Relativistic Alpha Field Theory Part II: Does a Gravitational Field Could be Without Singularity?

Transcription

1 Intnational Jounal of Nw Tchnology and Rsach (IJNTR) ISSN:5-116 Volum-1 Issu-5 Sptmb 15 Pags Rlativistic Alpha Fild Thoy Pat II: Dos a Gavitational Fild Could b Without Singulaity? Banko M Novakovic Abstact Gnal Rlativity Thoy (GRT) cannot b applid to th xtmly stong gavitational fild at th Planck s scal bcaus of th latd singulaity H w show that Rlativistic Alpha Fild (RAF) thoy xtnds th application of GRT to th xtmly stong filds at th Planck s scal This is th consqunc of th following pdictions of RAF thoy: a) no a singulaity at th Schwazschild adius b) th xists a minimal adius at = min = (GM/c ) that pvnts singulaity at = i th natu potcts itslf c) th gavitational foc bcoms positiv (pulsiv) if (GM/c ) > 1 that could b a souc of a dak ngy and d) unification of lctical and gavitational focs can b don in th standad fou dimnsions (D) Pdictions a) and b) a psntd in this (scond) pat of this thoy It has bn shown that th mtics of th lin lmnt is gula in th gion wh adius is gat o qual to min and lss than infinity Th pdictions c) and d) a considd in th thid pat of th thoy Th ky point fo th pdictions of RAF thoy is th solution of th fild paamts psntd in th fist pat of th thoy If RAF thoy is coct thn it could b applid to th both wak and stong filds at th Univs and Planck s scals giving th nw light to th gions lik black hols quantum thoy high ngy physics Big Bang thoy and cosmology Kay wods: Rlativistic alpha fild thoy Engy momntum tnsos Elctical fild Gavitational fild I INTRODUCTION As it is wll known Gnal Rlativity Thoy (GRT) 1-6 cannot b applid to th xtmly stong gavitational fild at th Planck s scal bcaus of th latd singulaity H w psnt a nw thoy that is calld Rlativistic Alpha Fild (RAF) thoy W show that RAF thoy xtnds th capability of th GRT fo th application to th xtmly stong filds at th Planck s scal This is th consqunc of th following pdictions of RAF thoy: a) no a singulaity at th Schwazschild adius b) th xists a minimal adius at = (GM/c ) that pvnts singulaity at = i th natu potcts itslf c) th gavitational foc bcoms positiv (pulsiv) if (GM/c ) > 1 that could b a souc of a dak ngy in th univs and d) unification of lctical and gavitational focs can b don in th standad fou dimnsions (D) Pdictions a) and b) a psntd in this (scond) pat of this thoy It has bn shown that th mtics of th lin lmnt is gula in th gion wh adius is gat o qual to min and lss than infinity This mans that th mtics of th lin lmnt is gula at th Schwazschild adius as wll as at th minimal adius This povs pdictions a) and b) of RAF thoy Th pdictions c) and d) a considd in th thid pat of th thoy Banko Novakovic FSB Univsity of Zagb Lucicva 5 POB 59 1 Zagb Coatia Th ky point fo pdictions of RAF thoy is th solution of th fild paamts psntd in th fist pat of th thoy This solution povids divation of th ngy-momntum tnso fo th lctical and gavitational filds as wll as thi unifid fild using of th gomtic appoach Futh w show that th mntiond fild paamts satisfy th Einstin s fild quations with a cosmological constant = In th cas of a stong static gavitational fild th quadatic tm (GM/c ) gnats th latd ngy-momntum tnso T η fo th static fild Fo that cas w do not nd to add by hand th latd ngy-momntum tnso T η on th ight sid of th Einstin s fild quations In th cas of a wak static gavitational fild lik in ou sola systm th quadatic tm (GM/c ) is clos to zo and can b nglctd Fo that cas th fild paamts satisfy th Einstin s fild quations in a vacuum (T η = = ) It is also wll known that fo unification of th lctowak and stong intactions