the iaginay at an the satial iensions as the eal at, an the intoution within it of a new onet, Existene Veloity, a Sae Tie Doain esignate D is eate. The sile oess of inue satial etilinea otion within this Doain then eits the eivation, using lassial analytial ethos, of the ain kineati an kineti elationshis extant within the Seial Theoy, togethe with a nube of new ones. As a eonstation of aliability, the onet is then extene to lana, an ental obital otions. inally, via onfoane to the aoiate iteia, the Doain D is subsequently shown in Aenix A, to be equivalent to Pseuo Euliean Sae Tie. This is augente by the eution of selete eivations to thei lassial equivalents in Aenix B. P.G.Bass R Vesion.4. Otobe 8

6 . The SaeTie Doain D. Definition The Doain D is efine as a sae tie of fou utually othogonal linea iensions, thee of whih, X, X an X 3 ae satial, while the fouth, X o, following Minkowski, is teoal. X o is efine, an will be shown, (Aenix A), to be the out of the tie t in D an a onstant veloity aaete, esignate the Satial Teinal Veloity of D. The Doain D is futhe haateise in that fo any satial teoal oint to exist within it, that oint ust at all ties ossess a veto veloity, esignate Existene Veloity, the agnitue of whih has the sae value as the Satial Teinal Veloity. Thus, fo any oint to exist within D,, it is neessay fo the agnitue of the vetoial su of its veloities along the fou ooinate axes, to be at all ties, equal to.. Existene within D The osition of any ano oint B within D, elative to soe hosen efeene an be exesse in satial teoal veto fo as i x l x k x S 3 j x o (.) whee x, x an x 3 ae eah a istane along the oesoning satial axes of D, fo whih i, l an k ae the aoiate unit vetos. x o is a istane along the teoal axis fo whih j is the unit veto. i, l an k eah have the usual agnitue of unity, while j has the agnitue of. As the teoal axis of D is the out of the onstant with the tie t, so then will the istane x o be a out of an soe funtion of the tie t in D. Aoingly (.) ay be ewitten as S (.) j t whee t is a funtion of the tie t in D an, also, whee the satial oonent of (.) has been elae with its esultant satial veto osition on a ola satial linea ooinate axis R. The veloity of suh a oint in D is efine by iffeentiating (.) with eset to t thus: whee V S/ an v / v j V (.3) Invoking the haateisti of existene in D, (.3) ust at all ties onfo to the following ientity, V (.4) whee V is the agnitue of V. Substitution of (.4) into (.3) gives, afte taking the agnitue, v t t (.5) P.G.Bass R Vesion.4. Otobe 8

7 so that v (.6) whee v is the agnitue of v. Substitution of (.6) into (.3) then gives an the following tes ae efine thus: v v j V (.7) V is the Existene Veloity of the the oint B in D / is the Teoal Rate of the the oint B in D / is the Teoal Veloity of the the oint B in D an, t is the Poe Tie of the oint B in D. Thus t is the tie easue by any obseve oving with a satial veloity v in D. o (.7) it is evient that V ossesses a satial teoal oientation in D whih is ietly eenent uon the satial veloity agnitue v. As v ineases fo zeo, teoal veloity unegoes a ootional eution suh that V, elative to the teoal ooinate of D otates though an angle in the X o R lane, elate to v by the exession Thus, fo futue efeene it is note that Cos v Sin θ v θ (.8) (.9) This oletes the efinition an haateisation of D. Its equivalene with Pseuo Euliean SaeTie is eonstate in Aenix A. The next setion foulates the kineatis an kinetis of etilinea otion within D fo oaison with that of the Seial Theoy. P.G.Bass 3 R Vesion.4. Otobe 8

8 3. The Mehanis of Sile Satial Retilinea Motion in D 3. Mass an The Equation of Motion The Seial Theoy of Relativity assets that the ass of a fixe quantity of atte, satially in otion with a onstant etilinea veloity in Pseuo Euliean Sae Tie, is geate than when it is at est. o this to be so the inease in ass an only take lae uing the tie that satial aeleation is in effet. This oess an be investigate by teating ass as a vaiable when analysing the hange in the Existene Moentu of a oint ass subjete to satial aeleation in D. If is the ass of the oint ass with Existene Veloity V in D, then its Existene Moentu will be given by: V v v j M (3.) whee V has been substitute fo (.7). If is the foe alie to effet aeleation then: M v vv v v j v (3.) Showing that thee ae fou kineti eation tes involve in this oess. If is uely satial, the teoal oonent of (3.) will be zeo, wheeby: v so that uon seaating vaiables an integating vv (3.3) v v ln k ln (3.4) The onstant of integation is obtaine fo initial onitions viz: when v, o, the ass of the oint ass when satially at est in D, i.e. the 'est ass'. Then: whih gives in (3.4) k ln (3.5) (3.6) v P.G.Bass 4 R Vesion.4. Otobe 8