with gavity on can us th following two possibilitis 1-6: a) tying to dscib gavity as a gaug thoy o b) tying to dscib gaug thois as gavity Th fist possibility (a) has attactd a lot of attntion but bcaus of th known difficultis this appoach st gavity apat fom th standad gaug thois Th scond possibility (b) is much mo adical Th initial ida has bn poposd by Kaluza-Klin thoy 7 8 which today has many vaiations 9-1 and taks th plac in th modn thois lik high ngy physics (supgavity and sting thois 18-9) Ths thois us fiv o mo xta dimnsions with th latd dimnsional duction to th fou dimnsions Manwhil w do not know th answs to th som qustions lik: can w tak th xta dimnsions as a al o as a mathmatical dvic? Following th solution of th two dimnsionlss (unitlss) fild paamts α and α fo unifid lctical and gavitational fild in th fist pat of RAF thoy 3 th unification of lctical and gavitational focs in th standad fou dimnsions (D) has bn psntd in th thid pat of RAF thoy 31 This unification is basd on th gomtic appoach RAF thoy stats with th main pposition: if th lctical gavitational and unifid filds (focs) can b dscibd by th gomtic appoach thn th fild paamts α and α of a paticl in th lctical gavitational and unifid filds should satisfy th Einstin s fild quations and th Einstin s godsic quations Th poposition latd to th satisfaction of th Einstin s fild quations is povd in this (scond) pat of RAF thoy Th poposition latd to th satisfaction of th Einstin s godsic quations is povd in th thid pat 31 of RAF thoy If RAF thoy is coct thn it could b applid to th both wak and stong filds at th Univs and Planck s scals giving th nw light to th gions lik black hols quantum thoy high ngy physics Big Bang thoy and cosmology 31 wwwijntog

2 Rlativistic Alpha Fild Thoy Pat II: Dos a Gavitational Fild Could b Without Singulaity? This pap is oganizd as follows Divation of th ngy-momntum tnso fo lctical and gavitational filds is psntd in Sc II Th poofs of th pdictions a) and b) of RAF thoy is also psntd in Sc II as a subsction A Sc III shows th pocdu of divation of ngy-momntum tnso fo unifid lctical and gavitational fild Finally th latd conclusion and fnc list a psntd in Sc IV and Sc V spctivly II ENERGY-MOMENTUM TENSOR FOR ELECTRICAL AND GRAVITATIONAL FIELDS Th basic poblm of this pap is to dtmin th ngy-momntum tnsos fo lctical gavitational and unifid fild in th Einstin s fou dimnsion (D) by using th gavity (gomtic) concpt In that sns w statd with th gnal lin lmnt ds in an alpha fild givn in th fist pat 3 of this thoy ds c dt cdt dx x cdt dy y cdt dz dx dy dz z (1) Following th wll known pocdu 1-6 this lin lmnt can b tansfomd into th sphical pola coodinats in th nondiagonal fom ds c dt c dt d d d sin d Th lin lmnt () blongs to th wll known fom of th Rimanns typ lin lmnt g dx g33 dx ds g dx g dx dx g dx Compaing th quations () and (3) w obtain th coodinats and componnts of th covaiant mtic tnso valid fo th lin lmnt (): 1 3 dx c dt dx d dx d dx d ( ) g g1 g 1 g 11 1 g g33 sin Stating with th lin lmnt () w mploy fo th convnint th following substitutions: / (5) In that cas th nondiagonal lin lmnt () is tansfomd into th nw lation ds c dt cdt d d d sin d () (3) () (6) 1 g (7) sin This tnso is symmtic and has six non-zo lmnts as w xpctd that should b Th contavaiant mtic tnso g μη of th nondiagonal lin lmnt (6) is divd by invsion of th covaiant on (7) 1 / ( ) / ( ) / ( ) / ( ) g 1 / 1 / sin (8) Th dtminants of th tnsos (7) and (8) a givn by th lations: dt g sin 1 dt g (9) sin (a) Poposition 1 If th lctostatic fild is dscibd by th lin lmnt (6) thn th solution of th Einstin fild quations givs th ngy momntum tnsot of that