9 as assete in the Seial Theoy. Howeve, it is lea fo the above eveloent that, in aition to a onstant satial veloity, (3.6) is also vali uing the tie that satial aeleation of a oint ass is in effet. o easons that will be isusse late will be efee to as Enegy Mass. Substitution of (3.6) into (3.3) yiels: vv (3.7) 3 v whih thus eesents the tie ate of hange of ass subjete to satial aeleation in D. This an also be obtaine by sily iffeentiating (3.6) with eset to the tie t. These last two tes, (3.6) an (3.7), ay now be insete into (3.), wheeuon the teoal oonent vanishes an, if etilinea otion only is being onsiee, an also be eue to a satial veto,, so that (3.) yiels, afte eution: v v 3 (3.8) As is abitay, (3.8) eesents the satial etilinea equation of otion of a oint ass in D. (Non etilinea otion is exaine in Setions 4 an 5). Note that the ight han sie of (3.8) is the out of the satial aeleation an a ass te. Putting a v 3 (3.9) then a is, fo (3.8), synonyous with inetial ass. Thee values of ass have thus been ientifie fo the sae oint ass i.e. o v a v 3 Rest Mass Enegy Mass Inetial Mass The latte two ae soeties efee to in the liteatue [], [], [4] as 'tansvese' an 'longituinal' ass, (see Setion 4). P.G.Bass 5 R Vesion.4. Otobe 8

10 o inteetation of these esults, efeene is ae to ig. 3.. whee it is shown that as a onsequene of the otation of V in D o, the alie satial veto foe ay be esolve fo two satial teoal veto oonents, a noal to V an e aallel to V. v V j( v / ) / a e IG. 3.: COMPONENTS O WITH RESPECT TO V The oonent a is ootional to the hange in Existene Veloity, while the oonent e is ootional to the hange in the ass. This is lea fo (3.) whee by insetion: an a v v v j v (3. ) e V (3. ) v v j. (3. ) V (3.3 ) o these equations, (3.) an now be inteete kinetially. o (3.) an (3.) it an be seen that the kineti eations to the two oonents of, eah oise a satial an teoal te. If the balane foe veto is iagaatially eesente as in ig. 3., the fou kineti eation tes of (3.) an be inteete as follows: P.G.Bass 6 R Vesion.4. Otobe 8

11 v/ e jvv/ ( v / ) / a v/ j( v / ) /./ IG 3.: THE BALANCED ORCE VECTOR (i) The satial te, v is the eation foe of the enegy ass to satial aeleation. (ii) The teoal te, teoal eeleation. j v v ( v ) is the eation foe of the enegy ass to (iii) The teoal te, j ( v ). is a eation foe geneate by the obination of ass ate an teoal veloity an ats in oosition to the te in (ii). (iv) The satial te, v is a eation foe geneate by the obination of ass ate an satial veloity an ats as an aitional eation to satial aeleation, theeby ausing the aaent ass to inease, fo to a, uing the eio of aeleation. A onsequene of this, is that this last te ust be elate to the iffeene between inetial an enegy ass, by the out of that iffeene an the satial aeleation. This ay be shown as follows. o (3.6) an (3.9). a v v v v 3 3 (3.4) whih with insetion of (3.7) gives the equie elationshi: ( ) v a v (3.5) inally, a oent uon two aitional oints that eege fo the above analysis. istly the fat that the two teoal eation tes, ites (ii ) an (iii ) above, ae, as shown by (3.) an in ig. (3.), to be equal in agnitue but oosite in sign, oes not ean that they o not seaately exist. Whilst they o in the above exale, anel, they ae quite iffeent in P.G.Bass 7 R Vesion.4. Otobe 8

12 natue, the fist being a ass eation to teoal eeleation an the seon a ass ate eation to teoal veloity. They ae equal in agnitue but oosite in sign solely beause, in this ase, thee is no net teoal oonent of iesse foe, i.e. is uely satial. The seon oint onens the inetial ass a. While a an be exesse solely as a funtion of the satial veloity, as is aaent fo (3.9), it is equally aaent fo (3.5) that its existene is entiely eenent uon the satial eation te v/. As this te only exists while satial aeleation is taking lae, so then an a only exist uing this eio. When satial aeleation eases, the te v/ vanishes an, the value of the ass instantly evets to that of the enegy ass,. 3. Enegy In lassial ehanis the aeleation of a oint ass is sai to esult in it gaining a kineti enegy equal to the out of the alie foe an the istane ove whih it ats. The anne in whih kineti enegy was stoe by suh an aeleate ass was not aesse. o the esults of the eeing setion howeve, this an now be eonstate as follows. Consie the hange in enegy of the oint ass as the satial aeleation oees. Integating (3.8) with eset to the satial istane tavelle, whih fo (3.9) beoes E k v a (3.6) vv E k (3.7) 3 v using sile substitution ethos this evaluates to E k k v (3.8) Initial onitions ae that E k when v so that k. Inseting this into (3.8) then gives a vesion of Einstein's well known equation fo the enegy of atte. E k (3.9) whee eah te ay be inteete, as in the liteatue, as follows: (i) is the total enegy of atte at soe instantaneous satial veloity v. (ii) E k is the enegy iate to atte via the ation of the alie foe ove the satial istane tavelle uing its aliation, i.e. kineti enegy. (iii) is the est ass enegy of atte P.G.Bass 8 R Vesion.4. Otobe 8

13 o (3.9) it is seen that the inease in ass, fo, that at est, to, that at veloity v, is as esibe in the liteatue, [], ue to the stoage of enegy iate fo the alie foe. Thus is the ass equivalent of the total enegy of atte at the instantaneous veloity v. It was fo this eason that was ealie esignate as Enegy Mass. Reution of all of the above elationshis involving the satial veloity, v, to nonelativisti fo is effete in the usual anne, by assuing the Satial Teinal Veloity,, to be infinitely lage, (see Aenix B). P.G.Bass 9 R Vesion.4. Otobe 8

14 4 CuviLinea Motion in D Piay Equations This onition is biefly investigate to illustate the effets of aeleative foes on the ietion of otion of a oint ass in D. In oing so howeve, it also enables the eason fo the oiginal esignations of "longituinal" an "tansvese" ass to be sily eonstate (see Set.3.). The satial situation an ost easily be esibe by ig.4.. whee, fo laity, satial atesian ooinates ae now eesente by X an Y. v a v y ψ ξ η v v x The oentu equations ae: ig. 4. oe/veloity/aeleation Diaga (Satial) NonRetilinea Aeleation in D o M M x y v (4.) v x y (4.) v M t (4.3) whee is the enegy ass of the oint ass an M x an M y ae the esetive satial, an M t the teoal, existene oentus. The initial veloity at t is v, the satial aeleation is a, an the othe tes ae as shown in ig.4.. Diffeentiating with eset to t gives the foe equations thus: P.G.Bass R Vesion.4. Otobe 8

15 x y v v x x (4.4) v v y y (4.5) v v v t. (4.6) v Deteination of an / If in (4.6), t is zeo, an be eteine by sile integation to give, as in Set. 3: v (4.7) so that also v v v 3 (4.8) but in this ase with ( x y ) v v v (4.9) then v v v x x v v y y v (4.) an this in (4.8) gives: v v x x v v v y y 3 (4.) P.G.Bass R Vesion.4. Otobe 8

16 Substitution fo (4.9) an (4.) fo v x / an v y / then gives: ( v x os ξ v ξ) y sin (4.) inally substitution fo v x an v y fo elationshis iliit in ig. 4., yiels: v os [ ( ξ η) ] (4.3) whih lealy shows that a vaiation in ass only esults fo that eleent of alie aeleative foe ating along the veloity veto. Deteination of v x / an v y / an Assoiate Inetial Masses. Substitution of (4.7) an (4.) into (4.4) an (4.5) yiels afte eution: x y v v 3 3 v v v v v y x x y y (4.4) v v v v v x y x y x (4.5) so that fo the y half of this equation: v y v x 3/ y x y x v v v v (4.6) With substitution of this into x, (4.5), thee is afte eution: x v x v x v v v y x y v x (4.7) but fo ig. 4., y x tanξ whih when substitute into (4.7) yiels: v x v v v v x x y os ξ sin ξ (4.8) P.G.Bass R Vesion.4. Otobe 8

17 inally substitution fo v x an v y fo elationshis iliit in ig. 4., gives: v v os ξ os os v x (4.9) ( ξ η) η v y v v ( ) sin ξ os ξ η sin η (4.) Inteetation of these equations is quite sile when (4.7) an (4.3) ae intoue. In both ases (4.9) an (4.) eue to (4.4) an (4.5), showing that the fist te insie the esetive bakets is the noal aeleation esulting fo the atio of foe to enegy ass, while the seon te is the etaation ue to the eation between ass ate an satial veloity. Consequently fo (4.9) an (4.), ax an ay, the inetial asses in the esetive ietions, ay be exesse thus: v x x v ax (4.) v os ηos ( ξ η) se ξ ay v y y v v sinηos ( ξ η) os eξ (4.) The iffeene between these two tes is solely ue to the iffeent ass ate eation foes geneate in the esetive ietions. Equality ous when the alie foe an veloity vetos ae oinient i.e. when ξ an η ae equal (but not zeo ). Then: ax ay v 3 (4.3) an is of ouse equal to the inetial ass of etilinea otion. Angula Relationshi Between Alie oe an Aeleation. This an ost easily be insete by oaing the esetive angula elationshis between both the a an ietions an the X axis. Thus fo (4.9) an (4.)ietly: tanψ v v y x v sinξ v os ξ sinηos os ηos ( ξ η) ( ξ η) (4.4) P.G.Bass 3 R Vesion.4. Otobe 8