fild in th following fom: T T T 1T 1 T 11T T33 GQ 1 sin 8G q Q G A m H q and m a an lctic chag and a st mass of th lcton whil A is a scala potntial and Q is an lctic point chag of th lctostatic fild Paamt G = q/m is a constant that mands us to th constant of motion in th godsic quation of th Kaluza-Klin thoy 7-1 (1) (b) Poof of th poposition 1 In od to pov of th poposition 1 w can stat with th scond typ of th Chistoffl symbols of th mtic tnsos (7) and (8) Ths symbols can b calculatd by mploying th wll known lation 1-6 g g g g 1 3 (11) Thus mploying (6) (7) (8) and (11) w obtain th scond typ Chistoffl symbols of th sphically symmtic non-otating body: Using th coodinat systm () th latd covaiant mtic tnso g μη of th lin lmnt (6) is psntd by th matix fom 3 wwwijntog

3 Intnational Jounal of Nw Tchnology and Rsach (IJNTR) ISSN:5-116 Volum-1 Issu-5 Sptmb 15 Pags / D / D / D / D ( sin ) / D / D / D / D / D ( sin ) / D sincos ctg D t t (1) 1 Fo a static fild th Chistoffl symbols and a ducd to th simplst fom: 1 (13) In a static fild th oth Chistoffl symbols in (1) a maining unchangd As it is wll known th dtminant of th mtic tnso of th lin lmnt (6) should satisfy th following condition dt g sin 1 (1) Including th nomalization of th adius = 1 and th angl θ = 9 in (1) w obtain th impotant lations btwn th paamts ν and λ: 1 1 (15) ' ' '' ' '' If w tak into account th lations (15) thn th Chistoffl symbols in (1) and (13) bcom th functions only of th paamt Fo calculation of th latd componnts of th Rimannian tnso R and Ricci tnso R of th lin lmnt (6) w can mploy th following lations 1-6: R (16) R R R 1 3 Applying th Chistoffl symbols (1) to th lations (16) w obtain th latd Ricci tnso fo th static fild of th lin lmnt (6) with th following componnts: ' R 1 ' '' ' R1 R 1 ' '' (17) ' R 11 ' '' R ' 33 R ' sin Th oth componnts of th Ricci tnso a qual to zo Th latd Ricci scala fo th static fild is dtmind by th quation R g R 1 3 ' ' (18) R ' '' In od to calculat th ngy-momntum tnso T η fo th static fild on should mploy Ricci tnso (17) Ricci scala (18) and th Einstin s fild quations 1-6 without a cosmological constant ( = ) 1 8G R gr kt k 1 3 (19) c H G is th Nwton s gavitational constant c is th spd of th light in a vacuum and T η is th ngy-momntum tnso Thus mploying th Einstin s fild quations (19) w obtain th following lations fo calculation of th componnts of th ngy-momntum tnso T μη : ' ' kt 1 kt 1 kt 1 ' 1 ' kt 11 kt ' '' ' 8G kt33 sin ' '' k c () Fo calculation of th componnts of th ngy-momntum tnso T μη by th lations () w should know th paamt and its divations ' and '' fo th latd static fild Paamt is dfind by (5) as th function of th fild paamts α and α / / 1 (1) In od to dtmin th fild paamts α and α in an lctostatic fild w nd to know th potntial ngy of th paticl in that fild Thus if a paticl is an lcton that is psnt in an lctostatic fild thn th potntial ngy of th lcton in that fild U is dscibd by th wll known lation U qv q A () H q is an lctic chag of th lcton and V = A is a scala potntial of that fild Fo calculation of th paamt in an lctostatic fild w nd to know th diffnc of th fild paamts (α-α ) givn by th gnal fom in th fist pat 3 of this thoy: U U i (3) U U i H m is a st mass of th lcton Including th substitution U = U into (3) w obtain th diffnc of th fild paamts (α-α ) fo an lcton in an lctostatic fild: 33 wwwijntog