18 whih shows that the esultant satial aeleation oes not lie in the sae ietion as the alie foe. The eason is again the iffeene in the ass ate eation tes, in the X an Y ietions. Deteination of v/ an v n / These tes ae efine as the aeleations oue both along an noal to the veloity veto. Substitution of (4.9) an (4.) into (4.) yiels: v 3 v os ( ξ η) (4.5) Note that inetial ass along the veloity veto, is, as woul be exete by vitue of (4.3), equal to that in etilinea aeleation. To eteine v n /, note that fo ig. 4. v n (4.6) v tan ( ψ η) afte exansion, substitution fo (4.4) oues, afte soe eution: v n v tan v ( ξ η) (4.7) so that substitution fo v/ fo (4.5 ) then yiels: v sin v n (4.8) ( ξ η) The oint about this esult is of ouse the aeaane of the enegy ass, thee being no ass ate eation involve beause the ietion onene is noal to the veloity veto. Othe tes, suh as aeleation along an noal to the ietion of alie foe, angula ate an angula aeleation of the veloity veto an, assoiate enegies, exhibit elationshis of a siila natue to the above. Bounay Conitions of ξ. The two bounay onitions of ξ ae, ξ an ξ π /, at whih the following aly: (i ) ξ e.g. lies aallel to the Veloity Veto, along the X axis Motion in the Y ietion is nonexistent while that in the X ietion is of ouse that of etilinea otion of the ain text, i.e. η is also zeo. (Note that this onition is a seial ase of ξ η use to obtain (4.3) above ). Of atiula inteest howeve is fo (4.) an (4.) P.G.Bass 4 R Vesion.4. Otobe 8

19 ax ay v v 3 (4.9) (4.3) ax is now the inetial ass of etilinea otion an alies only in the X ietion, i.e. in the ietion of the now oinient foe, aeleation an veloity vetos. It was fo this eason that ax, in the above fo was in the liteatue oiginally tee "longituinal" ass. Une this onition ay is the ass aliable noal to the aeleate otion an was onsequently oiginally tee "tansvese" ass. It is equal to the value of enegy ass beause of ouse no ass ate eation te exists in the tansvese ietion. (ii ) ξ π / e.g. lies noal to the Initial Veloity Veto. The ost inteesting esult fo this situation eeges fo (4.4) an (4.5) i.e. v v sin η os η v x (4.3) ax (4.3) This sily states that with all the aeleative foe alie noal to the ietion of initial veloity, a vey sall eeleation in that ietion ous ue to the eation of ass ate with v x. Inetial ass in the X ietion onsequently takes the `hyothetial' value of zeo as a eeleation ous without the aliation of an extenal foe in that ietion. Suay It is lea that the iay iffeene between the otion esibe heein an that of Classial Mehanis, aat fo the ain elativisti effets, is ue to the ass ate eation tes. Most atiula in this eset is the nonoiniene of the foe an aeleation vetos. In the next Setion it is shown that a iay esult of this effet uon a tajetoy is to ause it to otate. P.G.Bass 5 R Vesion.4. Otobe 8

20 P.G.Bass R Vesion.4. Otobe Plana Obital Motion Kineatis In D. In this Setion, the equation of an obital otion is fist eive an then solve as a seon illustation of the anne in whih the onet of Existene Veloity, ay be alie to elativisti obles of this natue in Pseuo Euliean Sae Tie. Deivation of the Geneal Cuvi Linea Equation of Motion in a Plane in D. Reeating (.7) fo onveniene: v j v V If the enegy ass is then existene oentu, fo uely lana otion, is: j t n M (5.) whee fo atheatial onveniene satial ola axes have been hosen an whee, with efeene to soe stationay oigin, is the aial istane of the obit. / is the aial veloity of the obit. ϕ/ is the angula ate of the obit. n an t ae the aial an aial noal unit vetos. Diffeentiating (5.) with eset to tie yiels the foe equation thus: ( ) [ ] ( ) [ ]t n M (5.) ( ) j If is uely satial, then the teoal at of (5. ) is zeo an an be eteine by sile integation to be: (5.3) whee o is the est ass. Thus the ass ate is:

21 P.G.Bass R Vesion.4. Otobe o (5.4) Substitution of (5.3) an (5.4) into (5.) then yiels afte eution, ( is now uely satial), ( ) ( ) ( ) ( ) t n 3 o (5.5) This is the ost geneal fo of the foe equation fo satially aeleate uvi linea otion in a lane in D an whih lealy ossesses a istint syety. The Case of a Puely Raial oe If is uely aial (onstant angula oentu ), then in (5.), in aition to the teoal oonent, the aial noal oonent will also be zeo. Thus ( ) (5.6) whih fo (5.3) an (5.4) beoes ( ) (5.7) whih an also be obtaine fo the aial noal oonent of (5.5). Substitution of (5.7) into (5.5) gives ( ) o ½ n (5.8) This is the equation of otion of a oint ass in a lane in D subjete to an abitay satial aial foe. Note that substitution of (5.7) into (5.4) yiels afte eution ( ) ½ o (5.9) whih when substitute into (5.8) gives

22 P.G.Bass R Vesion.4. Otobe 8 8 n (5.) This is lealy seen to be iential to the etilinea ase an ovies futhe onfiation that the ass ate effet only exists along oinient eleents of the foe an veloity vetos. Convesion of the Equation of Motion to Poe Tie To eteine the equation of the obit it is fist neessay to onvet the equation of otion to the oe tie of the oint ass. Convesion of (5.8) to the oe tie of the oint ass, is ahieve as follows. With (5.) then (5.) onsequently with / / (5.3 ) Substitution fo (5.) an (5.) gives (5.4) But fo (5.7) 4 (5.5) Whih, when substitute into (5.4) gives afte eution

23 P.G.Bass R Vesion.4. Otobe 8 9 (5.6) Also fo (5.8) afte taking the agnitue (5.7) an substitution of this into (5.6) then yiels (5.8) but fo (5.) ϕ ϕ (5.9) so that this gives in (5.8) (5.) It now only eains to onvet the te ( ) / / / to oe tie as follows; ewiting (5.) as (5.) an, fo (5.9), with (5.) then

24 P.G.Bass R Vesion.4. Otobe 8 (5.3) an eaangeent of this then gives / (5.4) so that (5.) finally beoes (5.5) an fo a uely satial aial foe, is the equation of lana otion in D exesse as a funtion of the oe tie. Deivation of the Equation of the Obit To obtain the equation of the obit fo (5.5), it is initially neessay to evaluate the fist integal of (5.6). Reaangeent of that equation yiels (5.6) Integating (5.6) gives k (5.7) whih then, afte the eteination of the onstant of integation, k fo initial onitions, an, when gives, togethe with (5.) (5.8) an in line with onvention this onstant is esignate, h. The equation of the obit ay now be obtaine in the usual way thus. Putting (5.9)

25 P.G.Bass R Vesion.4. Otobe 8 then ϕ ϕ h (5.3) an ( ) ( ) h h ϕ ϕ (5.3) Insetion of (5.8), (5.9) an (5.3) an (5.3) into (5.5) then gives the esie esult fo the equation of the obit. ϕ ϕ h h (5.3) Solution of the Equation of the Obit fo Two Oositely Chage Patiles In a Vauu. Assuing onitions ae suh that the only effet between the two atiles is an eletostati one an that thei elative size is suh that the salle has negligible effet uon the lage, an ignoing any sin effets, then the foe of attation between the ay be exesse as (5.33) The equation of the obit of the salle atile, fo (5.3) an (5.33) then beoes h h ϕ ϕ (5.34) to solve this equation ut Θ ϕ (5.35) this being the invese of the eeniula istane fo a foal oint of the obit to a tangent at any oint on the satial tajetoy. Diffeentiating (5.35) with eset to ϕ gives Θ Θ ϕ (5.36) Substitution of (5.35) an (5.36) into (5.34) then yiels h h Θ Θ Θ (5.37) This equation an now be solve using stana ethos to yiel

26 P.G.Bass R Vesion.4. Otobe 8 ( ) ϕ h h (5.38) whee the onstant of integation has, togethe with (5.35), been insete. Reaanging (5.38 ) fo /ϕ gives ϕ h h h h h h (5.39) This equation is also a stana tye that an be solve using onventional ethos to yiel, afte soe eution h h h h h h / osφ (5.4) whee ϕ Φ h (5.4) an whee initial onitions have been hosen suh that the onstant of integation is zeo. Equation (5.4) esibes the satial tajetoy of the salle atile about the lage an lealy, as in the liteatue [5], is seen to be a otating oni setion. o the seon at of (5.4), this otation is seen to be a funtion of the finite Satial Teinal Veloity,, within D, an also, that the eession angle is a onstant etogae one being, unlike the gavitational ase, ineenent of the te.