4 Rlativistic Alpha Fild Thoy Pat II: Dos a Gavitational Fild Could b Without Singulaity? qv qv i qv qv i () Futh fo th gaug fild on should us th wll known lctostatic ansatz 6 Thus including th lctostatic ansatz and applying th lations (1) and () w obtain th two solutions of th paamt : A A ( ) V( ) A A A t qv qv Q q i V A G m GQ GQ i 1 c c (5) H Q is a point chag of th lctostatic fild and G is th Kaluza-Klin constant 7 8 Th all itms ndd fo calculations of th componnts of th ngy-momntum tnso T μη in () a givn by th following lations: G Q GQ GQ GQ GQ ' i / c c c c c G Q GQ GQ ' c c c G Q G Q ' G Q G Q c c c c 3 3 GQ GQ ' '' ' ' 3 3 c c (6) Applying (6) to (18) and () w obtain th componnts of th ngy-momntum tnso T μη and Ricci scala R in an lctostatic fild: GQ GQ GQ kt kt kt c c c G GQ GQ k kt kt c c c GQ ' ' 33 kt sin R ' '' c GQ G Q c c (7) Fom th pvious lation w can s that th Ricci scala is qual to zo Finally includd paamt k into th lations (7) w obtain th componnts of th ngy-momntum tnso in an lctostatic fild: T T T T T T T GQ (8) q 1 sin G 8G m Bcaus th lation (8) is qual to th lation (1) th poof of th poposition 1 is finishd (c) Rmaks 1 In od to mak th solution (8) consistnt to th latd solution in a gavitational fild w should intoduc th paamt k 8G / c On th oth hand fo th consistnc to th Maxwll fild thoy this paamt should b k 8G / c : 8G k T T c T T T T T G GQ 1 sin 8 G 8 k T T T T T T T c Q sin 8 (8a) Futh th all itms givn by () (3) to (9) a also valid in a gavitational fild (d) Poposition If th gavitational static fild is dscibd by th lin lmnt (6) thn th solution of th Einstin fild quations givs th ngy momntum tnsot of that fild in th following fom T T T T T T T GM 1 sin 8G (9) H G and M a th gavitational constant and th gavitational mass spctivly () Poof of th poposition In od to pov of th poposition w should stat with th gnal lations givn by (1) () to (1) Fo dtmination of th fild paamts α and α in a gavitational fild on nd to know th potntial ngy of th paticl in that fild Thus if a paticl with st mass m is in a gavitational fild thn th potntial ngy of th paticl in that fild U g is dscibd by th wll known lation 1-6 mgm Ug = m Vg m A g (3) H V g =A g is a scala potntial of th gavitational static fild G is th gavitational constant M is a gavitational mass is a gavitational adius and m is a st mass of th paticl that is psnt in a gavitational static fild Fo calculation of th paamt in a gavitational static fild w nd to know th diffnc of th fild paamts (α-α ) 3 wwwijntog

5 Intnational Jounal of Nw Tchnology and Rsach (IJNTR) ISSN:5-116 Volum-1 Issu-5 Sptmb 15 Pags givn in th gnal fom by (3) Including th substitution U = U g into (3) w obtain th diffnc of th fild paamts (α-α ) fo a paticl in a gavitational static fild: GM GM c c GM GM c c (31) Applying th sults fom (31) to th lations in (1) w obtain th two solutions of th paamt in a gavitational static fild GM GM (3) c c Including (3) to () w obtain th all itms ndd fo calculations of th componnts of th ngy-momntum tnso T μη in a gavitational static fild GM GM GM GM GM ' / c c c c c GM GM GM ' c c c ' GM GM GM GM 3 3 c c c c GM 3 GM '' ' ' 3 c c ' (33) Now applying th lations (33) to th quations (18) and () w obtain th componnts of th ngy-momntum tnso and Ricci scala valid fo th gavitational static fild: GM GM 1 kt c c GM 8G kt1 kt 1 k c c GM GM 11 1 kt kt c c GM kt33 sin c ' ' R ' '' GM c c GM (3) Fom th pvious lations w can s that th Ricci scala is qual to zo Finally includd paamt k into th lations (3) w obtain th componnts of th ngy-momntum tnso in th gavitational static fild T T T 1T 1 T 11T T33 GM 1 sin 8G (35) Bcaus th lation (35) is qual to th lation (9) th poof of th poposition is finishd (f) Rmaks Th pvious lations show that th fild paamts (31) satisfy th Einstin s fild quations with a cosmological constant = In th cas of a stong static gavitational fild -37 th quadatic tm GM / c gnats th latd