27 6. Conluing Reaks The onet of Existene Veloity within the Relativisti Doain D, both as have been efine in this ae, has, using ethos iential to those of lassial ehanis, enable a silifie atheatial foulation of the kineatis an kinetis of the Seial Theoy of Relativity, fo one atiula ase, etilinea otion. In oing so, it has also ovie a leae insight into the elativisti natue of an alie satial foe, an the assoiate kineti enegy that it oues in a satially aeleate oint ass. In Setions 4 an 5 this silifie etho has been use to exaine two futhe elativisti kineati situations, linea lana, an ental obital otion, as eonstations of its aliation. It ust be note that these onets ae only vali within the oain D i.e. PseuoEuliean Sae Tie, (see Aenix A). o othe elativisti oains, suh as one ontaining gavitation, it is neessay to hange the haateistis of the Doain aoingly. This will be the subjet of the next ae whee a new theoe fo gavitation will be esente using a suitably oifie Relativisti Doain. This new theoe will iffe fo that of the Geneal Theoy of Relativity in that although the oifie Doain will iffe fo PseuoEuliean SaeTie, it will still be one exhibiting a linea ooinate syste. One final note. In the Seial Theoy, the liiting satial veloity is geneally aete to be the veloity of eletoagneti aiation in a vauu, i.e. light. In this ae the liiting veloity has been tee the "Satial Teinal Veloity" an no efeene has been ae to the veloity of light. This aoah has been aote beause it has not been onlusively oven that the liiting satial veloity in the Seial Theoy is inee the veloity of light. It ay well be that the tue liiting veloity is slightly iffeent fo this. This is believe ossible beause fo the Seial Theoy it is known that atte ossessing ass an thus enegy, annot be aeleate to the veloity of light within a finite tie. Yet it is also known that eletoagneti aiation ossesses enegy an ust theefoe also ossess ass, howeve sall. Thus it is onsiee obable that the veloity of light ay well be slightly lowe than the tue liiting satial veloity in the Seial Theoy an theefoe also in D. P.G.Bass 3 R Vesion.4. Otobe 8

28 APPENDIX A Equivalene of the Doain D with Pseuo Euliean Sae Tie To fully eonile the aliation of the onets esente in this ae with the Seial Theoy of Relativity, it is neessay to eonstate that the Doain D is equivalent to the sae tie in whih that theoy alies, Pseuo Euliean Sae Tie. It is theefoe neessay to show that D ossesses the following aitional haateistis to those aleay efine. (i ) The Teoal Cooinate X o is elate to the tie t in D by the exession, (afte Minkowski ), X j t (A.) (ii ) (iii ) The agnitue of the axiu theoetially attainable satial veloity in D, is equal to the Satial Teinal Veloity,. When the satial veloity of a oving oint in D is etilinea an onstant, easueents of tie an istane elate to axes assoiate with it, tansfo fo those of D aoing to the Loentz Tansfoations of the Seial Theoy. The Teoal CoOinate X o In (.7), if v is zeo, i.e. a oint is satially at est in D, its Existene Veloity is eue to: V j (A.) whih uon integation with eset to t gives: S j t (A.3) whee o, the onstant of integation, is the onstant satial osition in D fo soe stationay efeene. In this ase the tajetoy of otion is lealy, fo (A.), along the teoal axis Xo so that, in (A.3), the elationshi of (A.) is iliit. The Maxiu Theoetially Attainable Satial Veloity in D. Inseting (.8) an (.9) into (.7) gives fo Existene Veloity in satial teoal ola fo V s Sinθ j Cos θ (A.4) whee s is a unit veto in the ietion of v. Clealy the axiu theoetially attainable satial veloity ous when θ π, the Existene Veloity beoing sily: V s (A.5) At any othe value of θ, the satial oonent of V ust be less than. Tansfoation of The Axes P.G.Bass 4 R Vesion.4. Otobe 8

29 This is the oe olex of the thee iteia to ove. It is aolishe by eivation fo fist iniles, of the elationshi between the satial an teoal axes of a oint in D oving with onstant veloity, an those of D itself. The teoal axis elationshi is eive in the fo of a tie aaete in oe to fully eonstate ageeent with the Loentz Tansfoations. Duing this oess a elationshi fo the oe tie of the oint is also eive. Let B be a oint in D oving with a onstant satial etilinea veloity v. Let R' be a "sae like" ooinate assoiate with B, let Q be any fixe oint on R', an let t' q be the tie along a "teoal" axis X' q, assoiate with R' at the loation of Q. inally, let initial onitions be suh that at soe instant in D esignate t, the osition of B in D is efine as a efeene oint, (ig. A ay be usefully efee to in following this eivation). Consie fist the satial axis R'. At tie t the ositions of B an Q in D will be given by : S S b q b j t b q j t q (A.6) (A.7) o (A.6) an (A.7) the osition of Q on R' an be exesse in satial teoal veto fo as: Diffeentiating (A.8) with eset to t S q S j q b q b ( t t ) q b (A.8) q q b q j b (A.9) But with an In (A.9) this gives b b v v q v j q q v (A.) (A.) (A.) an taking the agnitue of (A.) q q q v v (A.3) P.G.Bass 5 R Vesion.4. Otobe 8