ngy-momntum tnso T η fo th static fild Fo that cas w do not nd to add by hand th latd ngy-momntum tnso T η on th ight sid of th Einstin s fild quations In th cas of a wak static gavitational fild lik in ou sola systm w obtain th quadatic tm GM / c Fo that cas th fild paamts (31) satisfy th Einstin s fild quations in a vacuum (T η = = ) This cosponds to th wll known Schwazschild solution of th lin lmnt Th scond intptation could b that th quadatic tm GM / c gnats th cosmological paamt as a function of a gavitational adius fo T η = It has bn shown in 5 that this solution of is valid fo both Planck s and cosmological scals Futh th mtics of RAF thoy 3 has bn applid to th divation of th gnalizd lativistic Hamiltonian 36 and dynamic modl of nanoobot motion in multipotntial fild 6 A Poofs of th Pdictions a) and b) of RAF Thoy RAF thoy pdicts that: a) no a singulaity at th Schwazschild adius and b) th xists a minimal adius at = (GM/c ) that pvnts singulaity at = i th natu potcts itslf In od to pov pdictions a) and b) w stat with th solution of th paamts and in a gavitational static fild givn by (15) and (3) and valid fo th lin lmnt (6): GM GM 1 1 c c GM GM GM c c c GM 1 3 sch sch sch c GM min 1 min im c GM 1 1 c (36) 35 wwwijntog

6 Rlativistic Alpha Fild Thoy Pat II: Dos a Gavitational Fild Could b Without Singulaity? Following th lations in (36) w can s that at th Schwazschild adius sch paamts and a gula This povs th pdiction a) no a singulaity at th Schwazschild adius Futh fom (36) w also can s that at th minimal adius min GM / c paamts and a also gula and fo paamt bcoms imaginay numb im This povs th pdiction b) th xists a minimal adius at = (GM/c ) that pvnts singulaity at = It sms that th xistnc of th minimal adius tll us that th natu potct itslf fom th singulaity Thus w can say that th mtics of th lin lmnt in (6) is gula fo a gavitational fild in th gion min On that way th poof of th popositions a) and b) is finishd min III ENERGY-MOMENTUM TENSOR FOR UNIFIED FIELD In od to dtmin of th fild paamts α and α fo th unifid lctical and gavitational static fild w nd to know th potntial ngy of a paticl in that fild Lt th souc of th unifid static fild is an objct with mass M lctic point chag Q and adius Thus if th paticl in th unifid fild is an lcton with st mass m and an lctic chag q thn th potntial ngy of th lcton in th unifid fild U is dscibd by th lation U U U q V m V g g qq mgm q A m A g (37) H V = A is a scala lctical potntial V g = A g is a scala gavitational potntial and G is a gavitational constant Now following (37) w can calculat th dimnsionlss tmu / U qq m GM G Q GM g m c m c m c c c c q G M G Q GM g m M (38) H paamt G = q/m is a constant wll known in Kaluza-Klin thoy 7-1 Th fou solutions of th fild paamts α and α fo th lcton in th unifid lctical and gavitational static fild can b obtaind by th substitution of th dimnsionlss tm (38) into th gnal solution of th fild paamts α and α givn in th fist pat 3 of this thoy: f (U ) U / m c U / m c g M / c M / c g 1 i f (U ) 1 i f (U ) i f (U ) 1 i f (U ) 3 3 (39) It is asy to pov that th all αα pais fom (39) satisfy th following invaiant lations: U qq mgm ii 1 = 1 mc mc Mg 1 i 1 3 ' c qq mgm Ec ' 1 mc mc g m c qv m V () H E c is th covaiant ngy of an lcton standing (v = ) in th unifid lctical and gavitational static fild Fo calculation som of th quantitis in that fild w oftn nd to know th diffnc of th fild paamts (α-α ) fo an lcton in th unifid lctical and gavitational static fild: Mg Mg i Mg Mg i (1) Th all itms givn by () (3) to (9) a also valid fo th unifid lctical and gavitational static fild (g) Poposition 3 Lt th souc of th unifid