30 but fo Q to exist in D q q (A.4) so that in (A.3) q q v v q (A.5) but ' q is onstant, theefoe ' q / ust be zeo. This gives in (A.5) q v q v (A.6) As v is onstant (A.6) an be integate ieiately to give: t v t q v q k (A.7) Substitution of (A.7) into (A.8) then yiels: ' q v( q vt) q v t j k v (A.8) Now as Q is any oint on the R' axis, it an be oinient with B to give (A.8) woul eue to: q q v t an k (A.9). In this ase Theefoe k ust be zeo fo all R' Consequently (A.7) an (A.8) esetively eue to: t q v t q v (A.) P.G.Bass 6 R Vesion.4. Otobe 8

31 an q q v t v q j v ( vt) (A.) The agnitue of (A.) yieling: q vt q v (A.) As Q is any oint on R', then (A.) an (A.) eesent the elationshi between the satial axis assoiate with B an that of D, an (A.) eesents a easue of the oe tie of Q in D. The aaetes ' q an q ay theefoe, in (A.) an (A.), be elae with the axis esignatos R', an R esetively, an the lengths ' q. an q theefoe eesent the elationshi between thei sales. Subsequent efeene to (A.) shows that R' ossesses both satial an teoal oonents an theefoe a eise oientation in D o. Substitution of (A.), (.8) an (.9) into (A.) gives: ( s os θ j θ) R R sin (A.3) Whee s is a unit veto in the ietion of v. This shows by oaison with (A.4) that R' is othogonal to the Existene Veloity Veto an, theefoe, the satial teoal tajetoy of B in D. Also, fo (A.) it is lea that units of length along R' (i.e. with t onstant ) ae geate than units of length along R, an that the inease is a iet esult of the oientation of R' elative to R in D. Now onsie the teoal ooinate assoiate with R' at the loation of Q. istly it is note that sine Q is fixe in elation to B, its only otion in axes assoiate with B is a teoal one. Theefoe the ie teoal axis along whih Q is in otion, X' q, ust lie along its tajetoy in D whih ust be aallel to that of B. As a onsequene, this axis ust be othogonal to R'. Now (A.), as state above, is a easue of the oe tie of Q in D an by vitue of (A.7) is theefoe ietly ootional to its osition on X o fo its initial osition at t o. In like anne howeve, the oe tie of Q on its ie teoal axis is ietly ootional to its osition on that axis fo its initial osition. Note howeve that, as Q ossesses only teoal otion in the ie axes, its oe tie along X' q will be iential to that of X' q itself. Theefoe, to eive the elationshi between tie on the two teoal axes fo any onstant value of q, a onetoone oesonene between ineental istanes on the an be establishe as follows. If X o is an ineental istane along the teoal axis of D, an X' q an ineental istane along the teoal axis assoiate with the oint Q on R' suh that they ae teoily oinient in D, then ue to thei elative oientation, they onfo to the following exession: P.G.Bass 7 R Vesion.4. Otobe 8

32 X q X v (A.4) As t' q is the tie along X' q, an sine the ie teoal veloity of all oints on R' is, i.e. equal to V, then by (A.), (A.4) ay be ewitten thus: q v (A.5) Integating (A.5) t tq v k (A.6) The onstant k is the initial onition that ensues teoal oiniene of the two ineentals within D, an k ust theefoe be suh that when t' q is zeo, the oe tie of Q in D is also zeo. Thus, fo (A.) when t q is zeo, t is given by whih gives in (A.6) when t' q is zeo v t q (A.7) k v q v (A.8) so that finally in (A.6) t tq v v q (A.9) This being the elationshi between tie on the X' q an X o axes. Note that (A 5 ) shows that the units of tie along X' q ae geate than those along X o by the sae fato an, fo the sae eason that the units of length along the satial axes iffe. Clealy all suh oints on R', inluing B, ust have assoiate with the a tie, along a unique teoal axis, of the fo of (A.9) in whih the satial te iffes aoiately. The lous of the efeene zeo on these axes lies along the satial axis of D at t. This togethe with the exansion of the units of tie along these axes ensues the siultaneity of existene of eah oint on R' in both faes of efeene. It is also note that the atheatial P.G.Bass 8 R Vesion.4. Otobe 8

33 elationshi between t' q an t is the sae as that between t q an t. They annot be equate howeve beause of the iffeene in the agnitue of the units. The above elationshis, seifially assoiate with the oint Q, an be iagaatially eesente as in ig. (A) below. The satial otion of the oint B, has, fo the above aguent, esulte in axes assoiate with it being exane an otate in the ietion of otion in D though the sae satialteoal angle θ as the Existene Veloity veto of B. This onus with stateents in the liteatue [3] that in Minkowski's `Wol' the Loentz Tansfoations "oeson to a otation of the ooinate syste". Clealy (A.) an (A.9) ae iential to the Loentz Tansfoations of the Seial Theoy an togethe with the evious esults of this Aenix, eonstates the equivalene of the Doain D with Pseuo Euliean Sae Tie. The aliation of the onet of Existene Veloity within the latte is theefoe a vali one. P.G.Bass 9 R Vesion.4. Otobe 8

34 X X' b B ' q X' q vt Q q at tie t in D o R' t b V t' q V t q B ( q v/)' ( v / ) / Q R R' ' q Q R' q at tie in D o IG. A.: DIAGRAMMATIC REPRESENTATION O THE RELATIONSHIP BETWEEN THE REERENCE AXES O D o AND THOSE ASSOCIATED WITH B AT THE POINT Q P.G.Bass 3 R Vesion.4. Otobe 8

35 APPENDIX B Reution of Selete Relativisti Equations to thei Classial Equivalents In all ases this is effete by allowing the onstant veloity aaete to beoe infinite. Only the ain equations fo whih a lassial equivalent exists is so teate. Tivial exales will be ignoe unless a seial onition is ilie. Setion (i) Eq. (.6), the teoal ate. When (B.) Hene in lassial theoy the oe tie of a oving boy is iential to the tie in D, PseuoEuliean SaeTie. (ii) Eq. (.7), Existene Veloity. When V v j (B.) Hene teoal veloity in lassial stuies is "infinite". Existene veloity oes not exist in lassial ehanis, an teoal veloity in suh stuies, is theefoe a eaningless onet beause it ilies that tie asses infinitely quikly. Whee a onet oes not exist in lassial ehanis et, elativisti eution geneally esults in an infinite o zeo value. Setion 3. (iii) Eqs. (3.6) an (3.9), Rest, Enegy an Inetial Mass. When a (B.3) Thus est, enegy an inetial ass ae iential in lassial ehanis. Hene any efeene to inetial ass in suh stuies is eaningless. Consequently, as is evient fo Eq (3.7), when (B.4) (iv) Eq (3.9), Kineti Enegy. To eteine the lassial exession fo kineti enegy ietly fo (3.9) woul be inoet beause (3.9) is a elationshi in atte enegy, a onet that oes not exist in lassial ehanis. The oet oeue is fist to inset (3.6) into (3.9) an exan the esult binoially to yiel E k 4 v 3v 5v (B.5) fo whih, when P.G.Bass 3 R Vesion.4. Otobe 8

36 v E k (B.6) the lassial esult. Setion 4. (v) Eq. (4.9) an (4.), the aeleation vetos along the ooinant axes, when v x osξ (B.7) v y sinξ (B.8) an lealy theefoe the foe an aeleation vetos ae oinient. (vi) Eq. (4.) an (4.), the ass on eah ooinant axis, when ax ay (B.9) (vii) Eq. (4.4), Angula elationshi between the aeleation veto an the X axis, when whih onfis the esult at (B.7) an (B.8). (viii) tan Ψ tanη Eqs. (4.5) an (4.8), aeleations along an noal to the veloity veto, when. an v v n so that fo (B.7), (B.8), (B.) an (B) os sin ( ξ η) ( ξ η) (B.) (B.) (B.) v x vy v v n (B.3) whih now also shows that the foe, aeleation an veloity vetos ae oinient. Setion 5. (ix) Eq. (5.5), geneal uvilinea equation of otion, when the lassial equation in ehanis. [( ) n ( ) t] (B.4) (x) Eq. (5.5), equation of lana otion in the oe tie, when P.G.Bass 3 R Vesion.4. Otobe 8

37 (B.5) the lassial equation in ehanis. (xi) whee Eq. (5.3), the equation of a ental obit, when ϕ h (B.6) h (B.7) the lassial equations in ehanis. (xii) Eq. (5.4), the equation of a ental obit tajetoy, when h os ϕ h (B.8) whih is lealy the equation of a sile oni setion. The seilatus etu an eentiity ae h L an e h (B.9) These ae again the lassial esults. P.G.Bass 33 R Vesion.4. Otobe 8

38 A Belate Aknowlegeent. Sine the fist ubliation of this ae, it has been leane that in his exellent book, "Relativity Visualise", Lewis Caol Estein, use a onet vey siila to that in this ae to exlain the iossibility of tavelling at the see of light. Refeenes [] Began P.G., Intoution to the Theoy of Relativity, Dove Publiations In., (976) [] Bon M., Einstein's Theoy of Relativity, Dove Publiations In, (965) [3] Einstein A., Relativity (The Seial an Geneal Theoy), Univesity Paebaks, (UP), (965) [4] Pauli W., Theoy of Relativity, Dove Publiations In., (98) [5] Bon M., Atoi Physis, Blakie an Son Lt., (957) P.G.Bass 34 R Vesion.4. Otobe 8