lctical and gavitational static fild is an objct with mass M lctic point chag Q and adius Futh lt a paticl is an lcton with st mass m and an lctic chag q that is psnt in this unifid lctical nd gavitational static fild If th unifid fild is dscibd by th lin lmnt (6) thn th solution of th Einstin fild quations givs th ngy momntum tnsot valid fo that fild T T T T T T T g M 1 sin 8 G q Mg GQ GM G m () Paamt G is a constant that mands us to th constant of motion in th godsic quation of th Kaluza-Klin thoy 7-1 and G is th gavitational constant (h) Poof of th poposition 3 In od to pov of th poposition 3 w should stat with th gnal lations givn by (1) () to (1) Thus applying (1) to (1) w obtain two solutions of th paamt valid in th unifid static fild M g M g i (3) c c Th all itms ndd fo calculations of th componnts of th ngy-momntum tnso T μη in () a givn by th following lations: 36 wwwijntog

7 Intnational Jounal of Nw Tchnology and Rsach (IJNTR) ISSN:5-116 Volum-1 Issu-5 Sptmb 15 Pags Mg Mg Mg Mg Mg ' i / c c c c c Mg Mg Mg ' c c c M M ' M M c c c c g g g g 3 3 Mg Mg ' '' ' ' 3 3 c c () Now applying (3) to (18) and () w obtain th componnts of th ngy-momntum tnso kt μη and Ricci scala R of th unifid static fild: g g 8 1 M M G kt k c c c Mg Mg kt kt kt c c Mg Mg 33 kt kt sin c c ' ' R ' '' g Mg M c c (5) Fom th pvious lations w can s that th Ricci scala is qual to zo Finally includd paamt k into th lations (5) w obtain th componnts of th ngy-momntum tnso T μη in th unifid lctical and gavitational static fild T T T T T T T M (6) g 1 sin Bcaus (6) is qual to () w conclud that th poof of th poposition 3 is finishd (i) Rmaks 3 Th ngy momntum tnso (6) is gnal in th following sns: a) putting M g = G Q on obtains th solution in an lctostatic fild b) putting M g = - GM on obtains th solution in a gavitational static fild Using th dimnsional analysis dim((g = q/m ) ) = dim(( G) ) = dim(g) th quation (8) can b tansfomd into th nw fom: 8 G T T T T T T T Q 1 sin K 8 q G G K dim( K ) 1 m G (7) Somtims (s 6) th componnts of th ngy-momntum tnso T μη in an lctostatic fild hav bn dscibd by th lations (7) IV CONCLUSION In this pap w povd that th fild paamts α and α of th lctical gavitational and unifid filds satisfy th Einstin s fild quations and automatically gnat th latd ngy-momntum tnso in th standad fou dimnsions (D) This mans that fo lctical gavitational and unifid filds w do not nd to add by hand th ngy-momntum tnso to th ight sid of th Einstin s fild quations In a stong static gavitational fild th quadatic tm (GM/c ) gnats th ngy - momntum tnso on th ight sid of th Einstin s fild quations In th cas of a wak static gavitational fild lik in ou sola systm w obtain th quadatic tm (GM/c ) clos to zo Fo that cas th fild paamts satisfy th Einstin s fild quations in a vacuum (T η = = ) This cosponds to th wll known Schwazschild solution of th lin lmnt Futh w also povd two pdictions of RAF thoy: a) no a singulaity at th Schwazschild adius b) th xists a minimal adius at = (GM/c ) that pvnts singulaity at = i th natu potcts itslf Th pdictions c) and d) a considd in th thid pat of th thoy If RAF thoy is coct thn it could b applid to th both wak and stong filds at th Univs and Planck s scals giving th nw light to th gions lik black hols quantum thoy high ngy physics Big Bang thoy and cosmology ACKNOWLEDGMENTS Th autho wishs to thank to th anonymous viws fo a vaity of hlpful commnts and suggstions This wok is suppotd by gants ( ) fom th National Scintific Foundation of Rpublic of Coatia V REFERENCES 1 A Einstin Ann Phys (1916) A Einstin Th Maning of Rlativity (Pincton Univ Pss Pincton 1955) 3 C San Spactim and Gomty: An intoduction to Gnal Rlativiy (Amazoncom Bookshtm Hadcov 3) S Winbg Gavitation and Cosmology: Pincipls and Application of th Gnal Thoy of Rlativity (Gbundn Ausgab RlEspWinbgpdf 197) 5 S W Hawking G F R Ellis Th Lag Scal Stuctu of Spac-Tim (Univ Pss Cambidg 1973) 6 M Blau Lctu Nots on Gnal Rlativity (A Einstin Cnt fo Fundamntal Physics Univ Bn Bn 1 1) 7 T Kaluza Zum Unitätspoblm in d Physik (Sitzungsb Puss Akad Wiss Blin 191) 8 O Klin Z Phys A (196) 9 E Wittn Nucl Phys B (1981) 1 T Applquist A Chodos and P G O Fund Modn Kaluza Klin Thois (Addison Wsly Mnlo Pak Cal 1987) 37 wwwijntog

8 Rlativistic Alpha Fild Thoy Pat II: Dos a Gavitational Fild Could b Without Singulaity? 11 M J Duff Kaluza Klin Thoy in Pspctiv (Poc of th Symposium: Th Oska Klin Cntnay Wold Scintific Singapo ) 1 J M Ovduin and P S Wsson Phys Rp (1997) 13 P S Wsson Spac-Tim-Matt Modn Kaluza - Klin Thoy (Wold Scintific Singapo 1999) 1 P SWsson Fiv-Dimnsional Physics: Classical and Quantum Consquncs of Kaluza-Klin Cosmology (Wold Scintific Singapo 6) 15 D Z Fdman and A Van Poyn Supgavity (Cambidg Univ Pss Cambidg 1) 16 J Wss B and A Zumino Phys Ltt B 9 5 (197) 17 M K Gaillad and B Zumino Nucl Phys B (1981) 18 M B Gn J H Schwaz and E Wittn Supsting Thoy (Cambidg Univ Pss Cambidg 1987) 19 J Polchinski Sting Thoy (Cambidg Univ Pss Cambidg 1998) R Bandnbg and C Vafa Nucl Phys B (1989) 1 N Akani-Hamd A G Cohn and H Gogi Phys Rv Ltt (1) C T Hill S Pokoski and J Wang Phys Rv D (1) 3 C Cshaki G D Kibs and J Tning Phys Rv D () E C Poggio H R Quinn and S Winbg Phys Rv D (1976) 5 T R Taylo and G Vnziano Phys Ltt B 1 17 (1988) 6 H C Chng B A Dobscu and C T Hill Nucl Phys B () 7 C Cshaki J Elich C Gojan and G D Kibs Phys Rv D () 8 N Akani-Hamd and M Schmaltz Phys Rv D () 9 M Gogbashvili Euophys Ltt () 3 B M Novakovic Rlativistic alpha fild thoy - Pat I To b publishd in IJNTR (15) 31 B M Novakovic Rlativistic alpha fild thoy-pat III To b publishd in IJNTR (15) 3 B M Novakovic Int J of Comput Anticip Syst IJCAS 7 p 93 (1) 33 S Gallot D Hullin and D J Lafontan Rimannian Gomty ( Sping-Vlag Blin Nw Yok d 3 ) 3 C T J Dodson and T Poston Tnso Gomty Gaduat Txts in Mathmatics (Sping-Vlag Blin Nw Yok d 1991) p M T Vaughin Intoduction to Mathmatical Physics (Wily-VCH Vlag GmbH & Co Winhim 7) 36 B M Novakovic in Pocdings of th Ninth Int Conf on Comp Anticip Syst Lig 9 ditd by D Dubois (Univsity of Lig Lig 9) AIP-CP 133 p 11 (1) DOI: 1163/ P A M Diac Dictions in Physics (Wily Nw Yok 1978) 38 I Supk Thotical Physics and Stuctu of Matt Pat I (Skolska knjiga Zagb 199) 39 I Supk Thotical Physics and Stuctu of Matt Pat II (Skolska knjiga Zagb 199) D H Pkins Intoduction to High Engy Physics (Cambidg Univ Pss Cambidg ) 1 D Shman t al Nat Phys (15) J Stinhau Nat Phys (1) 3 M Mckl t al Nat Phys (1) B M Novakovic D Novakovic and A Novakovic in Pocdings of th Sixth Int Conf on Comp Anticip Syst Lig 3 ditd by D Dubois (Univsity of Lig Lig 3) AIP-CP 718 p133 () DOI: 1163/ B M Novakovic D Novakovic and A Novakovic in Pocdings of th Svnth Int Conf on Comp Anticip Syst Lig 5 ditd by D Dubois (Univsity of Lig Lig 5) AIP-CP 839 p1 (6) DOI: 1163/ B M Novakovic Stojastvo 53 () (11) 7 R Ding t al Phys Rv D 9 (158) (15) Banko Novakovic is a Pofsso mitus at FSB Univsity of Zagb Coatia Pof Novakovic civd his PhD fom th Univsity of Zagb in 1978 His sach of intst includs physics contol systms obotics nual ntwoks and fuzzy contol H is autho of two books Contol Systms (1985) and Contol Mthods in Robotics Flxibl Manufactuing Systms and Pocsss (199) and co-autho of a book Atificial Nual Ntwoks (1998) H has publishd ov sach paps in his sach of intst 38 wwwijntog